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TRANSCRIPT
On the analogy between the representations of a reductive Liegroup and those of its Cartan motion group
Alexandre AfgoustidisCEREMADE, Université Paris-Dauphine
Version of March 17, 2018
Contents
1. Introduction..................................................................................................................................................... 1Part I. The Mackey-Higson correspondence ................................................................................................. 52. Representations of the Cartan motion group .................................................................................................. 53. The Mackey bijection ..................................................................................................................................... 8
3.1. Real-infinitesimal-character representations and a theorem of Vogan .................................................. 83.2. The Mackey-Higson correspondence for tempered representations...................................................... 93.3. Preservation of minimal K-types ........................................................................................................... 113.4. The Mackey-Higson correspondence for admissible representations ................................................... 13
Part II. On the contraction of tempered representations ............................................................................. 154. Deformation to the motion group ................................................................................................................... 155. Examples of Fréchet contractions: the discrete series and the spherical principal series............................... 19
5.1. Contraction of discrete series representations ....................................................................................... 195.2. On the principal series and generic spherical representations ............................................................... 24
6. Real-infinitesimal-character representations................................................................................................... 316.1. Reduction to the quasi-split case ........................................................................................................... 316.2. The case of quasi-split groups and fine K-types.................................................................................... 41
7. Contraction of an arbitrary tempered representation ...................................................................................... 457.1. Real-infinitesimal character representations for disconnected little groups .......................................... 457.2. Parabolic induction and the reduction to real infinitesimal character.................................................... 48
Part III. The Mackey analogy and the Connes-Kasparov isomorphism .................................................... 518. Introduction and strategy ................................................................................................................................ 519. Distinguished subquotients of the reduced C?-algebras ................................................................................. 5510. Deformation of the reduced C?-algebra and subquotients.............................................................................. 61
1. Introduction
1.1. Contractions of Lie groups and a conjecture of Mackey
When G is a Lie group and K a closed subgroup, one can use the linear action of K on the vector space V =
Lie(G)/Lie(K) to define a new Lie group: the semidirect product G0 = K nV , known as the contraction of Gwith respect to K. The notion first arose in mathematical physics (see Segal [55], Inönü and Wigner [32], Saletan[54] and the lecture by Dyson [21]): the Poincaré group of special relativity admits as a contraction the Galileigroup of classical inertial changes, and it is itself a contraction of the de Sitter group which appears in generalrelativity.
It is a classical problem to try to determine whether there is a relationship between the representation theoriesof G and G0: for instance, the unitary (irreducible) representations of the Poincaré group famously yield spaces of
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quantum states for (elementary) particles [74], and it is quite natural wonder about the existence of “nonrelativistic”analogues in the representation theory of the Galilei group [31]. In many cases of physical interest, the observationthat the contraction G0 can be seen as the special point of a one-parameter family (Gt)t∈R of Lie groups (where theparameter t usually has some physical significance, and Gt is isomorphic with G for all t 6= 0) has led to attemptsto exhibit some representations of G0 as “limiting cases” of representations of G; early examples include [31],[28], [48], [46].
For most Lie groups, unitary representations do not behave well under the contraction; the parameters neededto identify an irreducible representation of G are often visibly different from those determining an irreduciblerepresentation of G0, and some of the differences are important for physics − a consequence of the bad behaviourin the case of the Poincaré group is that the notion of (rest) mass has different meanings in special and Galileanrelativity.
Now, suppose G is a reductive Lie group and K is a maximal compact subgroup. The contraction G0 is known inthat case as the Cartan motion group of G, probably as a tribute to Élie Cartan’s study of symmetric spaces: thegroup G acts by isometries on the negatively curved G/K, while G0 acts by Euclidean rigid motions on the (flat)tangent space to G/K at the identity coset.
The algebraic structures of G and G0 and their representation theories are quite different. Denote the unitarydual of a Lie group Γ by Γ and its reduced dual (when Γ is type-I) by Γ; then George W. Mackey’s early workon semidirect products [43, 44] describes G0 in very simple and concrete terms, but describing G remains tothis day an extremely deep problem. A complete understanding of G was achieved in the early 1980s, crowningtremendous efforts of Harish-Chandra and others that had begun in 1945.
In 1975, however, Mackey noticed surprising analogies, for several simple examples of reductive groups G, be-tween his accessible description of G0 and Harish-Chandra’s subtle parametrization of G. In the examples studiedby Mackey, “large” subsets of G0 and G could be described using the same parameters, and the classical con-structions for some of the corresponding representations were reminiscent of one another. Quantum-mechanicalconsiderations led him to believe that there was more to his than chance, and he went on to conjecture that thereshould be a “natural” one-to-one correspondence between “large” subsets of G0 and G [45]:
The physical interpretation suggests that there ought to exist a “natural” one-to-one correspondencebetween almost all the unitary representations of G0 and almost all the unitary representations of G − inspite of the rather different algebraic structure of these groups.
Mackey’s conjecture seems to have been considered overenthusiastic at the time. It is true that in the yearsimmediately following the appearance of Mackey’s paper, studying the relationship between G-invariant harmonicanalysis on G/K and G0-invariant analysis on G0/K became a flourishing subject (see e.g. [25, 52]), with beautifulramifications for all Lie groups [19, 33], and the close relationship between the two kinds of harmonic analysisdoes call to mind Mackey’s observations on the principal series. It must be said, however, that few of the au-thors who pursued this subject explicitly referred to Mackey’s conjecture; two notable exceptions are papers byDooley and Rice [18], who established in 1985 that the operators for principal series representations of G doweakly converge, as the contraction is performed, to operators for a generic representation of G0, and Weinstein[73], who wrote down a Poisson correspondence between coadjoint orbits associated to minimal principal seriesrepresentations.
But as one moves away from the principal series of G to the deeper strata of the tempered dual, it must haveseemed difficult in the early 1970s to imagine that much could be extracted from Mackey’s suggestions beyond thesimplest examples of reductive groups that he had considered. In contrast to the fact that every unitary irreduciblerepresentation of G is infinite-dimensional save for the trivial representation, an obvious feature of Mackey’s de-termination of G0 is the existence of a countable family of finite-dimensional unitary irreducible representations.In fact the geometrical construction of representations in the deeper strata of G, for instance that of the discreteseries, was a burning subject in the 1970s, whereas the most degenerate parts of G0 look somewhat trivial. Mackey
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had of course noticed the absence, for these deeper strata, of any clear geometrical relationship, remarking that inthe examples he had studied:
Above all [the analogy] is a mere coincidence of parametrizations, with no evident relationship between theconstructions of corresponding representations.
1.2. Relationship with the Connes-Kasparov isomorphism; Higson’s work
In the late 1980s and early 1990s, Mackey’s conjecture found an important echo in the study of group C?-algebras.The Baum-Connes conjecture for G, in its “Lie group” version due to Connes and Kasparov, describes the K-theoryof the reduced C?-algebra C?
r (G) of a connected Lie group G in terms of the representation ring of a maximal com-pact subgroup and Dirac induction. For this article’s purposes it may be useful to view the operator K-theory ofC?
r (G) as a non-commutative-geometry version of the topological K-theory of the reduced dual G (see §8); PaulBaum, Alain Connes and Nigel Higson pointed out that in this perspective, the Connes-Kasparov conjecture canbe reinterpreted as a statement that G and G0 share algebraic-topological invariants, and thus can be viewed as acohomological reflection of Mackey’s conjecture ([7], section 4). As Connes emphasized ([13], § II.10.δ ):
It is of course desirable to find direct proofs of the surjectivity of [the Baum-Connes assembly map]. Inthat respect the ideas developed by Mackey in [45], or in the theoretical physics literature on deformationtheory, should be relevant.
Connes and Higson had crafted a precise tool to do so 1990, by pointing out that the Connes-Kasparov conjecturecan rephrased using the contraction of G with respect to K: associated to the deformation (Gt)t∈R is a continu-ous field of group C?-algebras interpolating between the reduced C?-algebras C?
r (G) and C?t (G0); developing the
deformation theory of C? algebras and its relationship with K-theory, Connes and Higson proved [14] that theConnes-Kasparov conjecture is equivalent with the assertion that this field has constant K-theory.
Nigel Higson later remarked that Mackey had in mind a measure-theoretic correspondence (which may havebeen defined only for “almost every” representation), but that the interplay with the Connes-Kasparov isomorphismbrought the topologies between G and G0 into the game, albeit at the level of cohomology. Because the relationshipbetween the Fell topologies and the Plancherel measure can be somewhat subtle, he noted [29, 30] that the simplestway to reconcile both points of view was to guess that there might exist a natural one-to-one correspondencebetween every irreducible tempered representation of G and every unitary irreducible representation of G0.
In 2008, Higson examined the case of complex semisimple groups, in which the reduced dual G is completelydescribed by the principal series, and constructed [29] a natural bijection between G and G0. He used this bijectionto analyze the structures of C?(G) and C?(G0), discovering building blocks for the two C?-algebras that fit togetherrigidly enough in the deformation from C?
r (G) to C?t (G0); this led him to a new proof of the Connes-Kasparov
isomorphism for complex semisimple group that is both natural from the point of view of representation theory,and very elementary from the point of view of K-theory.
1.3. Description of our results and organization of this paper
• In the first part of the present article, we construct a simple and natural bijection between G and G0 for anyreal reductive group G (for our precise hypotheses on G, see §3.2).
David Vogan’s notion of lowest K-types will play a crucial part: we shall imitate Mackey’s classical parametriza-tion of G0 (see §2) to construct a Mackey-like parametrization of G using Vogan’s results on tempered irreduciblerepresentations with real infinitesimal character (see §3.1). Our correspondence will turn out (see §3.3) to preservelowest K-types and be compatible with a natural notion of variation of continuous parameters. As we shall see (in§3.4), it extends naturally to a one-to-one correspondence between the admissible irreducible representations ofboth groups.
With Mackey’s conjecture established at the level of parameters, we turn to the deformation (Gt)t∈R (see §4.1)to try to better understand the reasons for, and implications of, the existence of the correspondence.
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• In the second part, we focus on individual representation spaces and describe a geometrical deformation ofevery tempered irreducible representation π of G onto the representation π0 of G0 that corresponds to it in thebijection of Part I.
Of course if one is to associate with π is a family (πt)t>0, where each πt is a representation of Gt and to tryto prove that “πt goes to π0 as t goes to zero”, one needs to give a meaning to the convergence. There is nouniversally acknowledged setting for doing so; in fact, the various notions of “contraction of representations” thathave been used in mathematical physics are somewhat disparate in nature, and often formulated at the infinitesimal(Lie algebra) level.
Our strategy, described in detail in §4.3, is to choose, among the geometric realizations of π , a convenient oneacting on a Fréchet space, then having each πt , t > 0, act on a subspace of a fixed Fréchet space E. We shallthen observe, as t goes to zero, how vectors evolve under the contraction in a way determined by equivarianceconstraints, and how the initial Fréchet spaces and Gt -actions then wear out to a carrier space and operators for therepresentation π0 of G0.
The most suggestive and important examples are technically much simpler to deal with than the general case:before turning to the contraction of an arbitrary tempered representation, we will take the time to inspect the con-traction of a discrete series representation to its (unique) lowest K-type (see §5.1), as well as the contraction ofminimal principal series representations (see §5.2) to generic representations of G0 − for the latter case, our resultsare related to earlier treatments in the physical literature and in [18]. Our general treatment, relying on succes-sive reductions of the general case to that of real-infinitesimal character representations of connected semisimplegroups, then to that of irreducible constituents of principal series representations of quasisplit groups with realinfinitesimal character, is perhaps technically less simple; but almost all of the important ideas and lemmas arealredy present in the two extreme examples (see §6 for real-infinitesimal-character representations and §7 for thereduction of the general case to that one).
We should mention that except in the case of minimal principal series where where there is some overlap withthe existing physical literature, the settings we will use (for instance the description of discrete series represen-tations using Dirac induction, or of real-infinitesimal-character representations in Dolbeault cohomology spaces)force upon us a choice of Fréchet space whose topology is quite different from the Hilbert space topology in-herited from the unitary structure of the representations. Thus, our geometrical analysis of Mackey’s analogy ismade possible only by the deep work of many mathematicians (among which Helgason, Parthasarathy, Atiyah,Schmid, Vogan, Zuckerman, Wong), ranging from the early 1970s to the late 1990s, on the geometric realizationof irreducible tempered representations.• In the third part of this paper, we focus on the topologies on G and G0 and the behavior of some matrix coeffi-
cients as the contraction is performed, and turn the study of reduced C?-algebras. We take up Higson’s analysis ofthe complex semisimple case to obtain a new proof of the Connes-Kasparov “conjecture” for linear connected re-ductive groups, based on some (far from all) results of Part II. Of course for reductive groups, Wassermann’s proofusing Harish-Chandra’s Plancherel formula and the Knapp-Stein theory of intertwining operators is now thirtyyears old, and our new proof cannot be compared, in breadth and depth, to the work of Lafforgue at the end of thelast century [42]. Our aim in including it here, aside from confirming Connes and Higson’s long-standing insightthat the Connes-Kasparov isomorphism is in the reductive case a simple cohomological reflection of Mackey’sanalogy, is to show that there are nontrivial facts on the reductive side to be deducted from Mackey’s conjecture,and to establish some simple topological properties of the correspondence. The bijection defined in Part I cannotbe a homeomorphism (the dual G0 is Hausdorff, but G is not); yet we shall see that there are natural partitionsof G and G0 into countable families of locally closed subsets, associated to minimal K-types, and that Mackey’scorrespondence induces a homeomorphism between the matching pieces of the partition, showing that the maindifferences G and G0 lie in the way these pieces are stitched together (with, for example, the existence of limits ofdiscrete series G making G more singular). Part III includes an introduction giving more details on the specificallyC?-algebraic aspects.
The analytical methods to be used in Parts II and III are hardly the final word on realizing Mackey’s analogy
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as a deformation, and much remains mysterious about the relationship between the representation theories of Gand G0. In the time elapsed since a first version of this article began circulating in 2015, Bernstein, Higson andSubag have suggested that the methods of algebraic geometry may provide a framework for studying (algebraic)deformations of groups and families of representations in a much broader context. Their recent work [8, 9, 60, 5],as well as Shilin Yu’s recent attempts to obtain an algebraic-geometric realization of Mackey’s correspondenceusing deformations of algebraic D-modules [61, 77], make it reasonable to hope that much insight will be gainedin this way on Mackey’s analogy. To mention only the simplest related issue, it will soon be obvious that ouranalytic methods cannot be well-suited to realizing as a deformation the natural correspondence between theadmissible duals to be introduced in section §3.4, and it seems reasonable to turn to algebraic methods for thatstudy. On a more general note, one can only hope that the next years will say if Mackey’s ideas on real reductivegroups have any relevance to the representation theory of nonreductive Lie groups, or on that of reductive groupsover other fields.
Acknowledgments
Daniel Bennequin, who was my PhD advisor, encouraged my interest for this subject and offered constantlygood advice, with great kindness and generosity. Several conversations with Michel Duflo and Michèle Vergnewere very useful and important to me, especially in the early stages of this investigation. Nigel Higson’s warmwelcome did much to encourage me in my endeavour, and Part III greatly benefited from his remarks. FrançoisRouvière and David Vogan read the manuscript of my thesis [2] very carefully; the present version owes muchto their suggestions and corrections, some of which were crucial. I could not have hoped to write section 6.1below without Nicolas Prudhon’s bibliographical help. Jeremy Daniel heard preliminary (and very false) versionsof many of my ideas; our friendly discussions have been very helpful. I am happy to say here how very muchindebted I am to all of them.
During my time as a PhD student, in which most of the results of this article were obtained, I benefited from ex-cellent working conditions at Université Paris-7 and the Institut de Mathématiques de Jussieu - Paris Rive Gauche.The results of §6 are much more recent; I am grateful to my colleagues at Université Paris-Dauphine and CERE-MADE for their support.
PART I. THE MACKEY-HIGSON CORRESPONDENCE
2. Representations of the Cartan motion group
2.1. Notations
Throughout this paper, we shall call a Lie group G reductive, short for the more usual linear reductive, when• G is a closed subgroup of GL(n,R), for some n > 0;• G is stable under transpose;• G has a finite number of connected components.
Suppose G is a noncompact reductive group with Lie algebra g. Fix a maximal compact subgroup K of G, write θ
for the Cartan involution of G with fixed-point-set K, and form the corresponding Cartan decomposition g= k⊕p
of G: here k is the Lie algebra of K and p is a linear subspace of g which, although not a Lie subalgebra, is stableunder the adjoint action Ad : K→ End(g). Recall that the semidirect product Knp is the group whose underlyingset is the Cartesian product K×p, equipped with the product law
(k1,v1) ·0 (k2,v2) := (k1k2,v1 +Ad(k1)v2) (k1,k2 ∈ K, v1,v2 ∈ p).
We will write G0 for this group and call G0 the Cartan motion group of G.Writing V ? for the space of linear functionals on a vector space V and viewing p? as the space of linear func-
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tionals on g which vanish on k, we note that in the coadjoint action Ad? : G→ End(g?), the subspace p? ⊂ g? isAd?(K)-invariant.
We fix once and for all an abelian subalgebra a of g that is contained is p and is maximal among the abeliansubalgebras of g contained in p; we write A for expG(a).
When Γ is a type-I Lie group, we will write Γ for its unitary dual, Γ for its reduced dual (see for instance [17],Chapter 18) and Γadm for its admissible dual (see for instance [66], §0.3). The motion group G0 is amenable, sothat G0 = G0; but for our reductive group G, a unitary irreducible representation π ∈ G is contained in the reduceddual G if and only if it is tempered, which means that its K-finite matrix elements lie in L2+ε(G) for every ε > 0(see [15]); the trivial representation of G is not tempered, so G 6= G.
2.2. Mackey parameters
When χ is an element of p?, we write Kχ for its stabilizer in the coadjoint action of K on p; this is a compact,usually disconnected group. In physics, it is often called the “little group at χ for the Mackey machine”.
We call Mackey parameter (or: Mackey datum) any pair (χ,µ) where χ is an element of p? and µ is anirreducible representation of Kχ .
We call two Mackey parameters (χ,µ) and (χ ′,µ ′) equivalent when there exists an element k in K suchthat• χ = Ad?(k) ·χ ′, and• µ is equivalent, as an irreducible representation of Kχ , with k?µ ′ : u→ µ ′(k−1u).
This defines an equivalence relation on the set of Mackey data; we will write D for the set of equivalence classesof Mackey data.
It may be useful to mention two elementary facts from structure theory (see [38], sections V.2 and V.3):(i) Every Mackey parameter is equivalent with a Mackey datum (χ,µ) in which χ lies in a?; furthermore, if
χ and χ ′ are two elements of a? and we consider Mackey data (χ,µ) and (χ ′,µ ′), then (χ,µ) and (χ ′,µ ′)
are equivalent if and only if χ and χ ′, on the one hand, µ and µ ′, on the other hand, are conjugate under anelement of the Weyl group W =W (g,a).
(ii) The dimension of the Ad?(K)-orbit of χ in p? is maximal (among all possible dimensions of Ad?(K)-orbitsin p?) if and only if χ is regular. All regular elements in a? have the same Ad?(K)-stabilizer, viz. thecentralizer M = ZK(a) of a in K.
2.3. Unitary dual of the motion group
We recall the usual way to build a unitary representation of G0 from a Mackey parameter (χ,µ).Consider the centralizer Lχ
0 of χ in G0 (for the coadjoint action). It reads Kχ n p, where Kχ is the little groupof section 2.2.
Out of δ and χ , build an irreducible representation of this centralizer: fix a carrier space V for µ , and definean action of L0
χ on V , where (κ,v) ∈ Kχ np acts through ei〈χ,v〉µ(k). Write σ = µ⊗ eiχ for this representation ofL0
χ .Now, observe the induced representation
M0(δ ) = IndG0L0
χ
(σ) = IndG0Kχnp
(µ⊗ eiχ) . (2.1)
It will be useful to record a construction of M0(δ ) and three elementary remarks.• Construction. Fix a µ(K)-invariant inner product 〈·, ·〉 on V . Equip the Hilbert space
H =
f ∈ L2(K,V ), ∀m ∈ Kχ ,∀k ∈ K, f (km) = µ(m−1) f (k)
(2.2)
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with the G0-action in which
g0 = (k,v) acts through π0(k,v) : f 7→[u 7→ ei〈Ad?(u)χ,v〉 f (k−1u)
]. (2.3)
Then the representation (H,π0) of G0 is equivalent with M0(δ ).• Remark 1. Suppose χ is zero. Then Kχ equals K, µ is an irreducible representation of K, and the representationof M0(χ,µ) of G0 is the trivial extension of µ where p acts by the identity on V . Thus M0(δ ) is finite-dimensionalin that case.• Remark 2. Suppose χ is nonzero and µ is the trivial representation of Kχ . Then another geometric realizationfor M0(δ ) may be obtained by considering the space E of tempered distributions on p whose Fourier transform(a tempered distribution on p?) has support contained in the orbit Ad?(K) · χ; since that orbit is compact, E is aspace of smooth functions on p. The functions in E are “combinations” of Euclidean plane waves on p whosewavevectors lie in the orbit Ad?(K) · χ . The realization (2.2)-(2.3) of G0 can be viewed as transferred from thequasi-regular action of G0 on E by taking Fourier transforms.• Remark 3. When the representation µ of Kχ is not trivial, it may be noted that (2.2) is a model for the space ofsquare-integrable sections of the K-equivariant bundle K×µ V over the orbit Ad?(K) · χ ' K/Kχ ; the G0-action(2.3) can then be assembled from the quasi-regular action of K on that bundle’s sections and the action of p givenby the same Fourier-induced multiplication factor as appeared for trivial µ .
We close this subsection with Mackey’s description of the unitary dual of G0.
Theorem 2.1 (Mackey, 1949 [43]).
(a) For every Mackey parameter δ , the representation M0(δ ) is irreducible.
(b) Suppose δ and δ ′ are two Mackey parameters. Then the representations M0(δ ) and M0(δ′) are unitarily
equivalent if and only if δ and δ ′ are equivalent as Mackey parameters.
(c) By associating to any Mackey parameter δ the equivalence class of M0(δ ) in G0, one obtains a bijectionbetween D and G0.
2.4. Admissible dual of the motion group
We now outline Champetier and Delorme’s description ot the admissible dual of G0adm
([12]; see also Rader [51]).Recall that the action Ad? of K on p? induces an action on the complexified space p?C, in which K acts separatelyon the real and imaginary parts. We will still write Ad? for it.
We call admissible Mackey parameter a pair (χ,µ) in which χ lies in a?C and µ is an irreducible representationof Kχ . Note that in contrast to the previous section, we require from the outset that χ lie in a?C, for it is no longertrue that Ad?(K) ·a?C coincides with p?C.
We define on the set of admissible Mackey parameters the same equivalence relation as described in section 2.2and write Dadm for the set of equivalence classes of admissible Mackey data.
When δ = (χ,µ) is an admissible Mackey datum, we write M0(δ ) for the representation IndG0Kχnp(µ⊗ eχ) of
G0 : it is still defined by (2.2)-(2.3). This representation is unitary if and only if the real part of χ is zero; in thatcase it coincides with the representation defined in section 2.3 for imaginary χ .
Theorem 2.2 (see [12] or [51]).
(a) For every Mackey parameter δ , the representation M0(δ ) is irreducible.
(b) Suppose δ and δ ′ are two admissible Mackey parameters. Then the admissible representations M0(δ ) andM0(δ
′) are equivalent if and only if δ and δ ′ are equivalent as admissible Mackey parameters.
(c) By associating to any admissible Mackey parameter δ the equivalence class of M0(δ ) in Gadm, one obtains a
bijection between Dadm and G0adm
.
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3. The Mackey bijection
In this section, we introduce a natural bijection between the reduced duals G0 and G and an extension to a natural
bijection between G0adm
and Gadm.The key step is to determine the subset of G that corresponds to the “discrete part” of G0 encountered in Remark
2.3.1, which is a copy of K in G0.For some reductive groups, there is a natural candidate for that subset of G: the discrete series. If G is a reductive
group with nonempty discrete series, it is well-known from the work of Harish-Chandra, Blattner, Hecht-Schmid(see [24]) that every discrete series representation π comes with a distinguished element of K, the minimal (or:lowest) K-type of π (see for example [20], section I.5); furthermore, inequivalent discrete series representations ofG have inequivalent minimal K-types.
There are at least two obvious reasons not to use the discrete series for out construction, however:
(i) there are reductive groups with no discrete series representations;
(ii) even when G has a nonempty discrete series, there are elements of K which cannot be realized as the minimalK-type of some discrete series representations.
To define a “natural candidate” for the class of representations to be associated in G to the copy of K in G0, weturn to David Vogan’s work on lowest K-types for other representations.
