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The Convex Hull of the Highest Weight Orbit and the
Caratheodory Orbitope
Nigel Redding
Thesis submitted to the Faculty of Graduate and Postdoctoral Studies in partialfulfillment of the requirements for the degree of
Master of Science in Mathematics1
Department of Mathematics and StatisticsFaculty of Science
University of Ottawa
c© Nigel Redding, Ottawa, Canada, 2017
1The M.Sc. program is a joint program with Carleton University, administered by the Ottawa-Carleton Institute of Mathematics and Statistics
Abstract
In this thesis, we study the polynomial equations that describe the highest weight
orbit of an irreducible finite dimensional highest weight module under a semisimple
Lie group. We also study the connection of the convex hull of this orbit and the
Caratheodory orbitope.
ii
Dedications
To my parents.
iii
Acknowledgement
I would like to thank my supervisor Dr. Hadi Salmasian for all his guidance during
the writing of this thesis. I have learned a great deal under his supervision. I would
also like to thank my parents for being supportive when I was writing my thesis.
iv
Contents
Introduction vii
1 Preliminaries 1
1.1 Basic Algebra . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1
1.2 Basic Semi-algebraic Geometry . . . . . . . . . . . . . . . . . . . 3
1.3 Some Facts about Lie Algebras . . . . . . . . . . . . . . . . . . . 6
2 Real Semisimple Structure Theory 10
2.1 Advanced Theory of Lie Algebras . . . . . . . . . . . . . . . . . 10
2.2 Facts about Lie Groups . . . . . . . . . . . . . . . . . . . . . . . 15
2.3 Structure of SL2(R) and sl2(R) . . . . . . . . . . . . . . . . . . . 18
3 Real Versus Complex Representations 22
3.1 Basic Representation Theory . . . . . . . . . . . . . . . . . . . . 22
3.1.1 Representations of Lie Groups . . . . . . . . . . . . . . . . . 22
3.1.2 Representations of Lie Algebras . . . . . . . . . . . . . . . . 24
3.2 Representations of SL2(R), sl2(R) and SLn(R) . . . . . . . . . . 26
3.2.1 Representations of SL2(R) and sl2(R) . . . . . . . . . . . . . 27
3.2.2 Representations of SLn(R) . . . . . . . . . . . . . . . . . . . 34
3.3 Representations of SO2(R) . . . . . . . . . . . . . . . . . . . . . 35
3.4 Extreme Points of Convex Hulls of Orbits . . . . . . . . . . . . . 41
v
CONTENTS vi
4 Defining Equations of the Highest Weight Orbit 48
4.1 Orbits in Complex Vector Spaces . . . . . . . . . . . . . . . . . . 48
4.2 Highest Weight Orbits . . . . . . . . . . . . . . . . . . . . . . . . 50
4.2.1 Kostant’s Theorem for SLn(R) . . . . . . . . . . . . . . . . . 51
4.2.2 Equations for the Highest Weight Orbit for SL2(R) . . . . . 60
5 The Convex Hull of the Highest Weight Orbit 64
5.1 The Relationship Between Convex Hulls of the G and K-orbits . 64
5.2 Spherical SL2(R) Modules . . . . . . . . . . . . . . . . . . . . . . 74
Bibliography 87
Index 87
Introduction
Since its birth, Lie theory has always had a strong connection to geometry. One
of the geometric structures of particular interest in Lie theory is the construction
of a homogeneous space. The study of homogeneous spaces (or, equivalently, orbits
of group actions) is important because they naturally arise in applications of Lie
theory such as invariant theory and quantization. In this thesis, we consider questions
related to the structure of the orbit of the highest weight vector of an irreducible
representation of a real semisimple Lie algebra from the view points of algebraic and
convex geometry.
Roughly speaking, the setting of this thesis is as follows. Let G be a real split
semisimple Lie group, and let Vλ be a finite dimensional irreducible representation
of G of highest weight λ. Let vλ ∈ Vλ be a highest weight vector. Set O = G · vλand let OC be the orbit of vλ under the action of the complexification GC of G
inside the corresponding complex highest weight module. Then OC ∪ {0} is indeed
a complex algebraic variety, and therefore O ∪ {0} is a semialgebraic set. From the
Tarski-Seidenberg Theorem (see Chapter 1), it follows that conv(O ∪ {0}) is a semi-
algebraic set, and therefore it is described by a system of polynomial inequalities.
While there is a constructive proof of the Tarski-Seidenberg theorem which provides
an algorithm to describe the constraints of conv(O ∪ {0}), this algorithm turns out
to be very inefficient, and it is rather difficult to gain insight into the structure of this
set from this algorithm alone. Instead, we focus on techniques from representation
vii
INTRODUCTION viii
theory. Our main goal in this thesis, which is achieved in Chapter 5, is to obtain a
concrete description of the latter semi-algebraic set.
We now elaborate on the content of every chapter. The first chapter outlines
the algebraic objects used in this thesis, some results from real algebraic geometry,
and finally some basic notions about Lie algebras. In Section 1.1, we define the
tensor algebra, symmetric algebra, and the space of symmetric tensors. These are
used extensively in this thesis as representation spaces for Lie groups and Lie alge-
bras. Following this, in Section 1.2, we discuss some basic notions from real algebraic
geometry. We define the notion of a semialgebraic set, and prove that the convex
hull of a semialgebraic set is semialgebraic. In this proof, we use the celebrated
Tarski-Seidenberg theorem, which states that the projection of a semialgebraic set is
semialgebraic. This result is the starting point of our understanding of the highest
weight orbitope, which appears in Chapter 5. Finally, in Section 1.3, we discuss some
basic Lie theory, such as the notion of the universal enveloping algebra, the Casimir
operator, and the Killing form, all of which play an important role in later chapters.
In the second chapter, we discuss some results about the structure theory of real
semisimple Lie algebras and Lie groups. In Section 2.1, we discuss the structure of
real semisimple Lie algebras. The most important notions from this section include
the Iwasawa decomposition for Lie algebras and the notion of a Cartan subalgebra.
In Section 2.2, we consider the Cartan and Iwasawa decompositions in the group
setting. In Section 2.3, we illustrate the theory from the previous two sections using
the examples of sl2(R) and SL2(R).
In the third chapter, we discuss the representation theory of Lie groups and
Lie algebras. Section 3.1 is devoted to some generalities and elementary facts. In
Section 3.2, we classify the finite dimensional representations of SL2(R) and sl2(R).
There are some subtleties associated with classifying the real representations of sl2(R),
as opposed to the complex representations, which arise due to the fact that R is
not algebraically closed. Standard references such as Humphrey’s book [10] only
INTRODUCTION ix
consider the classification problem over algebraically closed base fields, which is why
we include the details for the real case. Finally, in Section 3.3, we classify the real
finite dimensional representations of SO2(R).
In the fourth chapter, we concentrate on the highest weight orbit O. In Section
4.1, we restrict our attention to the complex orbit OC. We show that OC ∪ {0} is
a complex algebraic variety. We also show that OC is the only orbit of GC acting
on Vλ with the latter property. In Section 4.2, we consider Xλ := O ∪ {0}. We
prove a modified version of Kostant’s theorem. Kostant originally proved that for a
complex simple Lie group G, the orbit OC ∪ {0} is described by a set of quadratic
equations. In this thesis, we prove that for G = SLn(R), Xλ is a semialgebraic
set, given by the intersection of variety determined by quadratic equations and a
specific semialgebraic set Eλ. We focus on SLn(R), but our proof works (with some
modification) for any real split semisimple Lie group. The statment of this theorem
(Theorem 4.2.3) explicitly describes the constraints on Xλ, and we explicitly give
the equations for the 5-dimensional representation of SL2(R) (Example 4.2.14). Our
version of Kostant’s theorem is original.
In the fifth chapter, we introduce convex hulls of orbits. In Chapter 4, we proved
that Xλ is semialgebraic and we described its constraints. It is thus natural to ask
whether we can do the same for conv(Xλ), or at least describe conv(Xλ) in a more
explicit way. For this, we need to focus on the theory of convex hulls of orbits of
compact groups, known as orbitopes. In [20], there is an extensive discussion of
of SO2(R) orbitopes. In this paper, the authors give an explicit description of the
orbitopes of SO2(R) as a spectrahedra, i.e. sets in Rn which are described by a linear
matrix inequality. In this chapter we obtain results about G-orbits similar to those
in [20], where G = SL2(R) (the difference between the work here and that in [20] is
that G is non-compact). To this end, we first observe that the G-orbit O contains a
K-orbit in a natural way, where K is a maximal compact subgroup of G. In Section
5.1, we describe the convex hull of Xλ in terms of this K-orbit. Finally, in Section 5.2,
INTRODUCTION x
we prove that in the case of G = SL2(R), the convex hull of Xλ can be described as a
cone over a certain orbitope, which is known as the Caratheodory orbitope (Theorem
5.2.15). This work is original.
There are a number of questions which naturally arise after the work done in
this thesis. In particular, the following questions are interesting. Does a version
of Kostant’s theorem hold when G is compact? Do the combinatorial results of
[20] extend to groups which are not compact? Given a general semialgebraic set,
is there an efficient way to construct the constraints for the cone over that set?
More specifically, is it possible to efficiently construct the constraints for the cone
conv(Xλ) = R+ conv(K · vλ)?
Chapter 1
Preliminaries
1.1 Basic Algebra
In this section, we review some of the basic algebraic objects used in this thesis.
Unless stated otherwise, the ground field is a field F of characteristic 0.
Definition 1.1.1. Let V be a vector space over a field F. For each k ≥ 0, we define
T k(V ) := V ⊗ · · · ⊗ V︸ ︷︷ ︸k-times
.
Note that T 0(V ) = F and T 1(V ) = V . Then we define the tensor algebra of V to be
the algebra T (V ) defined by
T (V ) :=∞⊕k=0
T k(V )
where multiplication is defined by
(v1 ⊗ · · · ⊗ vr) · (w1 ⊗ · · · ⊗ ws) := v1 ⊗ · · · ⊗ vr ⊗ w1 ⊗ · · · ⊗ ws
for every r, s ≥ 1, v1 ⊗ · · · ⊗ vr ∈ T r(V ), and w1 ⊗ · · · ⊗ws ∈ T s(V ). We extend this
1
1. PRELIMINARIES 2
multiplication linearly to all of T (V ).
Definition 1.1.2. Let V be a vector space over a field F. Let I be the ideal of T (V )
generated by the set {v ⊗ w − w ⊗ v : v, w ∈ V }. Then we define the symmetric
algebra S(V ) to be the quotient algebra
S(V ) := T (V )/I.
Note that I is a homogeneous ideal, and therefore
T (V )/I =∞⊕k=0
T k(V )/(I ∩ T k(V )).
We denote the k-th graded component of S(V ) by Sk(V ). We remark that
Sk(V ) ∼= T k(V )/(I ∩ T k(V )).
For an element v1 ⊗ · · · ⊗ vk ∈ T k(V ), we denote its image in S(V ) by
v1 · · · vk. (1.1.1)
Thus, all elements of S(V ) are linear combinations of elements of the form (1.1.1).
We observe that the image of T k(V ) in S(V ) is Sk(V ).
Let k ≥ 0. Let Sk denote the permutation group on {1, 2, . . . , k}. Then there is an
action of Sk on T k(V ) defined by
σ · v1 ⊗ · · · ⊗ vk = vσ−1(1) ⊗ · · · ⊗ vσ−1(k) σ ∈ Sk, v1 ⊗ · · · ⊗ vk ∈ T k(V ).
We extend this action linearly to all of T (V ).
Definition 1.1.3. Let k ≥ 0. We say a tensor t ∈ T k(V ) is a symmetric k-tensor
1. PRELIMINARIES 3
if σ · t = t for all σ ∈ Sk. We let Symk(V ) denote the vector space of symmetric
k-tensors. We define the vector space Sym(V ) by
Sym(V ) :=∞⊕k=0
Symk(V ).
Proposition 1.1.4. For each k ≥ 0, there is a vector space isomorphism
Sk(V )→ Symk(V ) v1 · · · vk 7→1
k!
∑σ∈Sk
vσ(1) ⊗ · · · ⊗ vσ(k).
These isomorphisms induce a vector space isomorphism
S(V )→ Sym(V ).
Proof: See [5, §11.5, Proposition 40].
1.2 Basic Semi-algebraic Geometry
In this section, we review the basic theory of semi-algebraic sets needed for this thesis.
We follow [1, Chapter 2]. The most important result of this section is that the convex
hull of a semi-algebraic set is semi-algebraic. This is obtained as a corollary of the
Tarski-Seidenberg theorem.
Definition 1.2.1. We say S ⊆ Rn is a basic semi-algebraic set if S has the form
S =k⋂i=1
{x ∈ Rn : fi(x) ∗i 0}
where fi ∈ R[x1, . . . , xn] and ∗i is either the symbol = or >, depending on i. We call
{(f1, ∗1), . . . , (fk, ∗k)} the constraints of S. A semi-algebraic set is defined as a finite
1. PRELIMINARIES 4
union of basic semi-algebraic sets.
The following proposition is the celebrated Tarski-Seidenberg theorem.
Proposition 1.2.2. Let S be a semi-algebraic subset of Rn+1, and let π : Rn+1 → Rn
be the projection
π(x1, . . . , xn, xn+1) = (x1, . . . , xn).
Then π(S) is a semi-algebraic subset of Rn.
Proof: See [1, Theorem 2.2.1]
Corollary 1.2.3. Let S be a semi-algebraic subset of Rn. If 1 ≤ i1 < · · · < ik ≤ n
are integers, and π : Rn → Rk is the projection
π(x1, . . . , xn) = (xi1 , . . . , xik),
then π(S) is semi-algebraic.
Definition 1.2.4. For a subset S of a real vector space V , we define the convex hull
of S by
conv(S) :={t1v1 + · · ·+ tkvk : k ∈ N, ti ≥ 0,
∑ti = 1, vi ∈ S
}.
Equivalently, conv(S) is the smallest convex subset of V containing S.
Theorem 1.2.5. (Caratheodory, 1911) For any set S ⊆ Rd, and any point x ∈
conv(S), x is a convex combination of at most d+ 1 points in S.
Proof: See [18, §17].
1. PRELIMINARIES 5
Recall that for n ≥ 1, the n-simplex ∆n is the subset of Rn+1 defined by
∆n :={
(t0, t1, . . . , tn) : ti ≥ 0,∑
ti = 1}.
It is straightforward to verify that ∆n is semi-algebraic.
Proposition 1.2.6. Let S be a semi-algebraic subset of Rn. Then the convex hull
conv(S) is semi-algebraic.
Proof: Write S = S1 ∪ · · · ∪ Sk where each Si is a basic semi-algebraic set.
Now define
A =
{(s1, . . . , sn+1, t1, . . . , tn+1,
n+1∑i=1
tisi) : si ∈ S, (t1, . . . , tn+1) ∈ ∆n
}.
By Corollary 1.2.3 and Theorem 1.2.5, it suffices to show that A is a semi-algebraic
subset of Rn2+3n+1.
As ∆n is semi-algebraic as mentioned, one can write ∆n = D1 ∪ · · · ∪D` where
each Di is a basic semi-algebraic set. Now for each (n + 1)-tuple α = (α1, . . . , αn+1)
in {1, 2, . . . , k}n+1 and each β ∈ {1, 2, . . . , `}, define
Aαβ =
{(s1, . . . , sn+1, t1, . . . , tn+1,
n+1∑i=1
tisi) : si ∈ Sαi , (t1, . . . , tn+1) ∈ Dβ
}.
