Quaternionic Harmonic Analysisand Classical Special Functions

PhD thesis

Grégory Mendousse

Abridged and corrected version of the author’s PhD thesisAnalyse Harmonique Quaternionique et Fonctions SpécialesClassiques (Université de Reims Champagne-Ardenne,https://hal.archives-ouvertes.fr/tel-01833951, 2017).Next page: front cover of original version.

Reims, 2019

UNIVERSITÉ DE REIMS CHAMPAGNE-ARDENNE

ÉCOLE DOCTORALE SCIENCES TECHNOLOGIE SANTE (547)

THÈSE

Pour obtenir le grade de

DOCTEUR DE L’UNIVERSITÉ DE REIMS CHAMPAGNE-ARDENNE

Discipline : MATHÉMATIQUES

Présentée et soutenue publiquement par

Grégory MENDOUSSE

Le 15 décembre 2017

Thèse dirigée par Michael PEVZNER

ANALYSE HARMONIQUE QUATERNIONIQUE ET FONCTIONS SPÉCIALES CLASSIQUES

JURY

Mme Angela PASQUALE Professeur Université de Lorraine, Metz Présidente

M. Michael PEVZNER Professeur Université de Reims Champagne-Ardenne Directeur de thèse

M. Toshiyuki KOBAYASHI Professeur Université de Tokyo, Japon Rapporteur

M. Tomasz PRZEBINDA Professeur Université d’Oklahoma, USA Rapporteur

M. Pierre CLARE Professeur Assistant Université William & Mary, USA Examinateur

M. Rupert YU Professeur Université de Reims Champagne-Ardenne Examinateur

To my wife, my children,my parents and my brothers

Contents

Preface ix

Notation xi

1 Introduction 1

2 Preliminaries: Lie theory and quaternions 152.1 Basic definitions and fundamental properties . . . . . . 152.2 Symplectic groups . . . . . . . . . . . . . . . . . . . . . 202.3 Lie theory applied to Sp(n) and sp(n) . . . . . . . . . . 22

2.3.1 Roots . . . . . . . . . . . . . . . . . . . . . . . . 222.3.2 Highest weights . . . . . . . . . . . . . . . . . . . 252.3.3 Casimir operator . . . . . . . . . . . . . . . . . . 26

2.4 Quaternions . . . . . . . . . . . . . . . . . . . . . . . . . 312.4.1 One way to define quaternions . . . . . . . . . . 312.4.2 Quaternionic linear algebra . . . . . . . . . . . . 342.4.3 Complex and quaternionic coordinates: identifi-

cations . . . . . . . . . . . . . . . . . . . . . . . . 352.4.4 Symplectic matrices seen as quaternionic matrices 38

3 Degenerate principal series of Sp(n,C) 413.1 A specific parabolic subgroup . . . . . . . . . . . . . . . 413.2 Specific induced representations . . . . . . . . . . . . . . 45

3.2.1 Usual definitions . . . . . . . . . . . . . . . . . . 453.2.2 Changing the carrying spaces . . . . . . . . . . . 47

vii

4 Actions of Sp(n) and Sp(1) 554.1 Left action of Sp(n) . . . . . . . . . . . . . . . . . . . . 55

4.1.1 Preliminaries . . . . . . . . . . . . . . . . . . . . 554.1.2 Highest weight vectors . . . . . . . . . . . . . . . 604.1.3 Isotypic decomposition with respect to Sp(n) . . 63

4.2 Right action of Sp(1) . . . . . . . . . . . . . . . . . . . . 684.2.1 Isotypic decomposition with respect to Sp(1) . . 684.2.2 K-type diagram . . . . . . . . . . . . . . . . . . 724.2.3 Bi-invariant polynomials . . . . . . . . . . . . . . 75

5 Compact picture and hypergeometric equation 815.1 Quaternionic projective space . . . . . . . . . . . . . . . 815.2 Reducing the number of variables . . . . . . . . . . . . . 855.3 Hypergeometric equation . . . . . . . . . . . . . . . . . 87

6 Non-standard picture and Bessel functions 956.1 Non-standard picture . . . . . . . . . . . . . . . . . . . . 956.2 Aim of this chapter: specific highest weight vectors . . . 976.3 Bessel functions and useful formulas . . . . . . . . . . . 986.4 First transform . . . . . . . . . . . . . . . . . . . . . . . 1036.5 Second transform . . . . . . . . . . . . . . . . . . . . . . 1056.6 Statement of main result . . . . . . . . . . . . . . . . . . 1096.7 Two interesting properties . . . . . . . . . . . . . . . . . 111

6.7.1 A differential operator that connects certain K-types . . . . . . . . . . . . . . . . . . . . . . . . . 111

6.7.2 Underlying Emden-Fowler equations . . . . . . . 115

Bibliography 117

viii

Preface

This work is an abridged and corrected version of my PhD the-sis "Analyse Harmonique Quaternionique et Fonctions Spéciales Clas-siques" (Université de Reims Champagne-Ardenne, 2017). The partsthat I have taken out of the original version are just those parts thatwere written in French so as to comply with regulations in French uni-versities (translated introduction, summaries of all chapters and con-clusion). Many corrections have been made, but they are all minorcorrections and they have no effect on results.

I am deeply endebted to my PhD supervisor Pr. Michael Pevzner(Reims) for his guidance, kindness and enthusiasm. One of the manygood memories I’ll keep of my PhD years is the time we spent in Moscowat the Interdisciplinary Scientific Center J.-V. Poncelet to discuss fu-ture works whilst I was still finishing my dissertation.

As a PhD student, I had the opportunity to travel and meet peoplewho, each in their own way, would give me some insight on Lie theory.In particular, I met Pr. Toshiyuki Kobayashi (Tokyo) and Pr. PierreClare (William & Mary, USA) on several occasions and I am extremelygrateful to them for many enlightening and helpful discussions.

Finally, I have written a condensed and more "straight to the point"presentation of the main results of this work in an article entitled "Spe-cial Functions Associated with K-types of Degenerate Principal Seriesof Sp(n,C)", which is due to appear in the Kyoto Journal of Mathe-matics.

Grégory Mendousse

Reims, November 2019

ix

x

Notation

The choices made here apply to all integers n ≥ 1.

The standard Euclidean inner product of Rn is denoted by a simpledot and defined for x = (x1, . . . , xn) ∈ Rn and y = (y1, . . . , yn) ∈ Rnby the usual sum:

x · y =n∑i=1

xiyi .

Given any two vectors z = (z1, . . . , zn) and w = (w1, . . . , wn) of Cn, weset:

〈z, w〉 =n∑i=1

ziwi .

We point out that this does not correspond to the standard Hermitianinner product of Cn.

We identify real, complex and quaternionic coordinates (in Chapter 2,we explain all we need to know about quaternions and quaternioniclinear algebra). The identifications work as follows:

• a vector (x, y) ∈ Rn×Rn ' R2n will correspond to the vectorz = x+ iy ∈ Cn;

• a vector (z, w) ∈ Cn×Cn ' C2n will correspond to the vectorh = z + jw ∈ Hn.

Denote by K the field R, C or H and consider any integers r ≥ 1 ands ≥ 1. Then:

xi

• We write K× = K \ 0.

• Mr,s(K) denotes the K-vector space of r × s matrices (meaningmatrices with r rows and s columns) whose coefficients all belongto K.

• Given g ∈ Mr,s(K), tg ∈ Ms,r(K) denotes the transpose of g.

• M(n,K) denotes the K-vector space of n×n matrices whose coef-ficients all belong to K.

• GL(n,K) denotes the group of n×n invertible elements of M(n,K).

We always identify matrices with the linear maps they correspond towith respect to the canonical basis of the underlying vector space.

All zero matrices are simply denoted by 0. The n× n identity matrixis denoted by In.

Whatever kind of matrices we are working with, Er,s refers to the ma-trix whose terms are all 0 except for one term which is equal to 1 andsits in Row r and Column s; such matrices are called elementary ma-trices.

We denote by dσ the standard Euclidean measure of the unit sphereSn−1.

Because R4n ' C2n ' Hn, the unit sphere S4n−1 of R4n will also bedenoted by S4n−1 when it is seen as a subset of C2n or Hn.

Given a measurable space X:

• L1(X) denotes the complex Banach space of equivalence classesof integrable complex-valued functions on X.

• L2(X) denotes the complex Hilbert space of equivalence classesof square-integrable complex-valued functions on X.

xii

Given a topological space X, C0(X) denotes the complex vector spaceof continuous complex-valued functions on X.

We now fix once and for all an integer n ≥ 2 and we set:

m = n− 1 ; N = 2n.

As for the bibliography, the references in no way mean to cover thehuge literature connected with this work; they only appear if explicitlymentioned at some point.

xiii

xiv

Chapter 1

Introduction

In broad terms, the point of representation theory is to give tangiblepictures of abstract groups by identifying their elements with operatorsthat one can visualise. This procedure helps understand the structureof the given groups, but, more importantly perhaps, uses the knowledgewe have about these groups to solve important and difficult problems.For instance, one might look for solutions of a certain differential equa-tion; these solutions might form a vector space on which some relevantgroup acts by linear operators, thereby defining a representation; onemight then try to break the vector space into explicit irreducible in-variant subspaces in order to write down the desired solutions as com-binations of elements of these subspaces.

These ideas appear in Fourier’s works. In his famous book [14], Fourierstudies the heat equation and decomposes its solutions into sums oftrigonometric functions. Of course, to refer to a single person whendiscussing pioneering work is unfair, in the sense that ideas always de-pend on existing knowledge (for example, Euler had already workedon trigonometric series). To summarise two centuries of mathemati-cal research in a few pages is obviously impossible, so many importantnames and contributions will unjustly be left out of this introduction.But our intention here is merely to outline a few important stages thathave led to representation theory as we know it today, in order to seehow our own work fits in the global picture of this theory. Though

1

we have had to choose which facts and people we mention as part ofthis exciting mathematical story, we have nonetheless tried to be asaccurate as possible. The first pages of this introduction are based onMackey’s historical survey [41] and also on [24].

The ideas of Fourier were developed by Cauchy, Poisson, Dirichlet, Rie-mann, Cayley and others, leading for instance to spherical harmonicsin arbitrary dimensions and more applications to physics. In the late19th century, group theory, which had appeared in the works of Galoisand Abel, was placed at the heart of analysis and geometry. Kleinexposed this point of view in a famous lecture given at the Universityof Erlangen (see [26]). Lie studied the actions of continuous groups.Frobenius decided to concentrate on representations of finite groups,showing that they are unitarisable and completely reducible, studyingthe connections between characters and functions on the group, lookingat the regular representation and so forth. Weyl studied representa-tions of compact groups, obtaining similar results to those of Frobenius,but using more sophisticated notions, in particular the theory of inte-gration made available by Borel, Lebesgue and Haar. With his studentPeter, he proved in [47] the famous Peter-Weyl theorem, which is ageneralisation of the Plancherel formula for Fourier series.