3.1. Real-infinitesimal-character representations and a theorem of Vogan
Fix a reductive group G and a maximal compact subgroup K.Given a Cartan subalgebra t of k and and a choice of positive root system ∆+
c for the pair (kC, tC), Vogan defines([64]) a positive-valued function µ → ‖µ‖K on K: starting from an irreducible µ in K, one can consider its ∆
+C -
highest weight µ and the half-sum ρc of positive roots in ∆+c ; these are two elements of t?C, which comes with a
Euclidean norm ‖·‖t?C ; we set ‖µ‖K = ‖µ +2ρc‖t?C .When π is an admissible (but not necessarily irreducible) representation of G, the lowest K-types of π are the
elements of K that appear in the restriction of π to K and have the smallest possible norm among the elementsof K that appear in π|K . Every admissible representation has a finite number of lowest K-types, and these do notdepend on the choice of T and ∆+
c .Given an irreducible tempered representation π of G, we say π has real infinitesimal character when there exists
a cuspidal parabolic subgroup P = LN of G and a discrete series representation σ of L, so that π is equivalent withone of the irreducible factors of IndG
P (σ ⊗1). We will not need to explain the link with the notion of infinitesimalcharacter and refer the reader to [68].
We write GRIC for the subset of G whose elements are the equivalence classes of irreducible tempered repre-sentations of G with real infinitesimal character. The existence of this class of representations was pointed out tome by Michel Duflo.
We now assume G to be a reductive group (we remind the reader that in this paper reductive means “linearreductive”) whose Cartan subgroups are all abelian.
Vogan discovered ([66], see also [65]) that if π is an irreducible tempered representations of G with real in-finitesimal character, then π has a unique lowest K-type. In addition, he proved the following theorem.
Theorem 3.1 (Vogan 1981 [66] ; see [69], Theorem 1.2). The map from GRIC to K which, to an irreducibletempered representation π of G with real infinitesimal character, associates the minimal K-type of π , is a bijection.
When µ is (the class of) an irreducible representation of K, we will write VG(µ) for the real-infinitesimal-characer irreducible tempered representation of G with minimal K-type µ .
We close this subsection with two remarks.• Remark 1. Still assuming that G is a linear reductive group with abelian Cartan subgroups, if π is an irre-
ducible admissible representation, then the minimal K-types of π occur with multiplicity one in π|K . This is a key
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fact in Vogan’s work and we will use it repeatedly.• Vague remark 2. When G has a nonempty discrete series, “most” representations in GRIC are discrete series
representations.
3.2. The Mackey-Higson correspondence for tempered representations
In this section, we assume that G is a (linear) reductive group in Harish-Chandra’s class and has abelian Cartansubgroups (see §0.1 in [66]). We first proceed to build tempered representations of G by following as closely aspossible the procedure described in section 2.3 to build representations of G0 from Mackey parameters.
Fix a Mackey parameter (χ,µ).Consider, as we did in section 2.3, the centralizer Lχ of χ in G for the coadjoint action.This is a reductive group; its Cartan subgroups are all abelian (combine [68], lemma 3.4(4), with [66], page
142). Furthermore, the “little group” Kχ is a maximal compact subgroup of Lχ . Applying Vogan’s theoremfrom the previous subsection, we can build, out of the irreducible representation µ of Kχ , a tempered irreduciblerepresentation of Lχ : the representation VLχ
(µ).The centralizer Lχ turns out to be the θ -stable Levi factor of a parabolic subgroup of G (see [68], Lemma
3.4(4)). Write Lχ = Mχ Aχ for its Langlands decomposition. This is a direct product decomposition; the group Aχ
is abelian and contained in expG(p), so that χ defines an abelian character of Lχ .From our Mackey datum (χ,µ), we can thus build as in section 2.3 a tempered irreducible representation
σ = VLχ(µ)⊗ eiχ of the centralizer Lχ .
In order to obtain a tempered irreducible representation of G, it is however not reasonable to keep imitating theconstruction of section 2.3 and induce without further precaution from Lχ to G. Indeed, our centralizer Lχ = ZG(χ)
in G is despite appearances a poor geometric analogue of the centralizer L0χ = ZG0(χ): these two groups usually
do not have the same dimension and the contraction of Lχ with respect to Kχ is usually not isomorphic with L0χ .
The induced representation IndGLχ(σ) is also likely to be very reducible.
But we recalled that there exists at least one parabolic subgroup Pχ = Lχ Nχ of G whose θ -stable Levi factor isLχ ; in fact there is only a finite number of subgroups of G that contain Lχ as a proper subgroup and are minimalamong those that contain Lχ as a proper subgroup, and these minimal extensions of Lχ are all conjugate withPχ . Now, the subgroup Pχ is closer to being a good geometric analogue in G of the centralizer L0
χ : these twogroups have the same dimension, and whenever P′χ is a parabolic subgroup of G with θ -stable Levi factor Lχ , thecontraction of P′χ with respect to Kχ is none other than L0
χ .It seems, then, that the representation obtained by extending σ to Pχ is the simplest possible analogue of the
representation of L0χ used in section 2.3. This makes it tempting to form the induced representation
M(δ ) = IndPχ(σ) = IndG
Mχ Aχ Nχ(VMχ
(µ)⊗ eiχ ⊗1). (3.1)
This is a tempered representation. To keep this subsection short, we will avoid recalling its construction right here,but refer the reader to §5.2.1 below; for now, we simply indicate that the last equality in (3.1) comes from the factthat VLχ
(µ) coincides with the representation of Lχ obtained by extending VMχ(µ) to Lχ , because both belong to
(Lχ)RIC and have the same minimal Kχ -type.
We can now state one of our main results.
Theorem 3.2.
(a) For every Mackey parameter δ , the representation M(δ ) is irreducible and tempered.
(b) Suppose δ and δ ′ are two Mackey parameters. Then the representations M(δ ) and M(δ ′) are unitarilyequivalent if and only if δ and δ ′ are equivalent as Mackey parameters.
(c) By associating to any Mackey parameter δ the equivalence class of M(δ ) in G, one obtains a bijection betweenD and G.
9
By comparing the above theorem with Mackey’s description of G0 in section 2.3, we obtain a bijection
M : G0∼−→ G. (3.2)
We call this bijection the Mackey-Higson correspondence.
Proof of Theorem 3.2. Our result is a simple and straightforward consequence of the classification of temperedirreducible representations of G; in other words, it is a simple and straightforward consequence of an immensebody of work (the bulk of which is due to Harish-Chandra).
We first remark that given Remark 2.2.(i) and the properties of unitary induction, we may assume without losinggenerality that χ lies in a?.
In that case, Mχ is a reductive subgroup of G that contains the compact subgroup M = ZK(a) of Remark 2.2.(ii);in addition, Aχ is contained in A, it is related with its Lie algebra aχ through Aχ = expG(aχ), and the group Mχ
centralizes Aχ . We recall that Lχ is generated by M and the root subgroups for those roots of (g,a) whose scalarproduct with χ is zero; in the Langlands decomposition, the subgroup Aχ is by definition the intersection of thekernels of these roots (viewed as abelian characters of A).
In order to choose a precise Nχ , fix an ordering on a?: with it comes a notion of positive weight for (g,aχ); thesum of root subspaces corresponding to positive weights yields a subalgebra nχ , and we set Nχ = expG(nχ). Butof course another choice for Nχ and Pχ in (3.1) would yield an equivalent representation of G.
Recall that an element in a? is called aχ -regular when its scalar product with every root for the pair (gC,aχ,C)
is nonzero. We now call on a theorem due to Harish-Chandra, although Harish-Chandra never published the theo-rem; see [40], Theorem 4.11, for semisimple G, [38], Theorem 14.93, for linear connected reductive G, and [68],Lemma 3.2(5) (with proofs in [59]) for the current class of reductive groups:
If β ∈ a? is aχ -regular and if η is an irreducible tempered representation of Mχ with real infinitesimal charac-ter, then IndG
Mχ Aχ Nχ(VMχ
(η)⊗ eiβ ) is irreducible.
To show that χ is aχ -regular, consider a root γ of (gC,aC) whose scalar product with χ is zero. Then γ is aroot for (lχ,C,aχ,C) (see the remarks on the structure of Lχ above), so that aχ is contained in the kernel of γ (seethe same remarks). Thus, the restriction of γ to aχ must be zero, and γ cannot be a root of (gC,aχ,C), because thelatter are precisely the roots which do not vanish on aχ .
Using Harish-Chandra’s theorem above, we obtain part (a) in Theorem 3.2. Part (b) then follows from the usualcriteria for equivalence between parabolically induced representations (see for instance [40], Theorem 4.11(i)):if M1A1N1 and M2A2N2 are two parabolic subgroups containing a parabolic subgroup MAN, if β1 and β2 areelements of a?1 and a?2 which are respectively a1 and a2-regular, and if η1 and η2 are two real-infinitesimal-character irreducible tempered representations of M1 and M2, then the representations IndG
M1A1N1(η1⊗ eiβ1) and
IndGM2A2N2
(η2⊗ eiβ2 ) are equivalent if and only if there is an element in the Weyl group W =W (gC,aC) that sendsβ1 to β2 and η1 to η2.
We now turn to (c), which is a simple consequence of the Knapp-Zuckerman classification of irreducible tem-pered representations [36, 37]. Suppose π is an arbitrary irreducible tempered representation of G; then thereexists a parabolic subgroup P = MPAPNP of G, a tempered irreducible representation σ of MP with real infinitesi-mal character, and an element ν of (aP)
?, so that
IndGMPAPNP
(σ ⊗ eiν)
is irreducible and equivalent with π . See [68], Theorem 3.3; for proofs see of course [36, 37] in the connectedcompact-center case, and [56], §5.4, for disconnected groups in the present class.
We need to prove that such a representation reads M(δ ) for some Mackey parameter δ . We first remark that Pis conjugate to a parabolic subgroup contained in Pν : since ν lies in (aP)
?, the centralizer LP = MPAP of a?P for the
10
coadjoint action is contained in Lν ; given the above remarks on the Langlands decomposition and the constructionof parabolic subgroups, we deduce that LP is contained in Lν , so that Mν contains MP and that Aν is containedin AP, and finally that Nν is conjugate to a subgroup N′ contained in NP. So P is conjugate to Mν Aν N′, which iscontained in Pν .
Write A and N for the subgroups of G whose Lie algebras are the orthogonal complements of aν and nν in aP
and nP, respectively. Then AP = Aν A and NP = Nν N; from the fact that mν is orthogonal to aν ⊕nν , we deducethat A and N are contained in Mν . Since AP is abelian and NP normalizes A, we obain
IndGMPAPNP
(τ⊗ eiν ⊗1
)= Ind(MPAN)Aν Nν
((τ⊗ e0)⊗ eiν) .
As a consequence, P = MPAN, is a subgroup of Mν , MA is the centralizer A in Mν , so that P is a parabolicsubgroup of Mν . Finally, we note that σ = IndMν
MAN(σ ⊗ e0) is a tempered representation of Mν , that it has real
infinitesimal character (by double induction from the definition of real infinitesimal character), and that it is irre-ducible (otherwise π would not be). By double induction, we conclude that
IndGPν
(σ ⊗ eiν)= IndG
Mν Aν Nν
(IndMν
P
(σ ⊗ e0)⊗ eiν
):
thus π is equivalent with M(ν ,µ), where µ is the minimal Kν -type of σ ⊗ e0. This concludes the proof.
3.3. Preservation of minimal K-types
In our construction of the Mackey-Higson correspondence in section 3.2, the notion of minimal K-types for ad-missible representations of G is of critical importance. Since K is also a maximal compact subgroup in G0, thenotion of minimal K-type for admissible representations of G0 can similarly be brought in, and every irreducibleadmissible representation of G0 has a finite number of minimal K-types.
Still assuming that G is a (linear) reductive group with abelian Cartan subgroups, we now prove that theMackey-Higson correspondence M : G0→ G preserves minimal K-types.
Proposition 3.3. Suppose π0 is a unitary irreducible representation of G0. Then then the representation π0 of G0
and the representation M (π0) of G and have the same minimal K-types.
We recall that π0 is equivalent with M0(χ,µ) for some Mackey datum (χ,µ). From the description of M0(χ,µ)
in section 2.3 we know that as a K-module, M0(χ,µ) is isomorphic with IndKKχ
(µ). So we need to show that theminimal K-types of the G-representation M(χ,µ) are exactly the irreducible K-submodules of IndK
Kχ(µ) with
minimal norm.Now, the definition of induced representations shows that M(χ,µ) is isomorphic, as a K-module, with IndK
Kχ
(V(µ)|Kχ
).
Of course that K-module contains IndKKχ
(µ).Suppose then that α is a minimal K-type in IndK
Kχ
(V(µ)|Kχ
), but is not a minimal K-type in IndK
Kχ(µ). Then
there is an element µ1 of Kχ , distinct from µ , such that α is a minimal K-type in IndKKχ(µ1), and since α must
appear with multiplicity one in IndKKχ
(V(µ)|Kχ
), µ1 must appear with multiplicity one in V(µ)|Kχ
. Because thelatter has only one minimal Kχ -type, we know that ‖µ1‖Kχ
must be greater than ‖µ‖Kχ.
If α were a minimal K-type in IndKKχ
(V(µ1)|Kχ
), the representations IndK
Kχ
(V(µ)|Kχ
)and IndK
Kχ
(V(µ1)|Kχ
)would have a minimal K-type in common; but that cannot happen, because of the following reformulation of aresult by Vogan:
Lemma 3.4. Suppose MAN is a cuspidal parabolic subgroup of G, and µ1, µ2 are inequivalent irreducible K∩M-modules. Then the representations IndK
K∩M (V(µ1)) and IndKK∩M (V(µ2)) have no minimal K-type in common.
Proof: When V(µ1) and V(µ2) are in the discrete series of M, this follows from Theorem 3.6 in [67] (see alsoTheorem 1 in the announcement [64], and of course [65]). In the other cases, this actually follows from the sameresult, but we need to give some precisions.
11
Assume that both V(µ1) and V(µ2) are either in the discrete series or nondegenerate limits of discrete series.Then both IndG
MAN (V(µ1)) and IndGMAN (V(µ2)) are irreducible constituents of some representations induced from
discrete series, from a larger parabolic subgroup if need be (see [36], Theorem 8.7). If IndGM∗A∗N∗ (δ1) (with
δ1 in the discrete series of M?) contains IndGMAN (V(µ1)) as an irreducible constituent, it must contain it with
multiplicity one, and the set of minimal K-types are IndGM∗A∗N∗ (δ1) is the disjoint union of the sets of minimal K-
types of its irreducible constituents (which are finite in number): see Theorem 15.9 in [38]. If IndGMAN (V(µ1)) and
IndGMAN (V(µ2)) are constituents of the same representation induced from discrete series, then the lemma follows;
if that is not the case we are now in a position to use Vogan’s result to the two representations induced fromdiscrete series under consideration (Vogan’s disjointness-of-K-types theorem is true of reducible induced-from-discrete-series representations).
Now, if V(µ1), in spite of its real infinitesimal character, is neither in the discrete series of M nor a nondegen-erate limit of discrete series, then there is a smaller parabolic subgroup M?A?N? and a discrete series or nondegen-erate limit of discrete series representation ε1 of M? such that IndK
K∩M(V(µ1)) = IndKK∩M?
(ε1). If necessary, wecan rewrite IndK
K∩M(V(µ2)) in an analogous way; then we can use Vogan’s result again, after some embeddings inreducible representations induced from discrete series as above if necessary. This proves the lemma.
Coming back to the proof of Proposition 3.3, we now know that if α is a minimal K-type in IndKKχ
(V(µ1)|Kχ
),
then it cannot be a minimal K-type in IndKKχ
(V(µ1)|Kχ
). Let then α1 in K be a minimal K-type in IndK
Kχ
(V(µ1)|Kχ
);
we note that ‖α1‖K < ‖α‖K . If α1 were to appear in IndKKχ(µ1), it would appear in IndK
Kχ
(V(µ)|Kχ
), and that can-
not be the case because α is already a minimal K-type there.We conclude that there exists α1 in K and µ1 in Kχ such that— ‖α1‖K < ‖α‖K and ‖µ1‖Kχ
> ‖µ‖Kχ
— α1 is a minimal K-type in IndKKχ
(V(µ1)|Kχ
), but it is not a minimal K-type in IndK
Kχ(µ1).
This seems to trigger an infinite recursion, because the same argument can be used again, beginning with(α1,µ1) instead of (α,µ); however, there are not infinitely many K-types which are strictly lower than α . Thusour hypothesis that the minimal K-type α in M(χ,µ) is not a minimal K-type in IndK
Kχ(µ) leads to a contradic-
tion.To conclude the proof, notice from the compatibility of induction with direct sums that
M(χ,µ) = IndKKχ
(V(µ)|Kχ
)= IndK
Kχ(µ)⊕ M, (3.3)
with M induced from a (quite reducible) Kχ -module.The above argument shows that every minimal K-type in M(χ,µ) must occur in as a minimal K-type in
IndKKχ
(µ). Conversely, every minimal K-type in IndKKχ
(µ) does occur as a minimal K-type in M(χ,µ): from(3.3) we know that it must occur there, and from the above argument we already know that it is lower than everyK-type occuring in M, so that it is a lowest K-type in IndK
Kχ(µ). This proves Proposition 3.3.
We note that the proof yields a non-elementary argument for an elementary-looking fact on the restriction to Kof unitary representations of G0:
Corollary 3.5. In every unitary irreducible representation of G0, each minimal K-type occurs with multiplicityone.
Proof: Suppose π0 is a unitary irreducible representation of G0; choose a Mackey parameter (χ,µ) for π0; recallthat M0(χ,µ) is equivalent, as a K-module, with IndK
Kχ(µ). In (3.3) we see that the multiplicity of every minimal
K-type in π0 is lower than its multiplicity in M(χ,µ); every minimal K-type occurs at least once in π0 and exactlyonce in M(χ,µ); the corollary follows.
Remark. Recent attempts at defining the Mackey-Higson correspondence seem to have started from the require-ment that the correspondence preserve minimal K-types and tried to find analogies between the subsets of G and
12
G0 consisting of representations with a given minimal K-type or set of minimal K-types. See for instance theconclusion to [30], or the conjectures in chapter 4 of [22].
We should note that the Mackey-Higson correspondence is compatible with other natural features of G andG0. We mention an elementary one, related to the existence in each dual of a natural notion of renormalization ofcontinuous parameters for irreducible representations:• On the G0-side, there is for every α > 0 a natural bijection
RαG0
: G0→ G0
obtained from Mackey’s description of G0 in section 2.3 by sending a representation π0 'M0(χ,µ) of G0
to the equivalence class of M0(χ
α,µ).
• On the G-side, there is also for every α > 0 a natural bijection
RαG : G→ G (3.4)
obtained from the Knapp-Zuckerman classification (see the proof of Theorem 3.2) as follows: starting withπ in G, we know that π is equivalent with some representation IndG
MPAPNP
(σ ⊗ eiν
), where MPAPNP, σ
and ν are as in the proof of Theorem 3.2; the tempered representation IndGMPAPNP
(σ ⊗ ei ν
α
)has the same
R-group ([36], Section 10) as IndGMPAPNP
(σ ⊗ eiν
), so it is irreducible; we define Rα
G(π) as the equivalenceclass of that representation.
It is immediate from the construction of the Mackey-Higson correspondence that it is compatible with theserenormalization maps:
Proposition 3.6. For every α > 0, we have RαG M = M Rα
G0.
Together Propositions 3.3 and 3.6 may shed some light on our construction of the correspondence: any bijec-tion
B : G0→ G
must, if it is to satisfy Proposition 3.6, induce a bijection between the fixed-point-setsπ0 ∈ G0 / ∀α > 0, Rα
G0(π0) = π0
and
π ∈ G / ∀α > 0, Rα
G(π) = π
.
The first class is the copy of K discussed in Remark 2.3.1; the second is the class GRIC discussed in section 3.1.So B must send any K-type µ to an element of GRIC; if B is to satisfy Proposition 3.3 it must send µ to VG(µ),like our correspondence M .
Now, observing the Knapp-Zuckerman classification, Lemma 3.4 and Proposition 3.3, and inserting the aboveremark, we note that if π0'M(χ,µ) is a unitary irreducible representation of G with nonzero χ , any representationof G having the same set of minimal K-types as π0 must read
IndMχ Aχ Nχ
(VMχ
(µ)⊗ eiν)for some ν in a?χ . We may view our correspondence M as that obtained by choosing ν = χ .
3.4. The Mackey-Higson correspondence for admissible representations
We now exend the Mackey-Higson correspondence M : G0 → G to a natural bijection between the admissible
duals G0adm
and Gadm.Let us first describe a way to build, out of an admissible Mackey datum δ = (χ,µ) (see section 2.4: χ ∈ a?C,
13
µ ∈ Kχ ), an admissible representation of G.Write χ = α + iβ , where α and β lie in a?. Consider the centralizer Lχ of χ for the coadjoint action: Lχ is
the intersection Lα ∩Lβ , and we note that in the notations of the proof of Theorem 3.2, Lχ is the centralizer ofaα +aβ . Thus Lχ appears as the θ -stable Levi factor of a real parabolic subgroup Pχ = Mχ Aχ Lχ of G, in whichKχ is a maximal compact subgroup. As before, we can build from δ the representation VMχ
(µ)⊗ eχ of Pχ ; wedefine
M(δ ) = IndGPχ(σ) = IndG
Mχ Aχ Nχ(VMχ
(µ)⊗ eχ ⊗1),
an admissible representation of G.
Theorem 3.7.(a) For every admissible Mackey parameter δ , the representation M(δ ) has a unique irreducible subquotient
Madm(δ ).
(b) Suppose δ and δ ′ are two admissible Mackey parameters. Then Madm(δ ) and Madm(δ ′) are equivalent if andonly if δ and δ ′ are equivalent as admissible Mackey parameters.
(c) By associating to any admissible Mackey parameter δ the equivalence class of Madm(δ ) in Gadm, one obtainsa bijection between Dadm and Gadm.
Combined with the description of G0adm
in section 2.4, this yields a natural bijection
M adm : G0adm ∼−→ Gadm (3.5)
whose restriction to the reduced dual G0 is the Mackey-Higson correspondence (3.2).
Proof of Theorem 3.7. We use the Langlands classification to reduce the description of the irreducible admissiblerepresentations of G to that of the irreducible tempered representations of some reductive subgroups of G.
We will write (a⊥α )? for the set of linear functionals on a whose restriction to aα is zero. Decompose
χ = α + i(β1 +β2), where α ∈ a?, β1 ∈ a?α and β2 ∈ (a⊥α )?.
We now point out that it is possible to view (β1,µ) as a (tempered) Mackey parameter for the reductive group Mα :indeed, Mα admits Kα as a maximal compact subgroup, and if we view β1 as a linear functional on mα ∩p (wheremα is the Lie algebra of Mα ), then the stabilizer of β1 for the action of Kα on (mα ∩p)? is Kα ∩Kβ1 = Kχ (sinceKα is contained in Kβ2 ).
We now remark thatM(χ,µ) = IndG
Pα
(MMα
(β1,µ)⊗ eα+iβ2). (3.6)
To see this, recall that MMα(β ,µ) is built using the centralizer
ZMα(β ) = g ∈Mα ,Ad?(g)β = β= Mα ∩Lβ = Mχ(Aχ ∩Mα) ;
thus, it is induced from a parabolic subgroup P of Mα , which reads P = Mχ(Aχ ∩Mα)N where N normalizesZMα
(β ). The right-hand side of (3.6) can thus be rewritten as
IndGMα Aα Nα
[IndMχ (Aχ∩Mα )N
(VMχ
(µ)⊗ eiβ)⊗ eα+iγ
],
which, by double induction, can in turn be written with only one parabolic induction as
IndGMχ [(Aχ∩Mα )(Aα )](NNα )
(VMχ
(µ)⊗ eα+i(β+γ)).
Since (Aχ ∩Mα)(Aα) is none other than Aχ , the parabolic subgroup that appears in the final expression is conjugateto Pχ , which proves (3.6).
14
From (3.6) we see that M(χ,µ) is induced from a tempered representation; furthermore, in (3.6) we knowfrom the proof of Theorem (3.2) that α is aα -regular; thus M(χ,µ) meets the usual criterion for the existenceof a unique irreducible subquotient (see [38], Theorem 8.54). This proves part (a) in Theorem 3.7; part (b) isa straightforward consequence of the usual criteria for equivalence between Langlands subquotients (implicit in[38], Theorem 8.54).
To prove part (c), recall from the Langlands classification that when π is an admissible irreducible representationof G, there exists a parabolic subgroup P = MPAPNP of G such that AP ⊂ A, there is an aP-regular element α in a?Pand an element γ in a?P, as well as an irreducible tempered representation σ of Mα , so that π is equivalent with theunique irreducible subquotient of IndG
P(σ ⊗ eα+iγ
). Since α is aP-regular, the subgroups Pα and P are conjugate,
and from Theorem 3.2, we know that the tempered representation σ of Mα is equivalent with MMα(β ,µ), for some
linear functional β on a∩ (mα ∩ p) = (a⊥α )? and some irreducible representation µ of Kα ∩ZMα
(β ) = Kα+iβ+iγ .We conclude that π is equivalent with a representation of the kind that appears in the right-hand side of (3.6); thisproves (c) and Theorem 3.7.
PART II. ON THE CONTRACTION OF TEMPERED REPRESENTATIONS
4. Deformation to the motion group
4.1. A family of Lie groups
For every nonzero real number t, we can use the global diffeomorphism
ϕt : K×p→ G
(k,v) 7→ expG(tv)k
to endow K×p with a group structure which makes it isomorphic with G. We write Gt for that group, so that theunderlying set of Gt is K×p, and the product law reads
(k1,v1) ·t (k2,v2) = ϕ−1t (ϕt(k1,v1) ·ϕt(k2,v2)) .