Then clearly we have
A =⋃α
⋃β
Aαβ.
Now we only need to show each Aαβ is semi-algebraic. Let {(f iγ) ∗iγ} be the constraints
for Sαi and let {(gµ, ∗µ)} be the constraints for Dβ. Then for v ∈ Rn2+3n+1, denote v
by v = (x1, . . . , xn+1, y, z) where xi ∈ Rn, y ∈ Rn+1 and z ∈ Rn. Then the constraint
1. PRELIMINARIES 6
polynomials for Aαβ are given by
hiγ(v) := f iγ(xi)
kµ(v) := gµ(t1, . . . , tn+1)
l(v) := z − y1x1 − y2x2 − · · · − yn+1xn+1
and the constraints are given by {(hiγ, ∗iγ)}, {(kµ), ∗µ} and {(l,=)}.
1.3 Some Facts about Lie Algebras
In this section, we review some facts concerning semisimple Lie algebras which will
be used in this thesis. We assume the reader is familiar with the basic theory of Lie
algebras. In this section, F is either R or C.
Definition 1.3.1. A Lie algebra g is semisimple if g has no nonzero solvable ideals.
Remark 1.3.2. Equivalently, one sees that g is semisimple if and only if g has no
nonzero abelian ideals.
Definition 1.3.3. Let g be a Lie algebra over a field F, and let X ∈ g. We define
the map ad(X) : g→ EndF(g) by
ad(X)(Y ) = [X, Y ] Y ∈ g.
Alternatively, we sometimes use the notation
adX = ad(X).
1. PRELIMINARIES 7
Definition 1.3.4. Let g be a Lie algebra. We define the Killing form on g by
B(X, Y ) = tr(ad(X) · ad(Y )) X, Y ∈ g.
Proposition 1.3.5. The Killing form B satisfies the following properties.
(1) B is bilinear and symmetric.
(2) B is associative, i.e. B([X, Y ], Z) = B(X, [Y, Z]) for all X, Y, Z ∈ g.
(3) B is nondegenerate if and only if g is semisimple.
Proof:
(1) The fact that B is bilinear is a straightforward calculation. The fact that B
is symmetric follows from the identity tr(XY ) = tr(Y X) for any two linear
endomorphisms X and Y .
(2) This is a straightforward calculation.
(3) See [12, Theorem 1.45].
Recall that for an associative F-algebra A, there is an associated Lie algebra
structure which has A as the underlying set and with the Lie bracket defined by
[X, Y ] = XY − Y X for X, Y ∈ A.
Definition 1.3.6. Let g be a Lie algebra. A pair (U(g), σ), where U(g) is a unital
associative algebra and σ : g → U(g) is a Lie algebra homomorphism is called a
universal enveloping algebra if for every unital associative algebra A, and for every
1. PRELIMINARIES 8
Lie algebra homomorphism ϕ : g→ A, there exists a unique homomorphism of unital
associative algebras ϕ : U(g) → A such that ϕ = ϕ ◦ σ. This is expressed by the
following commutative diagram.
gϕ //
σ��
A
U(g)
ϕ
==
Remark 1.3.7. (Uniqueness of universal enveloping algebra) Let (U(g), σ) and (U(g), σ)
be two universal enveloping algebras of g. Then there exists an isomorphism ϕ :
U(g)→ U(g) of unital associative algebras satisfying ϕ◦σ = σ. See [9, Lemma 7.1.2].
We now provide a construction of the universal enveloping algebra (U(g), σ). Let
T (g) be the tensor algebra of g), i.e.
T (g) = F⊕ g⊕ (g⊗ g)⊕ · · · .
Then let J be the ideal generated by the set
{X ⊗ Y − Y ⊗X − [X, Y ] : X, Y ∈ g}.
Then define U(g) = T (g)/J , and define σ : g → U(g) by σ(X) = X + J . See [9,
Proposition 7.1.3] for the proof that this is indeed a universal enveloping algebra of
g.
Theorem 1.3.8. (Poincare-Birkhoff-Witt Theorem (PBW)) Let g be a Lie algebra
and let {X1, . . . , Xn} be a basis of g. Then the set
{Xµ11 · · ·Xµn
n : µi ≥ 0}
is a basis of U(g).
1. PRELIMINARIES 9
Proof: See [9, Theorem 7.1.9].
Definition 1.3.9. Let g be a semisimple Lie algebra, and let X1, . . . , Xn be a basis
of g. Let X1, . . . , Xn be another basis of g which satisfies
B(Xi, Xj) = δij.
Then define the element
C :=n∑i=1
XiXi ∈ U(g).
We call this the Casimir element of g.
Remark 1.3.10. The Casimir element is unique. That is to say, it does not depend
on the choice of basis {X1, . . . , Xn}. See [21, Chapter 6, §3].
Chapter 2
Real Semisimple Structure Theory
2.1 Advanced Theory of Lie Algebras
In this section, we compile some of the more advanced definitions and results from
the theory of Lie algebras which will be used in this thesis. In this section, we assume
that the base field is F = R. Our main reference is [12].
Definition 2.1.1. Let g be a real Lie algebra, and let θ : g → g be a Lie algebra
automorphism. Assume θ satisfies
(1) θ2 = id, and
(2) the bilinear form Bθ on g given by
Bθ(X, Y ) = −B(X, θ(Y )) (2.1.1)
is positive definite.
Then we say θ is a Cartan involution of g.
10
2. REAL SEMISIMPLE STRUCTURE THEORY 11
Remark 2.1.2. The bilinear from Bθ on g is symmetric. Moreover, if g is semisimple,
then the non-degeneracy of B (from Cartan’s criterion) implies that Bθ is nondegen-
erate as well.
Let g be a real Lie algebra and let θ be a Cartan involution of g. Since θ2 = id,
the eigenvalues of θ are exactly +1 and −1. Therefore, g decomposes as
g = k⊕ p
where
k = {X ∈ g : θ(X) = X} and p = {X ∈ g : θ(X) = −X}.
We call this the Cartan decomposition of g. We sometimes refer to this as the polar
decomposition.
The following result has significant importance in the theory of real semisimple
Lie algebras.
Proposition 2.1.3. Let g be a real semisimple Lie algebra. Then g has a Cartan
involution θ.
Proof: See [12, Corollary 6.18]
Corollary 2.1.4. Every real semisimple Lie algebra admits a Cartan decomposition
g = k⊕ p.
Remark 2.1.5. Let g be a real semisimple Lie algebra with a Cartan involution θ
and Cartan decomposition g = k⊕p. Let Bθ be the bilinear form on g given as above
in (2.1.1). By Definition 2.1.1 we see that Bθ is an inner product on g. Henceforth,
all references to the notion of orthogonality and adjunction are with respect to Bθ.
2. REAL SEMISIMPLE STRUCTURE THEORY 12
Lemma 2.1.6. Let g be a real semisimple Lie algebra, and let θ and Bθ be as above.
Then
ad(X)∗ = − ad(θ(X)) for all X ∈ g.
Proof: The following proof is essentially from [12, Lemma 6.27]. Let X, Y, Z ∈ g.
Then
Bθ(ad(X)∗Y, Z) = Bθ(Y, ad(X)Z) = −B(Y, θ(ad(X)Z))
= −B(Y, θ([X,Z])
= −B(Y, [θ(X), θ(Z)])
= −B([Y, θ(X)], θ(Z))
= B([θ(X), Y ], θ(Z))
= B(ad(θ(X))Y, θ(Z))
= −Bθ(ad(θ(X))Y, Z)
Since g is semisimple, B is nondegenerate and therefore Bθ is nondegenerate as well.
Thus ad(X)∗ = − ad(θ(X)).
Let a be a maximal abelian subspace of p. We know this exists since p is finite
dimensional. By the above lemma, the set
F = {ad(H) : H ∈ a}
is a commuting family of self-adjoint transformations on g. From linear algebra, we
know that this family is simultaneously diagonalizable (since the family commutes)
with real eigenvalues (since each transformation is self-adjoint).
Let X ∈ g be an eigenvector of the family F . Then let H,H ′ ∈ a and suppose
2. REAL SEMISIMPLE STRUCTURE THEORY 13
H and H ′ have eigenvalues λH and λH′ , respectively. Also let α ∈ R. Then we have
ad(αH +H ′)X = α ad(H)X + ad(H ′)X = αλHX + λH′X
so
λαH+H′ = αλH + λH′ .
Hence, our simultaneous eigenvalues are members of the dual space a∗. For λ ∈ a∗,
we write
gλ = {X ∈ g : ad(H)X = λ(H)X for all H ∈ a}.
If gλ 6= 0 and λ 6= 0, we call λ a restricted root of the pair (g, a). We denote the set
of restricted roots by Σ. For λ ∈ Σ, we call gλ a restricted root space.
Proposition 2.1.7. The restricted roots and the restricted root spaces have the fol-
lowing properties:
(a) With respect to Bθ, g is the orthogonal direct sum
g = g0 ⊕⊕λ∈Σ
gλ.
(b) For λ, µ ∈ a∗, [gλ, gµ] ⊆ gλ+µ.
(c) θgλ = g−λ. Thus, λ ∈ Σ implies −λ ∈ Σ.
(d) g0 = a⊕m, orthogonally, where m = Zk(a).
Proof:
(a) This follows directly from the above discussion.
(b) Let λ, µ ∈ a∗, X ∈ gλ, Y ∈ gµ and H ∈ a. Then given the Jacobi identity
[H, [X, Y ]] = −[X, [Y,H]]− [Y, [H,X]]
2. REAL SEMISIMPLE STRUCTURE THEORY 14
we get
ad(H)([X, Y ]) = [H, [X, Y ]]
= −[X, [Y,H]]− [Y, [H,X]] = [X, [H,Y ]] + [[H,X], Y ]]
= [X,µ(H)Y ] + [λHX, Y ]
= µ(H)[X, Y ] + λ(H)[X, Y ]
= (λ+ µ)(H)[X, Y ]
(c) Let X ∈ gλ and H ∈ a. Then
ad(H)(θ(X)) = [H, θ(X)]
= θ[θ(H), X]
= −θ[H,X] since H ∈ a ⊆ p
= −λ(H)θ(X)
(d) First note that g0 = (k∩g0)⊕ (p∩g0). It then suffices to show that k∩g0 = Zk(a)
and p ∩ g0 = a.
Showing k ∩ g0 = Zk(a) is trivial. To show p ∩ g0 = a first note that a ⊆ p ∩ g0.
Now suppose a ( p ∩ g0. Then there exists X ∈ (p ∩ g0) \ a. Then a ⊕ FX is a
larger abelian subalgebra of p than a, a contradiction.
As in the case of any root system, we choose an ordering on Σ and let Σ+ be the
set of positive roots in Σ. Define
n =⊕λ∈Σ+
gλ.
2. REAL SEMISIMPLE STRUCTURE THEORY 15
By Proposition 2.1.7, we see that n is indeed a nilpotent subalgebra of g.
Theorem 2.1.8. (Iwasawa decomposition for Lie algebras) With the notation as
above, the real semisimple Lie algebra g admits a vector space decomposition g =
k⊕ a⊕ n.
Proof: See [12, Proposition 6.43].
Definition 2.1.9. Let g be a Lie algebra. We say a subalgebra h of g is a Cartan
subalgebra of g if (1) h is nilpotent and (2) h = ng(h).
Proposition 2.1.10. If t is a maximal abelian subspace of m = Zk(a), then h = a⊕ t
is a Cartan subalgebra of g.
Proof: See [12, Proposition 6.47].
Definition 2.1.11. We say a real semisimple Lie algebra g is split if t = 0. Alterna-
tively, we say h is a split Cartan subalgebra.
Definition 2.1.12. Let g be a real semisimple Lie algebra, and let h be a split
Cartan subalgebra. Chose a set of positive roots Σ+ for (g, h). The fundamental
Weyl chamber in h is defined to be the set
h+ := {X ∈ h : α(X) > 0 for all α ∈ Σ+}.
2.2 Facts about Lie Groups
We assume the reader is familiar with the basic theory of Lie groups. We use this
section to review some more advanced definitions and propositions from the theory
of real semisimple Lie groups.
2. REAL SEMISIMPLE STRUCTURE THEORY 16
Definition 2.2.1. We say a Lie group G is semisimple if it is connected and its Lie
algebra is semisimple.
Theorem 2.2.2. Let G be a semisimple Lie group, and let θ be a Cartan involution
of its Lie algebra g. Let g = k⊕ p be the corresponding Cartan decomposition. Let K
be the analytic subgroup of G with Lie algebra k. Then
(a) There exists a Lie group automorphism Θ of G with differential θ and Θ2 = id.
(b) The subgroup of G fixed by Θ is K.
(c) The map K × p→ G given by (k,X) 7→ kexp X is a diffeomorphism.
(d) K is closed.
(e) K contains the center Z of G.
(f) K is compact if and only if Z is finite.
(g) When Z is finite, K is a maximal compact subgroup of G.
We refer to the decomposition in (c) as the polar decomposition of G.
Proof: See [12, Theorem 6.31]
Example 2.2.3. An example of a split semisimple Lie group with infinite center is
the universal cover SL2(R). See [15, Example 1.4.13] for more details.
Theorem 2.2.4. (Iwasawa decomposition) Let G be a semisimple Lie group, and let
g be its Lie algebra. Let g = k ⊕ a ⊕ n be its Iwasawa decomposition. Let K, A and
N be the analytic subgroups of G with Lie algebras k, a and n, respectively. Then the
2. REAL SEMISIMPLE STRUCTURE THEORY 17
map
K × A×N → G
(k, a, n) 7→ kan
is a diffeomorphism. Moreover, the groups A and N are simply connected.
Proof: See [12, Theorem 6.46]
Our next goal is to describe another important decomposition of a semisimple
Lie group known as the Bruhat decomposition. We only describe this decomposition
in the case of the group G = SLn(R). Recall that SLn(R) is the group of n × n
real matrices of determinant 1. Let H be the standard Cartan subgroup consisting
of diagonal matrices in G. Let B be the Borel subgroup of G consisting of upper
triangular matrices in G.
Recall that a matrix M over a field F is a called a monomial matrix if there
exists exactly one nonzero entry in each row and exactly one nonzero entry in each
column.
Lemma 2.2.5. The normalizer NG(H) is equal to the set of monomial matrices in
G.
Proof: Let g = (aij) ∈ NG(H), and let h = diag(1, 2, . . . , n). Then x := ghg−1 ∈
H. Since h and x have the same eigenvalues, there exists some σ ∈ Sn such that
x = diag(σ(1), . . . , σ(n)). Since x = ghg−1, we also have the equation xg = gh. A
simple calculation shows that this implies that
aij(σ(i)− j) = 0
2. REAL SEMISIMPLE STRUCTURE THEORY 18
for all 1 ≤ i, j ≤ n. Fix some 1 ≤ j ≤ n. Then the above equation implies that for
each i such that σ(i) 6= j, we have aij = 0. It’s not hard to see that g cannot have
any entirely zero columns. This proves our claim.
Recall that the Weyl group of a semisimple Lie group G with a chosen maximal
torus T is defined as W := NG(T )/T . Notice that H is a maximal torus in SLn(R).
Corollary 2.2.6. The Weyl group for SLn(R) is isomorphic to Sn.
Theorem 2.2.7. (Bruhat decomposition) We have the decomposition
G =∐w∈W
BwB.