The next step involved Lie algebras. Lie had studied continuous trans-formation groups from an infinitesimal point of view. Infinitesimal ge-nerators of one-parameter subgroups gave rise to what would be calledthe Lie algebra of the given group. Lie algebras were then studied intheir own right by Killing and E. Cartan, whose works led to the com-plete classification of finite-dimensional simple Lie algebras. E. Cartanstudied representations of Lie algebras and classified the irreduciblefinite-dimensional ones in terms of highest weights. This gave newtools to Weyl to study representations of compact groups in furtherdetail, obtaining for instance the formulas known as the Weyl charac-ter formula and the Weyl dimension formula.

Many scientists contributed to quantum physics in the early 20th cen-tury (Schrödinger, Planck, Heisenberg, Von Neumann and so on). Their

2

works built up the setting we know today: a physical system cor-responds to a certain Hilbert space, whose one-dimensional subspacesdefine the (pure) states of the system; observables correspond to self-adjoint operators, which define probability distributions. The role ofHilbert spaces here justified the general interest in unitary represen-tations: Weyl used the exponential map to assign to a self-adjointoperator a family of unitary operators, thereby defining a unitary re-presentation of the additive group of the real line; diagonalising theself-adjoint operator meant decomposing the unitary representation.

The study of unitary representations of arbitrary groups (includingnon-abelian and non-compact groups) began in the forties, with theworks of Gel’fand, amongst others. The general idea arose that thereexisted a correspondance between groups and the set of their unitaryrepresentations (see for instance [16]) and that one could study ac-tions of groups on topological spaces X in terms of representationsof these groups on spaces of functions on X. Chevalley published anaccount of Lie theory (see [7]), extending the theory and introducingnew ideas (for instance the notion of analytic subgroup). In the fifties,Harish-Chandra set to study the irreducible representations of generalsemisimple Lie groups on Banach spaces, while Mackey developed thetheory of induced representations (see [38], [39] and [40]). From thecontributions over the years of numerous mathematicians and physi-cists, two guidelines emerged:

• the orbit method of A. A. Kirillov (see [25]), which associatesunitary irreducible representations of a large class of groups withorbits of their co-adjoint actions;

• the Langlands classification theorem, which says that, given aminimal parabolic subgroup of a linear connected reductive groupG, there is a bijection between equivalence classes of irreducibleadmissible representations of G and certain triples; those triplesare called the Langlands parameters (see [27], Section 15 of Chap-ter VIII); our work comes under this approach.

A great many general results are well known, such as:

3

• Irreducible unitary representations of compact topological groupsare finite-dimensional; this result is part of the Peter-Weyl theo-rem (see [27], Chapter I).

• Given a compact linear connected reductive group, the equiva-lence classes of its irreducible representations are parametrised,as Cartan showed, by specific linear forms (the highest weights;see [27], Chapter IV).

• Non-trivial irreducible unitary representations of non-compactlinear semisimple groups are infinite-dimensional (see [23], Sec-tion 11.1).

• An irreducible unitary representation of a linear connected reduc-tive group, when restricted to a certain maximal compact sub-group, decomposes into a Hilbert sum of subrepresentations, eachof which consists of a finite direct sum of equivalent irreduciblerepresentations which are finite-dimensional (this result is due tothe Peter-Weyl theorem and to the works of Harish-Chandra -see [27], Section 2 of Chapter VIII); in other words, irreducibilityplus unitarity imply admissibility.

• Harish-Chandra introduced the notion of (g,K)-module (see [52],Chapter 2) and obtained a remarkable theorem, of which wegive a stronger version (the subrepresentation theorem) provedby Casselman and Miličić in [5] (Theorem 8.21): consider the Liealgebra g of a connected semisimple Lie group whose center is fi-nite (the class of groups this theorem applies to is in fact larger);consider a maximal compact subgroupK and a minimal parabolicsubgroup P ; then an irreducible admissible (g,K)-module is al-ways embedded into a representation which is induced from someirreducible finite-dimensional representation of P .

In fact, so much is known that one could almost feel that representationtheory is near to complete. Which of course is not true. For onething, understanding an object requires assumptions; changing theseassumptions changes the theory; one could decide to work with otherspaces than Hilbert spaces, with fields of positive characteristic and

4

so forth (we will not go in these directions). For another thing, thereare many aspects of representations one can wish to investigate. Forinstance:

• Give tangible examples of representations: the Stone von Neu-mann theorem (in the thirties) classifies unitary representationsof the Heisenberg group, those of SL(2,R) were studied by Gel’-fand, Naımark and Bargmann in the forties (see [1] an [15]), ac-tions of symplectic groups were studied in the sixties and led tothe metaplectic representation (one can find an account of this in[13], along with representations of the Heisenberg group); moreexamples will be discussed further on. The point of examplesis of course to justify one’s interest in a theory but also to givesome insight on various areas it is connected to; it so happensthat representation theory is connected to many.

• Study the way representations break into irreducibles and whatrepresentations become when restricted to various subgroups.Along those lines, one has Kostant’s general branching theorem,Howe’s study of certain representations of O(p, q), U(p, q) andSp(p, q), works of T. Kobayashi and Pevzner in terms of differen-tial operators (see [36]) and so forth.

• Amongst the representations given by the Langlands classifica-tion, determine the ones that are unitary; many people haveworked in this direction (Zuckermann, Adams, Vogan...).

• Study connections that exist between representations and otherobjects. The present work investigates the appearance of specialfunctions in representation theory. The fact that such functionsdo appear is hardly a discovery. They can emerge from the com-putation of matrix coefficients: for example, the role of Besselfunctions is shown in Chapter 4 of [50] (Section 4.1) for some ac-tion of ISO(2) on smooth functions on the circle. They can appearas spherical functions of Riemannian symmetric spaces. Also, onan infinitesimal level, enveloping algebras correspond to differen-tial operators that lead to differential equations which sometimescharacterise well known special functions: for instance, we use

5

K-types and the Casimir operator to obtain a hypergeometricequation. Special functions also play a key role in the construc-tion of symmetry breaking operators (see [29] and [36]).

The interesting thing about special functions is that they are solutionsof particular differential equations, that they can be expressed in va-rious ways (as series, integrals and through recursive methods) and thatthey sometimes define orthonormal bases of functional Hilbert spaces.

Bearing in mind the role of special functions in representation theory,here are a number of considerations that guided us towards the settingwe have chosen for our work.

First, the subrepresentation theorem justified, to us, the choice of in-duced representations.

Then, to study the appearance of explicit objects, it seems appropriateto select concrete groups: the notion of reductive group is a gener-alisation of a number of matrix groups that are preserved by someinvolution; why not choose straight away one of those groups expli-citly? Which brings us now to the actual choice of matrix group.

The unitary duals of GL(n,R) and GL(n,C) have been studied in greatdetail. For instance, it is shown in [49] how to use unitary characters ofR× (resp. C×) and, in the real case, certain irreducible representationsof GL(2,R), to construct all irreducible unitarisable representations ofGL(n,R) (resp. GL(n,C)) via tensor products and parabolic induction.Looking at subgroups of GL(n,R) and GL(n,C), the classical ones aredefined with respect to various non-degenerate bilinear or sesquilinearforms they preserve. Representations of O(n), U(n) and Sp(n) are tho-roughly understood, in terms of highest weights, because these groupsare compact (see for instance [53] and more specifically [44] for Sp(n));we shall come back to them in a moment. Representations of O(p, q)and U(p, q) have been studied extensively. Submodules of some repre-sentations of these groups on spaces of homogeneous functions are de-scribed in [22]. In a series of papers ([32], [33] and [34]), Kobayashi andØrsted study the minimal unitary representation of O(p, q), describing

6

the minimal K-types in terms of special functions. One can also looktowards symplectic groups, which preserve alternating non-degeneratebilinear forms – see for example [22], [35] and [8]. These works arebased on the K-type structure, which comes from the admissibilityassumption of the given representations. Therefore, not only are theirreducible representations of O(n), U(n) and Sp(n) well understood,they are essential ingredients in the general study of representationsof reductive groups. Added to which it is well known that they arerelated to special functions via spherical harmonics.

All these considerations have drawn us to study K-types of irreducibleunitary parabolically-induced representations of classical linear groups.But why focus on Sp(n,C) ?

In [35] and [8], the authors study degenerate principal series of Sp(n,R)and Sp(n,C) that are geometrically realised on spaces of functions de-fined on R2n \0 and C2n \0. Restricting those functions to the unitsphere connects them to spherical harmonics. The cases of Sp(n,R) andSp(n,C) are different but share a same feature: scalar multiplicationby numbers of modulus 1 guides us to irreducibles. To be more specific,any invariant subspace under the action of a subgroup that contains K(K denotes a maximal compact subgroup) is a step forward towardsa K-type decomposition. The direct sums below illustrate this. Inthese sums, the parameters correspond to the characters of O(1) (resp.U(1)) and the summands are invariant subspaces under the left actionof O(4n) (resp. U(2n)):

• L2(S4n−1) = L2even(S4n−1)⊕ L2

odd(S4n−1);

• L2(S4n−1) =⊕δ∈Z

L2δ(S4n−1), where each space L2

δ(S4n−1) is the

Hilbert sum of all subspaces Yα,β of spherical harmonics of ho-mogeneous degrees (α, β) such that δ = β − α.

So two actions seem to rule the decomposition of L2(S4n−1) we arelooking for: left action of matrices and scalar multiplication by numbers

7

of modulus 1. This diagram captures the situation:

O(4n) y L2(S4n−1) x O(1)⋃ ⋂U(2n) y L2(S4n−1) x U(1).

One instinctively wants to add a line with:

• the group of linear isometries of Hn, which is isomorphic to Sp(n)and thus embedded in U(2n);

• the group of unit quaternions, which is isomorphic to the groupSp(1) and contains an embedding of U(1).

In other words, the line would be:

Sp(n) y L2(S4n−1) x Sp(1).

This suggests an interesting interaction between the left action of Sp(n)and a scalar kind of action of Sp(1). It must be pointed out that thisscalar action is not as simple as one might expect: non-commutativityof the field of quaternions requires a choice of sides for scalar multi-plication: we choose right multiplication. This multiplication in thequaternionic setting must then be translated back into the complexsetting. We call this scalar action the right action of Sp(1).

Putting all these considerations together, the central question of thiswork is:

Via the left action of Sp(n) and the right action of Sp(1), how dospecial functions connect to K-types of degenerate principal seriesof Sp(n,C)?

Remark: what we are trying to do is generalise spherical harmonics bychanging fields, choosing the non-commutative field H instead of R orC.

8

Let us point out that principal series of classical groups over the fieldof quaternions are discussed in [9], [18] and [46].

Generally speaking, works such as [20], [21], [30], [31] and [35] serveas guidelines to obtain special functions: we make use of differentialoperators and Fourier transforms.

Throughout this work, we consider an integer n ≥ 2: the minimalvalue 2 comes from the fact that we wish to work with a non-minimalparabolic subgroup.

Chapter 2 introduces the background of this work: standard definitionsand properties of Lie groups, Lie algebras and their representations,quaternions, quaternionic linear algebra and various computations thatare required further on.

