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Fixed Point Theorem and Character Formula
Hang WangUniversity of Adelaide
Index Theory and Singular StructuresInstitut de Mathematiques de Toulouse
29 May, 2017
OutlineAim: Study representation theory of Lie groups from the pointof view of geometry, motivated by the developement of
K-theory and representation;
Harmonic analysis on Lie groups.
Representation theory Geometry
character index theoryof representations of elliptic operators
Weyl character formula Atiyah-Segal-Singer
Harish-Chandra character formula Fixed point theorem
P. Hochs, H.Wang, A Fixed Point Formula andHarish-Chandra’s Character Formula, ArXiv 1701.08479.
Representation and characterG: irreducible unitary representations of G (compact, Lie);
For (π, V ) ∈ G, the character of π is given by
χπ(g) = Tr[π(g) : V → V ] g ∈ G.
Example
Consider G = SO(3) with maximal torus T1 ∼= SO(2) → SO(3).
Let Vn ∈ SO(3) with highest weight n, i.e.,
Vn|T1∼=
2n⊕j=0
Cj−n
where Cj = C, on which T1 acts by g · z = gjz, g ∈ T1, z ∈ C.Then
χVn(g) =
2n∑j=0
gj−n g ∈ T1.
Weyl character formula
Let G be a compact Lie group with maximal torus T .
Let π ∈ G. Denote by λ ∈√−1t∗ its highest weight.
Theorem (Weyl character formula)
At a regular point g of T :
χπ(g) =
∑w∈W det(w)ew(λ+ρ)
eρΠα∈∆+(1− e−α)(g).
Here, W = NG(T )/T is the Weyl group, ∆+ is the set ofpositive roots and ρ = 1
2
∑α∈∆+ α.
Elliptic operatorsM : closed manifold.
Definition
A differential operator D on a manifold M is elliptic if itsprincipal symbol σD(x, ξ) is invertible whenever ξ 6= 0.
Dirac type operators ⊂ elliptic operators.
Example
de Rham operator on a closed oriented even dimensionalmanifold M :
D± = d+ d∗ : Ω∗(M)→ Ω∗(M).
Dolbeault operator ∂ + ∂∗ on a complex manifold.
Equivariant Index
G: compact Lie group acting on compact M by isometries.
R(G) := [V ]− [W ] : V,W fin. dim. rep. of G representationring of G (identified as rings of characters).
Definition
The equivariant index of a G-invariant elliptic operator
D =
[0 D−
D+ 0
]on M , where (D+)∗ = D− is given by
indGD = [kerD+]− [kerD−] ∈ R(G);
It is determined by the characters
indGD(g) := Tr(g|kerD+)− Tr(g|kerD−) ∀g ∈ G.
Example. Lefschetz number
Consider the de Rham operator on a closed oriented evendimensional manifold M :
D± = d+ d∗ : Ωev/od(M)→ Ωod/ev(M).
kerD± ↔ harmonic forms ↔ Hev/odDR (M,R).
Lefschetz number, denoted by L(g):
indGD(g) =Tr(g|kerD+)− Tr(g|kerD−)
=∑i≥0
(−1)iTr [g∗,i : Hi(M,R)→ Hi(M,R)] .
Theorem (Lefschetz)
If L(g) 6= 0, then g has a fixed-point in M.
Fixed point formula
M : compact manifold.
g ∈ Isom(M).
Mg: fixed-point submanifold of M .
Theorem (Atiyah-Segal-Singer)
Let D : C∞(M,E)→ C∞(M,E) be an elliptic operator on M .Then
indGD(g) = Tr(g|kerD+)− Tr(g|kerD−)
=
∫TMg
ch([σD|Mg ](g)
)Todd(TMg ⊗ C)
ch([∧
NC](g))
where NC is the complexified normal bundle of Mg in M .
Equivariant index and representation
Let π ∈ G with highest weight λ ∈√−1t∗. Choose M = G/T
and the line bundle Lλ := G×T Cλ. Let ∂ be the Dolbeaultoperator on M .