Proof: Let g ∈ G. Then there exists some b ∈ B such that every row of b · g
contains a different number of 0’s. Therefore, there exists nσ ∈ NG(H) such that
nσ ·b ·g is in upper triangular form, i.e. b′ = nσ ·b ·g ∈ B. So g = b−1 ·n−1σ ·b′ ∈ BWB.
Recall that the double cosets in the above union are disjoint since they are the
equivalence classes of the relation ∼ on G defined by x ∼ y iff there exists b, b′ ∈ B
such that bxb′ = y.
2.3 Structure of SL2(R) and sl2(R)
Recall that the group SLn(R) is defined to be the subgroup of GLn(R) consisting of
matrices of determinant 1. As in Section 2.2, we define the standard Cartan subgroup
of SLn(R) to be the subgroup H of SLn(R) consisting of diagonal matrices, and the
standard Borel subgroup B of SLn(R) to be the subgroup of SLn(R) consisting of
2. REAL SEMISIMPLE STRUCTURE THEORY 19
upper triangular matrices. Clearly H is a subgroup of B.
Also recall that the Lie algebra sln(R) is defined to be the subalgebra of gln(R)
consisting of matrices with trace 0. We let h be the standard Cartan subalgebra of
g and b the standard Borel subalgebra of g. We know that h consists of all diagonal
matrices in g and b consists of upper triangular matrices in g.
In particular,
sl2(R) :=
a b
c d
: a, b, c, d ∈ R and a+ d = 0
and we have a basis of sl2(R) given by
E =
0 1
0 0
, F =
0 0
1 0
, H =
1 0
0 −1
which satisfies the relations
[H,E] = 2E, [H,F ] = −2F, [E,F ] = H.
Concretely, the standard Cartan subalgebra of sl2(R) is
h =
a 0
0 b
: a, b ∈ R and a+ b = 0
and its standard Borel subalgebra is
b :=
a b
0 c
: a, b, c ∈ R and a+ c = 0
2. REAL SEMISIMPLE STRUCTURE THEORY 20
which is indeed a subalgebra of sl2(R).
Example 2.3.1. A Cartan involution on sl2(R) is given by θ(X) = −XT . A standard
Iwasawa decomposition for sl2(R) is given by sl2(R) = k⊕ a⊕ n where k is the set of
matrices of the form 0 x
−x 0
,
a is the set of all matrices of the formx 0
0 −x
,
and n is the set of all matrices of the form0 x
0 0
where x is an arbitrary element of R.
Proposition 2.3.2. The Casimir element of sl2(R) is given by
C =1
4EF +
1
4FE +
1
8H2.
Proof: Using the relations above, we see that the matrices for ad(E), ad(F ) and
ad(H) are given as follows
ad(E) =
0 0 −2
0 0 0
0 1 0
, ad(F ) =
0 0 0
0 0 2
−1 0 0
, ad(H) =
2 0 0
0 −2 0
0 0 0
.
2. REAL SEMISIMPLE STRUCTURE THEORY 21
An easy calculation then confirms that
E ′ =1
4F, F ′ =
1
4E, H ′ =
1
8H
is a basis of sl(2,R) which satisfies κ(E,E ′) = 1, κ(E,F ′) = κ(E,H ′) = 0, and
κ(F,E ′) = 0, κ(F, F ′) = 1, κ(F,H ′) = 0 and κ(H,E ′) = 0, κ(H,F ′) = 0, κ(H,H ′) =
1. Therefore, our Casimir element is given by
C = EE ′ + FF ′ +HH ′ =1
4EF +
1
4FE +
1
8H2.
Remark 2.3.3. Note that EF+FE = EF−FE+2FE = [E,F ]+2FE = H+2FE.
Therefore, one can also write
C =1
2FE +
1
4H +
1
8H2.
Chapter 3
Real Versus Complex
Representations
In this chapter, we outline some basic results on the representation theory of Lie
groups and Lie algebras. Following this, we classify the real finite dimensional repre-
sentations of SL2(R) and SO2(R).
3.1 Basic Representation Theory
In this section, we outline some basic definitions and results in the representation
theory of Lie groups and Lie algebras. We assume that the reader is familiar with
some basic results, such as Schur’s Lemma. We take F to be either R or C.
3.1.1 Representations of Lie Groups
Definition 3.1.1. Let G be a Lie group. An F-representation of G is a pair (π, V )
where π is a smooth group homomorphism
π : G→ GL(V )
22
3. REPRESENTATIONS OF LIE ALGEBRAS 23
and V is a finite dimensional topological vector space over F.
Remark 3.1.2. If the underlying field F of V is R, we say (π, V ) is a real repre-
sentation of G. Similarly, if F = C, we say (π, V ) is a complex representation of
G.
Example 3.1.3. Let G be a Lie group, and let g be its Lie algebra. Let e be the
identity element of G. Thus, g is the tangent space TeG. For each g ∈ G, define the
inner automorphism Ψg : G → G by Ψg(h) = ghg−1. Define Adg : g → g to be the
differential of Ψg at e. If we realize G as a matrix Lie group, then Adg acts on g by
conjugation, i.e.
Adg(X) = gXg−1 g ∈ G,X ∈ g.
We thus have the adjoint representation Ad given by
Ad : G→ GL(g) g 7→ Adg.
Definition 3.1.4. Let G be a Lie group and let (π, V ) be a representation of G. We
define the dual representation of G to be the representation (π∗, V ∗) given by
π∗(g) := π(g−1)T
where X 7→ XT denotes the transpose operation and V ∗ denotes the linear dual of
V . Explicitly, for g ∈ G, λ ∈ V ∗, and v ∈ V , we have
π∗(g)(λ)(v) = λ(π(g−1)v).
Proposition 3.1.5. Let G be a compact Lie group, and let (π, V ) be a finite dimen-
sional F-representation of G. Suppose 〈·, ·〉 is an inner product on V . Then there is
3. REPRESENTATIONS OF LIE ALGEBRAS 24
an inner product (·, ·) on V that is G-invariant, i.e. satisfies
(π(g)v, π(g)w) = (v, w) for all g ∈ G, v, w ∈ V.
Proof: We give a sketch of the proof. Define (·, ·) : V × V → F by
(v, w) :=
∫G
〈π(g)v, π(g)w〉dg
for all v, w ∈ G and where dg is the Haar measure on G. Then (·, ·) is G-invariant.
Remark 3.1.6. Typically, we would say that (·, ·) makes (π, V ) into a unitary rep-
resentation, but the notion of unitary representations will not be used in this thesis.
3.1.2 Representations of Lie Algebras
Definition 3.1.7. Let g be a Lie algebra. A representation of g is a pair (π, V ) where
V is a vector space and
π : g→ gl(V )
is a Lie algebra homomorphism.
Remark 3.1.8. If V is a real (respectively, complex) vector space, we say (π, V ) is
a real (complex) representation of g.
Example 3.1.9. Let G be a Lie group with Lie algebra g. Recall that we have the
adjoint representation Ad : G→ GL(g) from Example 3.1.3. Define ad : g→ End(g)
to be the differential of Ad at e, i.e. ad := d(Ad)(e). Then one sees that ad(X)(Y ) =
[X, Y ] for X, Y ∈ g.
Definition 3.1.10. Let g be a Lie algebra and let (π, V ) be a representation of g.
3. REPRESENTATIONS OF LIE ALGEBRAS 25
We define the dual representation of (π, V ) to be the representation (π∗, V ∗) given by
π∗(X) := −π(X)T X ∈ g.
Definition 3.1.11. Suppose g is a real semisimple Lie algebra with Iwasawa decom-
position g = k ⊕ a ⊕ n. Let λ : a → R be a linear functional, and let (π, V ) be a
real representation of g. We say a nonzero vector vλ ∈ V is a highest weight vector of
weight λ with respect to a and n if
π(H)vλ = λ(H)vλ, π(X)vλ = 0 for all H ∈ a, X ∈ n.
We say V is a highest weight module of weight λ if V is generated by a highest
weight vector vλ of weight λ.
Theorem 3.1.12. Let g be a real split semisimple Lie algebra (in the sense of Defi-
nition 2.1.11) and let V be an irreducible finite dimensional representation. Then V
is a highest weight representation.
Proof: See [9, Theorem 7.3.15].
Remark 3.1.13. There is a parallel theory in the complex case which is more stan-
dard. We will not review the complex case, and refer the reader to references such as
[10].
Theorem 3.1.14. Let G be a Lie group, and let F be R or C. Let (π, V ) be an
F-representation of G. Then for every X ∈ Lie(G), the map
t 7→ π(exp(tX)) t ∈ R
3. REPRESENTATIONS OF LIE ALGEBRAS 26
is smooth. Set
π(X) :=d
dt|t=0π(exp(tX)).
Then π(X) ∈ EndF(V ) and the map
Lie(G)→ EndF(V ) X 7→ π(X)
is a Lie algebra homomorphism.
Note that this notation is abusive, and several authors use the notation dπ for
the representation on g.
Proof: See [8, Chapter 2].
3.2 Representations of SL2(R), sl2(R) and SLn(R)
The goal of this section is to classify the real finite dimensional irreducible represen-
tations of SL2(R) and sl2(R) up to isomorphism. The classification is, in principle,
the same for the complex finite dimensional representations of SL2(C), but some sub-
tleties occur in the real case which need to be addressed. The main issue is that the
action of the Cartan subalgebra of sl2(R) is not obviously diagonalizable over R, and
therefore we need to modify the argument from the complex case to obtain weight
spaces which are defined over R.
For each d ≥ 0, there is exactly one (d + 1)-dimensional real irreducible rep-
resentation of SL2(R), which we denote by (πd, Vd). We begin by constructing this
family. We then show that this is an exhaustive family of representations of SL2(R)
and sl2(R). All representations in this section will be real. We end with a word on
representations of SLn(R).
3. REPRESENTATIONS OF LIE ALGEBRAS 27
3.2.1 Representations of SL2(R) and sl2(R)
Fix an integer d ≥ 0 and let Vd denote the space of homogeneous polynomials in
x and y of degree d with coefficients in R. Recall that we have the usual action of
SL2(R) on R2 given by
a b
c d
.(x, y) := (ax+ by, cx+ dy).
We then define the representation
πd : SL2(R)→ GL(Vd)
as follows. For P (x, y) ∈ Vd and g ∈ SL2(R), we set
πd(g)P (x, y) = P (g−1(x, y)).
Note that if
g =
a b
c d
∈ SL2(R)
then
g−1 =
d −b
−c a
so
πd(g)P (x, y) = P (dx− by,−cx+ ay).
Proposition 3.2.1. Let d ≥ 1. Then (πd, Vd) is a representation of SL2(R).
Proof: The fact that πd is a homomorphism is a straightforward calculation. The
smoothness of πd follows from the fact that the the map
SL2(R)× R2 → R2 (g, v) 7→ g−1v
3. REPRESENTATIONS OF LIE ALGEBRAS 28
is smooth.
We now differentiate (πd, Vd) to obtain a representation of sl2(R) which we still denote
by (πd, Vd) (by abuse of notation).
Let t ∈ R. Then
exp tE =
1 t
0 1
, exp tF =
1 0
t 1
, exp tH =
et 0
0 e−t
.
So
(exp tE)−1 =
1 −t
0 1
, (exp tF )−1 =
1 0
−t 1
, (exp tH)−1 =
e−t 0
0 et
.
Now let f ∈ Vd. Then
d
dt|t=0π(exp tE)f(x, y) =
d
dt|t=0π
1 t
0 1
f(x, y)
=d
dt|t=0f(
1 −t
0 1
(x, y))
=d
dt|t=0f(x− ty, y)
= −y∂f∂x
3. REPRESENTATIONS OF LIE ALGEBRAS 29
and
d
dt t=0π(exp tF )f(x, y) =
d
dt t=0f(
1 0
−t 1
(x, y))
=d
dt t=0f(x,−tx+ y)
= −x∂f∂y
and
d
dt t=0π(exp tH)f(x, y) =
d
dt t=0f(
e−t 0
0 et
(x, y))
=d
dt t=0f(e−tx, ety)
= −x∂f∂x
+ y∂f
∂y
So we have the following proposition.
Proposition 3.2.2. Let d ≥ 1. Let {v0, . . . , vd} be a basis of Vd, where vi = xd−iyi
for 0 ≤ i ≤ d. Then (πd, Vd) is given by the following relations:
(a) πd(H)(vi) = (2i− d)vi
(b) πd(E)(vi) = (i− d)vi+1
(c) πd(F )(vi) = −ivi−1
Here we assume v−1 = 0 and vd+1 = 0.
Proof: This immediately follows from the above calculations. See [23] for the
details.
3. REPRESENTATIONS OF LIE ALGEBRAS 30
Remark 3.2.3. Recall the Iwasawa decomposition sl2(R) = k⊕ a⊕ n from Example
2.3.1. Clearly a = RH and n = RE. In the module Vd, the vector vd satisfies
πd(H)vd = dvd πd(E)vd = 0.
Thus, if we define λ : a → R by λ(H) = d then we see that vd is a highest weight
vector of Vd with weight λ. Moreover, Vd is generated by vd, so Vd is a highest weight
module.
Remark 3.2.4. This is a specific case of a very general principle on the classification
of representations of split semisimple Lie algebras.
Lemma 3.2.5. Each module (πd, Vd) is irreducible.
The proof of the above lemma is identical to the complex case, and may be found in
standard books on Lie algebras (e.g. [6, Theorem 8.2]).
Lemma 3.2.6. Let n ≥ 0. Then
1. [H,F n] = −2nF n.
2. [E,F n] = n(H + (n− 1)id)F n−1.
3. [F,En] = −n(H − (n− 1)id)En−1.
Proof: We only prove (a) for n = 2. See [9, Lemma 6.2.2] for a full proof. By the
3. REPRESENTATIONS OF LIE ALGEBRAS 31
PBW theorem (Theorem 1.3.8) we obtain
[H,F 2] = HF 2 − F 2H
= (HF − FH)F + FHF − F 2H
= [H,F ]F + FHF − F 2H
= −2F 2 + FHF − F 2H
= F (−2F +HF )− F 2H
= F (−4F + FH)− F 2H
= −4F 2
Lemma 3.2.7. Let (ρ, V ) be a finite dimensional sl2(R) representation. Suppose
there exists v ∈ V such that ρ(E)v = 0 and ρ(H)v = λv for some λ ∈ R. Then
(i) λ is a non-negative integer.
(ii) v generates a submodule of V isomorphic to Vλ.
Proof:
(i) Let n ≥ 0. By Lemma 3.2.6, we have
ρ(H)ρ(F )nv = ([ρ(H), ρ(F )n] + ρ(F )nρ(H))v = (λ− 2n)ρ(F )nv.
Similarly, we have
ρ(E)ρ(F )nv = ([ρ(E), ρ(F )n] + ρ(F )nρ(E))v
= nρ(F )n−1(ρ(H)− n+ 1)v = n(λ− n+ 1)ρ(F )n−1v.
3. REPRESENTATIONS OF LIE ALGEBRAS 32
This shows that the submodule W generated by v is
W = spanR{ρ(F )nv : n ≥ 0}.
Since V is finite dimensional, ρ(H) only has finitely many eigenvalues on V .
Hence, there is a minimal N ≥ 0 with ρ(F )N+1v = 0. From the fact that
ρ(E)ρ(F )N+1v = 0, we obtain λ = N .
(ii) For each 0 ≤ k ≤ λ− 1, define
vk :=ρ(F )λ−kv
λ(λ− 1) · · · (k + 1).
Then W = spanR{v0, . . . , vλ}. A simple computation shows that W ∼= VN .