Chapter 3 introduces the actual degenerate principal series we are in-terested in. This series is obtained by parabolic induction which wemake explicit: choice of parabolic subgroup (its nilradical is isomorphicto the complex Heisenberg group), choice of character and geometricrealisations. We obtain a two parameter family of representations ofG = Sp(n,C), denoted by πiλ,δ with (λ, δ) ∈ R×Z and defined asfollows (denoting by Viλ,δ the carrying space of πiλ,δ):

• Consider the space V 0iλ,δ of all functions f ∈ C0(CN \0) that are

covariant, by which we mean that for all c ∈ C× and x ∈ CN \0:

f(cx) =(c

|c|

)−δ|c|−iλ−Nf(x).

This equality will be refered to as the covariance property. Con-sider the left action of G on this space, defined by

πiλ,δ(g)f(x) = f(g−1x)

for all (g, f, x) ∈ G× V 0iλ,δ × (CN \0).

9

• Viλ,δ and πiλ,δ are then obtained by completion of V 0iλ,δ with re-

spect to the norm ‖ · ‖ defined by:

‖f‖2 =∫S4n−1

|f(x)|2dσ(x).

Representations πiλ,δ are unitary. They are also irreducible if and onlyif (λ, δ) 6= (0, 0).

The above description of πiλ,δ is called the induced picture. By chang-ing the carrying space in a suitable way, one obtains other descriptions.In particular, the covariance property enables one to change the carry-ing space Viλ,δ into some subspace of L2(S4n−1); this gives the so-calledcompact picture of πiλ,δ (see Chapter 3) and leads us to study the wholespace L2(S4n−1).

In Chapter 4, we make the structure of L2(S4n−1) explicit with respectto both the left action of Sp(n) and the right action of Sp(1). Althoughthis double K-type structure appears in [22], the credit one can give tothe present work is to offer a personal treatment of the subject, totallyexplicit and self-contained. Moreover, we highlight polynomials whichare invariant under both actions and we show how to compute them.They are the ingredient that takes us to hypergeometric equations. Themain results of Chapter 4 are gathered in Theorems A and B.

Theorem A.

• With respect to the left action of Sp(n), the space L2(S4n−1) de-composes into the following Hilbert sum:

L2(S4n−1) =⊕k∈N

⊕(α, β) ∈ N×Nα + β = k

min(α,β)⊕γ=0

V α,βγ .

In this sum, V α,βγ is the irreducible invariant subspace generated

by the left translates of the restriction to S4n−1 of the polynomialPα,βγ which is defined by:

Pα,βγ (z, w) = wα−γ1 z1β−γ(w2z1 − w1z2)γ .

10

Here, (z, w) denote the coordinates on Cn×Cn ' C2n, takingz = (z1, . . . , zn) and w = (w1, . . . , wn).

• With respect to the right action of Sp(1), the space L2(S4n−1)decomposes into the Hilbert sum

L2(S4n−1) =⊕k∈N

E( k2 )⊕γ=0

dkγWkγ ,

where E(k2

)denotes the integer part of k

2 . In this sum, W kγ

is a finite-dimensional irreducible invariant subspace which con-tains the restriction P k−γ,γγ

∣∣S4n−1 as a highest weight vector; dkγ

is a positive integer and there are dkγ invariant subspaces that areequivalent to W k

γ .

Theorem B. Consider an even integer k ∈ N and set α = k2 . Denote

by Hk(R4n) the space of harmonic polynomials of 4n real variables withcomplex coefficients and which are homogeneous of degree k. Denoteby 1× Sp(n− 1) the group of matrices of Sp(n) that can be written, in

the quaternionic setting, as(

1 00 A

)with A ∈ Sp(n− 1). Then there

exists a unique (up to a constant) element of Hk(R4n) (we see thiselement as a polynomial of 2n complex variables and their conjugates)which is invariant under the left action of 1×Sp(n−1) and also underthe right action of Sp(1); such a polynomial is said to be bi-invariant(and we show how to compute it). The restriction of this polynomialto S4n−1 (also said to be bi-invariant) belongs to V α,α

α .

As mentioned earlier on, our interest lies in the link one can estab-lish between representations and special functions. What makes thechoice of unitary groups all the more appropriate for us is that it iswell known how to associate special functions to irreducible spacesof spherical harmonics by adding an additional invariance constraint(defining zonal functions) and using the Laplace operator (see Chap-ter 9 of [12]). To use similar methods with symplectic groups requiresa slight adjustment of the additional constraint; this is precisely thepoint of the right action of Sp(1). The special functions we end up

11

with, as for the orthogonal groups, are solutions of a hypergeometricequation, but this only works for specific K-types. These methods aredeveloped in Chapter 5; here is a partial statement of the main result:

Theorem C (Compact picture and hypergeometric equation). Con-sider any integer α ∈ N. Then the unique (up to a constant) bi-invariant function of V α,α

α (given by Theorem B) can, after a suitablereduction of variables, be written as a solution of the following hyper-geometric equation:

u(1− u)ϕ′′ + 2(1− nu)ϕ′ + (α2 + (2n− 1)α)ϕ = 0.

Another common description of parabolically induced representationsis the so-called non-compact picture (see Chapter 3). A way to ob-tain special functions can then be to apply Fourier transforms to thispicture (this leads to the so-called non-standard picture), simply be-cause special functions can be expressed in many ways, some of whichinvolve integrals. In [35], the authors consider the group Sp(n,R) andan analogue of our degenerate principal series πiλ,δ; a specific partialFourier transform enables them to use Bessel functions to write downinteresting formulas for elements of minimal K-types (when the pa-rameter λ is equal to 0).

In Chapter 6, we generalise this, not only by adapting it to a complexsetting, but also by finding explicit formulas of highest weight vectorsin terms of modified Bessel functions for all pairs (λ, δ) and for a muchwider set of K-types. Before we actually give the formulas, let us defineon L2(C2m+1) the partial Fourier transform F on which is based thenon-standard picture. It is defined for f ∈ L1(C2m+1) ∩ L2(C2m+1) by

F(f)(s, u, v) =∫C×Cm

f(τ, u, ξ) e−2iπRe(sτ+〈v,ξ〉) dτdξ,

where (s, u, v) denote the coordinates of C×Cm×Cm ' C2m+1. Wecan now state our main theorem, namely Theorem 6.12, but in a slightlydifferent and lighter way:

Theorem D (Non-standard picture and Bessel functions). Let n ∈ Nbe such that n ≥ 2 ; set N = 2n and m = n − 1. Let k ∈ N and

12

(α, β) ∈ N2 be such that α+β = k ; set δ = β−α. For λ ∈ R, considerthe representation πiλ,δ of Sp(n,C) and the function

f : Cn×Cn \ (0, 0) −→ C(z, w) 7−→

(‖z‖2 + ‖w‖2

)− iλ+k2 −n

wα1 z1β,

where z1 (resp. w1) denotes the first coordinate of z (resp. w). Then fgenerates a finite-dimensional subspace under the action of πiλ,δ

∣∣Sp(n);

f is a highest weight vector of this subspace and the non-standard formof f , meaning the function F

((τ, u, ξ) 7−→ f(1, u, 2τ, ξ)

), assigns to all

triples (s, u, v) ∈ C×Cm×Cm such that v 6= 0 and s 6= 0 the value

R(s, u, v) K iλ+δ2

(π√

1 + ‖u‖2√|s|2 + 4‖v‖2

),

where we set

R(s, u, v) = (−i s)α πiλ+β+n

2 iλ+k2 +1 Γ( iλ+k

2 + n)

(√|s|2 + 4‖v‖2π√

1 + ‖u‖2

) iλ+δ2

.

Chapter 6 ends with two interesting observations:

• there exists a simple differential operator that connects the non-standard forms of certain highest weight vectors to one another(Theorem 6.13);

• the formula given in Theorem D is linked to differential equationsknown as Emden-Fowler equations (Section 6.7.2).

13

14

Chapter 2

Preliminaries: Lie theoryand quaternions

2.1 Basic definitions and fundamental proper-ties

Given a set X, denote by Bij(X) the set of bijections from X onto X;it is a group with respect to composition. An action of a group G onX is a group homomorphism from G into Bij(X).

Given a vector space V , a representation of a group G on V is an ac-tion π of G on V such that all maps π(g) are linear isomorphisms of V ;one also says that (π, V ) is a representation of G. The space V is thecarrying space of π; the dimension of π is the dimension of V (possiblyinfinite). A subspace W of V is invariant (or stable) under the actionof π if π(g)W ⊆ W for all g ∈ G; in this case, the restriction π|W is asubrepresentation of π. The representation π is algebraically irreducible(or irreducible, for short) if it has no invariant subspaces other than 0and V itself.

Take K to be the field R, C or H and denote by V the vector spaceKn. Consider a subgroup G of GL(n,K). The natural action of G onV is simply defined by standard matrix multiplication: a matrix g ∈ Gassigns to an element x ∈ V the element gx ∈ V . Now consider a

15

subset S of V which is stable under the natural action of G and thenconsider the complex vector space F(S,C) = f : S −→ C. BecauseS is stable under the natural action of G, given f ∈ F(S,C) and g ∈ Gone can define a new element of F(S,C):

L(g)f : S −→ Cx 7−→ f

(g−1x

).

By varying f , we define a bijection:

L(g) : F(S,C) −→ F(S,C)f 7−→ L(g)f.

This induces another map:

L : G −→ Bij (F(S,C))g 7−→ L(g).

This map L is a representation of G on F(S,C); we will simply call itthe left action of G on F(S,C). We say that an element f of F(S,C)is left-invariant if f is invariant under the left action of G, meaningthat L(g)f = f for all g ∈ G.

Given a subspace F ′ of F(S,C) which is stable under the left action ofG, the map

G −→ Bij (F ′)g 7−→ L(g)

∣∣F ′

is of course also a representation; it will be referred to as the left actionof G on F ′ and, for simplicity, also denoted by L.

According to the kind of spaces one is interested in, some additionalstructure is necessary to be technically able to study group represen-tations on those spaces. Our setting will be that of Lie groups actingon complex Hilbert spaces.

A Lie group G is a group that has the structure of a smooth mani-fold such that multiplication and inversion are smooth. For instance,the group GL(n,K) and all its closed subgroups are Lie groups (see

16

[28], Theorem 0.15); they are in fact called linear Lie groups; whenconnected and stable under conjugate transpose, taking K to be R orC, they are called linear connected reductive groups and if their cen-ter is also finite they are linear connected semisimple groups. Being amanifold, G has tangent spaces at all of its elements. In particular,denote by g, or Lie(G), the tangent space at the identity element. Itcan be shown that g is a Lie algebra with respect to a bilinear skew-symmetric product, called the Lie bracket, which involves left-invariantvector fields and relates to G via the so-called exponential map. Wedo not enter the details here, because all we need to know in this workis that when G is a linear Lie group:

• The Lie bracket of its Lie algebra g is just the usual commutator[·, ·] of matrices, defined by [X,Y ] = XY − Y X; this in factdefines the adjoint representation ad of g: ad(X)Y = [X,Y ].

• The exponential map exp : g −→ G just assigns to each X ∈ gthe usual matrix expX = I +X + X2

2 + X3

3! + X4

4! + . . . .

The choice of Hilbert spaces is not a random choice: the scalar productenables one to use orthogonality to study invariant subspaces, fromthe corresponding norm arise continuity and differentiability, unitar-ity of operators has led to extensive results, quantum theory is basedon Hilbert spaces, finite-dimensional spaces can always be turned intoHilbert spaces and so on.