Recall from section 2.1 that the Cartan motion group G0 has underlying set K× p, and that the multiplicationin G0 reads (k1,v1) ·0 (k2,v2) = (k1k2,v1 +Ad(k1)v2).
It is well-known that the product law for gt goes to the product law for G0 as t goes to zero:
Lemma 4.1. Suppose (k1,v1) and (k2,v2) are two elements of K×p; then (k1,v1) ·t (k2,v2) goes to (k1,v1) ·0 (k2,v2)
when t goes to zero. The convergence is uniform on compact subsets of K×p.
Although this is a classical fact on Cartan motion groups, I have not been able to locate a complete proof in theliterature. Given the importance of this lemma for Parts II and II, it is perhaps reasonable to include a proof: thisone was communicated to me by François Rouvière.
Write (k1,v1) ·t (k2,v2) as (k(t),v(t)), where k(t) is in K and v(t) in p; what we have to prove is that when tgoes to zero, k(t) goes to k1k2 and v(t) goes to v1 +Ad(k1)v2. We start from the fact that
etv(t)k(t) =(etv1k1
)(etv2k2
)= etv1etAd(k1)v2k1k2. (4.1)
•Write
u : p→ G/K (4.2)
X 7→ expG(X)K
for the global diffeomorphism from the Cartan decomposition, and (g,X) 7→ g · x = geX K for the corresponding
15
action of G on p; then we deduce from (4.1) that u[tv(t)] = etv1 · u [tAd(k1)v2]. Now when t goes to zero, theright-hand side of the last equality goes to the origin 1GK in G/K, so that tv(t) must go to the origin of p; takingup (4.1) we deduce that k(t) goes to k1k2.• Let us turn to the convergence of v(t). Apply the Cartan involution θ to both sides of (4.1); we get e−tv(t)k(t) =e−v1e−tAd(k1)v2k1k2; taking inverses and multiplying by (4.1) yields
e2tv(t) = etv1e2tAd(k1)v2etv1 . (4.3)
From the Campbell-Hausdorff formula, we know that for small t, the right-hand side can be written as e2t(v1+Ad(k1)v2+r(t)),where r(t) is an element of g (the sum of a convergent Lie series) and r(t) = O(t) in the Landau “big O” notation.If t is close enough to zero, then both 2tv(t) and 2t (v1 +Ad(k1)v2 + r(t)) lie in a neighborhood of the origin g
over which expG is injective, so that (4.3) yields v(t) = v1 +Ad(k1)v2 + r(t). We deduce that for small enough t,r(t) lies in p, and of course since r(t) = O(t) we see that v(t) goes to v1 +Ad(k1)v2 as t goes to zero. The laststatement (the fact that the convergence is uniform on compact subsets) is now clear from the existence of explicitformulae for r(t) and k(t)(k1k2)
−1.
4.2. A family of actions and metrics on p
In this section, we describe a family of actions and a family of metrics on p. We shall use them in section 5 todescribe the contraction process for discrete series and spherical principal series represenations: these two caseswill introduce us to the main ideas and lemmas to be used for the general case in sections 6 and 7.
Suppose t is a real number (we do not exclude the case t = 0 here). Then Gt acts on the symmetric space Gt/K;because of the Cartan decomposition, the map
ut : pexpGt−→ Gt
quotient Gt/K (4.4)
is a global diffeomorphism. This yields a transitive action of Gt on p: if γ is an element of Gt , we write the actionof γ on p as x 7→ γ ·t x def
= u−1t[γ expGt
(x) K], where the product inside the bracket is that of Gt . Of course, we
obtain a similar action of G on p, defined through the diffeomorphism u from (4.2); we will write (g,x) 7→ g ·x forthe corresponding map from G×p to p.
The action of Gt on p induces a Gt -invariant riemannian metric on p: we choose a fixed K-invariant innerproduct B on p and write ηt for the (uniquely determined) Gt -invariant riemannian metric on p which coincideswith B at the origin of p. The metric ηt has constant scalar curvature, equal to−t2: thus (p,ηt) is negatively curvedfor every nonzero t, while (p,ηt) is Euclidean.
Since each Gt , t 6= 0, is obtained from G by a simple rescaling in the Cartan decomposition, there must be asimple relationship between actions and riemannian metrics we just defined on p. The precise relationship can bephrased using the dilation
zt : p→ p
x 7→ x/t
and the isomorphism ϕt : Gt → G from section 4.1:
Lemma 4.2. (i) For every x in p and g in G, we have ϕ−1t (g) ·t zt(x) = zt(g · x).
(ii) The metrics ηt and η1 are related through z?t ηt = t−2 ·η1.
Proof: For (i), fix x in p and g in G and observe the image of both sides of (i) by the diffeomorphism ut : on the
16
Figure 1 – This is a picture of (co)-adjoint orbits for G and Gt when G is SL(2,R). The horizontal plane is thespace p of trace-zero symmetric matrices, the vertical axis is the line k of antisymmetric matrices; the drawnmanifolds are the G-adjoint orbit and the Gt -coadjoint orbit of the point on the vertical axis. It is tempting, butwrong, to interpret the diffeomorphism u−1
t from (4.4) as the vertical projection; however, the vertical projectioncan be recovered as u−1
t τ , where τ : x 7→ sinh(‖x‖B)‖x‖B
Rot π2(x) accounts for the fact that the geodesics in (p,ηt) and
(p,η0) do not spread quite as quickly.
left-hand side,ut(ϕ−1t (g) ·t zt(x)
)= ϕ
−1t (g)expGt
(ztx)K = ϕ−1t (gexpG(x)K);
on the right-hand side, we see from the definitions of the actions that
ut (zt (g · x)) = expGt[zt(g · x)]K = ϕ
−1t (expG(g · x)K) = ϕ
−1t (gexpG(x)K) .
This proves (i). For (ii), we deduce from (i) that z?t ηt is a G1-invariant metric on p; but there are not many suchmetrics: given the fact that the derivative of zt at zero is multiplication by t−1, we know that z?t ηt and t−2 ·η1
coincide at zero, so these are two G1-invariant metrics on p which coincide at zero, and they must be equal.
We now give a precise statement for the idea that when t goes to zero the Gt action “goes to” the natural action ofG0 on p by Euclidean motions. Recall that each Gt , t ∈ R, has underlying set K× p, so the action of Gt on p isencoded by a map
At : (K×p)×p→ p
((k,v),x) 7→ (k,v) ·t x.
Lemma 4.3. For the topology of uniform convergence of compact subsets, At goes to A0 when t goes to zero.
Proof: For every ((k,v),x) in (K×p)×p and t in R, we first write
(k,v) ·t x =(ϕ−1t [ϕ1(k, tv)]
)·t x =
1t[ϕ1(k, tv)] · (tx) by lemma 4.2(i).
What we need to prove is that this goes to v+Ad(k)x when t goes to zero. But applying the diffeomorphism
17
u : p→ G/K, we can write
u(
1t
α(k, tv) · (tx))= expG
(1t
u−1 [ϕ1(k, tv)expG(tx)K]
)K
= expG
(1t
u−1 [expG(tv)k expG(tx)K]
)K
= expG
(1t
u−1 [expG(tv)expG(tAd(k)[x])K]
)K.
In the proof of Lemma 4.1, we saw that etvetAd(k)[x]) reads etβ (t)κ(t), where κ(t) lies in K and β (t) in p, withβ (t) = v+Ad(k)[x] + r(t), r(t) ∈ p, r(t) = O(t). We deduce that expG(tv)expG(tAd(k)[x]K = expG(tβ (t))K =
u [tβ (t)] and that
u(
1t
ϕ1(k, tv) · (tx))= expG
(1t[tβ (t)]
)K = u(v+Ad(k)[x]+ r(t)) ,
so 1t ϕ1(k, tv) · (tx) does go to v+Ad(k)x as t goes to zero. The fact that the convergence is uniform on compact
subsets is immediate from Lemma 4.1.
4.3. Fréchet contractions and a program for Part II
Suppose π is a tempered irreducible representation of G and π0 = M−1(π) is the corresponding representation ofG0 in the Mackey-Higson correspondence. We wish to give a meaning to the phrase “associated to the deformation(Gt) is a continuous family (πt) of representations, so that π1 = π and πt goes to π0 as t goes to zero”. Of coursea difficulty is that as t varies, the representations πt are likely to act on different spaces.
Here is the notion of “deformation of representations” to be used in this paper. For every irreducible temperedrepresentation π of G, we shall describe• a Fréchet space E, (A)• a family (Vt)t>0 of linear subspaces of E, (B)• for each t > 0, an irreducible representation πt : Gt → End(Vt) acting on Vt , (B’)• a family (Ct)t>0 of endomorphisms of E, to be called the contraction operators, (C)
with the folllowing properties.
• For every f in E and every t > 0, write ft for Ct f .— If f lies in V1, then for every t > 0, ft lies in Vt , (Vect1)— For every f in E, there is a limit (in E) to ft as t goes to zero. (Vect2)
• Set V0 =
limt→0
( ft), f ∈ V1
.
— Fix (k,v) in K×p. Fix F in V0 and choose f ∈ V1 such that F = limt→0
( ft).
Then there is a limit to πt(k,v) · ft as t goes to zero; the limit depends only on F. (Op1)— Write π0(k,v) for the map from V0 to itself obtained from (Op1). Then π0(k,v) is linear for every (k,v) in
K×p; the map π0 : K×p→ End(V0) defines a morphism from G0 to End(V0); the G0-representation(V0,π0) is unitary irreducible. Its equivalence class is that of the representation M−1(π) from theMackey-Higson correspondence. (Op2)
For each t > 0, our contraction operator Ct will be uniquely determined (up to a multiplicative constant) by anatural equivariance condition.
To simplify the notation, we identify G with G1 through the isomorphism ϕ1 : G1 → G of section 4.1 andidentify π with the representation π1 ϕ
−11 of G acting on V1. Then:
• if π has real infinitesimal character, we require that Ct intertwine the irreducible representations π andπt ϕ
−1t of G ; (NatRIC)
18
• for arbitrary π , we bring in the renormalization map R1/tG : G→ G from (3.6), and require that Ct intertwine
the irreducible representations R1/tG (π) and πt ϕ
−1t of G. (Natgen)
Of course we shall explain (in sections 5.2.1 and 5.2.2) why the renormalization in (Natgen) is necessary.
In the rest of Part II, we will implement this program for every irreducible tempered representation of G. Whenwe will have succeeded in carrying it our for a representation π , we shall say that the quadruple (E,(Vt)t>0,(πt)t>0,(Ct)t>0)
describes a Fréchet contraction of π onto π0.
5. Examples of Fréchet contractions: the discrete series and the spherical principal series
We here illustrate the program outlined in section 4.3 by implementing it for two particular classes of irreducibletempered representations of G.
There are at least three good reasons for beginning with these extreme and opposite examples before turningto more general results in sections 6 and 7, in spite of the fact that the later results will imply those of thissection:• All of the most important ideas in sections 6 and 7 will appear in these two examples, but they will be easier
to discern without the technicalities to be met in the later sections.• In our two examples, the representations admit geometric realizations as spaces of functions (or sections of
a finite-rank homogeneous vector bundle) on p: calling in the simple results of section 4.2 will make theproofs particularly simple.
• Most of the technical lemmas to be proven along the way will be used again in sections 6 and 7, making thelater sections a bit lighter.
5.1. Contraction of discrete series representations
Throughout section 5.1, we suppose G is a linear connected semisimple Lie group with finite center and assumethat G and K have equal ranks, so that G has a nonempty discrete series. We fix a discrete series representationπ of G and recall that the representation of G0 that corresponds to π in the Mackey-Higson correspondence isthe (unique) minimal K-type of π . We thus aim at describing a Fréchet contraction of π onto its minimal K-type.
5.1.1. Square-integrable solutions of the Dirac equation
We here recall some results of Parthasarathy, Atiyah and Schmid [4, 49] which provide a Hilbert space forπ .
Fix a maximal torus T in K. We will write Λ⊂ it? for the set of linear functionals that arise as the derivative ofa unitary character of T , ∆c for the set of roots of (kC, tC), and ∆ for the set of roots of (gC, tC); of course ∆c ⊂ ∆.If ∆+ is any system of positive roots for ∆, we can consider the associated half-sum ρ of positive roots, write ρc forthe half-sum of positive compact roots, and ρn for the half-sum of positive noncompact roots, so that ρc +ρn = ρ .We denote the ∆+
c -highest weight of µ by ~µ .We now call in the spin double cover Spin(p) of the special orthogonal group SO(p) associated with the inner
product B chosen in section 4.2, and its spin representation S. Recall that the condition that G and K have equalranks guarantees that dim(G/K) is an even integer, say 2q, that S has dimension 2q and splits into two irreducible2q−1-dimensional Spin(p)-submodules S+ and S−, with ρn a weight of S+ (for the action induced by the naturalmap from k to Spin(p)).
Suppose Vµ[ is the carrier space of an irreducible kC-module with highest weight µ[ = ~µ − ρn. Then we
can consider the tensor product Vµ[ ⊗S±, and although neither V
µ[ nor S± need be a K-module if G is not simplyconnected, the action of k on V
µ[ ⊗ S± does lift to K (the half-integral ρn-shifts in the weights do compensate).
So we can consider the equivariant bundle E = G⊗K
(V
µ[ ⊗S)
over G/K, as well as equivariant bundles E ± =
G⊗K(Vµ ⊗S±
).
19
Now, the natural G-invariant metric that G/K inherits from the Killing form of g and the built-in G-invariantspin structure of E make it possible to define a first-order differential operator D acting on smooth sections of E ,the Dirac operator. We will need a few immediate consequences of its definition the next subsection, so let usbriefly spell it out, referring to [49] for details.
We need the Clifford multiplication map c : pC → End(S). Whenever X is in p, c(X) sends S± to S∓; inaddition, the element X of g defines a left-invariant vector field on G/K, which yields a first-order differentialoperator XE acting (componentwise in the natural trivialization associated to the action of G on G/K) on sectionsof E . Suppose (Xi)i=1..2q is an orthonormal basis of p. The Dirac operator is then defined by
Ds =2q
∑i=1
c(Xi)XEi s (5.1)
for every section s of E . It splits as D = D++D−, with D± sending sections of E ± to sections of E ∓.Write H for the space of smooth, square integrable sections of E which are in the kernel of D. Since D is an
essentially self-adjoint elliptic operator, H is a closed subspace of the Hilbert space of square-integrable sectionsof E . Because D is G-invariant, H is invariant under the natural action of G on sections of E .
Theorem 5.1 (Parthasarathy, Atiyah & Schmid). The Hilbert space H carries an irreducible unitary represen-tation of G, whose equivalence class is that of π = VG(µ).
It is important for our purposes to point out that the details in Atiyah and Schmid’s proof show that the solutionsto the Dirac equation that are gathered in H do not explore the whole fibers, but that they are actually sections of asub-bundle of E whose fiber, a K-module, is irreducible and of class µ . To be precise, let W denote the isotypicalK-submodule of V
µ[ ⊗ S+ for the highest weight ~µ = µ[+ρn; the K-module W is irreducible; let W denote theequivariant bundle on G/K associated to W ; then:
Proposition 5.2 (Atiyah & Schmid). If a section of E is a square-integrable solution of the Dirac equation, thenit is in fact a section of W .
Atiyah et Schmid comment on this result as follows 1 :
We should remark that the arguments leading up to [the fact that the cokernel of the Dirac operatoris zero] are really curvature estimates, in algebraic disguise. The curvature properties of the bundlesand of the manifold G/K force all square-integrable, harmonic spinors to take values in a certain sub-bundle of E , namely the one that corresponds to the K-submodule of highest weight µ[+ρn in V
µ[⊗S+.
5.1.2. Contraction to the minimal K-type
We are now in a position to observe how, for every real number t 6= 0, the above construction yields a Gt -invariantDirac equation on p, and study the behaviour of solutions when the parameter t goes to zero.
Fix a real number t. Recall from section 4.2 the diffeomorphism ut : p→ Gt/K, the action of Gt on p and theGt -invariant metric ηt on p. Using the action of K on Vµ ⊗S, we use the results of the previous subsection to builda Gt -invariant spinor bundle Et sur Gt/K. The vector bundle u?t Et over p can be trivialized using the Gt -action:this yields a vector bundle isomorphism, say Tt , between u?t Et and the trivial bundle p× (V
µ[ ⊗S).By using the metric ηt to build a Gt -equivariant Dirac operator acting on sections of u?t Et , and trivializing using
1. In the quotation, the notations were changed in order to fit those of the present article : Atiyah and Schmid write µ for our µ[ andVµ ⊗S+ for our E . In [4], Proposition 5.2 is proven in a context where µ satisfies an additional degeneracy condition besides that guaranteeingthat VG(µ) lies in the discrete series; however, the rest of their paper shows that Proposition 5.2 does hold for every µ such that VG(µ) lies inthe discrete series.
20
Tt , we obtain a first-order differential operator D′t acting on C∞(p,Vµ[ ⊗S). Proposition 5.2 makes it tempting to
build from D′t a differential operator acting on
E = C∞(p,W ), (A)
so we set∆t := ProjW
[(D′t)
2∣∣C∞(p,W )
]where ProjW is the (orthogonal) isotypical projection V
µ[ ⊗S→W .Applying the above construction for each t (including t = 0), we obain a second-order G0-invariant differential
operator ∆0 on the flat space (p,η0), as well as Gt -invariant differential operators ∆t , t 6= 0, for the negatively-curved spaces (p,ηt). These operators do fit together pleasantly:
Lemma 5.3. The family (∆t)t∈[0,1] of differential operators on E is weakly continuous: if f lies in E, then(∆t f )t∈[0,1] is a continuous path in E.
Proof: Fix an orthonormal basis (Xi)i=1..2q of p as in (5.1) and use the 2q corresponding Cartesian coordinates onp; then from (5.1) we deduce that D′t
∣∣C∞(p,W )
reads
D′t∣∣C∞(p,W )
=2q
∑i=1
Ait∂i +Kt
where Ait , i = 1...2q and Kt are continuous maps from p to L (W,V
µ[ ⊗ S) (the space of all linear maps from
W to Vµ[ ⊗ S). Now from the definitions of ut and Tt , we easily see that each of the vector fields (Tt ut)
?XEti
defines a continuous endomorphism of C ∞(p×R) when this space is equipped with its usual Fréchet topology;given (5.1) we deduce that multiplication by (x, t) 7→ Ai
t(x) and (x, t) 7→ Kt(x) define continuous endomorphismsof C ∞(p×R), and Lemma 5.3 follows.
We know from the previous subsection that the L2 kernel of ∆t , t > 0, is relevant for realizing the discrete seriesrepresentation of Gt whose minimal K-type is µ . For every t > 0, we set
Vt =
f ∈ C∞(p,W ) / ∆t f = 0 and f ∈ L2(ηt ,W ). (B)
We introduce the action of Gt on E = C∞(p,W ) induced by the Gt action ·t on p of section 4.2 and the actionµ of K on W : if (k,v) is an element of K× p, interpreted as an element γ of Gt and f is an element of E, wedefine
πt(k,v) f : x 7→ µ(k) · f (γ−1 ·t x) (5.2)
where the inverse for γ is taken in Gt . This gives rise to a (very reducible) representation πt : GT → End(E); fromthe Gt -invariance of ∆t (and from the fact that D′t and its square have the same L2 kernel) we know that
The subspace Vt of E is πt -stable; the representation of πt on Vt is irreducible.
It is a discrete series representation of Gt with minimal K-type µ . (B’)
Recall from Lemma 4.2 that the dilationzt : x 7→ x
tintertwines the actions of G and Gt on p. We can use zt to transform functions on p, setting
Ct f := x 7→ f (t · x) (C)
for every f in E. As an immediate consequence of Lemma 4.2(ii) and the definition of the Dirac operator in (5.1),
21
we getC−1
t ∆tCt = t4 ·∆1.
Together with the fact that Ct f is square-integrable with respect to ηt as soon as f is square-integrable with respectto η1, this means that
If f lies in V1, then for each t > 0, the map Ct f lies in Vt . (Vect1)
We remark that our contraction operators (Ct)t>0 satisfy the naturality property (NatRIC) of section 4.3, and are(almost) uniquely determined by that constraint:
Lemma 5.4. Fix t > 0. Then
(i) The contraction operator Ct intertwines the representations (V1,π1 ϕ1) and (Vt ,πt ϕt) of G.
(ii) Our contraction operator Ct is the only map from V1 to Vt that satisfies (i) and preserves the value offunctions at zero, in that (Ct f )(0) = f (0) for every f in E.
Part (i) is immediate from Lemma 4.2, and part (ii) is a consequence of the irreducibility of π1 and πt andSchur’s lemma.
Let us turn to the convergence of the contraction process. Fix an f in V1, and set ft = Ct f . We first make thetrivial but crucial observation that in the Fréchet topology on E,
When t goes to zero, ft goes to the constant function on p with value f (0) ∈W . (Vect2)
In our current setting, more can be said. We know that the representation V1 splits as a direct sum according to thedecomposition of (π1)|K into K-types, so that we can write f as a Fourier series
f = ∑λ∈K
fλ
where, for each λ in K, fλ belongs to a closed subspace Vλ1 of V1 on which (π1)|K restricts as a direct sum of
copies of λ ; we recall from Harish-Chandra that Vλ1 is finite-dimensional.
Of course a parallel decomposition holds for Vt , t 6= 0, and ft , too, has a Fourier series
ft = ∑λ∈K
ft,λ .
For every λ in K, we note that the dimension of the λ -isotypical subspace Vtλ
is independent of t, and that thesupport of the above Fourier series does not depend on t.
Lemma 5.5. The Fourier component ft,λ is none other than Ct fλ .
Proof: For each λ ∈ K, the map
f 7→ Pλ f :=[
x 7→∫
Kξ?λ(k) µ(k) · f
(Ad(k−1)x
)dk]
has a meaning as a linear operator from C∞(p,W ) to itself. In the above formula ξλ is the global character of λ
− a continuous function from K to C − and the star is complex conjugation. Now, if f is an element of Vt , wecan view K as a subgroup of Gt and since the action of K on p induced from the Gt -action ·t is none other than theadjoint action, the formula for Pλ f turns out to be exactly the formula for the isotypical projection from Vt to Vλ
t .From Lemma 5.4, we deduce that Pλ commutes with Ct , which proves Lemma 5.5.
Lemma 5.6. If λ ∈ K is different from the minimal K-type µ , then fλ (0) = 0.
22
Proof: The origin of p is a fixed point for the action of K on p; so
fλ (0) = (Pλ f )(0) =∫
Kξ?λ(k) µ(k) · f (0)dk. (5.3)
Recall that f (0) is in W , which is an irreducible K-module of class µ: now, (5.3) is the formula for the orthogonalprojection of f (0) onto the isotypical component of W corresponding to λ ∈ K, and this projection is zero when-ever λ 6= µ .
Lemmas 5.5 and 5.6, together with property (Vect2) above, prove that each Fourier component of f goes to zeroas the contraction is performed, except for that which corresponds to the minimal K-type. We can thus refine(Vect1)-(Vect2) into the following statement. Set
V0 = constant functions on p with values in W .
Proposition 5.7.
(i) For each f ∈ E, there is as t goes to zero a limit f0 to Ct f for the topology of E.
(ii) This limit is the element of V0 with value f (0).
(iii) Suppose f is in V1; write fmin for the orthogonal projection of f onto the isotypical subspace Vµ
1 of V1 forthe minimal K-type µ . Then Ct( f − fmin) goes to zero as t goes to zero.
(iv) The correspondence f → f0 induces a K-equivariant isomorphism between Vµ
1 and V0.
We turn to the asymptotic behavior of the operators πt(k,v), when (k,v) is a fixed element of K×p − interpretedfor each t as an element of Gt .
Fix an element f of V1 and consider the associated family ( ft)t>0. To study πt(k,v) ft , remark that
πt(k,v) ft = πt(k,v)( ft − f0)+πt(k,v) f0, (5.4)
and insert the following observation:
Lemma 5.8. Fix (k,v) in K×p. The topology of E can be defined by a distance with respect to which each of theπt(k,v), t ∈ [0,1], is 1-Lipschitz.
Proof: Whenever A ⊂ p is compact, the subset Π(A) = (k,v) ·t A | t ∈ [0,1] is compact too (see Lemma 4.3).So there is an increasing family, say (An)n∈N, of compact subsets of p, such that Π(An) ⊂ An+1 for each n and∪n∈NAn = p.
For each (n,K) in N2 and f in E, set ‖ f‖n,K = sup|α|1≤K
supx∈An
‖∂ α f (x)‖W , where α ∈NK are multi-indices and | · |1
is the `1 height function on NK . Whenever f1 and f2 are in E, we have ‖πt f1−πt f2‖n,K ≤ ‖ f1− f2‖n+1,K . Recall
that a distance whose associated topology is that of E is d( f , f ′) = ∑n
‖ f− f ′‖n,K2n+K(1+‖ f− f ′‖n,K)
. Then d/2 has the desired
property.
Taking up (5.4), we can deduce that πt(k,v)( ft− f0) goes to zero as t goes to zero from the fact that ft− f0 doesand πt(k,v) does not increase the distance of Lemma 5.8. As for the second term πt(k,v) f0 in (5.4), it is equal toµ(k) f0 independently of t. In conclusion:
πt(k,v) · ft goes to µ(k) · f0 as t goes to zero. (Op1)
Since property (Op2) of section 4.3 is immediate when we set π0(k,v) = µ(k), we have proved:
23
Theorem 5.9. The quadruple (E,(Vt)t>0,(πt)t>0,(Ct)t>0) describes a contraction from the discrete representa-tion π to its minimal K-type µ .