Definition 3.2.8. Let V be a vector space over R. Consider the real vector space
VC = V ⊗R C. We define the action of C on VC by
α.(v ⊗ β) = v ⊗ (αβ) v ∈ V, α, β ∈ C.
We then view VC as a complex vector space with this scalar multiplication.
Remark 3.2.9. Any time we use the notation VC in this thesis, we are viewing VC
as a complex vector space.
Definition 3.2.10. Let g be a real Lie algebra. Then we define its complexification
gC to be the Lie algebra gC := g ⊗R C where the Lie bracket on gC is the unique
extension to gC of the Lie bracket on g.
3. REPRESENTATIONS OF LIE ALGEBRAS 33
Proposition 3.2.11. Let (ρ, V ) be an irreducible sl2(R)-module of dimension d + 1
for some d ≥ 0. Then
(ρ, V ) ∼= (πd, Vd).
Proof: Denote the Lie subalgebra of gl(V ) generated by ρ(E) and 12ρ(H) by b.
Note that b is solvable. It follows that the complexification bC is a solvable subalgebra
of gl(VC). Thus, by Lie’s theorem, there exists a basis of VC such that both π(E) and
12ρ(H) are upper triangular. But note that
[1
2ρ(H), ρ(E)] =
1
2ρ([H,E]) = ρ(E).
The commutator of two upper triangular matrices is strictly upper triangular. It thus
follows that ρ(E) is nilpotent. Let d ≥ 1 be the minimal positive integer such that
ρ(E)d = 0. By Lemma 3.2.6 we have
0 = [ρ(F ), ρ(E)d] = −d(ρ(H)− (d− 1)id)ρ(E)d−1.
Thus, any v0 ∈ ρ(E)d−1V is an eigenvector for ρ(H) with eigenvalue d−1. By Lemma
3.2.5 and Lemma 3.2.7, V ∼= Vd.
Theorem 3.2.12. The Casimir operator C of sl2(R) acts on a highest weight module
Vλ by the scalar 14λ+ 1
8λ2.
Proof: Let v ∈ Vλ be the highest weight vector. Note that since C is in the center
of U(sl2(R)) and Vλ = U(sl2(R))v, C must act by a scalar on all of Vλ. The scalar
3. REPRESENTATIONS OF LIE ALGEBRAS 34
may be computed by the action of C on v. We obtain
C.v =1
2(FE).v +
1
4H.v +
1
8(H2).v
= 0 +1
4λv +
1
8λ2v
= (1
4λ+
1
8λ2)v.
3.2.2 Representations of SLn(R)
Let G = SLn(R) and let g = sln(R) be the Lie algebra of G. We let B be the standard
Borel subgroup of G, and H be the standard Cartan subgroup of G. Similarly, we let
b be the standard Borel subalgebra of g and h be the standard Cartan subalgebra of
g. The reader may revisit the end of Section 2.2 for the definitions of G, H, B, g and
b. We have the triangular decomposition
g = n− ⊕ h⊕ n+
where n− and n+ are the subalgebras consisting of strictly lower triangular and strictly
upper triangular matrices, respectively.
Before ending this section, we take a quick look at the highest weight modules
for G and g, which will be used in the proof of Kostant’s theorem for G, in Section
4.2.1.
By 3.1.12, all irreducible finite dimensional representations of G and g are highest
weight representations. Let (π, Vλ) be a highest weight representation of G with
highest weight λ ∈ h∗ (we also denote the representation of g by (π, Vλ)). Let Γλ be
3. REPRESENTATIONS OF LIE ALGEBRAS 35
the set of weights of (π, Vλ), and decompose Vλ into h-weight spaces
Vλ =⊕µ∈Γλ
Vλ(µ).
Moreover, if X ∈ n+ and x ∈ B, then X.v = 0 and x.v ∈ (R \ {0})vλ.
Assume that Vλ is not the trivial G-module. Then λ 6= 0. Now, define εi ∈ h∗ by
εi(diag(t1, . . . , tn)) = ti.
Then one can write λ = λ1ε1 + · · ·+λnεn where λi−λi+1 ∈ N∪{0} for 1 ≤ i ≤ n−1.
Note that this representation of λ is not unique, as one can also write
λ = λ1ε1 + · · ·+ λnεn + α(ε1 + · · ·+ εn)
where α 6= 0. In particular, we can assume that λi ∈ Z for 1 ≤ i ≤ n. Since we have
ε1 + · · ·+ εn = 0, and we know that λ 6= 0, we must have λi > λj for some i < j.
The action of H on Vλ is given as follows. Suppose x = diag(t1, . . . , tn) ∈ G.
Then for v ∈ Vλ(λ),
x.v = t1λ1 · · · tnλnv.
3.3 Representations of SO2(R)
In this section, we classify the real finite dimensional representations of SO2(R). All
representations will be over real and finite dimensional vector spaces.
First recall that the group SOn(R) is defined to be the group of n×n orthogonal
3. REPRESENTATIONS OF LIE ALGEBRAS 36
matrices with determinant 1. When n = 2, we know that
SO2(R) =
cos θ − sin θ
sin θ cos θ
: θ ∈ [0, 2π)
.
We let (ρ0,R) be the trivial representation of SO2(R) on R, i.e.
ρ0(g)x = x for all g ∈ SO2(R), x ∈ R.
For k ∈ Z \ {0}, we define the representation (ρk,R2) by
ρk
cos θ − sin θ
sin θ cos θ
:=
cos kθ − sin kθ
sin kθ cos kθ
for all θ ∈ [0, 2π).
Proposition 3.3.1. The representations (ρ0,R) and (ρk,R2), k ∈ Z \ {0}, are irre-
ducible.
Proof: The fact that (ρ0,R) is irreducible is trivial. Now let k ∈ Z\{0}. To show
that (ρk,R2) is irreducible, suppose there exists an invariant 1-dimensional subspace
U ⊆ R2. Say U is spanned by some vector
v =
v1
v2
.
Then
ρk
cos π/2k − sin π/2k
sin π/2k cosπ/2k
v =
0 −1
1 0
v1
v2
=
−v2
v1
.
But the right hand side is not a member of U , because−v2
v1
and
v1
v2
3. REPRESENTATIONS OF LIE ALGEBRAS 37
always span R2. This is a contradiction.
Remark 3.3.2. Let X be an n×n matrix with real entries and an eigenvalue λ ∈ C,
with a corresponding eigenvector v ∈ Cn. If λ is non-real and v = u + iw where
u,w ∈ Rn, then w 6= 0.
Lemma 3.3.3. Let n ≥ 2 and let X ∈ SOn(R). Suppose λ = a+ ib ∈ C is a non-real
(i.e. b 6= 0) eigenvalue of X with eigenvector u = v + iw ∈ Cn, where v, w ∈ Rn.
Then the space U := spanR{v, w} has real dimension 2 and is invariant under X.
Proof: We first prove that U has real dimension 2. Indeed, suppose for a con-
tradiction that U has real dimension 1. Then there exists some scalar α ∈ R such
that v = αw. But notice that λ′ = a− ib is also an eigenvalue of X with eigenvector
u′ = v − iw. Since λ 6= λ′, and eigenvectors with distinct eigenvalues are linearly
independent, we know that {u, u′} is a linearly independent set over C. But the fact
that v = αw forces u = αw + iw = (α+ i)w and u′ = αw − iw = (α− i)w. This is a
contradiction. So U has real dimension 2.
We now show that U is invariant under X. Notice that Xu = Xv + iXw so
λu = (a+ ib)(v + iw) = (av − bw) + i(av + bu) thus
Xv = av − bw and Xw = aw + bv.
Since Xv and Xw are members of U , we obtain XU ⊆ U . Hence U is invariant under
X.
Proposition 3.3.4. Let V = Rn, and let F be a commuting subset of SOn(R). Then
one can write
V = U1 ⊕ · · · ⊕ Uk
3. REPRESENTATIONS OF LIE ALGEBRAS 38
where each Ui is either a 1-dimensional common eigenspace of F or a 2-dimensional
F-invariant subspace of V . Moreover, each Ui is irreducible, that is, it does not have
a proper F-invariant subspace.
Proof: Note that we may assume that F is a linearly independent finite set
{X1, . . . , Xd} of SOn(R) by taking a basis of the span of F over R. Moreover, since
each Xi is an orthogonal matrix, if U is F invariant, then so is U⊥. Thus, it suffices to
show that we may find a subspace U such that either (a) U is a 1-dimensional common
eigenspace of F or (b) U is a 2-dimensional F -invariant subspace of V . Indeed, one
can then write
V = U ⊕ U⊥
and notice that we may then restrict F to U⊥, and apply induction.
We proceed by induction on n = dimR V . For the base case, note that if n = 1,
then the claim is trivial.
Now suppose n ≥ 2. Recall that members of SOn(R) are diagonalizable over C,
so {X1, . . . , Xd} is a commuting family of matrices which are diagonalizable over C.
Thus, one can write
VC = V1 ⊕ V2 ⊕ · · · ⊕ Vk
where each Vi is a common eigenspace of {X1, . . . , Xd}. Note that there exist λ1, . . . , λd ∈
C such that
Xiv = λiv for all v ∈ V1.
There are two cases. In the first case, each λi is real. In the second case, at least one
λi is complex.
Case 1. Suppose each λi is real. Then there is an common eigenvector w ∈ V1 of
{X1, . . . , Xd} such that w ∈ Rn. Set U := spanR{w1}. It is clear that U is F -invariant
and has no proper F -invariant subspace.
3. REPRESENTATIONS OF LIE ALGEBRAS 39
Case 2. Suppose at least one of the λi’s is non-real. Without loss of generality,
we may assume that λ1 is non-real. Then by Remark 3.3.2, there is an eigenvector
w ∈ V1 such that w ∈ Cn \ Rn. There exist u, v ∈ Rn such that w = u +√−1v. Let
U := spanR{u, v}. For 1 ≤ i ≤ d, if λi is complex, then Lemma 3.3.3 implies that U
is invariant under Xi and is 2-dimensional. If λi is real, then note that w = u−√−1v
is also an eigenvector of Xi with eigenvalue λi. Thus, we obtain
Xi(u) = Xi(1
2(w + w)) = λiu
and
Xi(v) = Xi(1
2i(w − w) = λiv.
So u and v are both eigenvectors of Xi with eigenvalue λi. So U is invariant under
Xi. So U is F invariant. Moreover, U has no proper F -invariant real subspace, as
both u and v are complex eigenvectors of X1. This concludes our proof.
Lemma 3.3.5. Let (ρ, V ) be an n-dimensional representation of SO2(R). Then we
may choose an inner product on V such that each ρ(g), for g ∈ SO2(R), is an orthog-
onal map.
Proof: SO2(R) is compact, so we may simple take the SO2(R)-invariant inner
product (·, ·) on V from Proposition 3.1.5. Then for each g ∈ SO2(R), ρ(g) will be
orthongonal with respect to (·, ·).
Remark 3.3.6. Let (ρ, V ) be an n-dimensional representation of SO2(R). By Lemma
3.3.5, the image of ρ is a subgroup of On(R). Moreover, since ρ is continuous, and
3. REPRESENTATIONS OF LIE ALGEBRAS 40
SO2(R) is connected, the image of ρ is a connected subgroup of On(R). But the only
connected subgroup of On(R) is SOn(R), so ρ(SO2(R)) ⊆ SOn(R). Thus, we may
think of ρ as a map from SO2(R) to SOn(R).
Lemma 3.3.7. Let (ρ, V ) be a representation of SO2(R).
(1) If V is one-dimensional, then ρ ∼= ρ0.
(2) If V is two-dimensional, then ρ ∼= ρk for some k ∈ Z \ {0}.
Proof:
(1) By Remark 3.3.6, we may assume ρ is a map from SO2(R) to SO1(R) = {1}.
Thus ρ ∼= ρ0.
(2) By Remark 3.3.6, we may assume ρ is a map from SO2(R) to SO2(R). From [4,
Proposition 7.1.1], we know that the continuous endomorphisms of SO2(R) all
have the form ρk for some k ∈ Z. So ρ ∼= ρk for some k ∈ Z \ {0}.
Theorem 3.3.8. Let (ρ, V ) be a real representation of SO2(R). Then there exist
non-negative integers a0, . . . , ak such that
ρ = ρa0 ⊕ · · · ⊕ ρak .
Proof: Assume that V = Rn. Let
S = spanR{ρ(g) : g ∈ SO2(R)}.
By Lemma 3.3.7, it suffices to show that we may write
V = U1 ⊕ · · · ⊕ Uk
3. REPRESENTATIONS OF LIE ALGEBRAS 41
where each Ui is a one or two dimensional S-invariant irreducible subspace of V . Now,
let F be a basis of S. We may assume that
F = {X1, . . . , Xd}
where each Xi ∈ SOn(R) (by Remark 3.3.6), and is also a commuting family. We
apply Proposition 3.3.4, and this concludes our proof.
3.4 Extreme Points of Convex Hulls of Orbits
Now that we have discussed some of the theory of cones in representation spaces,
we devote this section to an interesting example. In this section, we study the set of
extreme points of the convex hull of SL2(R) highest weight orbits. In [20], the authors
prove that for a subgroup G of SOn(R), every point in an orbit G·v is an extreme point
in the convex hull conv(G · v). See [20, Proposition 2.2] for a precise formulation.
We prove this result for a specific representation of SL2(R). We will be interested in
the representations which contain a regular cone, i.e. spherical representations. As it
turns out, these are the odd dimensional representations, which are all isomorphic to
S2d(R2) ∼= Sym2d(R2). We first show that all of these representations are orthogonal.
Definition 3.4.1. Let V be a real vector space and let ω : V × V → R be a bilinear
form. We say ω is a symplectic form if ω is alternating (i.e. ω(v, v) = 0 for each
v ∈ V ) and non-degenerate (i.e. ω(v, w) = 0 for all w ∈ V implies v = 0). We say
the pair (V, ω) is a symplectic space.
Definition 3.4.2. Let G be a Lie group and let (π, V ) be a representation of G. If
3. REPRESENTATIONS OF LIE ALGEBRAS 42
there is a symplectic form ω on V such that
ω(π(g)v, π(g)w) = ω(v, w)
for all g ∈ G and v, w ∈ V then we say that (π, V, ω) is a symplectic representation
of G. When the representation π and symplectic form ω are clear from the context,
we sometimes say V is a symplectic module.
First we construct a symplectic form on the standard representation V1∼= R2 of
SL2(R). Define the real bilinear form ω on R2 by
ω
ab
,
cd
:= det
a c
b d
.
We let
e1 =
1
0
, e2 =
0
1
.
Notice that we have the relations
ω(e1, e2) = 1, ω(e2, e1) = −1, ω(e1, e1) = ω(e2, e2) = 0.
Remark 3.4.3. Note that ω is a symplectic form on R2.
Lemma 3.4.4. Let (π,R2) be the standard representation of SL2(R). Then (π,R2, ω)
is a symplectic representation of SL2(R).
Proof: In view of the above remark, it suffices to show that ω preserves the action
of SL2(R). Let
x =
a b
c d
∈ SL2(R), v =
x1
x2
, w =
y1
y2
∈ R2.
3. REPRESENTATIONS OF LIE ALGEBRAS 43
Then
ω(π(x)v, π(x)w) = ω
ax1 + bx2
cx1 + dx2
,
ay1 + by2
cy1 + dy2
= det
ax1 + bx2 ay1 + by2
cx1 + dx2 cy1 + dy2
= (ax1 + bx2)(cy1 + dy2)− (cx1 + dx2)(ay1 + by2)
= (ad− bc)x1y2 + (bc− da)x2y1
= x1y2 − x2y1
= ω(v, w)
where we use the fact that ad − bc = 1 and bc − da = −1 since detx = 1 (as
x ∈ SL2(R)).