Consider a Lie group G. From now on, when considering a representa-tion π of G on a Hilbert space H, we automatically assume that π iscontinuous, by which we mean that π satisfies the following continuityproperty: the map

G×H −→ H(g, x) 7−→ π(g)x

is continuous. In particular, this implies that each linear isomorphismπ(g) is continuous; the bounded inverse theorem (which is a conse-quence of the open mapping theorem) then implies that the inversemap

(π(g)

)−1 = π(g−1) is also continuous.

17

Consider a Hilbert space H. Let us slightly adjust the notion of re-ducibility: π is said to be irreducible if there are no closed invariantsubspaces of H other than 0 and H. Also, π is said to be unitary ifall operators π(g) are unitary operators of H. If one considers anotherrepresentation π′ of G, on a Hilbert space H ′, then π and π′ are equiv-alent if there exists a bounded linear isomorphism f : H −→ H ′ suchthat for all g ∈ G the following diagram commutes:

Hπ(g)−−−−→ Hyf yf

H ′ −−−−→π′(g)

H ′.

The set of equivalence classes of all irreducible unitary representationsof G is called the unitary dual of G and denoted by G. We identify Gwith any set of representatives of its equivalence classes: an equivalenceclass denoted by τ ∈ G implicitely means that τ is a representation ofG that belongs to this equivalence class; conversely, to say that a rep-resentation τ of G is irreducible and unitary we just write τ ∈ G,denoting its equivalence class also by τ .

One says that a vector v ∈ H is a smooth vector (or a C∞-vector) ofa representation π if the map g −→ π(g)v is C∞ on G. We denoteby C∞(π) the set of such vectors. It can be shown that it is a densesubspace ofH (see [27], Theorem 3.15). One can define on this subspaceC∞(π) the differential dπ of π at the identity element by

dπ(X)v = d

dt

∣∣∣∣t=0

(π(exp tX)v

)for all (X, v) ∈ g × C∞(π). Elements X are thereby seen as first or-der differential operators. In our work, we will in fact not need toworry about C∞-vectors, because calculations will always be done forrestrictions of representations to invariant subspaces which are finite-dimensional and whose elements are therefore all C∞-vectors (this fol-lows from Theorem 3.15 of [27]).

18

We finish this section with perhaps the most important example, then ahistorical and founding theorem and finally a major consequence; theyall revolve around compactness.

This proposition follows [27] (Chapter I, Section 3, fourth example):

Proposition 2.1 (Important example). Consider a compact Lie groupK, a left-invariant measure µ of K, the Hilbert space L2(K,µ) and theleft action of K on L2(K,µ), defined (as we saw earlier on) by

L(k)f(x) = f(k−1x)

for all (k, f, x) ∈ K × L2(K,µ) ×K. Then L is a continuous unitaryrepresentation of K on L2(K,µ), called the left-regular representationof K.

Then comes a theorem that is also proved in [27] (Theorem 1.12):

Theorem 2.2 (Peter-Weyl Theorem). Let K be a compact Lie groupand π a unitary representation of K on a Hilbert space H. Then:

1. For each τ ∈ K, let Hτ denote the sum of all invariant subspacesof H that define irreducible subrepresentations that are equivalentto τ (if there are none, just set Hτ = 0). There exists a count-able number of such subspaces that add up to Hτ ; we will denoteby nτ ∈ N∪∞ the lowest of such numbers (if Hτ = 0, onejust sets nτ = 0).

2. All representations τ ∈ K are finite-dimensional.

3. Given τ ∈ K, we denote by Eτ the orthogonal projection on Hτ ;elements h ∈ H can be written h =

∑τ∈K

Eτ (h). If two represen-

tations τ and τ ′ of K are inequivalent, then:

EτEτ ′ = Eτ ′Eτ = 0.

Another way to state Theorem 2.2 is to say that π decomposes into asum over τ ∈ K of subrepresentations π|Hτ such that each π|Hτ decom-poses into nτ finite-dimensional irreducible unitary subrepresentations

19

that are equivalent to τ . This kind of decomposition is written:

π ∼=∑⊕

τ∈K

nττ.

Theorem 2.2 can also be used to study representations of groups thatare not compact, simply by restricting to compact subgroups. Undercertain assumptions, multiplicities are finite:

Theorem 2.3. Suppose G is a linear connected reductive group andlet K be a maximal compact subgroup of G. Consider an element π ofG. Apply Theorem 2.2 to the restriction of π to the subgroup K:

π|K ∼=∑⊕

τ∈K

nττ.

Then each nτ is finite. More accurately:

nτ ≤ dimτ <∞.

The sum in Theorem 2.3 is called the isotypic decomposition of π. Anelement τ of K for which nτ 6= 0 is called a K-type of π; the numbernτ is its multiplicity.

Remarks:

• This theorem is part of Harish-Chandra’s work. For a proof, onecan read Section 2 of Chapter VIII in [27] (Theorem 8.1).

• We point out that all maximal compact subgroups of G are con-jugate (see for instance [4], Chapter VII, Theorem 1.2).

2.2 Symplectic groupsFirst of all, it will be convenient to split matrices into blocks. A 2n×2ncomplex matrix g, that is, an element of M(2n,C), will be split intofour n× n complex matrices A,B,C,D:

g =(A CB D

).

20

In particular, we shall be interested in the matrix:

J =(

0 −InIn 0

).

We denote by ω the standard symplectic form on C2n, defined by:

ω(X,X ′) = tXJX ′ =⟨y, x′

⟩−⟨x, y′

⟩,

where X = (x, y) and X ′ = (x′, y′) belong to Cn×Cn.

A 2n× 2n complex matrix g is said to be symplectic iftgJg = J.

This is equivalent to saying that g preserves ω, meaning:

∀(X,X ′) ∈ C2n×C2n : ω(gX, gX ′) = ω(X,X ′).

The group of such matrices is the complex symplectic group (or justsymplectic group)

G = Sp(n,C).

It is straightforward to check that a 2n×2n complex matrix(A CB D

)is symplectic if and only if:

tAB = tBA, tCD = tDC and tAD − tBC = I. (2.1)

The group G is a closed subgroup of GL(n,C) and is therefore a Liegroup. It is in fact a linear connected semisimple group. Its Lie algebrais the complex symplectic algebra (or just symplectic algebra) sp(n,C),that is, the complex vector space

g = sp(n,C) =X ∈ M(2n,C) / tXJ + JX = 0

or, expressed otherwise:

g =X =

(A CB −tA

)∈ M(2n,C) / B and C are symmetric

.

21

Define the compact group K = Sp(n) = Sp(n,C) ∩ U(2n) (as usual,U(2n) denotes the group of unitary complex 2n×2n matrices). Unitar-ity and preservation of the standard symplectic form imply immediatelythat K consists of all matrices of U(2n) that can be written(

A −BB A

),

where of course M denotes the (complex) conjugate of a given matrixM . It is well known (see for instance Theorem 6.5.2 in [42]) that Kis simply connected (and therefore connected). It can be shown thatK is a maximal compact subgroup of G, via what is called the Cartandecomposition of g (which we say nothing of, just referring the readerto [27], Section 1 of Chapter I). Being a closed subgroup of GL(n,C),K is itself a Lie group; denote by k = sp(n) its Lie algebra (it is a Liesubalgebra of g). This Lie algebra is the real vector space that consistsof all skew-hermitian (skew, for short) elements of g:

k =X =

(A −BB A

)∈ M(2n,C) / A is skew and B is symmetric

.

Because K is connected, exp k generates K (see [28], Corollary 0.20).

2.3 Lie theory applied to Sp(n) and sp(n)Let us point out that the complexification of k is g and that g is acomplex semisimple Lie algebra (for a definition of semisimplicity, werefer the reader to [48], Sections 1.6 and 1.10).

In Sections 2.3.1 and 2.3.2 we summarise some well known facts, ap-plying the general theory of complex semisimple Lie algebras to theparticular case of g. We follow [27] (Chapter IV) and [48] (Chapter 2),using the explicit setting of k = sp(n) and g = sp(n,C).

2.3.1 Roots

Let h be the usual Cartan subalgebra of g consisting of diagonal ele-ments of g. If r belongs to 1, ..., n, denote by Lr the linear form that

22

assigns to an element of h its rth diagonal term. Denote by ∆K theset of roots of g with respect to h; if ν is a root, the corresponding rootspace is (by definition of a root, this space is not reduced to 0):

gν = X ∈ g/∀H ∈ h : ad(H)(X) = ν(H)X .

The set ∆K consists of the following linear forms, where r and s denoteintegers that belong to 1, ..., n:

• Lr − Ls with r < s;

• −Lr + Ls with r < s;

• Lr + Ls with r < s;

• −Lr − Ls with r < s;

• 2Lr;

• −2Lr.

Because g is semisimple, the root spaces are complex one-dimensionalsubspaces. One can check that the root spaces of the above roots arerespectively generated by the following root elements:

• U+r,s = Er,s − En+s,n+r (with r < s);

• U−r,s = Es,r − En+r,n+s (with r < s);

• V +r,s = Er,n+s + Es,n+r (with r < s);

• V −r,s = En+s,r + En+r,s (with r < s);

• D+r = Er,n+r;

• D−r = En+r,r.

As settled in the notation section, each Er,s refers to an elementarymatrix. We will write

Hr = Er,r − En+r,n+r

23

for each integer r ∈ 1, ..., n. Let us point out that

Hrr∈1,...,n

is a C-basis of h and that

Hr, D+r , D

−r r∈1,...,n ∪ U+

r,s, U−r,s, V

+r,s, V

−r,sr,s∈1,...,n,r<s (2.2)

is a C-basis of g.

Denote by hR the real form of h that consists of real matrices of h.Denote by h′R the space of real-valued linear forms defined on hR. Anyelement σ of h′R can be written σ = ∑n

r=1 lrLr, where each lr is a realnumber; we then identify σ with its coordinates (l1, l2, · · · , ln). Thespace h′R is endowed with a natural Euclidean product < ·, · > definedby:

< σ, σ′ >=n∑r=1

lrl′r.

Remark: the inner product used in [27] (Chapter IV, Section 2) corre-sponds to 1

2 < ·, · >. The fact that we have omitted the coefficient 12

has no consequence in what we discuss further on: for instance, in theWeyl dimension formula (see Section 2.3.2), all the coefficients 1

2 justcancel out.

The restrictions of the various roots to hR are real-valued and thereforebelong to h′R; they also completely determine the corresponding roots.This is why we shall think of roots as elements of h′R. If r belongs to1, ..., n, we denote the restriction Lr

∣∣hR

again by Lr, for simplicity.

Consider the lexicographical order on h′R (with respect to the coor-dinates introduced above). Then the positive roots are the followinglinear forms:

• Lr − Ls with r < s;

• Lr + Ls with r < s;

• 2Lr.