Theorem (Borel-Weil-Bott)
The character of an irreducible representation π of G is equal tothe equivariant index of the twisted Dolbeault operator
∂Lλ + ∂∗Lλ
on the homogenous space G/T .
Theorem (Atiyah-Bott)
For g ∈ T reg,
indG(∂Lλ + ∂∗Lλ)(g) = Weyl character formula.
ExampleLet ∂n + ∂∗n be the Dolbeault–Dirac operator on
S2 ∼= SO(3)/T1,
coupled to the line bundle
Ln := SO(3)×T1 Cn → S2.
By Borel-Weil-Bott,
indSO(3)(∂n + ∂∗n) = [Vn] ∈ R(SO(3)).
By the Atiyah-Segal-Singer’s formula
indSO(3)(∂n + ∂∗n)(g) =gn
1− g−1+
g−n
1− g=
2n∑j=0
gj−n.
Overview of main results
Let G be a compact group acting on compact M by isometries.From index theory,
G-inv elliptic operator D → equivariant index → character
Geometry plays a role in representation by
R(G)→ special D and M → character formula
When G is noncompact Lie group, we
Construct index theory and calculate fixed point formulas;
Choose M and D so that the character of indGD recoverscharacter formulas for discrete series representations of G.
The context is K-theory:
“representation, equivariant index ∈ K0(C∗rG).”
Discrete series(π, V ) ∈ G is a discrete series of G if the matrix corficient cπgiven by
cπ(g) = 〈π(g)x, x〉 for ‖x‖ = 1
is L2-integrable.
When G is compact, all G are discrete series, and
K0(C∗rG) ' R(G) ' K0(Gd).
When G is noncompact,
K0(Gd) ≤ K0(C∗rG)
where [π] corresponds [dπcπ] (dπ = ‖cπ‖−2L2 formal degree.)
Note that
cπ ∗ cπ =1
dπcπ.
Character of discrete seriesG: connected semisimple Lie group with discrete series.T : maximal torus, Cartan subgroup.
A discrete series π ∈ G has a distribution valued character
Θπ(f) := Tr(π(f)) = Tr
∫Gf(g)π(g)dg f ∈ C∞c (G).
Theorem (Harish-Chandra)
Let ρ be half sum of positive roots of (gC, tC). A discrete seriesis Θπ parametrised by λ, where
λ ∈√−1t∗ is regular;
λ− ρ is an integral weight which can be lifted to a character(eλ−ρ,Cλ−ρ) of T .
Θλ := Θπ is a locally integrable function which is analytic on anopen dense subset of G.
Harish-Chandra character formula
Theorem (Harish-Chandra Character formula)
For every regular point g of T :
Θλ(g) =
∑w∈WK
det(w)ew(λ+ρ)
eρΠα∈R+(1− e−α)(g).
Here,
T is a manximal torus,
K is a maximal compact subgroup and WK = NK(T )/T isthe compact Weyl group,
R+ is the set of positive roots,
ρ = 12
∑α∈R+ α.
Equivariant Index. Noncompact CaseLet G be a connected seminsimple Lie group acting on Mproperly and cocompactly.
Let D be a G-invariant elliptic operator D.
Let B be a parametrix where
1−BD+ = S0 1−D+B = S1
are smoothing operators.
The equivariant index indGD is an element of K0(C∗rG).
indG : KG∗ (M)→ K∗(C
∗rG) [D] 7→ indGD
where
indGD =
[S2
0 S0(1 + S0)BS1D
+ 1− S21
]−[0 00 1
].
Harish-Chandra Schwartz algebraThe Harish-Chandra Schwartz space, denoted by C(G), consistsof f ∈ C∞(G) where
supg∈G,α,β
(1 + σ(g))mΞ(g)−1|L(Xα)R(Y β)f(g)| <∞
∀m ≥ 0, X, Y ∈ U(g).
L and R denote the left and right derivatives;
σ(g) = d(eK, gK) in G/K (K maximal compact);
Ξ is the matrix coefficient of some unitary representation.
Properties:
C(G) is a Frechet algebra under convolusion.
If π ∈ G is a discrete series, then cπ ∈ C(G).