We extend ω to T 2d(R2) in the following way, and continue to denote the form on
T 2d(R2) by ω. For simple tensors v1 ⊗ · · · ⊗ v2d, w1 ⊗ · · · ⊗ w2d ∈ T 2d(R2), define
ω(v1 ⊗ · · · ⊗ v2d, w1 ⊗ · · · ⊗ w2d) := ω(v1, w1) · · ·ω(v2d, w2d). (3.4.1)
The form ω induces a form on the subspace Sym2d(R2) of T 2d(R2).
Lemma 3.4.5. Sym2d(R2) is an orthogonal SL(R)-module with respect to ω.
Proof: We need to show that on Sym2d(R2), ω is symmetric, non-degnerate and
preserves the action of SL2(R). The fact that ω is symmetric on Sym2d(R2) follows
from the fact that ω is alternating on R2 and that 2d is even. The non-degeneracy of
ω on Sym2d(R2) follows from Equation 3.4.1 and the fact that ω is non-degenerate on
R2. Finally, ω preserves the action of SL2(R) on Sym2d(R2) by Equation 3.4.1 and
the fact that ω preserves the action of SL2(R) on R2.
3. REPRESENTATIONS OF LIE ALGEBRAS 44
Recall that for a real vector space V and a simple tensor x = v1 ⊗ · · · ⊗ vk ∈ T k(V ),
we define
Sym(x) =1
k!
∑σ∈Sk
vσ(1) ⊗ · · · ⊗ vσ(k).
For vectors v1, . . . , vk ∈ V , we also use the shorthand
v1 . . . vk = Sym(v1 ⊗ · · · ⊗ vk)
and for a vector v ∈ V , we write vk to denote vv · · · v (k-times).
Proposition 3.4.6. There is a basis w−d, . . . , wd of Sym2d(R2) such that ω(wi, wi) =
(−1)d+i and ω(wi, wj) = 0 for all i, j ∈ {−d, . . . , d} when i 6= j.
Proof: For each −d ≤ i ≤ d, define
vi := ed−i1 ed+i2 ± ed+i
1 ed−i2
where the sign is + if i ≤ 0 and − if i > 0. We claim that {v−d, . . . , vd} is an
orthogonal basis of Sym2d(R2) with respect to ω. Notice that for i 6= j we have
ω(vi, vj) = 0.
We claim that ω(vi, vi) 6= 0 for all −d ≤ i ≤ d. Indeed, fix −d ≤ i ≤ d. Then
ω(vi, vi) = ω(ed−i1 ed+i2 + ed+i
1 ed−i2 , ed−i1 ed+i2 + ed+i
1 ed−i2 )
= ω(ed−i1 ed+i2 , ed+i
1 ed−i2 ) + ω(ed+i1 ed−i2 , ed−i1 ed+i
2 )
= 2ω(ed−i1 ed+i2 , ed+i
1 ed−i2 )
3. REPRESENTATIONS OF LIE ALGEBRAS 45
For each σ ∈ S2d, let vσ denote the tensor in T 2d(R2) given by
vσ = σ · (e1 ⊗ · · · ⊗ e1 ⊗ e2 ⊗ · · · ⊗ e2)
where there are d − i e1’s, and d + i e2’s. Notice that for each σ ∈ S2d, there are
exactly (d − i)! · (d + i)! elements µ ∈ S2d such that ω(vσ, vµ) 6= 0. In this case,
ω(vσ, vµ) = (−1)d+i.
We compute ω(ed−i1 ed+i2 , ed+i
1 ed−i2 ) as follows. Notice that
ω(ed−i1 ed+i2 , ed+i
1 ed−i2 ) =
(1
(2d)!
)2 ∑σ∈S2d
∑µ∈S2d
ω(vσ, vµ)
=
(1
(2d)!
)2
(2d)!(d− i)!(d+ i)!(−1)d+i
=1
(2d)!(d− i)!(d+ i)!(−1)d+i
so
ω(vi, vi) = 21
(2d)!(d− i)!(d+ i)!(−1)d+i =
(2d
d− i
)−1
(−1)d+i.
So we normalize the basis and define
wi :=1√
|ω(vi, vi)|vi.
Then ω(wi, wi) = (−1)d+i.
We define the real quadratic form Q on Sym2d(R2) by Q(v) = ω(v, v). We
consider the basis {w−d, . . . , wd} of
Sym2d(R2)
3. REPRESENTATIONS OF LIE ALGEBRAS 46
above and define Q(x−d, . . . , xd) := Q(∑xiwi). Notice, then, that
Q(x−d, . . . , xd) = x2−d − x2
−d+1 + · · · ± x20 ∓ x2
1 + · · ·+ x2d.
We have thus proven the following result.
Proposition 3.4.7. Let d ≥ 1. Then the quadratic form Q on Sym2d(R2) given by
Q(x−d, . . . , xd) := x2−d − x2
−d+1 + · · · ± x20 ∓ x2
1 + · · ·+ x2d
is SL2(R)-invariant.
Example 3.4.8. Let d = 1 and letO be the SL2(R) orbit of the highest weight vector.
Then Proposition 3.4.7 implies that every point v = (x−1, x0, x1) in O satisfies
Q(v) = 0 i.e. x2−1 − x2
0 + x21 = 0.
In other words, every point v ∈ O lies in the light cone, i.e. the points (x−1, x0, x1)
satisfying
x2−1 − x2
0 + x21 = 0.
Moreover, since O does not contain 0 and O is connected, all points in O must lie
on one half of the light cone. But we may realize the representation on the space
of 2× 2 symmetric matricies with (x−1, x0, x1) corresponding to
x0 x−1
x−1 x1
, where
we know the highest weight vector is
1 0
0 0
. So all points on O satisfy x0 > 0.
Moreover, one can show by an elementary linear algebra argument that the action is
transitive, and O the half of the light cone with x0 > 0, i.e.
O = {(x−1, x0, x1) ∈ Sym2d(R2) : x2−1 − x2
0 + x21 = 0 and x0 > 0}.
3. REPRESENTATIONS OF LIE ALGEBRAS 47
−10 −5 0 510−10
0
10−5
0
5
Figure 3.1 The Light Cone
Chapter 4
Defining Equations of the Highest
Weight Orbit
The main goal of this chapter is to prove a real version of Kostant’s theorem, which
gives a set of quadratic equations which define the Zariski closure of the highest weight
orbit. This result is proven in Section 4.2.1.
4.1 Orbits in Complex Vector Spaces
We begin with some results on orbits in complex vector spaces. We do this because
we need results from algebraic geometry, which are only true over an algebraically
closed field.
For simplicity in this section, we assume that G = SLn(R) and GC = SLn(C),
but the results of this section hold in a much more general context which involves the
theory of semisimple algebraic groups. Let (π, V ) be a finite dimensional complex
representation of GC. Then there is an action of GC on the projective space P(V )
48
4. STRUCTURE OF HIGHEST WEIGHT ORBITS 49
given by
g.[v] := [π(g)v] g ∈ GC, v ∈ V.
Proposition 4.1.1. Let (π, V ) be an irreducible complex representation of GC, and
let v ∈ V be a highest weight vector. Then the complex orbit GC · [v] in P(V ) is Zariski
closed.
Proof: Let BC be the Borel subgroup of GC which stabilizes [v]. Such a Borel
subgroup exists because v is a highest weight vector. Then note that the map
f : GC → P(V ) given by f(g) = g.[v] is a morphism of algebraic varieties. We know
from [2, §11.1] that GC/BC is a projective algebraic variety. It follows from the univer-
sal property of quotients of algebraic groups that the induced map f : GC/BC → P(V )
is a morphism of algebraic varieties. Since f is a morphism, we know that its image
in P(V ) is Zariski closed (see [22, §5.2]).
In the proof of the following proposition, we use the fact that all the Borel
subgroups of GC are conjugate, i.e. if BC and B′C are two Borel subgroups of GC then
there exists g0 ∈ GC such that g0BCg−10 = B′C. See [11, p. 135] for a proof.
Proposition 4.1.2. Let (π, V ) be an irreducible complex representation of GC, and
let v ∈ V be nonzero. If the complex orbit GC · [v] is Zariski closed in P(V ), then v is
a highest weight vector.
Proof: Suppose that GC.[v] is Zariski closed. Then, let X = GC.[v], and let BC
be the Borel subgroup of GC consisting of complex upper triangular matricies. Then
BC acts on X and since X is Zariski closed, the Borel Fixed Point Theorem (see [16,
§3.4.3]) implies that there exists a fixed point g0.[v] in X of this action, i.e.
BC.(g0.[v]) = g0.[v]
4. STRUCTURE OF HIGHEST WEIGHT ORBITS 50
which implies
g−10 BCg0.[v] = [v].
Thus, v is a highest weight vector with respect to the Borel subgroup g−10 BCg0 of GC.
4.2 Highest Weight Orbits
We begin by defining the object of interest for this section. Let G = SLn(R), as
in Section 4.1 and let g = Lie(G). Let h denote the Cartan subalgebra of diagonal
matricies in g. Recall from Proposition 3.1.12 that if G is a Lie group and (π, V ) is
a real irreducible representation of G, then V is a highest weight representation.
Definition 4.2.1. Let G be a real split semisimple Lie group. Let (π, Vλ) be a real,
finite dimensional irreducible representation of G of highest weight λ. Let vλ ∈ Vλ be
a chosen highest weight vector. Then we set
Xλ := G.vλ ∪ {0} = {π(x)vλ : x ∈ G} ∪ {0} (4.2.1)
Recall that we say α ≺ β for two weights α, β ∈ h∗ if β−α is a linear combination
of simple roots with non-negative coefficients.
Definition 4.2.2. Let G be a real split semisimple Lie group. Let (π, Vλ) be a real
finite dimensional irreducible representation of G of highest weight λ, and let vλ ∈ Vλbe a highest weight vector. Let W denote the Weyl group of G. We choose a basis
B of Vλ consisting of weight vectors, such that w.vλ ∈ B for every w ∈ W . For
every v ∈ Vλ, we write v as a linear combination of vectors in B, and we denote the
coefficient of w.vλ by c(v, w). Next, we set
Eλ,w := {v ∈ Vλ : c(v, w) > 0 and c(v, w′) = 0 if wλ ≺ w′λ}.
4. STRUCTURE OF HIGHEST WEIGHT ORBITS 51
Finally, set
Eλ := {0} ∪⋃w∈W
Eλ,w.
It is clear that Eλ is a semialgebraic set.
We now state Kostant’s Theorem. Recall that C is the Casimir element of g and
for a weight µ, C(µ) is the scalar with which C acts on the irreducible highest weight
representation Vµ of weight µ.
Theorem 4.2.3. (Kostant) Let G be a real split semisimple Lie group and let g =
Lie(G). Let (π, Vλ) be a real irreducible representation of G of highest weight λ and
with a chosen highest weight vector of vλ. If −vλ ∈ G · vλ, then Xλ is given by
Xλ = {v ∈ Vλ : C(v ⊗ v)− C(2λ)v ⊗ v = 0}.
If −vλ 6∈ G · vλ, then Xλ is given by the intersection of the set given by the above
equations and the set Eλ.
For the sake of simplicity, we only prove this result for G = SLn(R). Our proof
is based on the proof given in [17, Chapter 10, §6.6], but as in Chapter 3, we need
to address the issues that arise from the difference between R and C.
Remark 4.2.4. From Example 3.4.8 it follows that in the case of d = 1, the inter-
section of the complex orbit SL2(C) · vλ with the real subspace Vλ is the entire light
cone, which is strictly larger than the real orbit SL2(R) · vλ. This example clarifies
the necessity of the constraints from Eλ.
4.2.1 Kostant’s Theorem for SLn(R)
The proof of Theorem 4.2.3 is an immediate consequence of Proposition 4.2.5, Propo-
sition 4.2.12 and Proposition 4.2.13, which will be proved below.
4. STRUCTURE OF HIGHEST WEIGHT ORBITS 52
Let E be a finite dimensional real vector space, and as usual, let P(E) denote the
projective space of E. We give E and P(E) the following topologies, both of which
we call the real Zariski topology. Recall that the topology of E is the one in which
the closed sets are precisely the loci of ideals of P(E). The topology of P(E) is the
one in which the closed sets are the loci of homogeneous ideals of P(E). Recall that
G = SLn(R) and that Xλ is defined in equation 4.2.1. In this section, H and B will
denote the standard Cartan and Borel subgroups of G, respectively. We let HC and
BC denote the standard Cartan and Borel subgroups of GC = SLn(C).
Proposition 4.2.5. Assume the setting of Theorem 4.2.3. If −vλ ∈ Xλ, then we
have GC · vλ ∩ Vλ = G · vλ. If −vλ 6∈ Xλ, then G · vλ is the subset of GC · vλ consisting
of vectors which satisfy the constraints given by Eλ.
Proof: Let V := Vλ and let VC := V ⊗R C be the complexification of V . Then
VC is a complex highest weight module of G with highest weight vector vλ. Let N−
and N−C denote the subgroups of lower unipotent elements in G and GC, respectively.
First assume that −vλ ∈ Xλ. We want to show that G · vλ = GC · vλ ∩ V .
Note that G and GC have the same Weyl group W . Moreover, we have the
Bruhat decompositions
G =∐w∈W
N−wB
and
GC =∐w∈W
N−C wBC.
Then it is enough to show that for each w ∈ W , we have
N−wBvλ = (N−C wBCvλ) ∩ V.
The inclusion N−wBvλ ⊆ (N−C wBCvλ)∩V is clear. For the other inclusion, note that
our assumption implies that −vλ ∈ G · vλ. This implies that B · vλ = (R \ {0})vλ =
4. STRUCTURE OF HIGHEST WEIGHT ORBITS 53
(C \ {0}vλ) ∩ V = BC · vλ ∩ V . Now note that
N−C wBCvλ =⋃α∈C×
αN−C w.vλ.
First assume α ∈ R. Then set v′ = αwvλ. By [14, Lemma 7.1], we have
N−C .v′ ∩ V = N−v′ = αN−wvλ. On the other hand, suppose α ∈ C \ R. We claim
that (αN−C wvλ) ∩ V = ∅. Suppose x ∈ αN−C .vλ ∩ V . Then x = αg.(wvλ) for some
g ∈ N−C . But g.(wvλ) = wvλ +∑
η 6=wλ vη where wvλ ∈ V (wλ) and vη ∈ VC(η). But
αwvλ /∈ V as well, a contradiction.
The reasoning in the case where −vλ 6∈ G · vλ is similar. The only difference is
that this time,
N−C wvλ ∩ V = N−wvλ ∪ (−N−wvλ).
The constraints from Eλ discard the extra piece −N−wvλ.
Let E be a finite dimensional real vector space. Let S(E), P(E) and D(E)
denote the symmetric algebra, polynomial algebra and algebra of constant coefficient
differential operators on E, respectively. Let Sm(E), Pm(E) and Dm(E) denote their
m-th graded components. There is a natural isomorphism Sm(E) ∼= Dm(E) given by
w1 · · ·wm 7→ ∂w1 · · · ∂wm
where
∂vf(x) = limh→0
f(x+ hv)− f(x)
h
for v ∈ E, f ∈ P(E) and x ∈ E. Furthermore, there is a non-degenerate bilinear form
〈·, ·〉 : D(E)× P(E)→ R
4. STRUCTURE OF HIGHEST WEIGHT ORBITS 54
given by
〈D, p〉 = Dp(0).