24

Denote by ∆+K the set of positive roots and by ρK their half sum. It is

straightforward to check the formula

ρK = nL1 + (n− 1)L2 + ...+ Ln,

which, in coordinates, reads:

ρK = (n, n− 1, . . . , 1). (2.3)

2.3.2 Highest weights

Consider a representation π ofK on a finite-dimensional complex vectorspace V . One can study its infinitesimal action, that is, the R-linearrepresentation dπ of k. It can always be thought of as a C-linear re-presentation $ of g = k⊕ ik on V by extending it to g in the followingnatural way:

$(X + iY ) := dπ(X) + idπ(Y )for all X,Y ∈ k. One refers to $ as the complexification of dπ and saysthat dπ has been complexified (one often simply writes dπ instead of$). The correspondance between the initial representation dπ and itscomplexification $ is clearly one-to-one.

One defines weights similarly to roots: weights are linear forms on hthat give the corresponding eigenvalues of joint eigenvectors of all linearmaps$(H), whereH runs through h; if σ is a weight, the correspondingweight space is (one also adds in the definition of a weight that thisspace must not be reduced to 0):

Vσ = v ∈ V/∀H ∈ h : $(H)(v) = σ(H)v .

The structure of semisimple Lie algebras and the properties of finite-dimensional representations of sl(2,C) imply that the restrictions ofthe various weights to hR are real-valued. This is why we can think ofweights as elements of h′R.

Consider a highest weight vector v of (highest) weight σ, that is, avector v that belongs to Vσ and that is cancelled by the action under$ of all elements of

n+ = VectCU+r,s, V

+r,s1≤r<s≤n ⊕VectCD+

r 1≤r≤n

25

(see Section 2.3.1 for notation). Applying $ to v for all X ∈ g thengenerates an irreducible invariant subspace Virr(v) of V ; the linear formσ is called a highest weight because basic Lie theory shows that if acertain weight vector belongs to Virr(v), its weight cannot be largerthan σ. Basic Lie theory also assures us that applying π(k) to v forall k ∈ K generates the same subspace Virr(v). Though it is in factdefined with respect to $, we shall say that the linear form σ (resp.the vector v) is a highest weight (resp. highest weight vector) of π.

The highest weight theorem (see [27], Theorem 4.28) applied to Ksets a correspondence between irreducible representations of K andtheir highest weights; these are the forms σ = ∑n

r=1 lrLr such that(l1, ..., ln) ∈ Nn and l1 ≥ l2 ≥ ... ≥ ln; as mentioned previously, weidentify σ with (l1, ..., ln).

If σ is the highest weight of an irreducible representation of K, thenthe dimension dσ of this representation is given by Weyl’s dimensionformula (see Theorem 4.48 of [27]):

dσ =∏α∈∆+

K< σ + ρK , α >∏

α∈∆+K< ρK , α >

. (2.4)

2.3.3 Casimir operator

As we have seen, the Lie subalgebra k consists of complex N × Nmatrices (

A −BB A

)such that A is a skew-Hermitian n×n matrix and B a symmetric n×nmatrix.

Define (r and s denote integers that belong to 1, ..., n):

• Ar = iEr,r − iEn+r,n+r;

• Br,s = Er,s − Es,r + En+r,n+s − En+s,n+r (when r 6= s);

• Cr,s = iEr,s + iEs,r − iEn+r,n+s − iEn+s,n+r (when r 6= s);

26

• Dr = En+r,r − Er,n+r;

• Er = iEn+r,r + iEr,n+r;

• Fr,s = En+r,s + En+s,r − Er,n+s − Es,n+r (when r 6= s);

• Gr,s = iEn+r,s + iEn+s,r + iEr,n+s + iEs,n+r (when r 6= s).

One easily sees that:

• If n ≥ 2 then

Ar, Dr, Err∈1,...,n ∪ Br,s, Cr,s, Fr,s, Gr,sr,s∈1,...,n,r<s

is a basis of k over R.

• If n = 1 then A1, D1, E1 is a basis of k over R.

Elements of k correspond to elements of K via the exponential map.One easily obtains the following table for n, r, s ∈ 1, ..., n:

Matrix M M2 M3

Ar −Ir −ArDr −Ir −Dr

Er −Ir −ErBr,s −Kr,s −Br,sCr,s −Kr,s −Cr,sFr,s −Kr,s −Fr,sGr,s −Kr,s −Gr,s

where we denote by Ir the matrix Er,r +En+r,n+r and by Kr,s the ma-trix Er,r + Es,s + En+r,n+r + En+s,n+s.

From this table we obtain the following exponentials for t ∈ R:

Lemma 2.4.

• For r ∈ 1, 2, ..., n and M ∈ Ar, Dr, Er:

exp(−tM) = I − (sin t)M + (cos t− 1)Ir.

27

• For r, s ∈ 1, ..., n such as r < s (here, we assume that n ≥ 2)and for M ∈ Br,s, Cr,s, Fr,s, Gr,s:

exp(−tM) = I − (sin t)M + (cos t− 1)Kr,s.

Define on k the bilinear form β by setting

β(X,Y ) = −12 Tr(XY )

for all (X,Y ) ∈ k × k, with Tr denoting the matrix trace form. Oneeasily checks:

Lemma 2.5.

1. The basis

Bk = Ar, Dr, Err∈1,...,n∪Br,s√

2,Cr,s√

2,Fr,s√

2,Gr,s√

2

r,s∈1,...,n,r<s

of k is orthonormal with respect to β.

2. The symmetric bilinear form β is real-valued and positive-definite;in other words it is an inner-product on k.

3. β is ad-invariant, meaning invariant under the adjoint action:

∀X,Y, Z ∈ g : β ([X,Y ], Z) = −β (Y, [X,Z]) .

Let us consider a finite-dimensional representation σ of K. Because βis a nondegenerate ad-invariant bilinear form on k and because Bk isan orthonormal basis of k, we can define (as in [12], see Section 7 ofChapter VI) the Casimir operator of σ:

Ωσ =dim(k)∑r=1

(dσ(Xr))2 , (2.5)

where, for the time being, we denote byXr the elements of Bk. Thinkingof this operator as an element of the enveloping algebra of k makes (2.5)somewhat lighter:

Ωσ =dim(k)∑r=1

X2r . (2.6)

28

As we explained in Section 2.3.2, we can always extend dσ to obtain, ina one-to-one fashion, a C-linear representation of g. This enables us towrite down elements of Bk as combinations of elements of the basis (2.2)of g and consider the action on these combinations of the complexifi-cation of dσ, which, in turn, enables us to use information we have onweights, in particular highest weights. The formulas for the Casimiroperator become quite simple when applied to a highest weight vector.If the representation σ is irreducible, then Schur’s lemma, combinedwith the well known fact (see for instance [12], Proposition 6.7.1) thatΩσ commutes with dσ, implies that Ωσ is a scalar multiple of the iden-tity map. This scalar is easy to determine when considering a highestweight vector; the aim of this section is precisely to compute it.

One easily checks that for all integers r and s in 1, ..., n (U±r,s, V ±r,sand D±r denote the root elements defined in Section 2.3.1):

• Ar = iHr;

• Br,s = U+r,s − U−r,s (r < s);

• Cr,s = iU+r,s + iU−r,s (r < s);

• Dr = D−r −D+r ;

• Er = iD−r + iD+r ;

• Fr,s = V −r,s − V +r,s (r < s);

• Gr,s = iV −r,s + iV +r,s (r < s).

Proposition 2.6. If σ is a finite-dimensional irreducible representa-tion of K on a complex vector space and if λ is its highest weight,then:

Ωσ = −(

n∑r=1

(λ2(Hr) + 2λ(Hr)

)+n−1∑r=1

2(n− r)λ(Hr))Id,

where Id denotes the identity map.

Proof:

29

By definition:

Ωσ =n∑r=1

(A2r +D2

r + E2r

)+ 1

2∑

1≤r<s≤n

(B2r,s + C2

r,s + F 2r,s +G2

r,s

). (2.7)

Using the formulas given just before this proposition, multiplying thebrackets out (one must be careful: the operators do not commute) andcancelling out various terms, we get:

Ωσ = −n∑r=1

(H2r + 2D−r D+

r + 2D+r D−r

)−

∑1≤r<s≤n

(U+r,sU

−r,s + U−r,sU

+r,s + V +

r,sV−r,s + V −r,sV

+r,s

). (2.8)

Apply Ωσ to a highest weight vector v. Then, by definition of a highestweight vector, all terms such as D+

r (v), U+r,s(v) and V +

r,s(v) are equal to0. So (2.8) applied to v becomes:

Ωσ(v) = −n∑r=1

(H2r (v) + 2D+

r D−r (v)

)−

∑1≤r<s≤n

(U+r,sU

−r,s(v) + V +

r,sV−r,s(v)

). (2.9)

One can check the following commutation rules:

[D+r , D

−r ] = Hr ; [U+

r,s, U−r,s] = Hr −Hs ; [V +

r,s, V−r,s] = Hr +Hs.

Using these rules in (2.9) we get

Ωσ(v) = −n∑r=1

(H2r (v) + 2Hr(v)

)−

∑1≤r<s≤n

(Hr(v)−Hs(v) +Hr(v) +Hs(v)) ,

30

which can be written:

Ωσ(v) = −n∑r=1

(H2r (v) + 2Hr(v)

)−n−1∑r=1

2(n− r)Hr(v). (2.10)

Because λ is a weight and v a corresponding weight vector, (2.10) finallygives:

Ωσ(v) = −(

n∑r=1

(λ2(Hr) + 2λ(Hr)

)+n−1∑r=1

2(n− r)λ(Hr))

(v).

One concludes with Schur’s lemma.

End of proof.

2.4 Quaternions

2.4.1 One way to define quaternions

For the time being, we regard the set of quaternions H as the set C2. Wewrite (1, 0) = 1H and (0, 1) = jH. We endow C2 with the usual complexvector space structure, denoting scalar multiplication simply by a dot.A pair (u, v) ∈ C2 corresponds to the element u · 1H + v · jH. Thefollowing rules define a multiplication on H (we call it quaternionicmultiplication):

• (i) ∀(u, v) ∈ C2 : (u · 1H) (v · 1H) = (uv) · 1H.

• (ii) ∀(u, v) ∈ C2 : (u · 1H) (v · jH) = (uv) · jH.

• (iii) ∀(u, v) ∈ C2 : (u · jH) (v · 1H) = (uv) · jH. In particular,this implies the fundamental formula for all v ∈ C:

jH (v · 1H) = v · jH.

• (iv) ∀(u, v) ∈ C2 : (u · jH) (v · jH) = (−uv) · 1H. In particular,this implies that jH jH = −1 · 1H, which we of course just writej2H = −1H.

31

• (v) ∀(h, h′, h′′) ∈ H3 :

– h (h′ + h′′) = (h h′) + (h h′′).– (h+ h′) h′′ = (h h′′) + (h′ h′′).

These rules clearly imply that 1H is the left and right identity elementfor quaternionic multiplication. Let us also point out that quaternionicmultiplication is not commutative, precisely because of rule (iii), butis associative:

∀(h, h′, h′′) ∈ H3 : (h h′) h′′ = h (h′ h′′).

Moreover, we obviously have:

∀(u, h) ∈ C×H : u · h = (u · 1H) h.