We close our discussion of the discrete series with a side remark. It may seem annoying, in view of Lemma 5.3,to find that the space Vt of L2-solutions to the Gt -invariant Dirac equation ∆t f = 0 should converge in such anatural way as indicated by Theorem 5.9 to the G0-module V0, but to note V0 is not the L2 kernel of ∆0, since thatkernel is zero. A simple way to disminish the annoyance is to consider extended kernels, setting, for each t in R(including t = 0),
Vt =
f ∈ C∞(p,W ) / ∆t f = 0 and there is a constant c ∈W such that f + c ∈ L2(ηt ,W ).
Because the riemannian metric ηt has infinite volume, when f is in Vt , there can be only one constant c such thatf + c is square-integrable.
Lemma 5.10. (i) For t 6= 0, the extended kernel Vt is none other than the L2 kernel Vt .
(ii) For t = 0, the extended kernel H0 is the space of constant W-valued functions on p.
Proof: Let us come back to G/K and the Dirac operator D defined in section 5.1.1. Because of Parthasarathy’sformula for its square [49], we know that there is a scalar σ such that
D2 := D−D+ =−Ω+σ
where Ω is the Casimir operator acting on sections of E .Suppose a G-invariant trivialization of E is chosen, so that D2 is viewed as acting on functions from G/K to
Vµ[ ⊗S, and suppose D2g = 0, with g = f +C, f ∈ L2(G/K,V
µ[ ⊗S) and C a constant in Vµ[ ⊗S. Then
Ω f = σ f +σC. (5.5)
When C is nonzero, this is in fact impossible. To prove that, we use Helgason’s Fourier transform for functions onG/K (see [27], Chapter 3; we shall meet Helgason’s analogue of plane waves in §5.2.2). The Fourier transformof a smooth function with compact support on G/K is a function on a?×K/M, and it turns out that the transformdefines a linear isometry F between L2(G/K) and L2(a?×K/M) for a suitable measure on a?×K/M. In addition,there is a notion of tempered distributions on G/K and a?×K/M, and when f is a smooth, square-integrablefunction both f and Ω f are tempered distributions. Using this, the equality Ω f = σ f +σC becomes an equalityof tempered distributions on a?×K/M, namely
F(Ω f )−σF( f ) = σCδ(0,1M)
with δ(0,1M) the Dirac distribution at the point (0,1M). Of course, there are convenient transformation propertiesof F with respect to the G-invariant differential operators, and F(Ω f ) is actually the product of F( f ) − an elementof L2(a?×K/M) with a smooth function on a?. So if f were a smooth, square-integrable solution of (5.5), σCδ0
would be the product of an element in L2(a?×K/M) with a smooth function on the same space. This can onlyhappen if C is zero, which proves Lemma 5.10.
5.2. On the principal series and generic spherical representations
In this section, we consider the representations associated with Mackey parameters of the form δ = (χ,µ) whereχ is a regular element of a?. The corresponding representations of G are (minimal) principal series representa-tions, and several existing results can be understood as giving flesh to Mackey’s analogy at the level of carrierspaces.
24
There are several very well-known function spaces carrying a representation of G with class M(δ ) − see forinstance section VII.1. in [38]. We will use two of these standard realizations: in the first, the functions are definedon K/M (where M = ZK(a) is the compact group of Remark 2.2(ii)), which has the same meaning in G and G0; inthe second, to be considered here only when µ is trivial, the functions are defined on G/K, or equivalently on p,and the geometrical setting in section 4.2 will prove helpful.
5.2.1. A contraction of (minimal) principal series representations in the compact picture
We first assume δ = (χ,µ) to be an arbitrary Mackey parameter and recall the “compact picture” realization forthe G-representation M(δ ) from (3.1).
To simplify the notation, we write P = MAN for the cuspidal parabolic subgroup Pχ to be induced from.Suppose Vσ is the carrier space for a tempered irreducible M-module of class σ = VM(µ), and suppose that anM-invariant inner product is fixed on Vσ . Consider
Hcompσ = f ∈ L2(K;Vσ ) / f (km) = σ(m)−1 f (k),∀(k,m) ∈ K×M. (5.6)
To endow Hcompσ with a G-action, we call in the Iwasawa projections κ , m, a, ν sending an element of G to the
unique quadruple
(κ(g),m(g),a(g),ν(g)) ∈ K× expG (m∩p)×a×N such that
g = κ(g)m(g)expG(a(g))ν(g) (5.7)
(this quadruple is unique, see [38]). We recall that if P is minimal, the map m is trivial. For every g in G, anoperator acting on Hcomp
σ is defined by setting
πcompχ,µ (g) = f 7→
[k 7→ e〈−iχ−ρ, a(g−1k)〉
σ(m(g−1k))−1 f(κ(g−1k)
)](5.8)
where ρ is the half-sum of those roots of (g,a) that are positive in the ordering used to define N. The Hilbert space(5.6) does not depend on χ , but that the G-action (5.8) does.
We now assume that χ is a regular element of a?, so that Pχ is a minimal parabolic subgroup of G, and describea Fréchet contraction of π = M(χ,µ) onto M0(χ,µ).
Recall from Remark 2.2.(ii) that in the decomposition Pχ = Mχ Aχ Nχ , we have Mχ = M = ZK(a) and Aχ =
A.Fix t > 0, λ in a? and σ in M. Then we can view (λ ,σ) as a Mackey parameter for Gt and run through the
constructions of section 2.3 for Gt : this leads to defining a parabolic subgroup Pλ ,t = MtAtNt of Gt , with Mt = M,and to considering the principal series representation π
t,compλ ,σ of Gt acting on
Hcompσ from (5.6)
viaπ
t,compλ ,σ (γ) : f 7→
[k 7→ exp〈−iλ −ρt ,at(γ
−1k)〉 f(κt(γ
−1k))]
(5.9)
where ρt is the half-sum of positive roots for (gt ,a) associated with the ordering used to define Nt , and κt : Gt→K,at : Gt → a are the Iwasawa projections.
At this point, given our Mackey parameter (χ,µ) it may seem natural to fix an element (k,v) in K×p, view itas an element of Gt for each t and try to study the convergence of π
t,compχ,µ (k,v) from (5.9) as t goes to zero. We
shall pursue this, but we must point out that this idea surreptitiously brings in the renormalization of continuousparameters from property (Natgen) in section 4.3. Indeed, define a representation of G as the composition
πt,compχ,µ (γ)ϕ
−1t : G
ϕ−1t−→ Gt
πt,compχ,µ−→ End(Hcomp
σ ).
25
Proposition 5.11. For each t > 0, πt,compχ,µ (γ)ϕ
−1t is equal to π
compχ/t,µ .
To prove Proposition 5.11, we only have to understand what happens to the half-sum of positive roots when wego from G to Gt , and to make the relationship between the Iwasawa decompositions in both groups clear. Here arestatements that apply to any parabolic subgroup.
Lemma 5.12. Suppose P = MAN is an arbitrary parabolic subgroup of G with A contained in expG(p). If α ∈ a?
is a root of (g,a), then t ·α is a root of (gt ,a).
Proof: When α is a root of (g,a), there is a nonzero X ∈ g such that [X ,H] = α(H)X for each H ∈ a. To keeptrack of X through the contraction, let’s write X = Xe +Xh with Xe ∈ k and Xh ∈ p. Then
[Xe,H]−α(H)Xh = [Xh,H]−α(H)Xe. (5.10)
The left-hand-side of (5.10) is in p and the right-hand-side is in k, so both are zero.Now, the group morphism ϕt : Gt → G from section 4.1 induces a Lie algebra isomorphism φt : gt → g. The
isomorphism φ−1t sends X to X t = Xe +
1t Xh ∈ gt , and for each H ∈ a,
[X t ,H]gt = [Xe,H]gt +1t[Xh,H]gt = [Xe,H]g+ t[Xh,H]g (5.11)
(for the last equality, recall that if U,V are in g, the bracket [U,V ]gt is defined as φ−1t [φtU,φtV ]g, so that [U,V ]gt =
[U,V ]g when U lies in k and V lies in p, whereas [U,V ]gt = t2[U,V ]g when they both lie in p).The right-hand side of (5.11) is α(H)Xh + t ·α(H)Xe = t ·α(H)X t , so X t is in the (gt ,a) root space for t ·α ,
whence Lemma 5.12.
The proof shows that the root space for t ·α is the image of gα under φ−1t ; we need to spell out a simple and
straightforward consequence.
Lemma 5.13. Suppose P = MAN is a parabolic subgroup of G. Then Pt := ϕ−1t (P) is a parabolic subgroup of Gt ,
with Iwasawa decomposition MtAtNt = (ϕ−1t M)A(ϕ−1
t N). The Isawawa projections for G and Gt are related asfollows: for every g in G,
κt(ϕ−1t g) = κ(g);
mt(ϕ−1t g) = m(g);
at(ϕ−1t g) =
a(g)t
.
We return to the proof of Proposition 5.11, assuming again that χ is regular. Inspecting the definitions of section3.2, we know from Lemma 5.13 that the parabolic subgroup Pχ,t of Gt used in (5.9) is ϕ
−1t (Pχ) = MAϕ
−1t (Nχ).
From Lemma 5.12 we know that ρt in (5.9) is tρ .Now fix g in G. Then
πt,compχ,µ (ϕ−1
t (g)) = f 7→[k 7→ exp〈−i
χ
t−ρ, t ·at(ϕ
−1t[g−1k
])〉 f(κt(ϕ
−1t[g−1k
]))]
= f 7→[k 7→ exp〈−i
χ
t−ρ,a(g−1k)〉 f
(κ(g−1k)
)](using Lemma 5.13)
= πcompχ
t ,µ(g).
With Proposition 5.11 and its proof in hand, discussing the contraction from G to G0 becomes particularlysimple. Set
E = Hcompσ from (5.6) (A)
26
and for every t > 0,
Vt = E (B)
πt = πt,compχ,µ from (5.9) (B’)
Ct = idE. (C)
Fix (k,v) in K×p. For every t > 0, the endomorphism πt(k,v) of E is because of Proposition 5.11 an operator fora principal series representation of G with continuous parameter χ
t ; but as t goes to zero it gets closer and closerto an operator for the representation of G0 with Mackey parameter (χ,µ):
Proposition 5.14. Fix (k,v) in K×p and f in E. Then there is is a limit to πt(k,v) f as t goes to zero; it is given byπ0(k,v) f from (2.3). The convergence holds both in the L2(K,Vµ) sense and in the sense of uniform convergencein C(K,Vµ).
Proof: Recall that when k is in K and v in p,
the element expGt(v)k of Gt is just (k,v) in the underlying set K×p. (5.12)
As a consequence, we can rewrite (5.9) as
πt,compχ,µ (k,v) = f 7→
[u 7→ exp〈−iχ− tρ,at
((k−1 ·t expGt
(−v) ·t u))〉 f(κt(k−1 expGt
(−v)u))].
where the symbols ·t indicate products in Gt . On the other hand, recall from section ?? that
π0(k,v) = f 7→[u 7→ exp〈iχ,Ad(u−1)v〉 f
(k−1u
)].
To make the two look more similar, notice that[π
t,compχ,µ (k,v) f
](u) = e〈−iχ−tρ,at((k−1u)·t expGt (−Ad(u−1)v))〉 f
(κt(k−1uexpGt
(−Ad(u 1)v))
= e〈−iχ−tρ,at(expGt (−Ad(u−1)v))〉 f((k−1u) ·κt(expGt
(−Ad(u−1)v))).
We now need to see how the Iwasawa projection parts behave as t goes to zero. Let us introduce, for every t > 0,the Iwasawa maps
Kt : p→ K and It : p→ a (5.13)
v 7→ κt(expGt
(v))
v 7→ at(expGt
(v)).
The Iwasawa map It from p to a is a nonlinear map, but as t goes to zero it gets closer and closer to a linearprojection :
Lemma 5.15. As t tends to zero,• It admits as a limit (in the sense of uniform convergence on compact subsets of p of every derivative) the
orthogonal projection from p to a, while• Kt tends to the constant function on p with value 1K .
Proof: •We check that It is none other than v 7→ 1t I(tv). Since ϕt is a group morphism from Gt to G, the definition
of group exponentials does imply that expG(tv) = expG(dϕt(1)v) = ϕt(expGt
v). Let us write expGt
v = keIt (v)nt
with k ∈ K and nt ∈ Nt , then ϕt(expGt
v)= ketIt (v)n, with n = ϕt(nt) in N. So we know that
expG(tv) = ketIt (v)n (5.14)
and thus that I(tv) = tIt(v), as announced.
27
But then as t goes to zero, the limit of It(v) is the value at v of the derivative dI(0). This does yield theorthogonal projection of v on a: although the Iwasawa decomposition of g is not an orthogonal direct sum becausek and n are not orthogonal to each other, they are both orthogonal to a with respect to the Killing form of g, so thedirect sum k⊕n is the orthogonal of a.• As for Kt , from (5.14) we see that Kt(v) = κt (expGt v) = κ (expG(tv)), and this does go to the identity uni-
formly on compact subsets as t goes to zero (here the topology used in the target space K is associated with thebi-invariant metric on K whose volume form is the normalized Haar measure).
The “convergence with respect to the topology of uniform convergence” part of Proposition 5.14 follows imme-diately, and of course since we are dealing with continuous functions on a compact manifold here, the statementon L2 convergence follows too.
Remark. Proposition 5.14 can be viewed as a reformulation of Theorem 1 in Dooley and Rice [18]. Our reason forincluding it to the present article is that in our treatment of the general case in section 7, we will take up the strategyof the proof and use the lemmas here proven. The interplay with Helgason’s waves (in the next subsection) mayalso throw some light on the phenomenon, for instance on the necessity of renormalizing continuous parameters.
5.2.2. A contraction of generic spherical representations using Helgason’s waves
Helgason’s waves and picture for the spherical principal series. We first recall how the compact picture ofthe previous subsection is related to the usual "induced picture", for which the Hilbert space is
Hindδ
=
f : G→Vσ
∣∣ f (gmeHn) = e〈−iλ−ρ,H〉σ(m)−1 f (g) for all (g,meHn) ∈ G×P
and f∣∣K ∈ L2(K;Vσ )
, (5.15)
the inner product is the L2 scalar product between restrictions to K, and the G-action is given by π indδ
(g) f =[x 7→ f (g−1x)
]for (g, f ) in G×Hind
δ. Because of the P-equivariance condition in (5.15), every element of Hind
δis
completely determined by its restriction to K, so that restriction to K induces an isometry (say I ) between Hindδ
and Hcompσ . The definition of π
compλ ,µ in (5.8) is just what is needed to make I an intertwining operator.
We now consider a Mackey parameter δ = (χ,µ) and assume that• χ is regular and lies in a?, so that Kχ is equal to M = ZK(a) ;• µ is the trivial representation of M.
There is then a distinguished element in Hcompσ : the constant function on K with value one. Under the isometry I ,
it corresponds to the functioneλ ,1 = keHn 7→ e〈−iλ−ρ,H〉
in Hindδ
; this in turn defines a function on G/K if we set eλ ,1(gK) = eλ ,1(g−1), and a function on p if we seteλ ,1(v) = eλ ,1(expG(v)K). Figure 5.2.2 shows a plot of eλ ,1 when G is SL2(R).
Now set eλ ,b(v) = eλ ,1(b−1v) for b in K/M and v in p. Then
L2(K/M) = Hcompσ → C∞(p)
F 7→∫
K/Meλ ,bF(b)db
turns out to intertwine πcompλ ,1 with the quasi-regular action of G on the Fréchet space
E = C∞(p) (A)
associated with the action (g,x) 7→ g · x defined below (4.4), and to be an injection (see [27], Chapter 3). We
28
Figure 2 – Plot of the real part of the Helgason wave e30,1. We used the mapping from R2 to the unit disk providedby the Cartan decomposition, and the explicit formulae availiable on the unit disk: see [26], chapter 0. The x- andy- range is [-1.5, 1.5] (this region is chosen so that the modulus varies clearly but within a displayable range, andthe choice of λ is to have enough waviness in the region).
write
VλHelgason =
∫K/M
eλ ,bF(b)db, F ∈ L2(K/M)
(5.16)
for the image of this map, a geometric realization for the spherical principal series representation of G withcontinuous parameter λ .
For each t > 0, we can repeat this construction to attach to each regular λ in a? a geometric realization for thespherical principal series representation of Gt with continuous parameter λ .
Of course the action of Gt on p described in section 4.2 gives rise to the quasi-regular action
πt : Gt → End(E) (5.17)
where an element γ = (k,v) in Gt acts through πt(k,v) : f 7→[x 7→ f (γ−1 ·t x)
].
After a patient inspection of definitions and Lemma 5.12, define, for every λ in in a? and b in K/M, a func-tion
εtλ ,b : p 7→ C
v 7→ e 〈 iλ+tρ , It (Ad(b)·v) 〉
where It is the Iwasawa projection from (5.13). Define
Vλt =
∫B
εtλ ,bF(b)db, F ∈ L2(B)
; (B)
this is a πt(Gt)-stable subspace of E. We note that Vλ1 coincides with Vλ
Helgason from (5.16).
Convergence of Helgason waves and contraction of spherical representations. Still assuming that χ is aregular element of a?, suppose π is the spherical representation of G with continuous parameter χ; then thegeometric realization for π described above is strongly reminiscent of the description we gave in Remark 2.3.2.for the representation M0(χ,1) of G0 associated to π by the Mackey-Higson correspondence. There G0 acted onthe subspace of E whose elements are combinations of plane waves whose wavevectors lie on the Ad?(K)-orbit ofχ in p?. Now, here is an immediate consequence of Lemma 5.15:
29
Figure 3 – Illustration of Lemma 5.16 : these are plots of ε1/2k
λ ,1 , k = 0,1,2,3, in the same domain as in Figure 5.2.2.Each of these waves is a building block for a principal series representation of G whose continuous parameter is2kλ , with λ = 30 here.
Lemma 5.16. Fix λ in a? and b in K/M. Then as t goes to zero, the Helgason waves ε tλ ,b converge (in the Fréchet
space E) to the Euclidean plane wave v 7→ e〈iλ ,Ad(b)·v〉.
This is illustrated in Figure 5.2.2.Of course Lemma 5.16 makes it tempting to use the spaces in (B) to write down a Fréchet contraction of π to
the representation of G0 on
V0 =
[v 7→
∫K/M
e〈iχ,Ad(b)·v〉F(b)db], F ∈ L2(K/M)
(5.18)
where an element (k,v) of G0 acts on E through the usual π0(k,v) : f 7→[x 7→ f (k−1x− k−1v)
].
The technical necessities have already been dealt with in sections 4.2 and 5.1. Recall the operator
Ct : E→ E (C)
f 7→ [x 7→ f (tx)]
from section 5.1.
Lemma 5.17. Suppose λ is an element of a? and t > 0.
(i) For every b in K/M, the operator Ct sends eλ ,b to ε ttλ ,b.
(ii) Fix f in E. Then Ct f lies in Vλt if and only f lies in V
λt
1 .
(iii) The map Ct inertwines the actions π and πt ϕ−1t of G on V
λt
1 and Vλt , respectively.
(iv) It is the only map satisfying (iii) which preserves the values of functions at zero.
Proof: For (i), we use Lemma 5.13 to find that for all v in p,
εttλ ,b(v) = e 〈 iλ+ρ , I(b·(tv)) 〉 = ε
1λ ,b(tv) = ε
1λ ,b(tv) = eλ ,b(tv).
Deducing (ii) is immediate. Part (ii) is from Lemma 4.2(i), and part (iv) is Schur’s Lemma.
It may be instructive to compare (i) and (ii) of the above result with Proposition 5.11 and property (Natgen) ofsection 4.3: starting from the fact, already used for the discrete series, that the zooming-in operator Ct intertwinesthe G- and Gt - actions on p, it is tempting to zoom-in on a Helgason wave for G to get a Helgason wave for Gt ;without any renormalization of the spacing between phase lines, however, it would be impossible to expect anykind of nontrivial convergence for the zooming-in process. In order to obtain a contraction from π 'M(χ,µtriv)
30
to π0 'M0(χ,µtriv), it is therefore natural to use, for the carrier space at time t,
Vt = Vχ
tt . (B, corrected)
Proposition 5.18. The quadruple (E,(Vt)t>0,(πt)t>0,(Ct)t>0) describes a Fréchet contraction from π to π0.
The only property that still needs proof is (Op1). We take up the reasoning in Lemma 5.4: fix (k,v) in K× p
and f =∫
B ε1λ ,bF(b)db in V1, set ft = Ct f =
∫B ε t
λ ,bF(b)db for every t > 0, write f0 : v 7→∫
B e〈iλ ,Ad(b)·v〉F(b)dbfor the limit of ft as t goes to zero, and remark that
πt(k,v) ft = πt(k,v)( ft − f0)+πt(k,v) f0. (5.19)
We now insert Lemma 5.8 (in the case of trivial fibers) to find that the first term goes to zero as t goes to zero, andLemma 4.3 to find that the second term goes to π0(k,v) f0. This proves propery (Op1); the other properties havealready been verified or are immediate.
6. Real-infinitesimal-character representations
We begin our search for a Fréchet contraction in the general case. This section describes a contraction of everyirreducible tempered representation of G with real infinitesimal character onto its lowest K-type, in case G isconnected semisimple. In §7, we will reduce the general case to that one.
6.1. Reduction to the quasi-split case
In this section, we assume that G is a connected semisimple Lie group with finite center.
6.1.1. Cohomological induction and real-infinitesimal-character representations
Fix an irreducible tempered representation π with real infinitesimal character and write µ for its lowest K-type. Abody of work by Vogan, Zuckerman, Schmid, Wong and others make it possible to build out of µ a pair (L,σ),where• L is a quasi-split Levi subgroup of G, the centralizer of an elliptic element in g? ;• σ is a tempered irreducible representation of L with real infinitesimal character,
and obtain from (L,σ) a geometric realization for π , based on the existence of a G-invariant complex structureon the elliptic coadjoint orbit G/L. Indeed, from (L,σ) we may form the holomorphic equivariant bundle V overG/L with fiber an irreducible L-module built from σ (the bundle usually has infinite rank); one of the Dolbeaultcohomology spaces H0,q(G/L,V ) then yields a geometric realization for π .
Here is a summary of the construction.Fix a maximal torus T in K. Vogan defines from µ an element λ (µ) in t? (see [53], Proposition 2.3 and
Corollary 2.4), as follows. Write ∆+c for a system of positive roots for ∆(kC, tC), ρc for the half-sum of positive
roots, ~µ for the corresponding highest weight of µ; in ∆ = ∆(gC, tC), fix a positive root system ∆+ making ~µ +ρc
dominant; form the corresponding half-sum ρ of posiive roots; finally, define λ (µ) ∈ t? to be the projection of~µ +2ρc−ρ on the convex cone of ∆+-dominant elements.
Consider the cenralizer L of λ (µ) in G for the coadjoint action. This is a quasi-split reductive group of G,which is usually not compact; it admits (K∩L) as a maximal compact subgroup.
The representation σ of L to be used in the construction is specified, for instance, in Theorem 2.9 of [53] :it is the inverse image of π under the bijection (2.2) from there (here are some precisions : π belongs to theclass denoted by Πλa
a (G) in the right-hand side of ([53], 2.2); the inverse image σ is an irreducible admissiblerepresentation of L ; from Corollary 4.4 in [53] we know that it is tempered; from the way the bijection ([53], 2.2)affects infinitesimal character, we know that it has real infinitesimal character). The unique lowest (K ∩L)-typeof σ is fine, in the sense of [66], Definition 4.3.9; as a consequence, σ appears as an irreducible constituent of a
31
principal series representation of L − we shall say more on this in section 6.2.In the complexification GC, there exists a parabolic subgroup Q with Levi factor LC (for an especially conve-
nient choice, see [69], section 13). Fix such a parabolic subgroup and write u for the unipotent radical of the Liealgebra q, so that q = lC⊕ u; the complex dimension of u is n = 1
2 (dim(G)− dim(L)). The action of L on q/lCinduces an action on u; thus, on the complex line ∧n(u) (top exterior power), there acts abelian character of L: wewill write e2ρ(u) for it.
From an irreducible (l,(K ∩ L))-module V with class σ , form the (l,(K ∩ L))-module V ] = V ⊗∧n(u); thegiven choice of Q equips G/L with a G-invariant complex structure inherited from GC/Q; we can then form theholomorphic vector bundle V ] = G×L (V ]) over G/L, then the Dolbeault cohomology groups H(p,q)(G/L,V ])
(for these, see [70], section 7, and of course [76], section 2). Wong’s work on the closed-range property for theDolbeault operator ([76], Theorem 2.4.(1)) equips H(p,q)(G/L,V ]) with a Fréchet topology and a (g,K)-modulestructure.
The inclusion K/(K∩L) →G/L is holomorphic; we write X for the (maximal) compact complex submanifoldK/(K∩L) of G/L, and
s = dimC(K/(K∩L)
)for the complex dimension of X .
Theorem (from work by Vogan, Zuckerman, Knapp, and Wong). The (g,K)-module H(0,s)(G/L,V ]) is irre-ducible and its equivalence class is that of π .
