The Convex Hull of the Highest Weight Orbit and the Carath … · 2018-03-05 · 3.1.1...

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.


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


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]]


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.


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.


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


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


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


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


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

.


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


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.


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


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).


3.2.1 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


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


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.


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


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.


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.


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


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


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


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


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


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.


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


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


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


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.


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.


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 )


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)


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}.


−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]


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}.


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.


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 =


(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


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.


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).


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.


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.


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.


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).


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.


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


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


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.


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}.


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


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


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


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).


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


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


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).


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

.


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


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).


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)}.


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.


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.


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


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


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).


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].


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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