In this sense, if we identify complex numbers u with quaternions u ·1H,we can say that quaternionic multiplication has absorbed complexscalar multiplication, which is therefore no longer needed: all the struc-ture of H lies in addition and quaternionic multiplication. For simpli-city, we now:

• just write u instead of u · 1H, given u ∈ C;

• just write j instead of jH;

• omit all multiplication symbols.

Rule (iii) implies that any quaternion h can be written in two ways forsuitable (u, v) ∈ C2: either u + jv or u + wj (with w = v). In otherwords, we could choose to write the second coefficients of quaternionseither on the right or on the left of j. We choose the first possibi-lity because we find it more convenient for quaternionic linear algebra.Finally, as we shall soon see, every non-zero quaternion has a uniqueinverse, which is both a left and right one. Now let us summarise thefacts and definitions we have seen so far:

• The set of quaternions is H =u+ jv / (u, v) ∈ C2

.

• H is a skew (non-commutative) field, with respect to addition andquaternionic multiplication.

32

• ∀v ∈ C : jv = vj.

• j2 = −1.

Given a quaternion h = u + jv, one defines its quaternionic conjugateh∗ (or conjugate for short):

h∗ = u− jv.

Lemma 2.7. Given any quaternions h, h1 and h2:

1. hh∗ = h∗h;

2. (h1 + h2)∗ = h∗1 + h∗2;

3. (h1h2)∗ = h∗2h∗1.

The modulus of a quaternion h is then defined by:

|h| =√hh∗ =

√|u|2 + |v|2.

This definition shows that every non-zero quaternion h, as announcedpreviously, has a unique left inverse, a unique right inverse, and thatboth of these inverses coincide:

h−1 = h∗

|h|2.

Also, one obviously has:

Lemma 2.8. Given any quaternions h1 and h2:

1. h1 = 0⇐⇒ |h1| = 0;

2. |h1 + h2| ≤ |h1|+ |h2|;

3. |h1h2| = |h1||h2|.

A quaternion h will be called a unit quaternion if |h| = 1. We point outthat the set of unit quaternions is a group with respect to quaternionicmultiplication; we shall denote it by UH; it is clearly diffeomorphic tothe three-dimensionnal sphere.

33

Remark: this complex version of quaternions corresponds exactly tothe traditional and historical real version (used by W. R. Hamiltonhimself - see [19]). In the real version, quaternions are usually written

h = a+ bi+ cj + dk,

with k = ij and (a, b, c, d) ∈ R4, their conjugates becoming

a− bi− cj − dk

and their moduli √|a|2 + |b|2 + |c|2 + |d|2.

The correspondance between the real and complex versions works viathe identifications u = a+ ib and v = c− id.

2.4.2 Quaternionic linear algebra

Given a positive integer p, we endow Hp with the usual addition andscalar multiplication (by quaternions). Multiplication can apply eitheron the left or on the right of coordinates; the only difference, comparedto Rp or Cp, is that left and right multiplication usually give differentresults (H is non-commutative). Considering left multiplication, we seethat all axioms of a vector space are satisfied, so we say that Hp is aleft quaternionic vector space; similarly, one can focus on right mul-tiplication and say that Hp is a right quaternionic vector space. In away, which point of view one chooses has no importance, as long asone’s choice is clear. However, linearity of functions must be specifiedaccordingly: left linearity does not coincide with right linearity, again,because H is not commutative. We shall always work with right li-nearity: for us, a linear map from some quaternionic vector space toanother will always mean a right-linear map; the reason for this choicelies in matrix multiplication.

Given positive integers r and s, we endow Mr,s(H) with the usual ma-trix addition. Scalar multiplication (by quaternions) can be defined onthe left or on the right, so Mr,s(H) is both a left and a right quater-nionic vector space. Multiplying matrices by matrices of suitable sizes

34

is defined as usual, but one must respect the order in which the coeffi-cients come.

Coming back to a positive integer p, a matrix A ∈ M(p,H) is invertibleif for some matrix B ∈ M(p,H) one has AB = BA = Ip, in which caseB is the inverse of A and is denoted by A−1. The set of invertible p×pmatrices is of course a group with respect to matrix multiplication andis denoted by GL(p,H).

2.4.3 Complex and quaternionic coordinates: identifica-tions

Again, consider any positive integer p. As we have seen, C2p identifieswith Hp: (u, v) ∈ Cp×Cp identifies with u+jv ∈ Hp. Given an elementx of Hp, we shall write:

• x = (x1, ..., xp) ∈ Hp;

• xr = ur + jvr, for all r ∈ 1, · · · , p, with (ur, vr) ∈ C2;

• u = (u1, · · · , up) ∈ Cp and v = (v1, · · · , vp) ∈ Cp;

• x = u+ jv.

We define for all vectors x and y in Hp:

• 〈x, y〉H = ∑pr=1 xry

∗r ∈ H;

• ‖x‖2 = 〈x, x〉H = ∑pr=1 xrx

∗r = ∑p

r=1 |xr|2.

Lemma 2.9. Given any vectors x, y ∈ Hp and any quaternion h:

• x = 0⇐⇒ ‖x‖ = 0;

• ‖hx‖ = ‖xh‖ = |h|‖x‖;

• ‖x+ y‖ ≤ ‖x‖+ ‖y‖.

Therefore ‖ · ‖ is a norm on Hp. It corresponds to the standard normof R4p ' Hp, turning Hp into a real Banach space as well as a complexBanach space.

35

A matrix M ∈ M(p,H) will be decomposed as M = A + jB, with Aand B two complex p× p matrices; M can then be identified with thefollowing element of M(2p,C):

M ′ =(A −BB A

).

We denote by MH(2p,C) the vector space that consists of all elementsof M(2p,C) that can be written likeM ′. This procedure defines a map:

ECH : M(p,H) −→ MH(2p,C)

M 7−→ M ′.

This map is bijective and respects matrix multiplication, meaning that,given any two elements M1 and M2 of M(p,H):

ECH(M1M2) = EC

H(M1)ECH(M2).

This implies that if M2 is a right inverse of M1, then M ′2 = ECH(M2) is

a right inverse of M ′1 = ECH(M1). The same goes for left inverses and,

because left and right inverses coincide in GL(2p,C):

Lemma 2.10. If a matrixM of M(p,H) has a right (resp. left) inverseM ′, then M ′ is also a left (resp. right) inverse of M .

This is why we grouped both kinds of inverses in a single definition whenwe introduced invertible matrices earlier on. We denote by GLH(2p,C)the image of GL(p,H) under the map EC

H; it is obviously a subgroupof GL(2p,C) and the restriction of EC

H to GL(p,H) is an isomorphismonto GLH(2p,C).

Applying a matrix M ∈ M(p,H) to a vector u + jv ∈ Hp correspondsto applying EC

H(M) = M ′ to the vector (u, v) ∈ C2p.

If one looks back at the expression of elements of the compact group Kgiven in section 2.2, one cannot miss the similarity with matrices suchas the matrix M ′ above. This tells us that, via the map EC

H, the groupSp(p) corresponds to the group of elements of GL(p,H) that preservethe norms of vectors of Hp, that is, the group of linear isometries of

36

Hp. We again denote this group by Sp(p). In particular: Sp(1) ' UH.

Left and right multiplication by quaternions in Hp can also be read inC2p, but not as simple scalar multiplications: they combine the complexcoordinates in a more subtle way. Indeed, multiplying a quaternionh = u + jv ∈ H by another quaternion q = a + jb ∈ H on the rightgives:

hq = (ua− vb) + j(va+ ub).

This formula also shows how left multiplication operates: h and q playsymmetric roles. Let us summarise how the actions of quaternionicmatrices and quaternionic scalars read in the complex setting:

Lemma 2.11. Given a vector x = u+jv ∈ Hp, a scalar q = a+jb ∈ Hand a matrix M = A+ jB ∈ M(p,H):

1. Mx ∈ Hn corresponds to(A −BB A

)(uv

)∈ C2p;

2. xq ∈ Hp corresponds to

(u1a− v1b, . . . , upa− vpb, v1a+ u1b, . . . , vpa+ upb) ∈ C2p .

Consider a quaternionic matrix M ∈ M(p,H) and denote by mrs itscoefficients. Write mrs = urs + jvrs and define:

‖M‖op = max|urs|, |vrs|1≤r,s≤n.

Obviously, ‖M‖op is equal to the standard norm of ECH(M) in M(2p,C)

which hands out the highest modulus of the coefficients of ECH(M).

Lemma 2.12. Given any matrices M,M ′ ∈ M(p,H) and any quater-nion h:

• ‖M‖op ≥ 0;

• M = 0⇐⇒ ‖M‖op = 0;

• ‖hM‖op = ‖Mh‖op = |h|‖M‖op;

• ‖M +M ′‖op ≤ ‖M‖op + ‖M ′‖op.

37

In other words, ‖ · ‖op is a norm on M(p,H). It turns M(p,H) into aBanach space and the map EC

H into a homeomorphism, allowing one todefine the exponential map exp on M(p,H) by the usual formula:

exp(M) =∞∑r=0

M r

r! .

One easily checks that the maps ECH and exp commute; in other words,

we have the following commutative diagram:

GL(p,H)EC

H−−−−→ GLH(2p,C)expx xexp

M(p,H) −−−−→EC

H

MH(2p,C).

2.4.4 Symplectic matrices seen as quaternionic matrices

Elements of Sp(n) (see Section 2.4.3) can be seen, via the embeddingmap EC

H, as quaternionic matrices; in this process of identification, forsimplicity, we will not change notation, writing in particular:

• Ar = iEr,r;

• Br,s = Er,s − Es,r (when n ≥ 2 and r 6= s);

• Cr,s = iEr,s + iEs,r (when n ≥ 2 and r 6= s);

• Dr = jEr,r;

• Er = jiEr,r;

• Fr,s = jEr,s + jEs,r (when n ≥ 2 and r 6= s);

• Gr,s = jiEr,s + jiEs,r (when n ≥ 2 and r 6= s).

Then Ar, Dr, Err∈1,...,n∪Br,s, Cr,s, Fr,s, Gr,sr,s∈1,...,n,r<s is a ba-sis over R of k (seen as a subspace of M(n,H)). Applying the exponen-tial map and Lemma 2.4, given t ∈ R, we have:

Lemma 2.13.

38

• For r ∈ 1, 2, ..., n and M ∈ Ar, Dr, Er:

exp(−tM) = I − (sin t)M + (cos t− 1)Er,r.

• For r, s ∈ 1, ..., n such as r < s (here it is assumed that n ≥ 2)and for any M ∈ Br,s, Cr,s, Fr,s, Gr,s:

exp(−tM) = I − (sin t)M + (cos t− 1)(Er,r + Es,s).

One can transport the inner product β defined in section 2.3.3 to ma-trices of k seen as quaternionic matrices: for such matrices X and Y ,one sets β(X,Y ) = β

(EC

H(X), ECH(Y )

). Again:

• β is an inner product on k (seen as a subspace of M(n,H)).

• The set

Bk = Ar, Dr, Err∈1,...,n∪Br,s√

2,Cr,s√

2,Fr,s√

2,Gr,s√

2

r,s∈1,...,n,r<s

is an orthonormal basis of k (seen as a subspace of M(n,H)) withrespect to β.