In the rest of §6, we will use this realization of π (thus, we shall view π as a (g,K)-module and stay away fromthe delicate questions related to unitary globalizations).
Keeping in mind that our objective is the description of a Fréchet contraction of π onto its lowest K-type µ , wemention an analogous geometrical realization for µ .
Recall that the representation (V,σ) of L has a unique lowest (K∩L)-type: let us write µ[ for it, and W for theµ[-isotypical subspace of V ; because µ[ occurs with multiplicity one in σ , the (K ∩L)-module W is irreducible.The actions of L on V and ∧n(u) induce an action of (K∩L) on W and another action of K∩L on ∧n(u); we writeW ] for the (K ∩L)-module W ⊗∧n(u). We should emphasize the importance of using ∧n(u) once more: we donot switch to ∧top(u∩ k), although ∧n(u) does not have any meaning “internal to K”.
We can as before form the holomorphic vector bundle V ] = K×K∩L (W ]) over X = K/(K ∩L) (this one hasfinite rank), and obtain the Dolbeault cohomology groups H(p,q)(X ,W ]).
Proposition. The K-module H(0,s)(X ,W ]
)is irreducible and its equivalence class is µ .
Proof: Using Wong’s work ([76], Theorem 2.4) and the isomorphism theorems in [39], Chapter VIII, the (g,K)-module H(0,s)
(X ,V ]
)defined above, the (g,K)-module R s(V ) defined in [39], Eq. (5.3b), and the (g,K)-module
Ls(V ) defined in [39], Eq. (5.3a), are all equivalent. To describe their common minimal K-type, we can proceedas follows: from Wong’s work we know that the K-module H(0,s)
(X ,W ]
)is isomorphic with that written LK
s (W )
in [39], (5.70); because of the remark beloc Corollary 5.72 in [39], the latter K-module is irreducible. Finally,the bottom-layer map (from [39], Section V.6) induces, as recalled for instance in Theorem 2.9.(a)-(b) in [53], aK-equivariant isomorphism between LK
s (W ) (viewed as a K-submodule of LKs (V )) and the µ-isotypical subspace
in Ls(V )' π .
6.1.2. Zooming-in operators and contraction in Dolbeault cohomology
In this section, we show that given a sufficiently pleasant Fréchet contraction of the representation σ of L onto itsminimal (K∩L)-type µ[, it is possible to build a Fréchet contraction of π onto its minimal K-type.
32
On a Mostow decomposition and the structure of elliptic coadjoint orbits. We first give a meaning to thenotion of “zooming-in on a neighborhood of K/(K∩L) in G/L”. A few remarks on the structure of the coadjointorbit G/L will help.
Fix a G-invariant nondegenerate bilinear form on g and write s for the orthogonal complement of (l∩p) in p;the dimension of s is an even integer. Of course the map
Ψ : K× s→ G/L
(v,u) 7→ k · expG(v) ·L
cannot be a global diffeomorphism, since the dimensions of the source and target spaces are dim(K ∩ L) unitsapart. Still, this map will be useful to us:
Lemma 6.1. (a) The map Ψ is surjective.
(b) Given (k,v) and (k′,v′) in K×s, the condition Ψ(k,v) = Ψ(k′,v′) is equivalent with the existence of some u inK∩L such that: v = Ad(u) · v′ and k = k′ ·u.
Proof: G. D. Mostow proved in 1955 ([47], Theorem 5) that the map
K× s× (l∩p)→ G
(k,v,β ) 7→ k · expG(v) · expG(β ) (6.1)
is a diffeomorphism. Part (a) immediately follows. For (b), fix two pairs (k,v) and (k′,v′) in K× s and assumeΨ(k,v) = Ψ(k′,v′), which means that k′ev′ ∈ kevL; then there is an element β in (l∩p) and an element u in K∩Lsuch that: k′ev′ = kevueβ = (k′u)eAd(u)v′eβ ; from this and Mostow’s decomposition theorem we get (b). We men-tion that the above is very close to the proof of Lemma 4.4 in [6].
Now, equip K× s with the K ∩L-action in which an element u of K ∩L acts through (k,v) 7→ (uk,uv); writeY for the quotient manifold (K× s)/(K ∩L) and projY : K× s→ Y for the quoient map. Given Lemma 6.1, theunique map ψ : Y →G/L such that ψ projY =Ψ is a global diffeomorphism. The inverse image of X =K/(K∩L)under the diffeomorphism ψ is a compact submanifold of Y with codimension dim(s), which we will still denoteby X .
For every t > 0, the map (k,v)→ (k, vt ) (from K× s to itself) induces a map from Y to itself; this “zooming-in
map” preserves the compact submanifold X .
Spaces of “differential forms” and models for the Dolbeault cohomologies. We now bring in the deformation(Gt)t∈R. Taking up notations from sections 4.3 and 6.1.1, we assume given a Fréchet contraction
(E,(Vt)t>0,(σt)t>0,(ct)t>0) (6.2)
of σ onto its minimal (K∩L)-type.For every t > 0, we can consider the subgroup Lt = ϕ
−1t (L) of Gt (where ϕt : Gt → G is the isomorphism
from section 4.1); the maximal compact subgroup K ∩ Lt does not depend on t. Setting Qt = ϕ−1t (Q) yields a
parabolic subgroup in the complexification of Gt ; write ut = ϕ?t u for the unipotent radical of its Lie algebra. From
the (lt,C,K ∩L)-module Vt and the abelian character e2ρ(ut ) of Lt acting on ∧n(ut), we can form the (lt,C,K ∩L)-module V ]
t =Vt ⊗∧n(ut) and the Dolbeault cohomologies H p,q(Gt/Lt ,V]
t ) from section 6.1.1.The results recalled in section 6.1.1 show that H0,s(Gt/Lt ,V
]t ) carries an irreducible tempered representation
of Gt with minimal K-type µ .
The program described in section 4.3 can only be realized if we embed all of the Fréchet spaces H0,s(Gt/Lt ,V]
t )
33
inside a common Fréchet space. To do so, we need a convenient definition for the notion of “differential form oftype (0,s) on Gt/Lt with values in V ]
t ”.To make it easier to obtain a common Fréchet space, it will be helpful to assume the given Fréchet contraction
(6.2) to have pleasant properties:
Assumption 6.2.i. The Fréchet space E is nuclear.
ii. For every t > 0, the irreducible representation σt : Lt → End(Vt) is the restriction to Vt of a (very reducible)smooth representation σt : Lt → End(E) ; furthermore, for every u in K∩Lt = K∩L, the operator σt(u) doesnot depend on t.
iii. Suppose B is a compact subset of p. Then the Fréchet topology of E may be defined by a distance with theproperty that there exists some κ > 0 so that
(a) each the of σt(k,v), k ∈ K∩L, v ∈B∩ lt∩, t ∈]0,1],(b) each of the σt(X), X ∈U(lt,C) (enveloping algebra),
(c) and each of the ct , t ∈]0,1],is κ-Lipschitz as an endomorphism of Vt .
Property iii. will be used at the end of this subsection, when we will need an analogue of Lemma 5.8 (seepage 40). Property ii, of the kind already encountered in (5.2) and (5.17), makes it possible for us to define forevery t > 0 a holomorphic vector bundle E ]
t over Gt/Lt . In order to circumvent, in our discussion of the space ofdifferential forms with values in E ]
t , some of the analytical difficulties related with the fact that E ]t is an infinite-
rank bundle, we will follow Wong [76] and define the Dolbeault complex directly: for every nonnegative integerq, set
Ωqt =
ϕ ∈ C ∞
(Gt , E]⊗∧q(u?t )
)/ ∀` ∈ Lt ,∀γ ∈ Gt , ϕ(γ`) = ςt(`)
−1 f (γ)
(6.3)
where ςt is the Lt -representation obtained from the action σt on E, the action of Lt on ∧n(ut) through an abeliancharacter denoted e2ρ(ut ), and the adjoint-action-induced representation ξt of Lt on ∧q(u?t ), by setting ςt = σt ⊗e2ρut ⊗ξt . The notion of smooth function on Gt with values in E]
t ⊗∧q(u?t ) is that from [62], Chapter 40 (see also[76], Section 2). In the present context, the Dolbeault operator ∂
Ωqt
: Ωqt →Ω
q+1t is defined in Wong [76], Section
2; we omit the details.Using the map
Ψt : K× s→ Gt/Lt
(k,v) 7→ k · expGt(v) ·Lt
from Lemma 6.1, we can now exhibit a Fréchet space isomorphic with Ωtq, but contained in a Fréchet space
independent of t.Note first that the derivative of Ψt at (1K ,0) induces a K∩L-equivariant isomorphism between
a= (k/(k∩ l))⊕ s
and gt/lt , and between the complexifications aC and gt,C/lt,C. Write ιt : aC→ gt,C/lt,C for that (K∩L)-equivariantisomorphism, and
ηt ⊂ aC (6.4)
for the inverse image of qt/lt,C under ιt . Identifying qt/lt,C with ut , we obtain a (K ∩ L)-equivariant isomor-phism
ιt : ηt → ut ; (6.5)
that one will make it possible to view ηt as the “antiholomorphic tangent space to the manifold Y from page 33 atprojY [(1K ,0)], when Y is equipped with the complex structure from Gt/Lt”.
34
We now define our spaces of “differential forms of degree q on Y with values in E ]” and “differential forms oftype (0,q) for the Gt -invariant complex structure on Y ”, setting
Aq :=
f ∈ C ∞
(K× s,E]⊗∧q(aC
?)) ∣∣ ∀u ∈ K∩L,∀(k,v) ∈ K× s,
f (ku,Ad(u)v) = λ (u)−1 f (k,v)
(6.6)
and
A(0,q)t :=
f ∈ C ∞
(K× s,E]⊗∧q(η?
t )) ∣∣ ∀u ∈ K∩L,∀(k,v) ∈ K× s,
f (ku,Ad(u)v) = λ (u)−1 f (k,v)
(6.7)
=
f ∈ Aq / f is E]⊗∧q(η?t )-valued
,
where λ : K∩L→ End(E]⊗∧q(a?C)
)denotes the action of K∩L on E]⊗∧q(a?C) induced by σt , by e2ρut and by
the natural representation of K∩L on a. It should be noted that λ does not depend on t and preserves E]⊗∧q(η?t ):
this is due to Assumption 6.2(i) and to the easily verified fact that e2ρ(ut )|(K∩Lt ) does not depend on t.We equip Aq with the Fréchet topology from [62], Chapter 40 (see [76], page 5); then A(0,q)t is closed in
Aq.Using Mostow’s decomposition (6.1), we can extend every element in A(0,q)t to an element Ω
qt , obtaining an
isomorphism of Fréchet spaces
It : A(0,q)t →Ωqt
f 7→Φtf : (6.8)
to be precise, when f is an element of A(0,q)t and γ is in Gt , define Φtf (γ) by writing γ = k expv
Gtexpβ
Gt(k ∈K, v∈ s,
β ∈ `∩ p) for the Mostow decomposition of γ , then setting Φtf (γ) = ςt(expGt
(−β )) f (k,v) (where we identified∧q(ηt
?) and ∧q(u?t ) through the isomorphism ιt from (6.5)). The map Φtf is an element of C ∞
(Gt ,E]⊗∧q(u?t )
),
and it is easily checked that it satisfies the Lt -equivariance condition from (6.3), so that Φtf ∈Ω
qt . To check that It
is an isomorphism, we need only add that every Φ in Ωqt is the image under It of the map (k,v) 7→Φ(k expGt
(v))(here we keep identifying ∧q(ηt
?) and ∧q(u?t ) through ιt ).By using It to transfer the group action and Dolbeault operator on Ω
qt to A(0,q)t , we obtain for every t > 0
• An action of Gt on A(0,q)t ; when f is an element A(0,q)t , we will write
act(k,v)[ f ] (6.9)
the image of f under the action of an element (k,v) in Gt .• A linear act(Gt)-invariant differential operator
∂t : A(0,q)t → A(0,q+1)t ,
whose range is closed thanks to Wong’s work [76].
We now define the “spaces of ∂t -closed and ∂t -exact forms”
Fqt =
f ∈ Aq / f is V ]
t ⊗∧q(ηt?)-valued and ∂t f = 0
, and
Xqt =
f ∈ Aq / f is V ]
t ⊗∧q(ηt?)-valued, and there exists ω ∈ Aq−1 so that ∂tω = f
.
For every t > 0, Fqt and Xq
t are act(Gt)-invariant closed subspaces of A(0,q)t , hence of Aq. The induced actionand topology on Fs
t /X st equip that quotient with the structure of an admissible representation of G; given the results
35
recalled in section 6.1.1, this is an irreducible tempered representation with real infinitesimal characer and minimalK-type minimal µ .
There is of course an analogous description of the (less subtle) notion of differential form on K/(K ∩L) withvalues in the (finite-rank) vector bundle W ] whose fiber is• the subspace W = limt→0 ctφ , φ ∈V1 of E obtained by going through the Fréchet contraction of σ onto
its lowest (K∩L)-type µ[,• equipped with the action µ[⊗ e2(ρu)|(K∩L) of K∩L.
The intersection ut ∩ kC does not depend on t and can be identified with the antiholomorphic tangent space toK/(K ∩L) at 1K(K ∩L); the inverse image ι
−1t (ut ∩ kC) is a vector subspace (k/(k∩ l)) that does not depend on t
and is contained in each of the ηt , t > 0: we will write η0 for it. The space of differential forms of type (0,q) onK/(K∩L) with values in W ] can thus be identified with
Bq =
f ∈ Aq / f is W ]⊗∧q(η0)?-valued and ∀(k,v) ∈ K× s, f (k,v) = f (k,0)
.
For every t > 0, we remark that Bq ⊂ A(0,q)t and that the action act induces an action of K on Bq, which does notdepend on t; we will write ac : K→ End(Bq) for it. For every q is defined, as before, an ac(K)-invariant Dolbeaultoperator ∂0 : Bq→ Bq+1 with closed range (the latter property is from [75]); we set
Fq0 = Ker
(∂0 : Bq→ Bq+1) and Xq
0 = Im(∂0 : Bq−1→ Bq) ,
so that Fq0 and Xq
0 are K-invariant subspaces of Aq; the quotient Fq0 /Xq
0 carries an irreducible K-module of classµ .
Contraction operators. We started this subsection by giving a meaning to the notion of “zooming-in on aneighborhood of K ∩ (K ∩L) in G/L”. Furthermore, we have assumed that contraction operators ct : E → E aregiven that act on the fibers of the holomorphic vector bundles considered in this section. For every t > 0, theoperator ct induces an endomorphism ct of E]⊗∧qa?C: using the decomposition a = k/(k∩ l)⊕ s and writingφt : a→ a for the map k+ v→ k+ tv induced by the isomorphism ϕt : Gt → G, and ∧pφ ?
t for the endomorphismof ∧qa?C induced by φt , we set ct = ct ⊗∧pφ ?
t .Combining these two operations on the base space and on the fibers, we obtain the linear automorphisms of Aq
to be used for our contraction purposes,
Ct : Aq→ Aq (6.10)
f 7→ Ct f = the map (k,v) 7→ ct · f (k, tv) (k ∈ K,v ∈ s).
We now remark that Ct intertwines the action of G1 on A(0,q)1 and that of Gt on A(0,q)t . In order to make thenotations in the statement and proof lighter, we write A(0,q) for A(0,q)1 and ac : G→ End(A(0,q)) for the G-actionobtained by composing ac1 and the isomorphism ϕ1 : G1→ G.
Lemma 6.3. Fix t > 0, f ∈ A(0,q) and g ∈ G. Then Ct [ac(g) · f ] = act(ϕ−1t (g)
)· [Ct f ].
Proof: When g is an element of G, we write its Mostow decomposition as g = κ(g)expX(g)G expβ (g)
G , gettingMostow maps κ : G→ K, X : G→ s and β : G→ l∩p. We write κt : Gt → K, Xt : Gt → s and βt : Gt → lt ∩p forthe analogous Mostow maps of Gt .
Now fix g in G and t > 0.• By definition, ac(g) · f is a map from K× s to E]⊗∧qu?, namely
ac(g) · f : (k,v) 7→ ς
(e−β [g−1k expG(v)]
)f(κ[g−1k expG(v)
],X[g−1k expG(v)
]).
36
Thus Ct [ac(g) · f ] is the map
(k,v) 7→ ct · [ac(g) · f ] (k, tv)
= ct · ς(
exp−β [g−1k expG(tv)]G
)f(κ[g−1k exptv
G], X
[g−1k exptv
G])
= ςt
(ϕ−1t (exp
−β [g−1k exptvG])
G
)ct · f
(κ[g−1k exptv
G], X
[g−1k exptv
G])
= ςt
(exp− 1
t β [g−1k exptvG])
Gt
)ct · f
(κ[g−1k exptv
G], X
[g−1k exptv
G])
(in the transition between the second and third line, we use property (NatRIC) for the contraction operatorct : it intertwines the representations σ and σt ϕ
−1t of L).
• We turn to act(ϕ−1t (g)
)· [Ct f ], which is the map
(k,v) 7→ ςt
(exp
βt(ϕ−1t (g)−1k expGt (v))
Gt
)· [Ct f ]
(κt(ϕ−1t (g)−1k expv
Gt
), Xt
(ϕ−1t (g)−1k expv
Gt
))= ςt
(exp
βt(ϕ−1t (g)−1k expGt (v))
Gt
)· ct · f
(κt(ϕ−1t (g)−1k expv
Gt
), t ·Xt
(ϕ−1t (g)−1k expv
Gt
)).
If γ = keveβ is an element of G, then ϕ−1t (γ) = k expX/t
Gtexpβ/t
Gt, so that κ(γ) = κt(ϕ
−1t (γ), X(γ) = t ·
Xt(ϕ−1t (γ)) and β (γ)= t ·βt(ϕ
−1t (γ)). Using this to compare the last lines of our descriptions of Ct [ac(g) · f ]
and act(ϕ−1t (g)
)· [Ct f ], we see that they are equal.
Our contraction map Ct also intertwines the Dolbeault operators ∂ := ∂1 and ∂t :
Lemma 6.4. For every t > 0, the operator C−1t ∂t Ct is none other than ∂ .
Proof: Recall that our Dolbeault operator ∂ on A(0,q) is defined by transferring to A(0,q), through the Fréchetisomorphism I : A(0,q)→Ω
q1 from (6.8), the Dolbeault operator acting on Ω
q1 defined in [76] section 2. For every
t > 0, the operator ∂t acting on A(0,q)t similarly comes from Wong’s Dolbeault operator on Ωqt , transferred to A(0,q)t
via the Fréchet isomorphism It : A(0,q)t →Ωqt .
We now remark that the linear automorphism Ct of Aq induces a linear isomorphism between Ωq1 and Ω
qt ,
namely the map It Ct I −11 . Inspecting definitions, we notice that for every F : G1→ E]⊗∧qu? in Ω
q1, the map
It Ct I −11 (F) is none other than
Ct(F) : Gt → E]⊗∧qu?
(k,v) 7→ (ct ⊗φ?t ) ·F(ϕt(k,v))
where φ ?t : ∧qu?→∧qu?t is the linear isomorphism induced by the derivative of ϕ
−1t at the identity.
This means that Ct(F) is the element Ωqt induced by the group isomorphism ϕ1 ϕ
−1t between G1 and Gt . That
isomorphism induces an isomorphism of holomorphic vector bundles between the bundle V ]1 over G1/L1 and the
bundle V ]t over Gt/Lt ; by naturality of the Dolbeault operator, that bundle map must intertwine the Dolbeault
operators acting on Ωq1 and Ω
qt . So
C−1t ∂
ΩqtCt = ∂
Ωq1,
which proves Lemma 6.4.
Corollary 6.5. If f lies in Fq, then Ct f lies in Fqt ; if f lies in Xq, then Ct f lies in Xq
t .
37
Representatives whose contraction is harmonic. Our aim of contracting π onto its minimal K-type µ nowseems within reach: at this stage, we can associate, to every vector in the carrier space H(0,s)(G/L,V ]) for π ,an element in the carrier space H(0,s)(K/(K ∩ L),W ]) for µ . Indeed if f ∈ Fs := Fs
1 is a representative of acohomology class in H(0,s)(G/L,V ]), then as t goes to zero, ft := Ct f goes (for the given topology on Aq) to anelement f0 in Aq that is constant in the s-directions and is W ]⊗ (k/(k∩ l))-valued. Given the definitions of theDolbeault operators, when f lies in Fs, the contraction f0 lies in Fs
0 ; furthermore, if f and g are two elements ofFs with the same cohomology class (meaning that f −g lies in X s := X s
1), then f0 and g0 are two elements of Fs0
with the same cohomology class (meaning that f0−g0 ∈ X s0): so the class of f0 in Fs
0 /X s0 depends only of that of
f in Fs/X s, and defines an element in H(0,s)(K/(K∩L),W ]).However, we cannot completely realize the program of section 4.3 without embedding all of the Fs
t /X st , t > 0,
inside a common Fréchet space. The need to mod out a closed subspace X st that depends on t will make things
slightly acrobatic, and we will have to choose a representative in Fst for each cohomology class in Fs
t /X st ; in other
words, we will choose for every t > 0 a “sufficiently pleasant” algebraic complement to X st within Fs
t .Now, the complex manifold K/(K ∩L) is compact; in the space Bq of differential forms with type (0,q) with
values in the Hermitian holomorphic vector bundle W ], there is in consequence a convenient notion of harmonicform (see for instance in [71] the Introduction and Section II.6): a ∂ -closed form of type (0,q) on K/(K∩L) withvalues in W ] is harmonic when it is orthogonal, for the inner product on forms associated with the K-invariantKähler metric on K/(K∩L) and a K-invariant inner product on W , to every ∂ -exact form. We thus have a notion ofharmonic form in Bs, and a harmonic form can be ∂0-exact only if it is zero; in addition, if f ∈ Bq is harmonic, thenac(k) · f is harmonic too. For background on harmonic forms, see [71], Chapter III, especially around Theorems5.22-5.24.
Write Hs0 for the subspace of Bs consisting of harmonic forms: this is an ac(K)-invariant subspace of Fs
0 , andFs
0 = Hs0⊕X s
0 . Now, defineHs := f ∈ Fs / f0 ∈ Hs
0 ,
a subset of A(0,s).
Lemma 6.6. The space Fs of closed forms decomposes as Hs +X s = Fs.
Proof: Fix f in Fs; the contraction f0 is ∂0-closed, so we can write f0 = a+b with a ∈ Hs0 a harmonic form and
b ∈ X s0 a ∂0-exact form. Write b = ∂0β , where β ∈ Bs−1; we can extend β trivially to K× s, obtaining an element
γ of As−1 constant in the s-directions. Set ω = ∂ γ; then the form ω lies in X s, and f −ω lies in Hs (becausef0−ω0 = a is harmonic). We have proved that f can be written as a sum of a form in Hs and another in X s; thatis the lemma.
With Lemma 6.6 in hand, choose a closed complement Σ for Hs∩X s in Hs, so that
Fs = Σ⊕X s.
The fact that a closed Σ may be chosen follows from the theory of nuclear spaces (see Treves [62], chapters 50and 51, and Grothendieck [23]): because of Assumption 6.2, the Fréchet space As is nuclear (see [62], Proposition50.1 and section 51.8), so the closed subspaces Hs, X s and Hs∩X s are both nuclear; because X s∩Hs is closed inHs, the identity Hs ∩X s→ Hs ∩X s is a nuclear linear map ([23], corollaire 2, p. 101); it admits ([23], théorème3, p. 103) a nuclear extension F : Hs → Hs ∩X s. We define Σ as the kernel of F : it is a closed complement toHs∩X s in Hs, as desired.
For every t > 0, the spaceΣt = Ct [ f ] / f ∈ Σ (6.11)
38
is a closed complement to X st in Fs
t ; since X st and Σt are closed in Fs
t , the projection
qt : Σt → Fst /X s
t (6.12)
is a Fréchet isomorphism. Thus, we have defined a subspace of As such that very class in H(0,s)(Gt/Lt ,V ]) has aunique representative in Σt .
Given the arbitrary choice made just after Lemma 6.6, it is unreasonable to hope that Σt is act(Gt)-stable. We canhowever use the isomorphism (6.12) to transfer to Σt the action acquo
t of Gt on Fst /X s
t inherited from act : recallthat the projection
pt : Fst → Fs
t /X st (6.13)
makes it possible to define for every γ in Gt a map from Fst /X s
t to itself by
if ω = pt(F) for some F in Fst , then acquo
t (γ) ·ω = pt(act(γ)F); (6.14)
to transfer that action to Σt and set
πt(k,v) ·F : = q−1t[acquo
t (k,v) ·qt(F)]
(6.15)
= q−1t [pt (act(k,v) ·F)] (6.16)
for every (k,v) in K×p and every F in Σt . The representation (Σt ,πt) is irreducible tempered and its lowest K-typeis µ .
Conclusion. We now have all the pieces for a Fréchet contraction of π onto µ . Consider
E = As; (A)
Vt = Σt from (6.11); (B)
πt from (6.15); (B’)
Ct from (6.10). (C)
Theorem 6.7. The quadruple (E,(Vt)t>0,(πt)t>0,(Ct)t>0) describes a Fréchet contraction of the real-infinitesimalcharacter representation π onto its minimal K-type µ .