The form 〈·, ·〉 induces isomorphisms Dm(E) ∼= Pm(E)∗ and Sm(E) ∼= Dm(E). Thus,
we have
Sm(E) ∼= Dm(E) ∼= Pm(E)∗.
Proposition 4.2.6. For any m ≥ 0, the isomorphism Sm(E) ∼= Pm(E)∗ is given by
symm(v) 7→ ηv where ηv(p) = p(v).
Proof: It is enough to verify the statement for polynomials x 7→ φ(x)m where
φ ∈ E∗. Then we get that the image of symm(v) in Dm(E) is 1m!∂mv and
∂vφ(x) = limt→0
1
t(φ(x+ tv)− φ(x)) = lim
t→0
1
t(φ(x) + tφ(v)− φ(x)) = φ(v).
And it therefore follows from the Leibniz rule that 1m!∂mv (φm) = φ(v)m.
Let
I = {φ ∈ P(Vλ) : φ|Xλ = 0}.
Recall that P(Vλ) is a G-module via the action
g.φ(v) = φ(g−1.v).
Lemma 4.2.7. I is a homogeneous and G-invariant ideal of P(Vλ).
Proof: We know that I is G-invariant since Xλ is G-invariant. We now show
that I is a homogeneous ideal. It suffices to show that each element φ ∈ I is a
sum of homogeneous polynomials in I. Write φ = φ0 + · · · + φk where each φi is a
homogeneous polynomial. We now show that each φi is in I.
4. STRUCTURE OF HIGHEST WEIGHT ORBITS 55
Let a1, . . . , an be integers such thatn∑i=1
= 0 andn∑i=1
aiλi = N ∈ Z\{0}. For every
g ∈ G and every ta = diag(ta1 , . . . , tan) ∈ H, we have
(gtag−1).g(vλ) = gta.vλ = t
∑λiaig.vλ = tNg.vλ.
Set v := g.vλ. Then
0 = φ((gtag−1).v) = φ(tNv) = φ0(v) +
k∑i=1
tiNφi(v).
Then define f : R → R by f(t) = φ((gtag−1).v). Note that f is a polynomial in t
with coefficients φ0(v), . . . , φk(v). But f(t) = 0 for all t ∈ R, so f = 0, i.e. φi(v) = 0
for each i. So each φi ∈ I.
For each k ≥ 0, let
Ak := {φ|Xλ : φ ∈ Pk(Vλ)} = Pk(Vλ)/(I ∩ Pk(Vλ)).
Since I is G-invariant, Ak is a G-module, and hence an h-module, where h = Lie(H).
Moreover, Ak is finite dimensional and thus has a weight space decomposition
Ak =⊕µ∈h∗
Ak(µ).
Lemma 4.2.8. Fix µ ∈ h∗. Suppose φ ∈ Ak(µ) is such that φ(vλ) 6= 0. Then
µ = −kλ.
Proof: Write µ = µ1ε1 + · · ·+ µnεn. Let x = diag(t1, . . . , tn) ∈ H. Then
φ(x−1.vλ) = (x.φ)(vλ) = t1µ1 · · · tnµnφ(vλ).
4. STRUCTURE OF HIGHEST WEIGHT ORBITS 56
Note that we can choose a1, . . . , an ∈ Z such that∑ai = 0 but N =
∑aiλi 6= 0. Let
t ∈ R∗ and set ta := diag(ta1 , . . . , tan). Then
t−kNφ(vλ) = φ(t−Nvλ) = φ(ta−1.vλ) = t
∑aiµiφ(vλ).
Since t ∈ R is arbitrary,∑aiµi = −kN . Now let t = (t1, . . . , tn) ∈ H be such that
t1λ1 · · · tnλn = 1. Then
t1 · · · tn = 1 and t1λ1 · · · tnλn = 1.
Hence we have proven that for any element diag(t1, . . . , tn) ∈ H, if t1λ1 · · · tnλn = 1
then t1µ1 · · · tnµn = 1. Notice that tn = 1
t1···tn−1. Therefore, if
tλ1−λn1 · · · tλn−1−λnn−1 = 1
then
tµ1−µn1 · · · tµn−1−µnn−1 = 1
for every diag(t1, . . . , tn) ∈ H. Next we write tj = e2πiθj , aj = λj − λn, and
bj = µj − µn. Then if e2πi∑ajθj = 1 then e2πi
∑bjθj = 1 for any θ ∈ [0, 2π),
which means that if∑ajθj is an integer, then so is
∑bjθj. Thus if
∑ajθj = 0
then∑bjθj = 0 and hence (b1, . . . , bn) = k(a1, . . . , kn) for some k ∈ Q (given that
all aj, bj ∈ Z). Thus for each j we have µj−µn = k(λj−λn) so µj = kλj+(µn−kλn).
Lemma 4.2.9. As a G-module,
Sk(Vλ) = Vkλ ⊕⊕µ≺kλ
Vµ.
Proof: We prove the result for k = 2. The idea is the same for larger k.
4. STRUCTURE OF HIGHEST WEIGHT ORBITS 57
We first show that each weight in S2(Vλ) is at most 2λ in the ordering of the
weights. Let {v1, . . . , vn} be a basis of weight vectors of Vλ, where each vi has weight
λi. Let v =∑aijvivj be a weight vector of weight µ in S2(Vλ). Let H ∈ h∗. Then
H · v =∑
aijµ(H)vivj
and
H · v =∑
aij(λi + λj)(H)vivj.
Since H is arbitrary and there must exist i, j such that aij 6= 0, we have µ = λi + λj
for some i, j. Since λi, λj � λ, we get that µ � 2λ.
We now show that the 2λ weight space of S2(Vλ) is one-dimensional. Indeed, let
v =∑aijvivj be a weight vector in S2(Vλ) of weight 2λ. Then for any H ∈ h∗,
H · v =∑
aij(λi + λj)(H)vivj =∑
aij2λ(H)vivj.
Hence, each a11 6= 0 and aij = 0 for (i, j) 6= (1, 1). So v = a11v1v1 ∈ Rvλvλ. So the
2λ weight space has dimension 1 in S2(Vλ).
Finally, by Weyl’s theorem of complete reducibility ([21, p. 46]), we see that we
may decompose S2(Vλ) as
S2(Vλ) = V2λ ⊕ Vµ1 ⊕ · · · ⊕ Vµl
where Vµ1 , . . . , Vµl are highest weight modules of weights µ1, . . . , µl, respectively. By
the above work, for each 1 ≤ i ≤ l, we have µi � 2λ. But the 2λ weight space is
one-dimensional, and hence each µi ≺ 2λ. This concludes our proof.
4. STRUCTURE OF HIGHEST WEIGHT ORBITS 58
From the isomorphism Pk(Vλ) ∼= Sk(Vλ)∗ we make the identification
Pk(Vλ) = V ∗kλ ⊕⊕µ≺kλ
V ∗µ . (4.2.2)
Lemma 4.2.10. For every V ∗µ in (4.2.2) with µ 6= kλ, the image of V ∗µ under the
map
Pk(Vλ)→ Ak
is zero.
Proof: We have a weight space decomposition
Vµ∗ =
⊕V ∗µ (η).
From Lemma 4.2.9, we see that each η in this decomposition satisfies η 6= −kλ. Thus
if φ ∈ Vµ∗(η) we have φ(vλ) = 0. Since V ∗µ is G-invariant, the set
T := {x ∈ Vλ : φ(x) = 0 for all φ ∈ V ∗µ }
is also G-invariant. Since vλ ∈ T , we then obtain Xλ ⊆ T .
Proposition 4.2.11. As a G-module, Ak ∼= V ∗kλ.
Proof: From Lemma 4.2.10, we know that the map P k(Vλ)→ Ak induces a map
V ∗kλ → Ak. Since V ∗kλ is irreducible and this map is surjective, either Ak = 0 or the
map is an isomorphism. But Ak 6= 0, since one can choose an element f ∈ Ak such
that f(vλ) 6= 0.
4. STRUCTURE OF HIGHEST WEIGHT ORBITS 59
Proposition 4.2.12. Let v ∈ Vλ and let Xλ be as in Theorem 4.2.5. Then v ∈ Xλ if
and only if v ⊗ v ∈ V2λ.
Proof: Suppose v ∈ Xλ. Then v = g.vλ for some g ∈ G. Then v ⊗ v =
(g.vλ)⊗(g.vλ) = g.(vλ⊗vλ). But vλ⊗vλ ∈ V2λ, and V2λ is G-invariant. So v⊗v ∈ V2λ.
Conversely, suppose v ∈ Vλ and v ⊗ v ∈ V2λ. Then we need to show that for all
φ ∈ I we have φ(v) = 0. But we know from the last proposition that
I =∞⊕k=0
(I ∩ Pk(Vλ)) =∞⊕k=0
⊕µ≺kλ
V ∗µ .
Thus, to show that v ∈ X, we let φ ∈ V ∗µ where V ∗µ ⊆ Pk(Vλ) is one of the components
above, and we need to show φ(v) = 0. To see this, recall we have the G-invariant
non-degenerate pairing
Sk(Vλ)× Pk(Vλ)→ R.
Then φ(v) = 〈symk(v), φ〉. But we know that this form, when restricted to Vkλ× V ∗µ ,
must vanish. Therefore φ(v) = 0.
The following theorem shows that Xλ is a locus of quadratic equations.
Proposition 4.2.13. Let C be the Casimir element of sln(R) and let (π, Vλ) be an
irreducible highest weight G-module with highest weight λ. Let V2λ be the highest
weight representation of sln(R) with weight 2λ. Let C(2λ) be value that C acts on
V2λ by (see Theorem 3.2.12). Then we have
V2λ = {a ∈ Vλ ⊗ Vλ : Ca = C(2λ)a}.
Proof: We first note that if µ ≺ λ are dominant weights, then C(µ) < C(λ).
4. STRUCTURE OF HIGHEST WEIGHT ORBITS 60
Indeed, C(λ)−C(µ) = (λ+ρ, λ+ρ)− (µ+ρ, µ+ρ). One can write µ = λ−γ, where
γ is a positive combination of positive roots. Then (λ + ρ, λ + ρ)− (µ + ρ, µ + ρ) =
(λ + µ + 2ρ, γ). Since λ + µ + 2ρ is a regular dominant weight, and γ is a nonzero
sum of positive roots, we get (λ+ µ+ 2ρ, γ) > 0. Thus, C(µ) < C(λ).
Now, we can decompose Vλ ⊗ Vλ = V2λ ⊕⊕µ≺λ
Vµ. This shows that the elements
of Vλ⊗Vλ which are eigenvectors of C (recall Definition 1.3.9) with eigenvalue C(2λ)
are precisely the vectors in V2λ.
As mentioned in the begining of this section, Kostant’s theorem (Theorem 4.2.3)
is an immediate consequence of Proposition 4.2.5, Proposition 4.2.12 and Proposition
4.2.13.
4.2.2 Equations for the Highest Weight Orbit for SL2(R)
In this section we use Proposition 4.2.13 to write down the quadratic equations which
determine Xλ for the 5-dimensional irreducible highest weight module of SL2(R).
Let Vd be the highest weight representation of SL2(R) as defined in Section 3.2.
Recall that our Casimir operator for SL2(R) is given by
C =1
2FE +
1
4H +
1
8H2.
Let {v0, . . . , vd} be the basis of Vd as given in Proposition 3.2.2. Let 0 ≤ i, j ≤ d.
4. STRUCTURE OF HIGHEST WEIGHT ORBITS 61
Then
FE(vi ⊗ vj) = F (Evi ⊗ vj + vi ⊗ Evj)
= F ((i− d)vi+1 ⊗ vj + (j − d)vi ⊗ vj+1)
= (i− d)(Fvi+1 ⊗ vj + vi+1 ⊗ Fvj) + (j − d)(Fvi ⊗ vj+1 + vi ⊗ Fvj+1)
= ((d− i)(i+ 1) + (d− j)(j + 1))vi ⊗ vj
+ (d− i)jvi+1 ⊗ vj−1 + (d− j)ivi−1 ⊗ vj+1
and1
4H(vi ⊗ vj) = 1
2(i + j − d)vi ⊗ vj and
1
8H2(vi ⊗ vj) = 1
2(i + j −m)2vi ⊗ vj. so
we get
C(v ⊗ v) =∑∑
xixjC(vi ⊗ vj)
=1
2
∑∑xixj(2ij − id− jd+ d2 + d)vi ⊗ vj
+∑∑
xixj(d− i)jvi+1 ⊗ vj−1
+∑∑
xixj(d− j)ivi−1 ⊗ vj+1
=∑∑ 1
2(xixj(2ij − id− jd+ d2 + d) + xi−1xj+1(d− i+ 1)(j + 1)
+ xi+1xj−1(d− j + 1)(i+ 1))vi ⊗ vj
so our equation C(v ⊗ v)− C(2λ)(v ⊗ v) = 0 reduces to
∑∑1
2(xixj(2ij − id− jd+ d2 + d− C(2λ))+
xi−1xj+1(d− i+ 1)(j + 1) + xi+1xj−1(d− j + 1)(i+ 1))vi ⊗ vj = 0
4. STRUCTURE OF HIGHEST WEIGHT ORBITS 62
For each 0 ≤ i, j ≤ d let
αij =1
2(2ij − id− jd+ d2 + d− C(2λ))
βij =1
2((d− i+ 1)(j + 1))
γij =1
2((d− j + 1)(i+ 1))
For each 0 ≤ i, j ≤ d, let
Pij(x0, . . . , xd) := αijxixj + βijxi−1xj+1 + γijxi+1xj−1.
When working in S2(Vd), we see the following. If i 6= j, then the coefficient of vivj in
C(v ⊗ v)− C(2λ) is
Pij(x0, . . . , xd)+Pji(x0, . . . , xd) = (αij+αji)xixj+(βij+γji)xi−1xj+1+(γij+βji)xi+1xj−1.
If i = j, then the coefficient of vivj = vivi is Pij(x0, . . . , xd).
Example 4.2.14. Let d = 4. Then the equations for Xλ are given by
1. v0v0 : 0 = 0
2. v0v1 : 0 = 0
3. v0v2 : 3x21 − 8x0x2 = 0
4. v0v3 : 2x1x2 − 12x0x3 = 0
5. v0v4 : x1x3 − 16x0x4 = 0
6. v1v1 : 8x0x2 − 3x21 = 0
7. v1v2 : 12x0x3 − 2x1x2 = 0
8. v1v3 : 4x22 + 16x0x4 − 10x1x3 = 0
9. v1v4 : 2x2x3 − 12x1x4 = 0
4. STRUCTURE OF HIGHEST WEIGHT ORBITS 63
10. v2v2 : 9x1x3 − 4x22 = 0
11. v2v3 : 12x1x4 − 2x2x3 = 0
12. v2v4 : 3x23 − 8x2x4 = 0
13. v3v3 : 8x2x4 − 3x23 = 0
14. v3v4 : 0 = 0
15. v4v4 : 0 = 0
and the constraints from Eλ are given
x4 > 0 or (x4 = 0 and x0 > 0).
Chapter 5
The Convex Hull of the Highest
Weight Orbit
In the previous chapter, we have seen that Xλ is a semialgebraic over R, and thus,
by Proposition 1.2.6, the set conv(Xλ) is semi-algebraic. The goal of this section is
to study this set for G = SL2(R).