The following proposition will come out useful when studying stabilisersof actions of K:

Proposition 2.14.

1. Consider a quaternionic matrix M =(h L0 T

), with h ∈ H,

L ∈ M1,n−1(H) and T ∈ M(n − 1,H). Then M belongs to K ifand only if L = 0, h ∈ Sp(1) and T ∈ Sp(n− 1).

2. Consider a quaternionic matrix M =(h 0C T

), with h ∈ H,

C ∈ Mn−1,1(H) and T ∈ M(n − 1,H). Then M belongs to K ifand only if C = 0, h ∈ Sp(1) and T ∈ Sp(n− 1).

Proof:

39

Both items work similarly, so we just prove Item 1 (in fact, each itemimplies the other via matrix transpose).

Suppose M belongs to K. One can decompose M into M = U + jV ,with

U =(h1 L10 T1

)∈ Mn(C)

andV =

(h2 L20 T2

)∈ Mn(C).

Recalling the definition of the map ECH, we have:

ECH(M) =

h1 L1 −h2 −L20 T1 0 −T2h2 L2 h1 L10 T2 0 T1

.The matrix M belongs to K, so EC

H(M) is unitary:

tECH(M)EC

H(M) = ECH(M)tEC

H(M) = I2n.

By looking at the top-left coefficients of these products, one finds (here,the symbol ‖ ·‖ denotes the norm of Cn−1 and we identify row matriceswith vectors):

|h1|2 + ‖L1‖2 + |h2|2 + ‖L2‖2 = 1

and|h1|2 + |h2|2 = 1.

This implies:‖L1‖2 + ‖L2‖2 = 0.

Consequently, L = 0. From this it follows that h and T must belongrespectively to Sp(1) and Sp(n− 1).

This proves one implication; the converse is straightforward.

End of proof.

40

Chapter 3

Degenerate principal seriesof Sp(n,C)

The heart of our work is the study of certain induced representations ofG = Sp(n,C). Why it is interesting to concentrate on such representa-tions was discussed in the introduction. They are based on the choiceof a parabolic subgroup. The theory (usually refered to as structuretheory of reductive Lie groups) that defines such subgroups would taketoo long to introduce here, so we will just make the specific parabolicsubgroup we work with explicit, along with the corresponding inducedrepresentations, referring the reader to [27] (Sections 2 and 5 of ChapterV and Section 1 of Chapter VII) for a thorough lecture on the theorythat underlies Section 3.2.1.

3.1 A specific parabolic subgroupWe write 2n× 2n matrices in the following way:

g =

∗ ∗ ∗ ∗∗ E ∗ G∗ ∗ ∗ ∗∗ F ∗ H

,where E,F,G,H are m ×m matrices and where the stars denote sui-table numbers, row matrices or column matrices. We list below the

41

subgroups of G that we need to define our induced representations.

• We denote by M the subgroup of G consisting of all matrices ofthe following type:

m =

eiθ 0 0 00 A 0 C

0 0 e−iθ 00 B 0 D

,

with θ ∈ R and(A C

B D

)∈ Sp(m,C). We then write θ = θ(m).

• We denote by A the subgroup of G consisting of all matrices ofthe following type:

a =

α 0 0 00 Im 0 00 0 α−1 00 0 0 Im

,with α ∈]0,∞[. We then write α = α(a).

• We denote by N the subgroup of G consisting of all matrices ofthe following type:

n =

1 tu 2s tv0 Im v 00 0 1 00 0 −u Im

,with (s, u, v) ∈ C×Cm×Cm. The coefficient 2 in front of s is notreally necessary in the definition, but we keep it, as customary,to signify the isomorphism between N and the so-called complexHeisenberg group H2m+1: matrices n as above correspond toelements (s, u, v) of H2m+1. Though we will say nothing more ofit, this isomorphism underlies the non-compact picture to come.

• We define the parabolic subgroup Q as the group of elements qof G that can be written as q = man for some triple (m, a, n)

42

of M × A × N . We write Q = MAN , calling this equality theLanglands decomposition of Q (it defines a diffeomorphism of Qonto M ×A×N).

• We denote by N the subgroup of G that consists of all matricestn for n ∈ N .

Remark: we are aware that symbols m, n and N also refer to dimen-sions, but with context there can be no confusion. We also point outthat N and N are swapped in [35]; we have not followed the choice ofthe authors of [35] in order to stay close to Chapter VII of [27].

When defining induced representations, one can wish to change thecarrying spaces. In order to do so, the following proposition will proveuseful:

Proposition 3.1. Q is the subgroup of G consisting of all matrices ofG in which the entries of the first column are all 0 except the one atthe top.

Proof:

If q belongs to MAN then one can write:

q =

α 0 0 00 A 0 C0 0 α−1 00 B 0 D

1 tu s tv0 I v 00 0 1 00 0 −u I

,

where α ∈ C \0,(A CB D

)belongs to Sp(m,C) and where (s, u, v)

belongs to C×Cm×Cm. This gives the following form of q (we’ll saythat a matrix of this form is a Q-form matrix):

q =

α αtu αs αtv0 A Av − Cu C0 0 α−1 00 B Bv −Du D

.43

We see that the first column of q satisfies the required condition.

Suppose now that the first column of a matrix q of G has all its entriesequal to 0 except the one at the top; let us write:

q =

α tk t tl0 A x C0 te b tf0 B y D

,where A, B, C and D are m×m matrices, where α is non zero, (t, k, l)and (b, e, f) belong to C×Cm×Cm and (x, y) to Cm×Cm. Workingwith the properties of symplectic matrices given in formulas (2.1), onecan prove that:

• e = f = 0;

• b = α−1;

•(A CB D

)belongs to Sp(m,C).

Then, again using the conditions that a symplectic matrix must satisfy,one shows that:

• (i) kα−1 + tAy = tBx;

• (ii) lα−1 + tCy = tDx.

Comparing with the Q-form matrix obtained in the first part of theproof, we can identify s, u, v:

• s = α−1t;

• u = α−1k;

• v = α−1l.

So that (i) and (ii) become:(uv

)=(

tB −tAtD −tC

)(xy

).

44

Inverting this gives:(xy

)=(−C A−D B

)(uv

).

So we see that q =

α αtu αs αtv0 A Av − Cu C0 0 α−1 00 B Bv −Du D

is indeed a Q-form

matrix, that we can write

q =

α 0 0 00 A 0 C0 0 α−1 00 B 0 D

1 tu s tv0 I v 00 0 1 00 0 −u I

and that finally does belong to Q = MAN .

End of proof.

3.2 Specific induced representations

Throughout this section we consider any λ ∈ R and δ ∈ Z.

3.2.1 Usual definitions

Induced picture

Consider (λ, δ) ∈ R×Z and the character χiλ,δ of the parabolic sub-group Q = MAN (introduced in section 3.1) defined by

χiλ,δ(man) = eiδθ(m) (α(a))iλ+N

.

Consider the complex vector space V 0iλ,δ of all functions f ∈ C0(G) such

that for all (g,m, a, n) ∈ G×M ×A×N :

f(gman) = χ−1iλ,δ(man) f(g).

45

The induced representation πiλ,δ = IndGQ χiλ,δ is obtained by conside-ring the left action of G on V 0

iλ,δ and completing V 0iλ,δ (extending the

action of G accordingly) with respect to the norm ‖ · ‖ defined by

‖f‖2 =∫K|f(k)|2 dk,

where dk denotes the Haar measure (unique up to a constant) of K.Functions of V 0

iλ,δ are said to be (λ, δ)-covariant (or just covariant, forshort).

The representations πiλ,δ obtained by varying the parameters λ and δare continuous, unitary and form a family called the degenerate princi-pal series of G (the word degenerate refers to the fact that the parabolicsubgroup we have chosen is not minimal). They were studied in [18],where it is proved that πiλ,δ is irreducible if and only if (λ, δ) 6= (0, 0).

One can change the carrying space and define representations that areequivalent to πiλ,δ. To specify which one of them is considered, one usesthe word picture, by which we mean the description of the action andthe carrying space. The above description is called the induced picture.The equivalences between different pictures are due to structure theory,which, as mentioned previously, we have chosen not to discuss.

Compact picture

The compact picture is obtained by restricting functions of V 0iλ,δ to K.

One thus considers the complex vector spacef ∈ C0(K) / ∀k ∈ K, ∀m ∈M ∩K : f(km) = e−iδθ(m)f(k)

,

denoting it again by V 0iλ,δ and completing it with respect to the norm

‖ · ‖ defined by‖f‖2 =

∫K|f(k)|2 dk.

Identification of the carrying spaces of the induced and compact pic-tures is based on a specific decomposition of elements of G: structuretheory shows that G = KMAN and restriction to K then defines a

46

one-to-one correspondance between functions of V 0iλ,δ seen in the in-

duced picture and functions of V 0iλ,δ seen in the compact picture. The

action of G in the compact picture is not as simple as one might think,because multiplying elements of K by elements of G can lead to ele-ments that do not belong to K anymore. However, the restriction ofthe action to K is just the left action of K.

Non-compact picture

One can show that restricting functions of V 0iλ,δ (in the induced picture)

to N gives continuous elements of L2(N, dn), where dn denotes theHaar measure (unique up to a constant) of N . So, in the non-compactpicture, one chooses as carrying space the whole of L2(N). Here aswell, the action of G is not as simple as one might think, becausemultiplying elements of N by elements of G can lead to elements thatdo not belong to N anymore; but the restriction of the action to Nis just the left action of N . Identification of the carrying spaces ofthe induced and non-compact pictures is slightly more subtle than forthe compact picture, because based on a decomposition of G that onlyworks for almost all g ∈ G: NMAN is dense in G.

3.2.2 Changing the carrying spaces

Another way to change the carrying spaces is to make use of the naturalaction of the group G on CN . Each of the three pictures describedpreviously then leads to another picture; in the process, for simplicity,we do not rename the pictures and we keep on writing πiλ,δ for theinduced representations.

Another induced picture

Definition 3.2 (Induced picture of πiλ,δ). It is obtained by consideringthe left action of G on the complex vector space

V 0iλ,δ =

f ∈ C0

(CN \0

)/∀c ∈ C× : f(c·) =

(c

|c|

)−δ|c|−iλ−Nf(·)

47

and completing this space (extending the action of G accordingly) withrespect to the norm ‖ · ‖ defined by

‖f‖2 =∫S2N−1

|f(x)|2 dσ(x).

The completion of V 0iλ,δ is denoted by Viλ,δ. Functions of V 0

iλ,δ are alsosaid to be (λ, δ)-covariant (or just covariant, for short). Let us pointout that covariance makes ‖ · ‖ a proper norm, in the sense that thenorm of f ∈ V 0

iλ,δ is equal to 0 if and only if f vanishes everywhere.

Naturally, we have the same properties as in the initial induced picture:• πiλ,δ is continuous and unitary;

• πiλ,δ is irreducible if and only if (λ, δ) 6= (0, 0).It is explained in [8], amongst many other things, how π0,0 decomposesinto the direct sum of two irreducible invariant subspaces.

It is both interesting and important to understand how the correspon-dence works between both versions of the induced picture.