We have already checked properties (Vect1) and (Vect2) of section 4.3, and property (NatRIC) follows fromLemma 6.3 togeher with the fact that X s
t is act(Gt)-stable. We now need to prove properties (Op1) and (Op2).Let us consider the outcome
V0 = f0 / f ∈ Σ
of the contraction; this is a subspace of Fs0 . The contraction of every element in Σ is by definition a harmonic form,
so that V0 ⊂ Hs0. The reverse inclusion is true:
Lemma 6.8. Every harmonic form in Bs is the contraction of a form in Σ: we have V0 = Hs0.
Proof: Since the subspace Σ0 of Fs0 is ac(K)-invariant, the projection Σ0→ Fs
0 /X s0 is a morphism of K-modules;
since every element in V0 is harmonic and no nonzero harmonic form can be exact, this morphism is injective.Now, the K-module Fs
0 /X s0 is irreducible, so that morphism must also be surjective, unless V0 be zero. But if that
were the case, for every f in Fs = Σ⊕X s, the contraction f0 would lie in X s0 and be exact.
To prove that this is impossible, we call in the bottom-layer map of [39], section V.6 (see also [53], section 2,and especially [1], section 9). When applied to the (K∩L)-module W ], that map induces an injective morphism ofK-modules from Fs
0 /X s0 to Fs/X s, whose image is the subspace W of Fs/X s carrying the minimal K-type of that
39
G1-representation. Inspecting definitions reveals that the map sending an element of W to its inverse image by thebottom-layer map coincides with the map induced by the contraction f 7→ f0. If f0 were exact for every f in V1,we could deduce Fs
0 = X s0 , and that is not true.
We can now be sure that the K-module V0 is irreducible of class µ .Finally, fix an element F in V0 and an element f in Σ1 satisfying F = f0, and set ft = Ct f for every t > 0. To
complete the proof of Theorem 6.7, we need only check that πt(k,v) ft goes to ac(k) · f0 as t goes to zero.Fix (k,v) in K×p. Given Lemma 6.3 and the definition of our subspaces Σt in (6.11), we may write
πt(k,v) ft = q−1t [pt (act(k,v) ·Ct f )]
= q−1t ptCt [(ac(k expG(tv)) · f )]
= Ct[q−1
1 p1 (ac(k expG(tv)) · f )]
= Ct[q−1
1 p1 (ac(k) · f )]+(Ct q−1
1 p1) [ac(k expG(tv)) · f −ac(k) · f ] .
Recall that f lies in Σ1, and note that Σ1 is ac(K)-invariant, so that p1 (ac(k) · f ) = q1 (ac(k) · f ); applying Lemma6.3 again and inserting the fact that act(k) = ac(k) for all t, we conclude
πt(k,v) ft = ac(k) · ft +(Ct q−11 p1) [ac(k expG(tv)) · f −ac(k) · f ] . (6.17)
When t goes to zero, the first term in (6.17) goes to ac(k) · f0; what we need to check is that the second termgoes to zero. We now insert the following two remarks (compare Lemma 5.8):
Lemma 6.9. (i) The topology of E may be defined by a distance with respect to which each of the Ct , t ∈]0,1],is 1-Lipschitz.
(ii) When E is equipped with that distance, the Fréchet topology on Fs/X s may be defined with a distance suchthat the projection p1 : Fs→ Fx/X s is 1-Lipschitz and the inverse projection q−1
1 : Fs/X s→ Σ is 2-Lipschitz.
Proof: • For (i), build a countable family of semi-norms and a metric whose topology is that of E, following therecipe in [62], page 412: using the notations of the first sentence after Definition 40.2 in that textbook, we use thesubsets Ω j = K×B(0, j), j ∈ Z+, of K× s (where B(0, j) is the open ball s with center 0s and radius j), and useAssumption 6.2iii.(c) to obtain a countable family of semi-norms on the Fréchet space E]⊗∧sa? with respect towhich the operators ct , t ∈]0,1], are all 1-Lipschitz.• For (ii), we note that given a Fréchet distance on Fs, the usual way to define a Fréchet topology on Fs/X s is
through a distance for which p1 : Fs→ Fs/X s is obviously 1-Lipschitz (see section 12.16.9 in [16]); what needsproof is the statement on the inverse q−1
1 . Now, the inclusion Σt → Fst and the projection Fs
t → Fst /X s
t are 1-Lipschitz, so the composition q1 is 1-Lipschitz; the open mapping theorem for Fréchet spaces then shows that q−1
1is actually 2-Lipschitz with respect to the mentioned distances on Σ and on Fs/X s (see [16], §12.16.9 for the openmapping theorem and §12.16.8.2 for the Lipschitz statement).
Returning to the proof of Property (Op1) in Theorem 6.7, we take up (6.17) and use the fact that ac(k expG(tv)) ·f − ac(k) f goes to zero as t goes to zero and Lemma 6.9 to deduce that πt(k,v) ft goes to ac(k) · f0. Lemma 6.8yields (Op2); the proof of Theorem 6.7 is now complete.
We close this section by mentioning, for future use, that our Fréchet contraction satisfies a weakend analogue ofAssumption 6.2:
Lemma 6.10. Suppose U is a compact subset of p. Then there is a positive number K with the property that
(i) the Fréchet topology of E may be defined by a distance with respect to which each of the πt(k,v), k ∈ K,v ∈U, t ∈]0,1], and each of the Ct , t ∈]0,1], is K-Lipschitz as an endomorphism of Vt .
40
(ii) for every t > 0 and every (k,v) in K ×U, the operator πt(k,v) on Vt may be extended to a K-Lipschitzoperator acting on all of E. This may be done so that the resulting map πt : Gt → End(E) is continuous.
Lemma 6.10 is a rather poor substitute for Assumption 6.2, because the extension mentioned in part (ii) issimple-minded: we do not try to define it so that πt : Gt → GL(E) is a group morphism, and our Lipschitz esti-mates are crude: what will matter in section 7 is the existence of an extension to all of E and of a Lipschitz estimateuniform in k, v and t in the chosen domain.
Proof: For (i), we first unfold the definition of act(γ) in (6.9) and find that for every F in A(0,s)t and γ in Gt , wehave
act(γ) ·F = I −1t[x 7→ (ItF)(γ−1x)
]= (k,v) 7→ ςt
(exp−βt (γ
−1k expvGt)
Gt
)F[κt(γ
−1k expvGt), Xt(γ
−1k expvGt)]
where κt : Gt → K, Xt : Gt → s and βt : Gt → lt ∩p are the Mostow maps of Lemma 6.3.We now observe, as in the proof of Lemma 5.8, that there exists an increasing family (Ωn)n∈N of relatively
compact open subsets of K× s such that for every γ in K expGt(U) and every t in ]0,1], the image of Ωn under
the map (k,v) 7→(
κt(γ−1k expv
Gt), Xt(γ
−1k expvGt))
is contained in Ωn+1. We can in fact choose Ωn of the formK×B(0,Rn), where (Rn)n∈N is an increasing sequence of positive radii and B(0,Rn) is the ball with radius Rn in s.We then use those and Assumption 6.2.ii.(a)-(b) to follow the proof of Lemma 6.9.(i) and build the desired metricon E.
For (ii), fix a closed complement Z to Fs in E = As, and for all t > 0, set Zt = CtZ, so that
E = Σt ⊕X st ⊕Zt .
Write projΣt, projXs
tand projZt
for the corresponding linear projections, and for every (k,v) in Gt , extend theoperator πt on Σt by having the extension act as the identity on X s
t and Zt , setting
πt(k,v)F = πt(k,v)[projΣt
F]+projXs
t(F)+projZt
(F) (6.18)
for all F in E. Of course this extension is so simple-minded that it cannot define a representation of Gt on E, but wenote, taking up some arguments from the proof of Lemma 6.9(ii), that the projections projΣt
, projXst
and projZtare
all 2-Lipschitz. Indeed, if A is a Fréchet space equipped with a compatible distance and B,C are closed subspacesso that A = B⊕C, then the projection A→ C is 2-Lipschitz: it can be obtained by first applying the projectionA→ A/B, which as we already noted is 1-Lipschitz if the usual distance on A/B is chosen, then applying theisomorphism A/B→ C, which is the inverse of the 1-Lipschitz isomorphism C→ A/B and is as a consequence2-Lipschitz.
Returning to (6.18), we know from Lemma 6.9 that πt(k,v) is 4-Lipschitz, and use the preceding remarks todeduce that πt(k,v) is 10-Lipschitz. The continuity of (k,v) 7→ πt(k,v) is straightforward from that of πt .
6.2. The case of quasi-split groups and fine K-types
In this section, we consider a pair (G,π) where• G is a quasisplit linear group with abelian Cartan subgroups,• π is a tempered irreducible representation of G with real infinitesimal character and a fine minimal K-type
µ ,and we set out to build a Fréchet contraction of π onto µ
In the present context, the representation π can be realized as an irreducible factor in a (usually nonspherical)principal series representation of G, as in Chapter 4 in [66]: suppose P = MAN is a minimal parabolic subgroup
41
in G, then in our quasisplit case• M is abelian (and compact, but not connected),• and there exists a character σ of M such that µ is one of the minimal K-types in
Π = IndGMAN (σ ⊗1) .
The representation Π is a finite direct sum of irreducible tempered subrepresentations, each of which has realinfinitesimal character and a unique minimal K-type; the irreducible factor that contains µ is equivalent with π asa G-representation.
Recall that in the compact picture from section 5.2.1, the representation Π acts on
L2σ (K) :=
Φ : K→ C / ∀(u,m) ∈ K×M, Φ(um) = σ(m)−1
Φ(u)
through the action obtained by choosing λ = 0 in (5.9). The subspace
H = irreducible subspace of L2σ (K) containing the K-type µ (6.19)
thus carries a (not very explicit) realization of π on a space of complex-valued functions on K.Because the induced representation Π is reducible and has continuous parameter zero, the results of §5.2.1 can
not be applied here; yet some of the interplay between sections 5.2.1 and 5.2.2 will be helpful.
6.2.1. A Helgason-like realization
We now describe another realization, in which G acts on a space of sections of an equivariant vector bundleover G/K; it mimics the realization of spherical principal series representations in section 5.2.2 using Helgasonwaves.
For the rest of section 6.2.1, we fix an irreducible K-module V µ of class µ , write E for the equivariant bundleG/K×K V µ over G/K and Γ(E ) for the space of smooth sections of E .
We first describe the special sections E which will play in our construction the part Helgason waves did play insection 5.2.2.
From Frobenius reciprocity and the fact that the K-type µ occurs with multiplicity one in H, we know thatthe restriction µ|M contains σ with multiplicity one. We fix a vector v ∈ V µ belonging to the one-dimensionalσ -isotypical subspace and introduce
γ : G → V µ
keHn 7→ e−〈ρ, H〉µ(k)−1v.
Lemma 6.11. For all (g,m,H,n) in G×M×a×N, we have γ(gmeHn) = e−〈ρ,H〉σ(m)−1γ(g).
Proof: We use the Iwasawa decomposition (5.13) to find
γ(gman) = γ
(κ(g)eH(g)
ν(g)man)
= γ
(κ(g)meH+H ′ n
)for some n ∈ N (recall that N normalizes MA and M centralizes A)
= e−〈ρ,H+H ′〉µ(κ(g)m)−1v
= σ(m)−1e−〈ρ,H〉[e−〈ρ,H(g)〉
µ(κ(g))−1v]
(since µ(m)−1v = σ(m)−1v and σ(m) is scalar)
= e−〈ρ,H〉σ(m)−1γ(g).
42
Now set γ : g 7→ γ(g−1). A simple calculation shows that γ(gk) = µ(k−1)γ(g) for every k in K and g in G. Asa consequence, γ induces a section γ of the vector bundle E over G/K.
We will take up γ as an analogue in Γ(E ) of the Helgason “wave” eλ ,1K with (not very wavy) frequencyλ = 0.
For every b in K, the map γb : g 7→ γ(b−1g) induces another element γb in Γ(E ), which we will take up asan analogue of the Helgason “wave” e0,b. From Lemma 6.11, we know that γbm = σ(m)−1γb for every (b,m) inK×M.
We now describe a vector subspace of Γ(E ), invariant under the usual action of G on Γ(E ), by associating asection of E with each of the functions on K belonging to the Hilbert space H from (6.19).
Define
T : H→ Γ(E )
Φ 7→∫
KγbΦ(b)db;
note from the M-equivariance properties of Φ and b 7→ γb that the integrand is invariant under the change ofvariables b← b ·m.
Lemma 6.12. The map T is G-equivariant and injective.
Proof: Fix x in G and write x = κ(x)ea(x)n(x) for its Iwasawa decomposition. To prove that T is G-equivariant,we must prove that for every Φ in H, the sections
T (π(x)φ) = T[b 7→ e−〈ρ,a(x
−1b)〉 ·Φ(κ(x−1b)
)]=
(gK 7→
∫K
e−〈ρ,a(x−1b)〉
γb(gK) ·Φ(κ(x−1b)
)db)
=
(gK 7→
∫K
e−〈ρ,a(x−1b)〉e−〈ρ,a(g
−1b)〉 ·Φ(κ(x−1b)
)(µ(κ(g−1b)
)· v)
db)
(6.20)
and
λ (x)T (φ) =
(g 7→
∫K
γb(x−1gK) ·Φ(b)db)
=
(g 7→
∫K
e〈−ρ,a(g−1xb)〉 ·Φ(b)(µ(κ(g−1xb)
)· v)
db)
(6.21)
are equal. Going from (6.20) to (6.21) by substituting b← κ(x−1b) in (6.21) is made possible by the fact that theIwasawa maps κ , a, n satisfy cocycle relations. The necessary steps are identical with those taken by Camporesifor his proof of Proposition 3.3 in [10], so we omit the details.
The injectivity of T commes from the irreducibility of π: from the equivariance we know that T −1(0) isa G-invariant subspace of H. It is either 0 or H. If it were equal to H, the map T would be identically zero.Fixing a K-invariant inner product on Vσ , we notice that the matrix element ξv : k 7→ 〈v,µ(k−1)v〉Vσ
lies in L2σ (K)
and recall from the Peter-Weyl theorem that it transforms under Π|K according to µ , so that ξv lies in H; the valueof T (ξv) at the origin of p is the vector
∫K(µ(k
−1)v) · ξv(k)dk in V µ , whose scalar product with v is nonzero be-cause of Schur’s orthogonality relations. This proves that T (ξv) is not identically zero, so T must be injective.
We have obtained a geometric realization for π in which G acts on the subspace∫K
γb ·Φ(b) db, Φ ∈H
43
of Γ(G/K,V µ), through the usual action of G on Γ(G/K,V µ).
6.2.2. Contraction to the minimal K-type
We can now describe a Fréchet contraction of π onto µ using the same zooming-in operators as we did for thediscrete series in section 5.1 and for the Helgason-wave picture for spherical principal series in section 5.2.2.Set
E = C ∞(p,V µ). (A)
For every t > 0, we transfer to functions on p the sections of Γ(Gt/K,V µ) which the construction in section 6.2.1yields for Gt . Calling in the Iwasawa maps It : p→ a and κt : p→ K from (5.13), we define
Γ tb : p → V µ
u 7→ e〈tρ, It (Ad(b)·v)〉µ−1(κt(u))v.
Given the previous subsection, we define
Vt =
∫K
Γt
b ·Φ(b) db, Φ ∈Hπ
. (B)
The action of Gt on Γ(Gt/K,V µ) yields a Gt -action on E, viz.
Gt ×E→ E
((k,v),F) 7→ πt(k,v)F = (x 7→ µ(k) ·F [(k,v) ·t x]) (B’)
where ·t is the Gt -action on p defined in section 4.2. The subspace Vt of E is then πt -stable.Bringing in the zooming-in map zt : u 7→ u
t from p to itself, we recall from Lemma 5.4 that
Ct : f 7→ f z−1t (C)
intertwines the G1- and Gt -actions on E; from Lemma 5.17 we deduce that Ct sends Γ1b to Γt
b (the main differencewith the situation in §5.2.2 is that we need no renormalization of spatial frequencies here, because we are dealingwith representations with continuous parameter zero). Thus,
Ct sends V1 to Vt . (Vect1)
Now if f =(∫
K Γ1bΦ(b)db
)is any element in V1, then Ct f converges in E as t goes to zero to
f0 = the constant function with value∫
K(µ(b) · v)Φ(b)db. (Vect2)
If (k,v) is a fixed element in K×p and f is an element in V1, we have already proven in section 5.1 (see (5.4)and Lemma 5.8 that
As t goes to zero, πt(k,v) ft goes to µ(k) · ( the value of f0) . (Op1)
From the Schur orthogonality relations and formula (Vect2), we know that• if Φ lies in a K-isotypical subspace of H for a K-type other than µ , then f0 = 0;• but the map f 7→ f0 induces a K-equivariant isomorphism between the µ-isotypical subspace of H (which
is spanned, if we fix an orthogonal basis (v j) for V µ , by the matrix elements k 7→ 〈v j,µ(k)−1v〉) and thefiber V µ .
44
Thus
The outcome space V0 = f0, f ∈H1 naturally identifies, as a K-module, with V µ ; (Op2)
for every (k,v) in K×p, the limit of πt(k,v) ft then identifies with µ(k) · f0.
We have proved:
Theorem 6.13. The quadruple (E,(Vt)t>0,(πt)t>0,(Ct)t>0) describes a Fréchet contraction of π onto µ . Assump-tion 6.2 is satisfied by this Fréchet contraction.
(Concerning the part about Assumption 6.2, i. and ii. in Assumption 6.2 are clearly satisfied here; for iii., take upthe proof of Lemma 5.8 using star-shaped subsets of p in the construction of seminorms.)
7. Contraction of an arbitrary tempered representation
7.1. Real-infinitesimal character representations for disconnected little groups
Our analysis of real-infinitesimal-character representations in section 6 crucially used Wong’s work on the closed-range property for the Dolbeault operator. The setting in Wong’s paper [76] is that of a linear connected semisimplegroup: that was ourreason for the assuming G to be linear connected semisimple in §6.
Now suppose G is a linear connected reductive group. When (χ,µ) is an arbitrary Mackey parameter, therepresentation of G built from (χ,µ) in section 3.2 is induced from a parabolic subgroup Pχ = Mχ Aχ Nχ , using thereal-infinitesimal-character representation σ of Mχ with minimal Kχ -type µ .
Of course Mχ is usually disconnected, although the disconnectedness is limited in that Mχ satisfies the axiomsin section 1 of [35]. While it is quite possible that the theorem of Wong mentioned in 6.1.1 is true for linear(disconnected) reductive groups such as Mχ , I have not been able to find a reference; if we are to use the resultsof section 6 in a future discussion of parabolic induction from Pχ , we need to check that they extend to reasonablydisconnected groups.
For the rest of section 7.1, we consider a reductive group G satisfying the axioms of section 1 in [35].Consider the identity component G0 of G. It is a non-semisimple, connected Lie group and can be decomposed
as G0 = Gss (ZG)0, with Gss a connected semisimple Lie group with finite center. The abelian group (ZG)0 iscompact and central in G0 ([38], section V.5).
Suppose we start with a tempered irreducible representation of G0 with real infinitesimal character. Then theelements in (GM)0 will act as scalars, defining an abelian character of (GM)0. It is easy to check that the restrictionto Gss of our representation will then be irreducible and have real infinitesimal character.
A representation π0 of G0 in the class (G0)RIC is thus uniquely specified by a discrete series representation πss
of Gss and an abelian character ξ of (ZG)0 whose restriction to Gss∩ (ZG)0 coincides with (πss)∣∣Gss∩(ZG)0
. Givenπss and ξ , any carrier space for πss furnishes a carrier space for π0, with g = gssg(ZG)0
acting through the operatorξ (g(ZG)0
)πss(gss).
Now that we know how to describe the class (G0)RIC, let us write G] for the subgroup G0ZG of G; because of[38], Lemma 12.30, G0 has finite index in G]; in fact there is a finite, abelian subgroup F of K (it is the subgroupcalled F(B−) in [38]), lying in the center of G (hence of G]), such that
G] = G0F.
The arguments we used for G0 go through here, so that a representation π] in the class (G])RIC is uniquely specifiedby a discrete series representation π0 of G0 and an abelian character χ of F whose restriction to G0∩F coincideswith (π0)
∣∣G0∩F . As before, given π0 and χ , any carrier space for π0 furnishes a carrier space for π], with g = g0 f
acting through the operator χ( f )π0(g0).To obtain a tempered representation of G, we can start from a representation π] in the class (G])RIC and
45
setπ = IndG
G]
(π]).
It turns out (using parabolic induction from the result of [38], Proposition 12.32, concerning discrete series)that π is irreducible, has real infinitesimal character, and that π] 7→ π maps the class (G])RIC bijectively onto GRIC.In addition, the restriction of π to G] decomposes as
π∣∣G] = ∑
ω∈G/G]
ωπ]
where ωπ] is mg 7→ π](ω−1gω). Since G] has finite index in G (see (12.74) in [38]), the sum is finite here.
With the above description in hand, it is a simple matter to reduce the contraction problem for representations inthe class GRIC to the already solved problem for representations in the class (Gss)RIC.
Fix a representation π in GRIC; the above discussion says how π may be constructed from a triple (χ,ξ ,σ),where σ is a representation of Gss in the class (Gss)RIC and χ , ξ are characters of (ZG)0 and F .
Write µ[ for the minimal Kss-type of σ and assume given a Fréchet contraction
(E,(Vt)t>0,(σt)t>0,(ct)t>0) (7.1)
of σ onto µ[.Remark that (ZG)0 and F are contained in K, so that that there is a direct product decomposition
K = (ZG)0 ·F ·Kss. (7.2)
and that for each t > 0 the group G]t = ϕ
−1t (G]) decomposes as
G]t = (ZG)0 ·F ·ϕ
−1t (Gss).
For every t > 0, a represenation of Gt may be obtained by following the above procedure: write σ]t for the
irreducible tempered representation of G]t obtained from the representation σt of ϕ
−1t (Gss), so that every element
γ(ZM)0γF γss of G]
t acts on Vt through the operator ξ (γ(ZM)0)χ(γ f )σt(γss). For each t > 0, the representation
πt := IndGt
G]t
(σ]t
)(7.3)
is irreducible tempered, has real infinitesimal character; its minimal K-type is the minimal K-type µ of π .Turning to K-types, there is (applying the above remarks to the reductive group K) an analogous description of
µ as induced from a representation of K]: from the representation µ[ of Kss and the given characters ξ , χ , formthe representation µ\ := (µ[)] of K] where any k] = k(ZM)0
kF kss acts through ξ (k(ZM)0)χ(kF)σt(kss), then induce
to K: the result is an irreducible K-module, and we do have
µ ' IndKK]
(µ\). (7.4)
It is now a simple matter to construct a Fréchet contraction of π onto µ .The representation space for πt is a finite direct sum of copies of Vt , indexed by Gt/G]
t . Remarking that theinclusion of K in Gt induces a bijection
K/K] ∼−→ Gt/G]t , (7.5)
46
we will thus write the direct sum of copies of Vt as
Vt = ∑ω∈K/K]
Vt,w. (B)
Recall from the definition of induced representations that the representation πt of Gt on Vt may be described bychoosing a section of the projection K→ K/K], that is, a finite collection (κω)ω∈K/K] of elements of K. Indeed,
for every γ in Gt , there is a collection (γ]a)a∈K/K] of elements of G]t satisfying
∀ω ∈ K/K], γκω = κ[γω]t γ][γω]t
, (7.6)
where the action ω→ [γω]t of γ on K/K] is that induced from the action of γ on Gt/G]t through the bijection (7.5).
One can then setπt(γ) = the operator ∑
ω
xω 7→∑ω
σ]t (γ
][γω]
)x[γω] (B’)
to obtain a realization for the representation (7.3) of Gt .Now, each of the Vt , t > 0, is contained in a finite direct sum of copies of the Fréchet space E from (7.1), which
we will denote byE = ∑
ω∈K/K]
Eω , (A)
and equip with the direct product topology.For each t > 0, the contraction operator ct acts on every summand; if we write cω
t for the action on the summandwith index ω , we obtain an operator
Ct : V1→ Vt (C) ∑ω∈K/K]
fω
7→ ∑
ω∈K/K]
cωt fω
.
Proposition 7.1. The quadruple (E,(Vt)t>0,(πt)t>0,(Ct)t>0) describes a Fréchet contraction of π onto µ .
Proof: We start with property (NatRIC):
Lemma 7.2. The map Ct intertwines πt ϕ−1t and π1 ϕ
−11 .
Proof: Fix g in G, write γ1 for the element ϕ−11 (g) of G1 and γt for the element ϕ
−1t (g) of Gt . Suppose F = ∑
ω
fω
is any element of V1.Taking up (B’), we use the intertwining relation for ct to find
Ct (π1(γ1) ·F) = Ct
(∑ω
σ]t (γ
][γ1ω]
) f[γ1ω]
)= ∑
ω
cωt
[σ]1(γ
][γ1ω]
) f[γ1ω]1
]= ∑
ω
σ]1(ϕ
−1t ϕ1(γ
][γ1ω]1
)) ·[cω
t f[γ1ω]1
]. (7.7)
• For every ω in K/K], the isomorphism ϕt ϕ−1t sends the class of γ1κω in G1/G]
1 to the class of γtκω in Gt/G]t .