5.1 The Relationship Between Convex Hulls of the
G and K-orbits
Throughout this section, G is a split real semisimple Lie group with finite center,
and K is a maximal compact subgroup of G (see Section 2.2). In this section, we
state and prove some preliminary results about the convex hull of Xλ. In doing this,
we explain how G.vλ is related to the convex hull of the orbit K.vλ, where K is the
maximal compact subgroup of G. This will be useful, since orbits of compact groups
can be easier to understand than those of non-compact groups. Our main reference
is [7]. All representations will be real. We let g = Lie(G) and we let h be a chosen
Cartan subalgebra of g. Also, we denote the Cartan involution of G by Θ.
64
5. THE CONVEX HULL OF THE HIGHEST WEIGHT ORBIT 65
Lemma 5.1.1. Let (π, V ) be an irreducible representation of G. Then there exists
an inner product (·, ·) on V such that
π(g)∗ = π(Θ(g)−1)
for all g ∈ G.
Proof: Let gc = k⊕ ip, and let gC = g⊗R C. Then we know that gc corresponds
to a compact group Gc, since the Killing form on gc is negative definite. The real
representation π of g on V induces a complex representation π′ : gc → gl(VC) where
VC = V ⊗RC and π′ is defined by π′(X+ iY ) = π(X)+ iπ(Y ) for all X ∈ k and Y ∈ p.
We can consider π′ as a representation of Gc as well, and hence, by the compactness
of Gc, there exists a K-invariant inner product 〈·, ·〉 on VC. This implies that for all
X ∈ k, Y ∈ p, v, w ∈ V we have
〈π′(X + iY )v, w〉 = −〈v, π′(X + iY )w〉
i.e.
〈π′(X)v, w〉 = −〈v, π′(X)w〉 and 〈π′(Y )v, w〉 = 〈v, π′(Y )w〉.
Then define the inner product (·, ·) on V by (v, w) = <〈v, w〉. This inner product
satisfies the desired relation.
When given a representation (π, V ) of G or g, we always equip V with the inner
product (·, ·).
Proposition 5.1.2. Let (π, V ) be a finite dimensional irreducible representation of
G. Then dimV K ≤ 1.
5. THE CONVEX HULL OF THE HIGHEST WEIGHT ORBIT 66
Proof: This argument is similar to [19, Proposition 4.2]. Suppose dimV K ≥ 2.
Since V is irreducible, it is a highest weight module of some weight λ ∈ h∗. Let
V ′ =⊕µ6=λ
Vµ.
Then since dimVλ = 1 and dimV K ≥ 2, we must have V K ∩ V ′ 6= 0. Now let
v ∈ V K ∩ V ′ be a nonzero weight vector. Then the PBW theorem (Theorem 1.3.8)
and the Iwasawa decomposition (Theorem 2.1.8) imply that
V = U(g)v = U(n− ⊕ h⊕ k)v = U(n−)v ⊆ V ′
which is a contradiction.
Definition 5.1.3. We say a finite dimensional irreducible representation (π, V ) of G
is spherical if dimV K = 1.
By Proposition 5.1.2, we see that (π, V ) being spherical is equivalent to the
existence of a nonzero K-fixed vector u ∈ V .
Definition 5.1.4. Let V be a real vector space. We say a set C ⊆ V is a cone if, for
any x ∈ C and any r > 0, rx ∈ C.
Definition 5.1.5. Let V be a vector space, and let C ⊆ V be a cone. We say C is
pointed if C ∩ (−C) = {0}. We say C is generating if span(C) = Rn. If C is pointed,
generating, and closed, we say C is regular.
Definition 5.1.6. For a cone C ⊆ Rn, we define the dual cone of C to be the set
C∗ := {v ∈ V : (v, w) ≥ 0 for all w ∈ C \ 0}.
5. THE CONVEX HULL OF THE HIGHEST WEIGHT ORBIT 67
Remark 5.1.7. If G is a Lie group and (π, V ) is a representation of G, we say C is
G-invariant if π(g)v ∈ C for all g ∈ G and v ∈ C. If the group G and representation
(π, V ) are understood from the context, we simply say C is invariant.
Proposition 5.1.8. Let (π, V ) be an irreducible representation of G. If V contains
an invariant regular cone, then (π, V ) is spherical.
Proof: Equip V with the inner product of Proposition 5.1.1. Let C be an invariant
regular cone in V . There exists v ∈ C∗ such that (u, v) > 0 for all u ∈ C \ {0}. Fix
u ∈ C \ 0. Then (π(k)u, v) > 0 for all k ∈ K. Thus, the vector
uK :=
∫K
π(k)udk
is a member of C and is K-fixed (we are integrating with the Haar measure on K).
Note that uK 6= 0 since
(uK , v) = (
∫K
π(k)u, v) =
∫K
(π(k)u, v)dk > 0
so uK 6= 0. Thus (π, V ) is spherical.
Corollary 5.1.9. Let (π, V ) be an irreducible representation of G with the inner
product of Proposition 5.1.1. If C is an invariant regular cone in V , then C contains
a K-fixed unit vector.
Proof: This is the vector uK from Propositon 5.1.8.
If (π, V ) is spherical, then there are exactly two unit vectors in V K . Fix one,
and call it u0. The other unit vector is −u0. We say a cone C ⊆ V is maximal if
it is maximal with respect to inclusion in the collection of all invariant regular cones
5. THE CONVEX HULL OF THE HIGHEST WEIGHT ORBIT 68
containing u0. Similarly, we say a cone is minimal if it is minimal with respect to
inclusion in the collection of all invariant regular cones containing u0. Since V is
irreducible, it is a highest weight module with some highest weight λ ∈ h∗. We fix a
highest weight vector vλ ∈ V which satisfies (u0, vλ) > 0.
Theorem 5.1.10. Let (π, V ) be a spherical irreducible representation of G. Then
there exists a unique invariant minimal cone Cmin given by
Cmin = R+conv(G · u0).
Proof: Let C0 = R+conv(G · u0). We first show that C0 is a regular invariant cone.
Firstly, it is clear that C0 is closed and invariant. Note that since V is irreducible,
and span(C0) is an invariant subspace of V , we have that span(C0) = V , i.e. C0
is generating. We now show that C0 is pointed. Now let g ∈ G. By the polar
decomposition G = KP , we can write g = kp where k ∈ K and p = eX where X ∈ p.
Thus,
(π(g)u0, u0) = (π(k)π(eX)u0, u0)
= (π(eX)u0, π(k−1)u0)
= (π(eX)u0, u0)
= (π(e1/2Xu0, π(e1/2X)u0) > 0
Let v1 = g1 · u0 and v2 = g2 · u0 be members of G · u0, where g1, g2 ∈ G. Let
5. THE CONVEX HULL OF THE HIGHEST WEIGHT ORBIT 69
x = Θ−1(g2). Then
(v1, v2) = (π(g1)u0, π(g2)u0)
= (u0, π(Θ−1(g1)g2)u0)
= (u0, π(Θ−1(g1x)u0)
= (π(g1x)u0, u0) > 0
where the inequality follows from the above fact that (π(g)u0, u0) > 0 for all g ∈ G.
So if v ∈ C ∩ (−C), then (v,−v) ≥ 0, which implies v = 0. So C0 is pointed. Thus,
C0 is a regular invariant cone. Therefore, Cmin ⊆ C0. To see the opposite inclusion,
note that Cmin contains G · u0, is closed and convex, and therefore C0 ⊆ Cmin.
Lemma 5.1.11. Let (π, Vλ) be an irreducible representation of G with highest weight
λ, and let vλ ∈ Vλ be a highest weight vector. Then
conv(G · vλ ∪ {0}) = R≥0conv(K.vλ).
Proof: Let C0 = conv(G · vλ ∪ {0}) and C1 = R≥0conv(K.vλ). We prove the
inclusions C1 ⊆ C0 and C0 ⊆ C1. To begin, notice that we have the Iwasawa decom-
position G = KAN . Recall that AN.vλ = R+vλ. Therefore, the nonzero points of
conv(G · vλ ∪ {0}) have the form
x =∑i
λicixi
5. THE CONVEX HULL OF THE HIGHEST WEIGHT ORBIT 70
where∑i
λi ≤ 1, λi ≥ 0, ci ∈ R+ and xi ∈ K.vλ. One can write such a point as
(∑i
λici
)(∑i
µixi
)where µi =
λici∑λici
which shows that x ∈ R≥0conv(K.vλ). The other inclusion is clear.
Lemma 5.1.12. Let (π, Vλ) be an irreducible representation of G with highest weight
λ, and let vλ ∈ Vλ be a highest weight vector. The cone conv(G · vλ ∪ {0}) is closed,
generating, and invariant.
Proof: Let C0 = conv(G · vλ ∪ {0}). We have the Iwasawa decomposition KAN ,
and we know that AN · vλ = R+vλ. It thus follows that C0 is indeed a cone. It is also
clear that C0 is invariant, and thus span(C0) is an invariant subspace of V . Since V
is irreducible and C0 6= {0}, it follows that span(C0) = V , so C0 is generating.
We now show that C0 is closed. Indeed, let (xn) be a sequence in C0 converging
to a point x ∈ V . By Lemma 5.1.11, for each n, we can write
xn = αnyn
where αn ≥ 0 and yn is a sequence in conv(K · vλ). For each n, we can write
yn =∑ti,nπ(ki,n)vλ where ti,n > 0,
∑ti,n = 1 and ki,n ∈ K. Since K is compact, so
is K · vλ (as our action is continuous). Moreover, we know that the convex hull of a
compact set is compact, so conv(K ·vλ) is compact. Thus, by passing to a subsequence
of (yn), we may assume that (yn) converges to a point y ∈ conv(K · vλ).
5. THE CONVEX HULL OF THE HIGHEST WEIGHT ORBIT 71
Now, since (xn) converges, we know it is bounded. Moreover,
(xn, u0) = αn∑
ti,n(π(ki,n)vλ, u0)
= αn∑
ti,n(π(ki,n)vλ, π(ki,n)u0)
= αn∑
ti,n(vλ, u0)
= αn(vλ, u0).
Thus, by the Cauchy-Schwarz theorem,
|αn| =|(an, u0)||(vλ, u0)|
≤ ||xn||1/2||u0||1/2
|(vλ, u0)|
and so |αn| is bounded. Thus, by the Bolzano-Weierstrass theorem, we may pass to
a subsequence of (αn) and assume that (αn) converges to a point α ≥ 0. Thus, since
both (αn) and (yn) converge, the sequence (xn) defined by xn = αnyn converges to
αy. So C0 is indeed closed.
Theorem 5.1.13. Let (π, Vλ) be an irreducible spherical representation of G with
highest weight λ, and let vλ ∈ Vλ be a highest weight vector. Then the minimal cone
is given by Cmin = conv(G · vλ ∪ {0}).
Proof: Let C0 = R≥0conv(K.vλ). By Corollary 5.1.9, we know that C0 contains
the spherical vector u0. Moreover, we know that C0 is closed (by Lemma 5.1.12) and
is invariant. Thus,
R+conv(G · u0) ⊆ C0
i.e. Cmin ⊆ C0. It remains to prove the opposite inclusion.
We fix an orthonormal basis of h-weight vectors e1, . . . , ed for V , such that e1 =
vλ, where each ei has weight λi, and λ1 = λ. Then for H ∈ h+ (the fundamental
5. THE CONVEX HULL OF THE HIGHEST WEIGHT ORBIT 72
Weyl chamber) we have
0 < eλi(H) < eλ1(H)
for i ≥ 2. Now we write
u0 =d∑i=1
(u0, ei)ei.
Thus
π(eH)u0 =d∑i=1
(u0, ei)eλi(H)ei.
Therefore,
limt→∞
e−tλ(H)π(etH)u0 = (u0, vλ)vλ
and consequently vλ ∈ Cmin, as Cmin is closed. This implies the desired inclusion by
the invariance of Cmin.
Remark 5.1.14. In Example 5.1.15 and Proposition 5.1.16, we use the fact that for
a spherical representation (π, V ), we have Cmax = C∗min. See [7, Theorem II.2.2] for
a proof.
Example 5.1.15. Let G = SLn(R) and let V = Sym(n,R) be the space of n × n
symmetric matrices, equipped with the inner product (·, ·) given by
(u, v) = tr(uv).
Let g = sl(n,R) and let h be the Cartan subalgebra of g consisting of diagonal
matricies in g. Let (π, V ) be the representation given by
π(g)v = gvgt
where gt is the transpose of g. Recall that we have the maps εi ∈ h∗, 1 ≤ i ≤ n, given
5. THE CONVEX HULL OF THE HIGHEST WEIGHT ORBIT 73
by
εi(diag(a1, . . . , an)) := ai.
We can see that this representation is irreducible with highest weight 2ε1 (see [7,
p. 25]). The subgroup K = SO(n) is a maximal compact subgroup of G. The
representation π is spherical with a K-fixed vector of u0 = In (identity matrix). It is
not too difficult to see that Cmin = R+conv(G · u0) consists of all positive semidefinite
matricies in Sym(n,R). Moreover, this cone is self-dual, i.e.
Cmin = Cmax.
The next proposition shows that the situation of Example 5.1.15 is special.
Proposition 5.1.16. Let d ≥ 4 be even. Then the minimal and maximal cones of
the irreducible (d+ 1)-dimensional representation V of SL2(R) satisfy
Cmin 6= Cmax.
Proof: Recall that Cmax = C∗min, i.e.
Cmax = {v ∈ V : (v, w) ≥ 0 for all w ∈ Cmin}.
Our strategy is to find an element v ∈ Cmax \ Cmin.
Recall that we have the basis {v0, . . . , vd} of V given by vi = xd−iyi. We first
show that if v =∑d
i=0 civi ∈ Cmin, then if c0 6= 0 and cd 6= 0, then cd/2 6= 0. It
is sufficient to prove this for each vector v ∈ X = (SL2(R).yd) ∪ {0}. Indeed, let
g ∈ SL2(R) and suppose
g−1 =
α β
γ δ
∈ SL2(R).
5. THE CONVEX HULL OF THE HIGHEST WEIGHT ORBIT 74
Then
g.yd = (γx+ δy)d =d∑
k=0
(d
k
)γkδd−kxkyd−k.
Setting v := g.yd, we get that for 0 ≤ k ≤ d, ck =(dk
)γkδd−k. If c0 6= 0 and cd 6= 0, then
γd 6= 0 and δd 6= 0. Thus, γ 6= 0 and δ 6= 0. Consequently, cd/2 =(dd/2
)γd/2δd/2 6= 0.
From the above reasoning, we note that if v ∈ X, then either c0 > 0 or cd > 0.
This also holds if v ∈ Cmin.
We now find a vector x =∑d
i=0 xivi ∈ Cmax which satisfies x0 6= 0 and xd 6= 0,
but not xd/2 6= 0. Indeed, set x0 = xd = 1 and xi = 0 for 0 < i < d. Then, for
v =∑d
i=0 civi ∈ Cmin, we get
d∑i=0
xici = x0c0 + xdcd = c0 + cd > 0.
So x ∈ Cmax but x 6∈ Cmin.
5.2 Spherical SL2(R) Modules
Recall from Section 3.2 that for each d ≥ 0, there is an SL2(R)-module (πd, Vd), where
Vd is the space of degree-d homogeneous polynomials in x and y with coefficients in
R, and πd is given by πd(g)f(x, y) = f(g−1(x, y)) where g ∈ SL2(R).