C(G) ⊂ C∗r (G) and the inclusion induces
K0(C(G)) ' K0(C∗rG).
Character of an equivariant index
Definition
Let g be a semisimple element of G. The orbital integral
τg : C(G)→ C
τg(f) =
∫G/ZG(g)
f(hgh−1)d(hZ)
is well defined.
τg continuous trace, i.e., τg(a ∗ b) = τg(b ∗ a) for a, b ∈ C(G),which induces
τg : K0(C(G))→ R.
Definition
The g-index of D is given by τg(indGD).
Calculation of τg(indGD)If GyM properly with compact M/G, then∃c ∈ C∞c (M), c ≥ 0 such that
∫G c(g
−1x)dg = 1,∀x ∈M.
Proposition (Hochs-W)
For g ∈ G semisimple and D Dirac type,
τg(indGD) = Trg(e−tD−D+
)− Trg(e−tD+D−
)
where
Trg(T ) =
∫G/ZG(g)
Tr(hgh−1cT )d(hZ).
When G,M are compact, then c = 1 and Str(hgh−1e−tD2)
= Str(gh−1e−tD2h) =Tr(ge−tD
−D+)− Tr(ge−tD
+D−)
=Tr(g|kerD+)− Tr(g|kerD−).
⇒ τg(indGD) = vol(G/ZG(g))indGD(g).
Fixed point theorem
Theorem (Hochs-W)
Let G be a connected semisimple group acting on M properlyisometrically with compact quotient. Let g ∈ G be semisimple.Ifg is not contained in a compact subgroup of G, or if G/K isodd-dimensional, then
τg(indGD) = 0
for a G-invariant elliptic operator D.If G/K is even-dimensional and g is contained in compactsubgroups of G, then
τg(indGD) =
∫TMg
c(x)ch([σD](g)
)Todd(TMg ⊗ C)
ch([∧
NC](g))
where c is a cutoff function on Mg with respect to ZG(g)-action.
Geometric realisation
Let G be a connected semisimple Lie group with compactCartan subgroup T. Let π be a discrete series withHarish-Chandra parameter λ ∈
√−1t∗.
Corollary (P. Hochs-W)
Choose an elliptic operator ∂Lλ−ρ + ∂∗Lλ−ρ on G/T which is
the Dolbeault operator on G/T coupled with
the homomorphic line bundle
Lλ−ρ := G×T Cλ−ρ → G/T.
We have for regular g ∈ T ,
τg(indG(∂Lλ−ρ + ∂∗Lλ−ρ)) = Harish-Chandra character formula.
Idea of proof
[dπcπ] is the image of [Vλ−ρc ] under the Connes-Kasparovisomorphism
R(K)→ K0(C∗rG).
indG(∂Lλ−ρ + ∂∗Lλ−ρ) = (−1)dimG/K
2 [dπcπ].
(−1)dimG/K
2 τg[dπcπ] = Θλ(g) for g ∈ T.τg(indG(∂Lλ−ρ + ∂∗Lλ−ρ)) can be calculated by the main
theorem and be reduced to a sum over finite set (G/T )g.
Summary
We obtain a fixed point theorem generalizingAtiyah-Segal-Singer index theorem for a semisimple Liegroup G acting properly on a manifold M with compactquotient;
Given a discrete series Θλ ∈ G with Harish-Chandraparameter λ ∈
√−1t∗, the fixed point formula for the
Dolbeault operator on M = G/T twisted by the linebundle determined by λ recovers the Harish-Chandra’scharacter formula.
This generalizes Atiyah-Bott’s geometric method towardsthe Wyel character formula for compact groups.
Outlook
The expression∫TMg
c(x)ch([σD|Mg ](g)
)Todd(TMg ⊗ C)
ch([∧
NC](g))
can be obtained for a general locally compact group usinglocalisation techniques.
It is important to show that it factors through K0(C∗rG),
i.e., equal to τg(indGD).
Fixed point formulas and charatcer formulas can beobtained for more general groups (e.g., unimodular Lie,algebraic groups over nonarchemedean fields).
Could the nondiscrete spectrum of the tempered dual G ofG be studied using index theory?
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