As a consequence, [γ1ω]1 and [γtω]t coincide.• From (7.1) and the previous remark, we deduce
ϕtϕ−11
(γ][γ1ω]1
)= ϕtϕ
−11
(γ1κω κ
−1[γ1ω]
)= ϕtϕ
−11 γ1
(κω κ
−1[γ1ω]1
)= γtκω κ
−1[γt ω]t
= γ][γt ω]t
.
47
Inserting this into (7.7), we find
Ct (π1(γ1) ·F) = ∑ω
σ]t (γ
][γt ω]t
) ·[cω
t f[γt ω]
]= πt(γt) · [CtF ] ,
as desired.
Of course for each vector f in V1, there is a limit f0 to ft := Ct f as t goes to zero. The set of limits thusobtained is
V0 := f0 | f ∈ E= ∑ω∈K/K]
V0,ω ,
where V0,ω :=
limt→0
ct f , f ∈V1,ω
carries an irreducible representation of K] with class µ\.
Turning to the convergence of operators, fix F = ∑ω
f0,ω in V0 and f = ∑ω
fω in V1 so that F = f0, fix (k,v) in
K× p; to observe πt(k,v) ft , we need to insert γ = expGt(v)k in (B’). Now for every ω in K/K], we remark that
γ][γω]t
from is equal to expGt(v)k]kω
, where k]kωis the element of K] satisfying kκω = κkω k]kω
(compare (7.1); here
kω is just the class of kκω in K/K]). This indicates that
πt(k,v) ft = ∑ω∈K/K]
π]t (k
]kω,v) ft,ω .
As t goes to zero, the fact that we started with a contraction from σ onto µ[ means that for every ω , π]t (k
]kω,v) ft,ω
goes to µ\(k]kω) f0,ω . Thus,
πt(k,v) ft goes to π0(k,v)F := ∑ω∈K/Kχ
µ\(k]kω
) f0,ω .
In the last formula, we recognize the formula for the K-action in IndKK](µ
\); from (7.4) we deduce that the G0-representation (V0,π0) is the extension to G0 of an irreducible representation of K isomorphic with µ , provingProposition 7.1.
We note that if the contraction 7.1 from σ onto µ[ satisfies Lemma 6.10, then the contraction from π onto µ webuilt in Proposotion 7.1 obviously satisfies Lemma 6.10, too.
7.2. Parabolic induction and the reduction to real infinitesimal character
We return to an arbitrary Mackey parameter (χ,µ), call in the parabolic subgroup Pχ =Mχ Aχ Nχ from (3.1) and thetempered irreducible representation σ = VMχ
(µ) with real infinitesimal character and minimal Kχ -type µ (recallthat Mχ ∩K = Kχ ). The representation of G that corresponds to (χ,µ) in the constructions of section 3.2 is
π = IndMχ Aχ Nχ
(σ ⊗ eiχ) , (7.8)
and the representation of G0 paired wih π in the Mackey-Higson correspondence is
π0 = IndKχnp
(µ⊗ eiχ) . (7.9)
We start from a Fréchet contraction
(E,(Vt)t>0,(σt)t>0,(ct)t>0) (7.10)
of σ onto µ , assume that it has the properties in Lemma 6.10, and set out to build a Fréchet contraction of π .
48
For every t > 0, the space
Vt =
f : K continuous−→ Vt | ∀u ∈ Kχ , ∀k ∈ K, f (ku) = σt(u−1) f (k)
(7.11)
can be equipped with a family of irreducible representations of Gt : recall from Lemma 5.13 that the inverse imageof Mχ Aχ Nχ under the isomorphism ϕt : Gt → G is a parabolic subgroup Mt,χ Aχ Nt,χ of Gt . For every λ in a?χ , theIwasawa maps
κt : Gt → K,
mt : Gt →Mt,χ ∩ expGt(p)
at : Gt → aχ
as well as the half-sum ρt of positive roots in the ordering used to define Nt,χ , may be used to define an operatoracting on Vt , setting
πt,compλ ,µ (γ) := f 7→
[k 7→ exp〈−iλ −ρt ,at(γ
−1k)〉σ(mt(γ−1k)) f
(κt(γ
−1k))]
(7.12)
for every f in Vt and every γ in Gt . See the discussion of the compact picture in §5.2.1.Embedding these spaces into a fixed Fréchet space is easy: they are all vector subspaces of
E = continuous functions from K to E , (A)
a Fréchet space when equipped with the topology of uniform convergence. Now for t > 0, consider
Vt from (7.11) (B)
πt = πt,compχ,µ from (7.12) (B’)
Ct = pointwise composition with ct from (7.10) . (C)
We are ready to complete the task we set ourselves for Part II.
Theorem 7.3. Suppose π is an arbitrary irreducible tempered representation of G, and π0 is the representation ofG0 that corresponds to π in Theorem 3.2. The quadruple (E,(Vt)t>0,(πt)t>0,(Ct)t>0) above describes a Fréchetcontraction of π onto π0.
It is perhaps simplest to start by establishing the naturality condition (Natgen) of section 4.3: we need onlymimic the case of principal series representations and use the facts encountered in the proof of Proposition 5.11concerning the normalization of continuous parameters.
For every λ in a?χ , use the isomorphism ϕ1 : G1 → G to define a representation πλ
of G acting on V1, thecomposition
πλ : Gϕ−11−→ G1
π1,compλ ,µ−→ End(V1).
Consider the representation of G transferred from πt through the isomorphism ϕt : Gt → G,
πt ϕ−1t : G
ϕ−1t−→ Gt
πt,compχ,µ−→ End(Vt).
Lemma 7.4. . The map Ct intertwines πt ϕ−1t and πχ/t .
Proof: Fix f in V1 and g in G. We need to compare (πt ϕ−1t )(g) · [Ct f ] with Ct
[πχ/t(g) f
]. Lemma 5.12 proves
that ρt is equal with tρ1, so given the definition of Ct ,
(πt ϕ−1t )(g) · [Ct f ] is the map k 7→ e〈−iχ−tρ,at ([ϕ−1
t g]−1
k)〉σt(mt(
[ϕ−1t g
]−1k)) [ct f ]
(κt([ϕ−1t g
]−1k))
49
for every g in G. Lemma 5.13 identifies the maps κt ϕ−1t , mt ϕ
−1t and at ϕ
−1t ; inserting, from property (NatRIC)
of the Fréchet contraction from σ onto µ , the fact that ct intertwines σ1 and σt ϕ−1t , some rearranging leads to
πt,compλ ,σ (ϕ−1
t (g)) [Ct f ] =[k 7→ e〈−i λ
t −ρ, t·at (ϕ−1t [g−1k])〉σt(mt(
[ϕ−1t g
]−1k))(ct f )
(κt(ϕ
−1t[g−1k
]))]
=[k 7→ e〈−i λ
t −ρ, a(g−1k)〉ctσ(g−1k)
f(κ(g−1k)
)]= Ct
[πχ/t(g) f
],
which proves the lemma.
Among the properties of section 4.3, (Vect1) and (Vect2) are once again obvious from the fact that (E,(Vt)t>0,(σt)t>0,(ct)t>0)
contracts σ onto µ: if f lies in V1 and we set ft = Ct f for every t > 0, then of course ft converges to an elementof
V0 =
f : K continuous−→ V0 | ∀u ∈ Kχ , ∀k ∈ K, f (ku) = σt(u−1) f (k)
(7.13)
where V0 is the outcome of the Fréchet contraction (E,(Vt)t>0,(σt)t>0,(ct)t>0), an irreducible Kχ -module carryingan irreducible representation k 7→ σ0(k) of class µ . Of course every F in V0 arises as the limit of Ct f for some fin V1, because every element of V0 arises by definition as the contraction of some element in V1.
The space V0 comes already equipped with an irreducible G0-representation equivalent with π0, that describedin section 2.3 and Eq. (2.3): for every F in V0 and (k,v) in G0, we set
π0(k,v) ·F =[u 7→ ei〈Ad?(u)χ,v〉F(k−1u)
]. (7.14)
The only thing that still needs proof is the convergence of operators.
Proposition 7.5. Fix F in V0, fix f in V1 so that F = limt→0
ft , and (k,v) in K×p. Then πt(k,v) · ft goes to π0(k,v) ·Fas t goes to zero.
Proof: We first rearrange πt(k,v) ft := πt(expGt(v)k) ft as
u 7→ e〈−iλ−tρ,at
[k−1uexp−Ad(u−1)v
Gt
]〉σt
(mt
[k−1uexp−Ad(u−1)v
Gt
])−1ft(
κt(k−1uexp−Ad(u−1)v)Gt
).
Calling in the Iwasawa maps It : p→ aχ and Kt : p→ K from (5.13), notice that
κt
(k−1uexp−Ad(u−1)v
Gt
)= k−1u Kt
[−Ad(u−1)v
],
which makes it possible to rewrite πt(k,v) ft as
u 7→ e〈−iλ−tρ,It (−Ad(u−1)v)〉σt
(mt
[k−1uexp−Ad(u−1)v
Gt
])ft(k−1u Kt
[−Ad(u−1)v
]).
Now recall that for all u in K,
mt
[k−1uexp−Ad(u−1v)
Gt
]= mt
[exp−Ad(u−1)v
Gt.]
(7.15)
Set β (u,v) =−Ad(u−1)v, an element of the compact orbit Ad(Kχ)v in p. Then we just saw that
πt(k,v) ft = u 7→ exp〈−iχ− tρ,It(β (u,v))〉 (7.16)
×σt
(mt
[expβ (u,v)
Gt
])ft(k−1u Kt [β (u,v)]
). (7.17)
When t goes to zero, we already know from Lemma 5.16 that (7.16) goes to ei〈Ad?(u)χ,v〉, uniformly in u becauseof Lemma 5.15. To complete the proof of Proposition 7.5, it is enough to establish that (7.17) goes to F(k−1u)
50
as t goes to zero. For this, we call in Lemma 6.10 and use the extension of σt to all of E to rearrange (7.17). Tosimplify the notations, write
Σt(u,v) = σt
(mt
[expβ (u,v)
Gt
]);
then
σt
(mt
[expβ (u,v)
Gt
])ft(k−1u Kt [β (u,v)]
)= ft(k−1u) (Term 1)
+[Σt(u,v)− IdE ][
ft(k−1u)− f0(k−1u)]
(Term 2)
+Σt(u,v)[
ft(k−1u Kt [β (u,v)]
)− ft(k−1u)
](Term 3)
+[Σt(u,v)− IdE ][
f0(k−1u)]. (Term 4)
As t goes to zero,• Term 1 goes to f0(k−1u) = F(k−1u).• Term 2 goes to zero because of the Lipschitz estimate in Lemma 6.10: since β (u,v) lies on a compact orbit
in p, there is a number K independent of t and u so that Σt(u,v) is K-Lipschitz for all t,u (recall that v isfixed here); since ft − f0 goes to zero as t goes to zero, so does Term 2.
• Term 3 also goes to zero: it can be rewritten as (Σt(u,v) ct)[
f(k−1u Kt [β ]
)− (k−1u)
]; from the second
part of Lemma 5.15 and the compactness of β (u,v),u ∈ K, we know that u 7→ f(k−1u Kt [β (u,v)]
)−
(k−1u) goes to zero as t goes to zero, and we use the independent-of-t-and-u Lipschitz estimate of Lemma6.10 to conclude as we did for Term 2.• To prove that Term 4 goes to zero, we need to prove that Σt(u,v) f0(k−1u) goes to f0(k−1u). Now, f0 is by
definition a ct -invariant element of E, so for all u in K,
Σt(u,v) f0(k−1u) = Σt(u,v) · ct · f0(k−1u)
= σt
(mt
[expβ (u,v)
Gt
])· ct · f0(k−1u)
= σ1
(ϕ−11 ϕt
mt
[expβ (u,v)
Gt
])f0(k−1u)
= σ1
(m1
[expt·β (u,v)
G1
])f0(k−1u).
The transition between the second line and the third uses the intertwining relation (NatRIC) for ct , and thatbetween the third line and the fourth uses the fact that β (u,v) lies in p.As t goes to zero, the element m1
[expt·β (u,v)
G1
]of G1 goes to the identity of G1; from the continuity of σ1
(see Lemma 6.10) we deduce that Σt(u,v) f0(k−1u) goes to f0(k−1u), as desired.All convergences are uniform in u; Proposition 7.5 and Theorem 7.3 follow.
PART III. THE MACKEY ANALOGY AND THE CONNES-KASPAROV ISOMORPHISM
8. Introduction and strategy
When G is a second countable locally compact group, the Fell topology on G is in general very wild: studying thetopological space G directly is usually difficult. An indirect, often fruitful approach is to study a suitable comple-tion of the convolution algebra C∞
c (G) of continuous and compactly supported functions on G. The completionwill be a noncommutative algebra, and in the spirit of Connes’ noncommutative geometry, the various completionsmay be thought of as noncommutative replacements for the (not very helpful) space of continuous functions onG.
We shall be concerned with the reduced C? algebra of G. When f is an element of C∞c (G), it can be viewed as
51
a bounded (convolution) operator on the Hilbert space L2(G), and that operator has a norm, say ‖ f‖; the reducedC? algebra C?
r (G) is the completion of C∞c (G) with respect to ‖·‖. The spectrum of C?
r (G) is the reduced dualG.
The Baum-Connes conjecture describes the K-theory of C?r (G), to be thought of as a "non-commutative" re-
placement for the Atiyah-Hirzebruch K-theory of G (see [7] for the formulation of the conjecture, and [34] forthe K-theory). Although we will consider only reductive groups and their Cartan motion groups in this paper,we should mention that the interest of the conjecture partly stems from its generality (it encodes very nontrivialfeatures of both Lie groups and discrete groups, and the existence of an analogue for groupoids makes it suitablefor the study of foliations), and its deep connections with, and important implications in, geometry, index theoryand topology.
The Connes-Kasparov isomorphism for connected Lie groups. When G is a connected Lie group, the con-jecture is equivalent with the assertion that the reduced dual of G can be accounted for, at least at the level ofK-theory, with the help of Dirac operators. Suppose K is a maximal compact subgroup of G, and R(K) is therepresentation ring of K − whose underlying abelian group is freely generated by the equivalence classes of irre-ducible K-modules. Starting from an irreducible K-module, and after going up to a two-fold covering of G and K ifnecessary, one can build an equivariant spinor bundle on G/K and a natural G-invariant elliptic operator (the Diracoperator) acting on its sections; this operator has an index which can be refined into an element of the K-theorygroup K [C?(G)]. This produces a morphism of abelian groups
R(K)µ−→ K [C?(G)] (8.1)
called Dirac induction; the Connes-Kasparov conjecture is the statement that µ is an isomorphism.Of course when G is reductive, the current understanding of G is in many respects complete. As a topological
space, G is in addition much more reasonable than the reduced duals studied by the full-blown Baum-Connes con-jecture: it is (roughly speaking) close to being a real affine variety. Yet the reductive case is a very important one: amajor source for the formulation of the Baum-Connes conjecture was the Partasarathy-Atiyah-Schmid realizationof the discrete series (see section 5.1 and of course [49, 4]), and understanding the reductive case is the key tounderstanding the case of general Lie groups.
Previous proofs for reductive Lie groups. When G is linear connected reductive, the Connes-Kasparov “con-jecture” has been a classical result for thirty years. The two celebrated proofs are quite different in spirit:•Wassermann’s short note of 1987 [72] used the comprehensive knowledge of C?
r (G) extracted by Arthur [3]from Harish-Chandra’s work, and his proof consists in an explicit calculation of the right-hand side of (8.1) andthe arrow therein. This followed earlier treatment of special cases from the same perspective: the important butsimpler case of complex semisimple groups had been covered by Penington and Plymen [50], and the case of realrank-one groups by Valette [63]. Some rather delicate properties of Knapp-Stein intertwining operators are needed(see [72], and for comments, §6 in ??).• Lafforgue obtained in 1998 a very different proof as a consequence of a deep study of the actions of groups
on Banach spaces and the properties of K-theory ([42]; see also [57]); he found a way to the Baum-Connesisomorphism that is not only very well-suited to reductive Lie groups, but also encompasses reductive p-adicgroups as well as some discrete subgroups which had resisted every approach before his. His strategy is almostorthogonal to that of Wassermann, replacing most of the arsenal of representation theory by a few simple (butfar-reaching) facts on the distance to the origin in G/K and on Harish Chandra’s elementary spherical functions.This is not to say that Lafforgue turned away from representation theory: he proved that the Connes-Kasparovconjecture actually implies that the discrete series can be accounted for with the help of Dirac equations on G/K[41]. His general framework proved flexible enough to lend itself to the extensions needed to prove that (8.1) is
52
also an isomorphism when G is an arbitrary (connected 2) Lie group. For general Lie groups, Chabert, Etcherhoffand Nest proved the Connes-Kasparov conjecture in 2003 [11] by using the decomposition of an arbitrary Liegroup as a semidirect product of a reductive and a nilpotent group: calling in the Mackey machine for studyinggroup extensions [44] enabled them to reduce the general case to Lafforgue’s results on the reductive case.
The present article will describe a third way to the Connes-Kasparov isomorphism, based on Parts I and II andthe deformation (Gt)t∈R from section 4.1.
As we recalled in the Introduction, Connes and Higson had observed in the 1990s (see [14], section 4 in[7] and section 10.β in Chapter 2 of [13]) that the Connes-Kasparov conjecture is equivalent with the fact thatthe continuous field (C?(Gt))t>0 of C?-algebras has constant K-theory, although there are of course importantdifferences between C?
r (G0) and C?r (G). We here recall a precise formulation of their observation.
Connes and Higson’s deformation interpretation of the assembly map (8.1). We defined in section 4.1 afamily Gtt∈R of Lie groups, together with an isomorphism
ϕt : Gt → G
for each t 6= 0. For every t ∈ R, the group Gt is equal as a topological space with K× p, and there is a naturalsmooth family of measures on K×p giving a Haar measure for each group.
We now setG :=
⊔t∈R
Gt .
There is a natural bijectionG → G0∪
(G×R×
)(8.2)
which is the identity on G0 and sends gt ∈ Gt to (ϕt(gt), t) ∈ G×R×. It is recalled in section ?? of [29] howG0∪ (G×R×) has a natural structure of smooth manifold; using the above bijection, we transfer this structure sothat G becomes a smooth manifold, too.
We can consider the reduced C? algebra of each Gt once we equip Gt with the Haar measure in the familymentioned above. Since we chose the manifold structure on G to make it diffeomorphic with the version used inHigson’s paper [29], lemma 6.13 there shows that the field
C?r (Gt)t∈R
is a continuous field of C?-algebras.We will consider
C := C? algebra of continuous sections of the restriction of C?r (Gt) to [0,1]. (8.3)
Evaluation at t = 0 and t = 1 induce C?-algebra morphisms from C to C?r (G0) and C?
r (G), respectively, and in turnthese induce two homomorphisms α0 : K (C )→ K (C?
r (G0)) and α1 : K (C )→ K (C?r (G1)). Because the field
C?(Gt)t∈]0,1] (with t = 0 excluded) is trivial, α0 is an isomorphism. Now, the composition
α1 α−10 : K (C?(G0))→ K (C?(G)) (8.4)
has an important connection with the Connes-Kasparov conjecture. Because G0 = Knp is an extension of K by avector abelian group, there turns out to be a natural isomorphism between K (C?(G0)) and R(K); in fact, viewed
2. It is important for the proof in [11] that the connectivity hypothesis be dropped, but then the Connes-Kasparov conjecture can no longerbe formulated with (8.1).
53
through this isomorphism, (8.4) is none other than the Connes-Kasparov assembly map (8.1):
Theorem 8.1 (Connes and Higson, 1990 ??). If the map α1, induced by evaluation at t = 1, is an isomorphism,then the Connes-Kasparov assembly map (8.1) is an isomorphism, too.
Higson’s argument for complex semisimple groups. Higson decided, about ten years ago, to re-examine theconnections between the Mackey analogy and K-theory. His discovery of the Mackey bijection in the specialcase of complex semisimple groups revealed the structures of C?
r (G0) and C?r (G) to be close enough to lead to a
surprisingly simple proof of the Connes-Kasparov conjecture .We first outline his argument; although the correspondence for semisimple groups can be defined without any
reference to lowest K-types, minimal K-type do come in as a crucial tool in the study of the field of C?-algebras.Higson’s proof runs as follows:
(a) When G is a complex semisimple Lie group, there is a natural bijection between the reduced duals of G andG0, and it is compatible with minimal K-types.
(b) To a given (set of) minimal K-type(s), one can associate a subquotient of the reduced C? algebra of G anda subquotient of that of G0; these subquotients turn out to have the same spectrum (viewed as a topologicalspace), and each is Morita-equivalent with the algebra of continuous functions (vanishing at infinity) on thecommon spectrum.
(c) Associated with the deformation (Gt)t∈R of groups, there is a continuous field interpolating between the sub-quotients in (b); an analysis of the behavior of matrix elements for representations of Gt as t varies reveals thatthis subquotient is homotopically trivial, in the sense of strong Morita equivalence with a constant field.
(d) K-theory is cohomological in nature, and it is homotopy- and Morita-invariant.
This way to the Connes-Kasparov isomorphism does takes one through the fine structure of representationtheory, but rather than using it for a direct calculation of (8.1), it expresses the Connes-Kasparov isomorphismas a K-theoretic reflection of the relationship between C?(G) and C?(G0) induced by the analogy between G andG0. An appealing feature in this perspective is that only simple and quite general facts about K-theory are needed,and that no K-theory group need be written down explicitly. Higson’s results have been generalized to Lie groupswith a finite number of connected components and a complex semisimple identity component in a recent paper ofSkukalek [58].
Now the representation theory of complex semisimple groups is famously much simpler than that of generalreductive groups: for instance, the existence of the discrete series (and with it the need for the bulk of Harish-Chandra’s work) is a specific feature of the real case. Higson’s analysis of C?
r (G0) and C?r (G) accordingly rests
on representation-theoretic facts which, on the surface, may look quite special to the complex case; at least two ofthem, the existence of a unique minimal K-type for every irreducible tempered representation and the connected-ness of all the “little groups” of section 2.3, plsy an important role in [29].
With the results of Parts I and II established, we set out to adapt Higson’s method and to obtain a proof ofthe Connes-Kasparov “conjecture” for real reductive Lie groups. We should say very clearly that our main taskis to use the results of Parts I and II to see whether the complex-case-dependent details of Higson’s work can beadapted in order to cover to the case of real groups, rather than introducing new methods on the C?-algebra side.As a consequence, we will use many of the notations and lemmas in [29].
Point (a) has been established for real groups in sections 3.2 and 3.3, and (d) does not depend on the groupunder consideration. Our task for part III is thus to prove that in spite of the non-uniqueness of minimal K-typesin the real case, points (b) and (c) can be carried over to real reductive groups without any major hurdle. Whatis perhaps most surprising in our results is that one should meet no serious obstacle on the way from complexgroups to real groups except those which come from the non-uniqueness of minimal K-types and will very easilybe overcome, in spite of the fact that matrix elements of discrete series representations (or limits of such) will enterthe picture at some point.
Section 9 below establishes (b); it follows [29] closely − because the subquotients to be defined have a slightlymore complicated dual, we merely add some observations furnished by the Knapp-Zuckerman classification of
54
tempered representations and Vogan’s results on minimal K-types. Section 10 turns to the deformation from G toG0 and uses the results of Part II to obtain, for matrix elements, the kind of continuity as t goes to zero necessaryto complete the proof of the Connes-Kasparov isomorphism through Theorem 8.1.
9. Distinguished subquotients of the reduced C?-algebras
9.1. Notations on matrix coefficients
• Suppose K is any compact Lie group. When λ is a class in K, we shall write d(λ ) for the dimension of theirreducible K-modules with that equivalence class.
Suppose Vλ is an irreducible K-module with equivalence class λ , and write 〈·, ·〉 for a K-invariant inner producton it. When v is a nonzero vector in Vλ , set
pvλ= k 7→ 〈v,λ (k−1)v〉. (9.1)
When K is connected, the highest-weight vectors cut out a privileged one-dimensional subspace, and any unithighest-weight vector can be chosen for v, yielding a canonical choice for the corresponding matrix element, as in[29]. For our present purposes, it will be useful to delay the choice of matrix element until §9.3.1.•We now take up some notations from Higson’s work. Suppose G is a connected unimodular Lie group, K is a
compact subgroup, s is a smooth function on K and f is a smooth and compactly supported function on G. Choosea Haar measure on G and define two convolutions between s and f ,
s ?K
f = g 7→ 1Vol(K)
∫K
s(k) f (k−1g)dk , f ?K
s = g 7→∫
Kf (kg)s(k−1)dk.
These are two smooth and compactly supported functions on G.Now suppose K is any compact Lie group, K1 is a closed subgroup, (V,τ) is an irreducible representation of K
with orthonormal basis vα and W is a K1-invariant irreducible subspace of V . Write eαβ for the matrix elementk 7→ dim(V )〈τ(k)vβ ,vα〉 (this is a smooth function on K), and dαβ for k 7→ dim(W )〈τ(k)vβ ,vα〉 when vα and vβ
lie in W ; the restriction of dαβ to K1 is a matrix element of (W,τ|W ). Then the Schur-Weyl orthogonality relationsyield
eαβ ?K
eβγ = eαγ
dαβ ?K1
eβγ = eαγ = dαβ ?K1
eβγ (9.2)
(for the second equality, it is to be assumed that vα , vβ and vγ lie in W ). We also note that eαβ (k) = eβα(k−1) forall k.