We may restrict this representation to SO2(R). For θ ∈ R, define
gθ =
cos θ − sin θ
sin θ cos θ
.
5. THE CONVEX HULL OF THE HIGHEST WEIGHT ORBIT 75
Note that
SO2(R) = {gθ : 0 ≤ θ < 2π}.
We may restrict the action of SL2(R) on Vd to SO2(R) on Vd, and we obtain
πd(gθ)f(x, y) = f((cos θ)x+ (sin θ)y,−(sin θ)x+ (cos θ)y)
for all gθ ∈ SO2(R).
Note that SO2(R) also acts on the complexification Vd ⊗R C. For each 0 ≤ k ≤ d,
define
qk(x, y) := (x+ iy)k(x− iy)d−k
and note that for 0 ≤ k ≤ d, qd−k = qk. For each 0 ≤ k ≤ d, qk ∈ Vd ⊗R C. Then
πd(gθ)qk(x, y) = eiθ(d−2k)qk(x, y).
There exist Ak, Bk ∈ Vd such that qk = Ak+iBk, where a, b ∈ R. Let eiθ(d−2k) = a+ib.
Then
πd(gθ)qk = (aAk − bBk) + i(aBk + bAk).
Hence,
πd(gθ)Ak = πd(gθ)(1
2(qk + qd−k))
= aAk − bBk
and
πd(gθ)Bk = πd(gθ)(1
2i(qk − qd−k))
= aBk + bAk
5. THE CONVEX HULL OF THE HIGHEST WEIGHT ORBIT 76
We define
Wk = spanR{Ak, Bk}
and see that the above calculations imply that Wk is SO2(R)-invariant. Note that if
d is even, then q d2(x, y) = A d
2(x, y), so B d
2(x, y) = 0.
Lemma 5.2.1. We may decompose Vd as
Vd = W0 ⊕ · · · ⊕Wb d2c.
Proof: Note that {q0, . . . , qd} is a basis of Vd ⊗R C. Thus, {q0, . . . , qd} is linearly
independent over C. If d is even, define S := {A0, . . . , A d2, B0, . . . , B d
2−1} and if d is
odd, define S := {A0, . . . , A d−12, B0, . . . , B d−1
2}. In both cases, |S| = d+ 1 and
spanCS = spanC{q0, . . . , qd}.
Thus, S is a basis of Vd ⊗R C and hence is linearly independent over C. Thus, S
is linearly independent over R. Since S is linearly independent over R and |S| =
d + 1 = dimR Vd, we get that S is a basis of Vd. Since Wk = spanR{Ak, Bk} for each
0 ≤ k ≤ bd2c, our claim is proven.
Proposition 5.2.2. Let d ≥ 2 be even. Then the highest weight vector yd ∈ Vd has a
non-zero component in each Wk, 0 ≤ k ≤ d2.
Proof: We may write yd as
yd =1
id2d((x+ iy)− (x− iy))d
=1
id2d
d∑k=0
(d
k
)qk(x, y).
5. THE CONVEX HULL OF THE HIGHEST WEIGHT ORBIT 77
But(dk
)=(
dd−k
)and qk = qd−k for all 0 ≤ k ≤ d. Moreover, q d
2(x, y) = A d
2(x, y).
So
yd =1
id2d−1
d2−1∑k=0
(d
k
)Ak +
1
id2d
(dd2
)A d
2(5.2.1)
which has a non-zero component in each Wk, for 0 ≤ k ≤ d2.
Remark 5.2.3. Recall from Section 3.3 that the irreducible real representations of
SO2(R) have the form (ρk, Uk) where k ∈ Z, and Uk = R if k = 0 and Uk = R2 if
k 6= 0. If k 6= 0, then ρk is defined by ρk(gθ) := gkθ, for θ ∈ [0, 2π). We define ρ0 to
be the trivial representation.
Definition 5.2.4. For every (d + 1)-tuple A = (a0, . . . , ad) of integers satisfying
0 ≤ a0 ≤ · · · ≤ ad, we define
ρA := ρa0 ⊕ · · · ⊕ ρad
on UA := Ua0 ⊕ · · · ⊕ Uad .
Definition 5.2.5. Let A = (a1, . . . , ad) be a d-tuple of integers where 0 < a1 ≤ · · · ≤
ad, and consider the representation (ρA, UA) of SO2(R). Let (1, 0)d be the vector
((1, 0), (1, 0), . . . , (1, 0)) ∈ UA. We define CA by
CA := conv(ρA(SO2(R)) · (1, 0)d) ⊆ UA = R2 ⊕ · · · ⊕ R2 ∼= (R2)d.
If A = (1, 2, . . . , d) then we write Cd instead of CA, and we call Cd a universal
Caratheodory orbitope.
Remark 5.2.6. One can easily see that CA is the convex hull of the set
{(cos(a1θ), sin(a1θ), . . . , cos(adθ), sin(adθ)) : θ ∈ [0, 2π)}.
5. THE CONVEX HULL OF THE HIGHEST WEIGHT ORBIT 78
Proposition 5.2.7. Let d ≥ 2 be even and consider the realization of the represen-
tation (πd, Vd) of SL2(R) on the space of homogeneous polynomials in the variables x
and y with coefficients in R. Then the convex set
C = conv(SO2(R).yd)
is affinely isomorphic to C d2.
Proof: Let O1 = SO2(R).yd, let O2 be the orbit of (1, 0)d2 under ρ(2,4,...,d) and
let O3 be the orbit of (1, 0)d2 under ρ(1,2,..., d
2). We note that if two subsets of finite
dimensional vector spaces are affinely isomorphic, then their convex hulls must be
affinely isomorphic as well. Thus, it suffices to show that O1 is affinely isomorphic to
O2, and O2 is affinely isomorphic to O3.
We first show that O1 is affinely isomorphic to O2. Write Vd = W d2⊕ · · · ⊕W0.
By the proof of Proposition 5.2.2, we may write
yd =1
id2d−1
d2−1∑k=0
(d
k
)Ak +
1
id2d
(dd2
)A d
2.
For each 1 ≤ k ≤ d2, we have an isomorphism of SO2(R)-modules
ϕk : W d2−k → U2k
given by by
ϕ(Ak) :=id2d−1(
dd2−k
) (1, 0) ϕ(Bk) :=id2d−1(
dd2−k
) (0, 1).
For any point (x0, x1, . . . , x d2) ∈ O1, we have x0 = 1
id2d
(dd2
)A d
2. Define the map ϕ :
O1 → O2 given by ϕ(x0, x1, . . . , x d2) = (ϕ1(x1), . . . , ϕ d
2(x d
2)). Note that since each ϕk
is SO2(R)-equivariant, so is ϕ. We thus have ϕ(yd) = (1, 0)d and ϕ(O1) = O2. Since
each ϕk is injective, so is ϕ. Thus, ϕ is an isomorphism.
5. THE CONVEX HULL OF THE HIGHEST WEIGHT ORBIT 79
It now suffices to show that O3 is affinely isomorphic to O2. Indeed, for each
1 ≤ k ≤ d, we have a linear isomorphism µk : Uk → U2k given by
µk(ρk(gθ)(1, 0)) := ρ2k(gθ)(1, 0).
Then define the map µ : O3 → O2 by
µ(x1, . . . , x d2) = (µ1(x1), . . . , µ d
2(x d
2)).
Finally, Ψ d2
:= µ−1 ◦ ϕ is an affine isomorphism from O1 to O3.
Definition 5.2.8. Let Ψ d2
denote the affine isomorphism we constructed in the proof
of Proposition 5.2.7 from conv(SO2(R).yd) to C d2.
Definition 5.2.9. Let C ⊆ Rd be a convex set. We define C by
C := {(δ, a1, . . . , ad) : δ +d∑i=1
aibi ≥ 0 for all (b1, . . . , bd) ∈ C}.
Remark 5.2.10. Let C ⊆ Rd be a convex set of dimension d. Then we observe that
a point (a1, . . . , ad) ∈ Rd belongs to C if and only if
δ +d∑
k=1
akck ≥ 0
for all (δ, c1, . . . , cd) ∈ C. This follows from the separating hyperplane theorem. See
[3, §2.5.1].
We know from Proposition 1.2.6 that Cd is a semialgebraic set. It turns out that
there is an easy way of determining the inequalities which define this set. Before
proving this claim, we prove a preliminary proposition.
5. THE CONVEX HULL OF THE HIGHEST WEIGHT ORBIT 80
Proposition 5.2.11. Let d ≥ 1, and let δ, c1, . . . , cd ∈ C. Define the Laurent poly-
nomial
R(z) =d∑
k=−d
ukzk
with the coefficients defined as follows. Let u0 = δ, and if k > 0, define uk = ck and
u−k = uk. Further, we assume that R(z) ≥ 0 when |z| = 1. Then there exists some
H ∈ C[z] of degree d such that
R(z) = H(z−1) ·H(z).
Proof: We begin by noticing that R(z) = R(z−1). Note that the roots of R come
in pairs α, α−1. Thus, once we have shown that the roots lying on the unit circle T
have even multiplicity, we will be done.
Indeed, suppose z0 = eit0 is a root of R having odd multiplicity m. Let φ : R→ R
be defined by φ(t) = R(eit). Then by Taylor’s theorem, in some neighborhood of t0,
φ is represented by
φ(t) = φ(t0) + φ′(t0)(t− t0) + · · ·+ φm−1(t0)
(m− 1)!(t− t0)m−1 +
φm(t0)
m!(t− t0)m
for some ξ ∈ [t0, t]. Since the multiplicity of z0 is m, we have φ(k)(t0) = 0 for
0 ≤ k ≤ m − 1. Thus φ(t) =φm(ξ)
m!(t − t0)m. Since m is odd, this function changes
sign around t0, a contradiction.
Theorem 5.2.12. The Caratheodory orbitope C = Cd is equal to the set of all vectors
(s1, t1, . . . , sd, td) ∈ R2d such that the matrix
5. THE CONVEX HULL OF THE HIGHEST WEIGHT ORBIT 81
Md =
1 s1 +√−1t1 · · · sd−1 +
√−1td−1 sd +
√−1td
s1 −√−1t1 1 · · · sd−2 +
√−1td−2 sd−1 +
√−1td−1
...
sd−1 −√−1td−1 sd−2 −
√−1td−2 · · · 1 s1 +
√−1t1
sd +−√−1td sd−1 −
√−1td−1 · · · s1 −
√−1t1 1
is positive semidefinite.
Before beginning the proof, let us say a few words about the inner product
structure on our spaces. The first space we are working with is C2d+1. We equip this
space with the basis e−d, . . . , e0, . . . , ed, where ei is the column vector in C2d+1 with
0’s everywhere except the ith position. It will be made clear in the proof why we use
this notation. We equip C2d+1 with the standard inner product given by
〈x, y〉 =d∑
k=−d
xkyk.
Let
e−d, . . . , e0, . . . , ed (5.2.2)
be the dual basis with respect to this inner product. Then, of course, ei = (ei)t for
each i.
Secondly, we have the space Md+1(C) of (d + 1) × (d + 1) matrices with entries
in C. We give Md+1(C) the basis Eij, 0 ≤ i, j ≤ d, where Eij is the matrix with
1 in the (i, j)−th position and 0 everywhere else. This space is equipped with the
inner product 〈A,B〉 = tr(AB∗) where B∗ is the conjugate transpose of B. For a
matrix X ∈Md+1(C), let X ∈Md+1(C)∗ be the linear functional defined by X(A) =
〈A,X〉. From linear algebra, we know that Eij behaves in the following way. For
5. THE CONVEX HULL OF THE HIGHEST WEIGHT ORBIT 82
X ∈Md+1(C),
Eij(X) = xij
where xij is the (i, j)-th entry of X.
We are now ready to give the proof of Theorem 5.2.12.
Proof: Recall from Definition 5.2.9 that
Cd =
{(δ, a1, b1, . . . , ad, bd) : δ +
d∑k=1
(ak cos(kθ) + bk sin(kθ)) ≥ 0
}.
We identify each point (δ, a1, b1, . . . , ad, bd) ∈ R2d+1 with the Laurent polynomial
R(z) =d∑
k=−d
ukzk ∈ C[z, z−1]
where u0 = δ, uk = 12(ak −
√−1bk) and u−k = uk for 1 ≤ k ≤ d. Note that R ∈ Cd
if and only if R is nonnegative on the unit circle T ⊆ C. By Proposition 5.2.11, we
have a factorization
R(z) = H(z−1) ·H(z).
Now let γd : C → Cd+1 be defined by γd(z) = (1, z, . . . , zd)T . Thus, there is a vector
h ∈ Cd+1 such that
R(z) = γd(z−1)T · hhT · γd(z).
Now recall that a point (c1, s1, . . . , cd, sd) ∈ R2d belongs to Cd if and only if
δ +d∑
k=1
akck + bksk ≥ 0 for all (δ, a1, b1, . . . , ad, bd) ∈ Cd.
Let ζ = (x, 1, y), with xk = sk +√−1tk and yk = sk −
√−1tk. Now there is a linear
map π : Md+1(C)→ C2d+1 such that u = π(hhT ), where u = (u−d, . . . , u−1, δ, u1, . . . , ud).
5. THE CONVEX HULL OF THE HIGHEST WEIGHT ORBIT 83
Indeed, we see that R(z) =∑∑
hkh`z`−k and
hhT =
|h0|2 h0h1 · · · h0hd
h1h0 |h1|2 · · · h1hd...
......
hdh0 hdh1 · · · |hd|2
.
Thus, π is given by
π(A)j =∑`−k=j
ak,`
where −d ≤ k ≤ d. Next, in the standard basis of (C2d+1)∗ we write
ζ =−1∑i=−d
x−iei + e0 +
d∑i=1
yiei.
By the above definition of π, one then sees that for −d ≤ i ≤ −1, π∗(ei) is a
matrix with 1’s on the i-th super-diagonal, and for 0 ≤ i ≤ d, π∗(ei) is a matrix with
1’s on the i-th sub-diagonal. It then follows that Md = π∗(ζ). Thus we have
δ +d∑
k=1
akck + bksk = 〈ζ, π(hhT )〉
= 〈π∗(ζ), hhT 〉
= tr(Md · hhT )
= hT ·Md · h
So (s1, t1, . . . , sd, td) ∈ Cd if and only if hT ·Md · h ≥ 0 for all h ∈ Cd+1, i.e. if
and only if Md is positive semidefinite.
The following proposition is from [13, p. 566].
5. THE CONVEX HULL OF THE HIGHEST WEIGHT ORBIT 84
Proposition 5.2.13. (Sylvester’s Criterion) A Hermitian matrix M is positive semidef-
inite if and only if all of its principal minors are non-negative.
Remark 5.2.14. Note that Sylvester’s criterion gives us polynomial inequalities in
the variables s1, . . . , sd, t1, . . . , td.
Theorem 5.2.15. Let d ≥ 2 be even and consider the representation (πd, Vd) of
SL2(R), realized on the space of homogeneous polynomials of degree d in x and y
with coefficients in R. Let v = yd be the highest weight vector corresponding to the
standard Borel subgroup of SL2(R). In addition, let Ψ d2
be the affine isomorphism of
Definition 5.2.8. Then we have
conv(Xλ) = {x ∈ Rd : x = αy for some α ∈ R+, y ∈ Ψ−1d2
(C d2)}.
Proof: We have
conv(Xλ) = conv(SL2(R) · yd) = R+ conv(SO2(R) · yd)
where the last equality follows from Lemma 5.1.11. The result then follows from the
definition of Ψ d2.
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