9.2. Subquotients of the reductive group’s algebra
9.2.1. Ideals associated with a set of minimal K-types
We return to studying our linear connected reductive group G and maximal compact subgroup K. Define aset
K ⊂
finite subsets of K
by declaring that C is in K when there is an irreducible tempered representation of G whose set of minimal K-typesis exactly C . Note that in this case, ‖·‖K (from section 3.1) takes the same value on all the elements of C .
We will associate a subquotient of the reduced C∗-algebra C?r (G) to every element of K . Later on it will be
convenient that the family subquotients obtained in this way be associated to an increasing sequence of ideals inC?
r (G), so let us choose first a linear ordering
K = C1,C2, ...
55
in such a way that• if the value of ‖·‖K on Cp is (strictly) smaller than that on Cq, then p < q,• if the values of ‖·‖K on Cp and Cq agree but the number of elements in Cp is (strictly) larger than that in Cq,
then p < q.As soon as we choose choose an arbitrary nonzero vector vλ in an irreducible K-module with class λ for each
λ , and define a matrix element pλ = pvλ
λas above, we can define a closed ideal in C?
r (G) by setting
J[p] =⋂
λ∈Cp
C?r (G)pλC?
r (G),
and a subquotient of C?r (G) by setting
C[p] = (J[1]+ ...+J[p])/(J[1]+ ...+J[p−1]) .
These closed ideals correspond to locally closed subsets of G:• The spectrum of J[p] is the set of irreducible tempered representations of G whose restriction to K contains
every class in Cp.• The spectrum of C[p] is the set of irreducible tempered representations of G whose set of minimal K-types is
exactly Cp.
Remark. If G is a complex semisimple Lie group, then every C in K contains only one K-type. This leads to someimportant simplifications in [29].
9.2.2. Morita-equivalence with a commutative algebra
Recall that minimal K-types occur with multiplicity one in irreducible tempered representations of linear connectedreductive groups. Suppose λ is in K and π is an irreducible tempered representation of G, say on H , containing λ
as a minimal K-type. Then for every vector v in the subspace of H carrying the K-type λ , pvλ
defines a projectionin the multiplier algebra of C[p], and the projection π(pv
λ) has rank one.
Lemma 9.1 (Lemma 6.1 in [29]). Let C be a C∗-algebra and p be a projection in the multiplier algebra of C. Iffor every irreducible representation π of C the operator π(p) is a rank-one projection, then
(i) CpC = C;
(ii) pCp is a commutative C∗-algebra;
(iii) the spectrum C is a Hausdorff locally compact space;
(iv) The map a 7→ a from pCp to C0(C) defined by
∀π ∈ C, π(a) = a(π)π(p)
is an isomorphism of C∗-algebras.
9.3. Some precisions on the subquotient’s spectrum
Part of Lemma 9.1 says that the spectrum of C[p] is a locally compact Hausdorff space. We give some precisionsby recording a simple consequence of Lemma 3.4 (Vogan’s work) and the Knapp-Zuckerman classification.
Some additional notations will be necessary. Choose an family P1, ...Pr of nonconjugate cuspidal parabolicsubgroups in G, with one member in the family for each conjugacy class of cuspidal parabolic subgroups. WriteMiAiNi for the Langlands decomposition of Pi, and Ki for the maximal compact subgroup K∩Mi in Mi. Anticipat-ing a later need for further notation, write Mp
i for expG (mi∩p) and recall that, with the obvious notations for theLie algebras, the Iwasawa map from K× (mi∩p)×ai×ni to G is a diffeomorphism.
56
Now, fix a linear ordering Ki = λ1,λ2, ... in such a way that if ‖λn‖Ki< ‖λm‖Ki
, then n < m. By discrete pa-rameter, we will henceforth mean a couple (i,n) with i in 1, ..r and n in N. From Part I, we extract the followinginformation on the dual of C[p] (see the proof of Theorem 3.2 and the discussion of K-types in §3.3):
Lemma 9.2. Fix an element C of K . There exists a discrete parameter (i0,n0) and a subset a[p] of a?i0 , whichintersects every Weyl group orbit at most once, such that• VMi0
(µn0) is a discrete series or nondegenerate limit of discrete series representation of Mi0 ,• Every irreducible tempered representation of G whose set of minimal K-types is C is equivalent with exactly
one of theIndG
Mi0
[VMi0
(µn0)⊗ eiχ+ρ
],χ ∈ a[p].
• For every χ in a[p], Pχ ⊃ Pi0 (in fact Mχ ⊃Mi0 , while Aχ ⊂ Ai0 and Nχ ⊂ Ni0 ).
This identifies the spectrum of C[p] with a[p] as a set; we shall see that when a[p] is equipped with the topologythat it inherits from Euclidean space, the identification becomes a homeomorphism.
Example. When G = SL(2,R), the spherical principal series representation with nonzero continuous parameterhave the same minimal K-types as the (irreducible) spherical principal series representation with continuous pa-rameter zero, but the nonspherical principal series representation with nonzero continuous parameter have twodistinct minimal K-types and the nonspherical principal series representation with continuous parameter zero isreducible. So in that case a[p] is either a closed half-line, an open half-line or a single point. In general a[p]consists of an open Weyl chamber in a subspace of a, together with part of one of its walls, and that part-of-wallis itself stratified analogously until one reaches a minimal dimension (which might be zero, but might not). Notethat with the topology inherited from that of the Euclidean space a?, a[p] is Hausdorff and locally compact.
9.3.1. An explicit formula for the isomorphism in Lemma 9.1.
Suppose a discrete parameter (i0,n0), and thus a class C = Cp, are fixed, and let us start with χ in a[p].In the Hilbert space V n0 for a representation σn0 whose equivalence class is VMi0
(µn0), consider the Ki0 -isotypical subspace W which corresponds to the Ki0 -type µn0 , and choose a basis vα for it.
Choose one K-type λp in the class Cp. Since p will remain fixed until the end of section 9.2, we will usually
remove the subscript p from λ . We write Hχ
i0,n0for the Hilbert space carrying IndG
Mi0
[VMi0
(µn0)⊗ eiχ+ρ
]in the
usual induced picture: the completion ofξ : G smooth−→
comp. supp.V n0 | ξ (gman) = a−iχ−ρ
σn0(mp)−1
ξ (g) for (g,mp,a,n) ∈ G×Mpi0×Ai×Ni
in the norm associated to the inner product 〈ξ1,ξ2〉 =∫
K〈ξ1(k),ξ2(k)〉V n0 dk. We shall denote for the usual mor-
phism from G to the unitary group of Hχ
i0,n0as write πn0χ : for every g in G and ξ in H
χ
i0,n0, πn0χ(g)ξ is the map
x 7→ ξ (g−1x).Upon decomposing the λ -isotypical K-invariant component in H
χ
i0,n0, say V , into Ki0 -invariant parts, Frobenius
reciprocity says the Ki0 -type µn0 appears exactly once. Fix a K-equivariant identification between the correspond-ing Ki0 -irreducible subspace and W , write v for the vector in the λ -isotypical subspace V which the identificationassigns to any v in W . Now, choose an arbitrary v0 in W (in order to keep the notation reasonable, choose it amongthe vα s), and introduce as in (9.1) a matrix coefficient of our irreducible K-module of type λ as
pλ = pv0λ= k 7→ 〈v0,πn0χ(k)v0〉
This is our choice of v in (9.1); henceforth we shall assume that it is this matrix element that is chosen in thedefinition of J[p] and C[p]. Note that the smooth function on K which we just defined does not depend onχ .
57
(It may be useful to comment on the relationship with the treatment of complex semisimple groups in [29]: wehere choose a vector in W rather than any nonzero element the λ -isotypical subspace. For complex semisimplegroups the two choices could be made to coincide because the highest-weight vector for V automatically did lie inW , but that is a priori not the case here).
The projection πn0χ(pλ ) has rank one, and to make Lemma 9.1 explicit, we now imitate Higson’s analysis ofthe complex semisimple case, and use the calculations he made in [29] for the Cartan motion group. A first step isto identify the range of πn0χ(pλ ). Define a function from K to W as
ζn0χ : k 7→d(µn0 )
∑α=1〈πn0χ(k)vα , v0〉vα =
1d(λ )
d(µn0 )
∑α=1
eα0α(k).
We can define a vector in the representation space Hχ
i0,n0by extending ζn0χ to G, setting
ξn0χ(kmpan) =d(λ )1/2
vol(K)1/2d(µn0)1/2 e−iχ−ρ(a)∑
α
〈πn0χ(k)vα , v0〉 ·σn0(mp)−1 [vα ] .
Lemma 9.3. The operator π(pλ ) on Hχ
i0,n0agrees with the orthogonal projection on ξn0χ .
This is easily proved using the formula for the action of K and the inner product on the representation space,and a repeated application of the Schur-Weyl orthogonality relations.
Now put
f [p] : a[p]→ C
χ 7→∫
Gf (g)〈ξn0χ ,πn0χ(g)ξn0χ〉 (9.3)
as soon as f is a smooth and compactly supported function on G. If pλ ?K
f ?K
pλ = f , then πn0χ( f ) is proportional
to πn0χ(pλ ), and given the definition of πn0χ( f ), we see that πn0χ( f ) = f [p](χ) πn0χ(pλ ). This is a first step inmaking Lemma 9.1 explicit, and if we add that f [p] is continuous and vanishes at infinity as a function of χ (thiswill be obvious from the calculation below), we can summarize the above in the following statement (compareLemma 6.10 in [29]).
Proposition 9.4. By associating to any smooth and compactly supported function f on G such that pλ ?Kf ?
Kpλ = f ,
the element f [p] of C0(a[p]), one obtains a C?-algebra isomorphism between pλ C[p]pλ and C0(a[p]).
A consequence which we already mentioned is that the correspondence identifies the dual of C[p] with a[p]homeomorphically.
It will be important later on to have a completely explicit formula for f [p], so we now record a closed form for(9.3), to be used in section 10 below.
Write α0 for the one α such that v0 = vα . For a,b in 1, ..,d(µn0) and χ in a[p], set
f pa,b := χ 7→ Vol(K)
d(λ )
∫Ni0
dn∫
Ai0
da aiχ+ρ
∫Mp
i0
dm(
f ?K1
dα0a ?K1
dbα0
)(nam)〈vb,σn0(m
−1) [va]〉. (9.4)
Lemma 9.5. The element f [p] of C0(a[p]) can be expressed as
f [p] =1
d(µn0)
d(µn0 )
∑a,b=1
f pa,b.
Proof: We expand (9.3), closely following the calculations on page 15 of [29]. We start from the fact that
58
〈ξn0χ ,πn0χ(g)ξn0χ〉 =∫
K〈ξµn0 χ(k) , ξµn0 χ(g−1k)〉dk; after a change of variables g ← g−1k, and inserting the
necessary normalizations to have the eαβ appear, we find
f [p] =∫
G
(∫K
f (kg−1)〈ξµn0 χ(k),ξµn0 χ(g)〉dk)
dg
=∫
G
(∫K
f (kg−1)1
Vol(K)1/2d(λ )1/2d(µn0)1/2 ∑
α
eα0α(k)〈vα ,ξµn0 χ(g)〉dk)
dg
=
(1
Vol(K)d(λ )1/2d(µn0)
)1/2
∑α
∫G
Vol(K)(eαα0 ? f
)(g−1)〈vα ,ξµn0 χ(g)〉dg
=1
d(λ )d(µn0)∑α,β
∫Nχ
dn∫
Aχ
da∫
Mpχ
dm∫
Kdu (eαα0 ? f )(n−1a1m−1u−1)eα0β (u)〈vα ,σ(m−1)vβ 〉a−iχ−ρ
=Vol(K)
d(λ )d(µn0)∑α,β
∫Nχ
dn∫
Aχ
da a−iχ−ρ
∫Mp
χ
[eαα0 ? f ? eα0β
](n−1a−1m−1)〈vα ,σ(m−1)vβ 〉
in which the stars are convolutions over K. To relate this to (9.4) we need to have the dαβ enter the formula inplace of the eαβ . We use (9.2) to observe that
eαα0 ?Kf ?
Keα0β = dαα0 ?
Ki0
(eα0α0 ?K
f ?K
eα0α0
)?
Ki0
dα0β ;
because of our hypothesis on f this is equal to dαα0 ?Ki0
f ?Ki0
dα0β .
To get two convolutions on the right of f instead of one on each side of f , we shorten the above formula bywriting Γ
n0αβ
(m) for 〈vα ,σ(m−1)vβ 〉, and we use the structure properties of the parabolic subgroup Pi0 , along withthe fact that χ is Kχ -invariant, thus Ki0 -invariant because of the last point in Lemma 3.2.3, to obtain∫
Ni0
dn∫
Ai0
da a−iχ−ρ
∫Mp
i0
[dαα0 ? f ?dα0β
](n−1a−1m−1)Γ
n0αβ
(m) =∫Ni0
dn∫
Mpi0
dmΓn0αβ
(m)∫
Ki0
dk1
∫Ki0
dk2
∫Ai0
da a−iχ−ρ dαα0(k1) f (k−11 n−1m−1a−1k2)dα0β (k2) =∫
Ni0
dn∫
Mpi0
dmΓn0αβ
(m)∫
Ki0
dk1
∫Ki0
dk2
∫Ai0
da a−iχ−ρ dαα0(k1) f (n−1m−1a−1k−11 k2)dα0β (k2) =∫
Ni0
dn∫
Ai0
da a−iχ−ρ
∫Mp
i0
[f ?dαα0 ?dα0β
](n−1a−1m−1)Γ
n0αβ
(m)
(between the first and second line, we used the fact that Mi0 centralizes Ai0 ; between the second and third line, weused the fact that Ki0 is contained in Mi0 , and thus leaves Ai0 invariant and normalizes Ni0 , to perform the changeof variables m← k−1
1 mk1, n← k−11 nk1, a← k−1
1 ak1). The lemma is now proved.
Remark. In contrast to the situation for complex groups, the presence of a noncompact part in Mi0 makes thedependence of the transform f [p] on p somewhat nontrivial. But we shall see that the more complicated terms willactually vanish as we perform the contraction to the Cartan motion group.
9.4. Subquotients of the motion group’s algebra
Taking up the notations of sections 9.1 and 9.2.1, define a closed ideal in the reduced C? algebra C?r (G0) by setting,
as before,J0[p] =
⋂λ∈Cp
C?r (G0)pλC?
r (G0),
59
and a subquotient of C?r (G0) by setting
C0[p] = (J[1]+ ...+J[p])/(J[1]+ ...+J[p−1]) .
As before, the spectrum of C0[p] is the set of unitary irreducible representations of G0 whose set of minimal K-types is exactly Cp. Because of Proposition 3.3 from Part I, we know that the dual of C0[p] can be identified as aset with a[p].
Lemma 9.1 is still applicable here of course, and C0[p] is thus Morita-equivalent with the algebra of continuousfunctions, vanishing at infinity, on its spectrum − viewed as a topological space. To show prove that the analogybetween C0[p] and C[p] runs deeper, the next step is to identify the spectrum of C0[p] with a[p] not only as aset, but also as a topological space, writing down an analogue of Proposition 9.4. The explicit calculations inHigson’s paper are actually sufficient for this, so instead of imitating the results of section 5.3 of [29], we shallinvoke them.
For every χ in a[p], Lemma 9.2 and Proposition 3.3 show that there is exactly one element µp,χ in Kχ for whichthe set of minimal K-types of the G0 representation M0(χ,µp,χ) (see §2.3) is exactly Cp. Let us write W for thecarrier vector space of µp,χ (beware that the meaning of W is no longer the same as in the previous section).
We now use Lemma 5.8 in [29]: we extend χ to p by setting it equal to zero on the orthogonal of aχ , thenset
f (χ) =vol(K)
d(λ )
∫p
f (x)ei〈χ,x〉dx
as soon as f is a smooth and compactly supported function on G0, and realize M0(χ,µp,χ) as the completion H0
of ξ : K smooth−→ V
∣∣ ξ (gkx) = µp,χ(k)−1χ(x)−1
ξ (g) for (k,x,g) ∈ Kχ ×p×G0
in the norm induced by the scalar product between restrictions to K. We write π0
χ,µ for the G0-action on inducedby left translation.
At this point, we cast a look backwards to the situation for reductive groups. When the λp-isotypical subspaceV of H
χ
i0,n0is viewed as a K-module, remark that it must contain the Kχ -type µ exactly once as before, because
the class of πn0,χ in G is the representation M(χ,µp,χ) defined in section 3.2; as a consequence, we can obtaina unit vector ζpχ in H0 by copying the definition of ζn0χ (the only difference is that what W stands for haschanged).
The projection π0χ,µ(pλ) has rank one because of Corollary 3.5 on minimal K-type theory for G0. As detailed in
Lemma 5.8 of [29], that projection agrees with the orthogonal projection on ξp,χ , and the condition pλ ?K
f ?K
pλ = f
leads to the equalityπ
0χ,µ( f ) = f (χ) π
0χ,µ(pλ).
Lemma 9.1 then says that is we view a[p] as the dual of C0[p] and equip it with the corresponding (locallycompact and Hausdorff) Fell topology, then f becomes a continuous function of χ that vanishes at infinity. Butof course f is the ordinary Fourier transform of f , so that it is also a continuous function on a[p] when the latteris equipped with the topology inherited from that of Euclidean space. We can summarize the situation with thefollowing statement.
Proposition 9.6. By associating to any smooth and compactly supported function f on G0 such that pλ ?K
f ?K
pλ ,
the element f of C0(a[p]), one obtains a C?-algebra isomorphism between pλ C0[p]pλ and C0(a[p]).
We note that the transform at χ involves, on the G0-side, the representation π0χ,µ , and on the G-side the rep-
resentation πn0χ . These two do correspond to one another in the correspondence introduced in section 3.2, soPropositions 9.4 and 9.6 together yield:
Corollary 9.7. Fix a class C ∈K of minimal K-types, write G[C ] (resp. G0[C ]) for the set of irreducible tempered
60
representations of G (resp. unitary irreducible representations of G0) whose set of minimal K-types is exactly C .The Mackey-Higson correspondence induces a homeomorphism between G0[C ] and G[C ]
10. Deformation of the reduced C?-algebra and subquotients
In writing the above for each Gt , t > 0, and trying to observe the form of rigidity necessary to view Propositions9.6 and 9.4 as deformations of one another, the presence of a noncompact M in the parabolic subgroups whichgive rise to (9.4) will make it necessary to follow the matrix elements 〈vb,σn0(m
−1) [va]〉 through the deformation;not surprisingly, some of the technical lemmas in Part II will prove useful. Of course, as usual in Part III, [29] willshow the way (see [29], §6.2-6.3).
10.1. Subquotients of the continuous field and their spectra
We turn to the continuous field C from (8.3), and as before, let J [p] be the ideal⋂
λ∈Cp
C pλ C of C , then define a
subquotient of C asC [p] = (J [1]+ ...J [p])
/(J [1]+ ...+J [p−1]) .
We recall from [29], section .., that the dual of C [p] can be identified with a[p]× [0,1] as a set: the algebra Z ofcontinuous functions on the closed interval [0,1] lies in the center of the multiplier algebra of C [p], so that if Zt
is the subalgebra of functions which vanish at t for t in [0,1], the dual of C [p] can be identified as a set with thedisjoint union over t of the duals of the quotient algebras C [p]/(ZtC [p]): with each t, we need only associate thesubset of the dual of C [p] gathering the representations which restrict to zero on ZtC [p] .
But of course, the algebra C [p]/(ZtC [p]) is isomorphic with the subquotient Ct [p] of the group algebra C?r (Gt),
and we saw that it can be identified with a[p].To proceed further, we need to see how the dual of C [p] can be identified with a[p]× [0,1] as a topological space.
Now, suppose f is a smooth and compactly supported function on G . Write ft for its restriction to a smooth andcompactly supported function on Gt . Write Mi0,tAi0,tNi0,t for the cuspidal parabolic subgroup subgroup ϕ
−1t P of Gt ,
and note that it comes with an ordering on the ai0,t -roots and an associated half-sum ρt of positive roots. Supposeσ t
n0is an irreducible representation of Mi0,t with equivalence class VMi0 ,t
(µn0). Our treatment of the contraction oftempered representations in Part II furnishes a Ki0 -equivariant identification between the Ki0 -invariant subspace ofthe Hilbert space for σ t
n0which carries its minimal Ki0 -type, and the subspace W of σn0 : in the language of Part II
one contracts onto the other, and the contraction process induces isomorphisms on the level of minimal Ki0 -types.Write (vt
a) for the basis of that subspace thus associated to (va). Gather the transforms considered in sections 9.2.1and 9.4, and define, for every for χ in a[p],
f pa,b(χ, t) =
δa,b
∫p
f0(x)χ(x)dx (δab = Kronecker delta) if t = 0,
∫Ni0 ,t
dnt
∫Ai0
dat aiχ+ρtt
∫Mp
i0,t
dmt(
ft ?dα0a ?dbα0
)(ntatmt)〈vt
b,σtn0(m−1
t )[vt
a]〉 if t 6= 0.
(10.1)
Last, define
f p = (χ, t) 7→d(µn0 )
∑a,b=1
f pa,b(χ, t), (10.2)
a map from a[p]× [0,1] to C.
Lemma 10.1. Formula (10.2) defines a smooth function on a[p]× [0,1].
Proof: The most obvious inconvenience in applying the usual theorems on the continuity of parameter integrals isthat the subgroups on which the integrals are taken depend on t. In order to write down (10.1) as an integral over
61
a space which does not depend on t, let us write yi0 for p∩(ai0 ⊕mp
i0
)⊥and
βt : yi0 → ni0,t
for the inverse of n 7→ n+nθ (the Cartan involution of gt does not depend on t). Then using exponential coordinatesfor expGt
(p), remarking that mpi0
and ai0 are dϕt(0)-invariant and inserting the fact that ρt is none other than tρ
(Lemma 5.12 in part II), we can rewrite f pa,b(χ, t) as∫
yi0
dy∫ai0
da ei〈χ,a〉 e〈ρ,ta〉∫mp
i0
dm(
f ?dα0a ?dbα0
)(expβt y
Gtexpa
Gtexpm
Gt
)〈vt
b,σtn0(exp−m
Gt)[vt
a]〉
for t 6= 0. We now remark that f is a smooth, compactly supported function on G if and only if there is a smoothand compactly supported function F on K×mp
χ ×aχ ×yχ ×R such that
F(k,m,a,y) =
f0(k,m+a+ y) if t = 0,
ft(
k expmGt
expGtexpβt y
Gt
)if t 6= 0,
where ft is the restriction of f to Gt (compose (8.2) with the proof of Lemma 6.17 in [29]). Once we insert this,as well as the relationship between σ t
n0and σn0 detailed in Part II (sections 5.1 and 6), into (10.1), we can rewrite
f pa,b(χ, t) as
∫yi0
dy∫ai0
daei〈χ,a〉 e〈ρ,ta〉∫mp
i0
dm[F ?dα0a ?dbα0
](1,m,a,y, t)〈vb,σn0(expG(−tm)) [va]〉
for t 6= 0.Recall that for fixed p, we chose above a λ ∈ Cp above, and chose a definition for pλ . To check that (10.1)
does define a continuous function as soon as f induces an element of C[p], it is now enough to remember thatp=mp
i0+ai0 +yi0 , and to use the Lebesgue (dominated convergence) theorem, the fact that (va) is an orthonormal
basis, and the Schur-Weyl relations on the matrix elements d.
Combining Lemma 10.1 and the results in §9, we obtain a statement which summarizes point (c) in the list of§8:
Proposition 10.2. If f is a smooth and compactly supported function on G0 and if pλ f pλ = f , then its transfom f p
is a continuous function on a[p]× [0,1], and it vanishes at infinity. This determines an isomorphism of C?-algebras
pλ C [p]pλ → C0 (a[p]× [0,1]) .
10.2. Proof of the Connes-Kasparov isomorphism
We now have gathered all the ingredients needed to prove the Connes-Kasparov “conjecture”. The argument is notonly close to, it is identical with, that in section 7 of [29], so that the contents of this subsection are really due toNigel Higson.
Because⋃
p∈N(J [1]+ ...J [p]) is dense in C and because K-theory commutes with direct limits, we need only
prove that evaluation at t = 1 yields an isomorphism between K (J [1]+ ..++J [p]) and K (J[1]+ ..++J[p]) forevery p. By standard cohomological arguments, this will be attained if we prove that evaluation at t = 1 inducesan isomorphism between K (C [p]) and K (C[p]).
At this point we recall that for fixed p, the algebras pλpC[p]pλp and C[p]pλpC[p] are Morita-equivalent; there-fore the inclusion from pλpC[p]pλp to C[p] induces an isomorphism in K-theory. The problem then reduces toshowing that evaluation at t = 1 induces an isomorphism between K (pλpC [p]pλp) and K (pλpC[p]pλp).
62
Observing Proposition 10.2 and Proposition 9.4, we need only recall the homotopy invariance of K-theory tofind that between K (a[p]× [0,1]) and K (a[p]), evaluation at t = 1 does induce an isomorphism. With this theproof that α1 is an isomorphism is complete.
Remark. Skukalek’s results which generalize Higson’s work to almost connected Lie groups with complex semisim-ple identity component [58] can be used to extend the above proof to almost connected Lie groups with reductiveidentity component. These groups may be nonreductive, or may get out of Harish-Chandra’s class if they are.
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