Though we said we would not change notation, it will be helpful for ourexplanations (till the end of the present subsection) to write V 0

iλ,δ (resp.πiλ,δ and d·e) instead of V 0

iλ,δ (resp. πiλ,δ and ‖ · ‖) when consideringthe initial induced picture.

Consider the vector e1 = (1, 0, . . . , 0) of CN . Consider the naturalaction of G on vectors of CN \0 (matrices times column vectors).This action is transitive and the stabiliser S of e1 is the subgroup ofmatrices of G such that the coefficients of the first column are all zeroexcept the one at the top that has to be 1. Looking back at the proof ofProposition 3.1, we see that this stabiliser is a subgroup of the parabolicsubgroup Q = MAN ; more accurately, elements of S can be writtens = man with:

• m =

1 0 0 00 A 0 C

0 0 1 00 B 0 D

with(A C

B D

)∈ Sp(m,C);

48

• a = IN ;

• n =

1 tu 2s tv0 Im v 00 0 1 00 0 −u Im

with (s, u, v) ∈ C×Cm×Cm.

Consider a function f that belongs to the space V 0iλ,δ.

Elements man of S satisfy eiθ(m) = α(a) = 1, so covariance of f impliesthat f is constant when restricted to representatives of a same coset ofG/S and thus defines a function fS on G/S by:

fS(gS) = f(g).

SinceG/S is homeomorphic to CN \0, one can then define a (complex-valued) function f on CN \0 by

f(x) = fS(gS) = f(g),

where one chooses any element g that satisfies g(e1) = x.

Take c ∈ C \0 and set

C =

c 0 0 00 Im 0 00 0 c−1 00 0 0 Im

.

This matrix C belongs to Q because it can be written C = man with

m =

c|c| 0 0 00 Im 0 00 0

(c|c|

)−10

0 0 0 Im

, a =

|c| 0 0 00 Im 0 00 0 |c|−1 00 0 0 Im

49

and n = IN . Now if g ∈ G is such that g(e1) = x, one can write

f(cx) = f(cg(e1)

)= f

(g(ce1)

)= f

(gC(e1)

)= f(gC)=

(c|c|

)−δ|c|−iλ−N f(g)

and consequently

f(cx) =(c

|c|

)−δ|c|−iλ−Nf(x). (3.1)

Maps fS and f are continuous, so we have, so far, associated in a one-to-one fashion covariant functions f on G to continuous functions f onCN \0 that satisfy (3.1) for all non-zero complex numbers c. Thiscorrespondence makes representations and identifications compatible,in the sense that we have a commutative diagram for each element gof G:

f −−−−→ πiλ,δ(g)fy yf −−−−→ πiλ,δ(g)f.

Now, what is a relevant norm for V 0iλ,δ?

Because K is a subgroup of U(N), the sphere S2N−1 and its Euclideanmeasure σ are invariant under the natural action of K. The restrictionof the natural action of K to S2N−1 is transitive and the stabiliser ofe1 is SK = S ∩K. Thus the map (taking, as above, any k such thatke1 = x)

Ψ : S2N−1 −→ K/SKx 7−→ kSK

is a homeomorphism between S2N−1 and the quotient K/SK ; let usdenote equivalence classes by [k] (k refering to any representative ofthe coset kSK). Consider the function F = f ψ−1 on K/SK . Then:

f(x) = F ([k]) = f(k),

50

where k is any element of K such that x = ke1.

The natural action of K on S2N−1 induces a natural action (simplydenoted here by a dot) on K/SK defined on equivalence classes by(taking k1 and k2 in K)

k1 · [k2] = [k1k2].

The Euclidean measure σ can be transported to K/SK so as to definea K-invariant measure µ (the image measure of σ):

µ(B) = σ(Ψ−1(B)),

for all measurable subsets B of K/SK . One has:∫S2N−1

|f(x)|2 dσ(x) =∫K/SK

∣∣F ([k])∣∣2 dµ([k]). (3.2)

Because SK is a closed subgroup of K and because K is compact, thehomogeneous space K/SK has a unique left invariant Borel measure µ(up to a constant). This follows from a standard integration theoremthat one can find for instance in [28] (Theorem 8.36 – this theorem isin fact stated for real-valued functions, but it passes on to complex-valued ones). So measures µ and µ are proportional. Moreover, thesame theorem says that µ can be normalised so that for all continuousfunctions φ on K one has:∫

Kφ(k) dk =

∫K/SK

(∫SK

φ(ks)ds)dµ([k]),

where ds denotes the Haar measure of SK (induced by the normalisedleft Haar measure of K). Applying this to φ =

∣∣∣f |K ∣∣∣2 and denoting byVol(SK) the volume of SK with respect to the Haar measure dk, weget: ∫

K|f(k)|2 dk = Vol(SK)

∫K/SK

∣∣F ([k])∣∣2 dµ([k]). (3.3)

Because µ and µ are proportionnal, (3.2) and (3.3) imply that thenorms d·e and ‖ ·‖ are proportionnal. These norms therefore define thesame topology and unitarity in one picture corresponds to unitarity inthe other picture. These considerations justify our choice of norm ‖ · ‖.

51

Another compact picture

Definition 3.3 (Compact picture of πiλ,δ). Here, the carrying spaceis the Hilbert space

Viλ,δ =f ∈ L2(S2N−1) / ∀θ ∈ R : f(eiθ·) = e−iδθf(·)

with respect to the norm ‖ · ‖ defined by:

‖f‖2 =∫K|f(k)|2 dk.

We say that elements of Viλ,δ are δ-covariant (or just covariant, forshort). Again, the action of G is not as simple as one might think, butits restriction to K is just the left action of K.

The parameter λ does not explicitly appear in the compact picture,but is hidden in the following observation. Restriction of functions Fof V 0

iλ,δ to S2N−1 (in the new induced picture) establishes a one-to-onecorrespondence with continuous elements f of the space Viλ,δ in thenew compact picture. This correspondence works as follows:

F (x) = ‖x‖−iλ−Nf(x

‖x‖

),

for x ∈ CN \ 0.

Another non-compact picture

This picture is based on two observations:

• The image of e1 = (1, 0, . . . , 0) ∈ CN under the natural action ofthe subgroup N is the (affine) hyperplane

P = 1 × Cm×C×Cm .

Indeed, given an element n ∈ N written as in Section 3.1, theelement tn ∈ N assigns to e1 the point (1, u, 2s, v) ∈ P.

• The Haar measure of N is the Lebesgue measure ds du dv.

52

Restricting functions of V 0iλ,δ to N , in the initial induced picture, cor-

responds to restricting functions of V 0iλ,δ to the hyperplane P, in the

new induced picture, thereby obtaining continuous functions of L2(P),with respect to the Lebesgue measure ds du dv. These considerationsnaturally lead us to the following choice:

Definition 3.4 (Non-compact picture of πiλ,δ). The carrying space isL2(P), with respect to the Lebesgue measure ds du dv, where (1, u, 2s, v)denote the coordinates on P. Again, the action of G is not as simpleas one might think, but the action restricted to N is just the left actionof N .

One has to be careful with the way the induced picture and the non-compact picture correspond to one another. As much as one can alwaysrestrict functions of V 0

iλ,δ, in the induced picture, to obtain continuouselements of L2(P), the reverse procedure via covariance only gives func-tions defined on CN ∩(z1 6= 0). In more detail, a continuous functionf on P defines a continous function F on S2N−1 ∩ (z1 6= 0):

F (z, w) =(z1|z1|

)−δ|z1|−iλ−Nf

(1, z2z1, . . . ,

znz1,w1z1, . . . ,

wnz1

).

The measure of S2N−1 ∩ (z1 = 0) is 0 in S2N−1, so this procedure en-ables one to obtain the whole of Viλ,δ in the new compact and inducedpictures.

Via the bijection

C×Cm×Cm −→ 1 × Cm×C×Cm(s, u, v) 7−→ (1, u, 2s, v)

we will identify the spaces L2(P) and L2(C2m+1) (again, with respectto the Lebesgue measures ds du dv).

Remark: the parameters λ and δ do not actually appear in this picture;they are hidden in the restriction/extension process.

53

54

Chapter 4

Actions of Sp(n) and Sp(1)

In this section, we work with the compact picture of the induced repre-sentations πiλ,δ. Because the restrictions πiλ,δ

∣∣K

coincide with the leftaction of K on Viλ,δ, we now simply write L instead of πiλ,δ

∣∣K, what-

ever the value of δ. By changing the values of δ, one can reconstructthe whole of V = L2

(S2N−1

), as we will see. We intend to study

how this Hilbert space decomposes into irreducible invariant subspacesunder two actions: the left action L of K = Sp(n) and the right actionof Sp(1) (which we shall define later on).

4.1 Left action of Sp(n)

4.1.1 Preliminaries

For the time being, denote the coordinates of R2N by (x1, . . . x2N ).Consider the Laplace operator ∆R defined by:

∆R =2N∑i=1

∂2

∂x2i

.

For k ∈ N, denote by Hk the complex vector space of polynomial func-tions f defined on R2N , with complex coefficients and such that:

1. f is homogeneous of degree k;

55

2. f is harmonic, that is, ∆R(f) = 0.

Denote by Yk the complex vector space whose elements are the restric-tions to S2N−1 of elements of Hk; these restrictions are called sphericalharmonics. It is well known that (see for example [12], Chapter 9):

L2(S2N−1

)=⊕k∈NYk, (4.1)

where the various spaces Yk are orthogonal to one another in L2(S2N−1).It is also well known that the subspaces Yk are stable under the leftaction of SO(2N) and that they define irreducible pairwise inequivalentrepresentations of SO(2N) (see Chapter 9 of [51]).

Because K can be seen as a subgroup of SU(N) which can itself beseen as a subgroup of SO(2N), the sphere S2N−1 is stable under thenatural actions of K and SU(N) and one can therefore consider the leftactions of K and SU(N) on Yk. It so happens that the right actionof Sp(1) that we will define later also preserves Yk. So, to understandhow L2(S2N−1) decomposes under the left action of K and the rightaction of Sp(1), one just needs to concentrate on each Yk.

Let us switch to complex coordinates. Put x = (x1, . . . , xN ) ∈ RN ,y = (xN+1, . . . , x2N ) ∈ RN and z = x + iy = (z1, . . . , zN ) ∈ CN .Because x = z+z

2 and y = z−z2i , a function f of x and y can be written

as a function F of the complex variable z and its conjugate z:

f(x, y) = F (z, z).

An element u of SU(N) can be viewed as a matrix u of SO(2N), throughthe following embedding of the set of complex N ×N matrices into theset of real 2N × 2N matrices (the way we have identified R2N and CNis precisely designed to match this embedding): one decomposes u intotwo real matrices A and B, writing u = A+ iB, then defines

u =(A −BB A

)

56

and checks that u belongs to SO(2N) and that the mapping u −→ u isan injective group morphism. The left action of u on f then transfersto F in the obvious way:

L(u)P (z, z) = P (u−1z, u−1z) = P (u−1z, tuz). (4.2)

The last equality holds because u is unitary and therefore u−1 = tu.In the coordinates (z, z), the Laplace operator becomes:

∆C = 4N∑i=1

∂2

∂zi∂zi.

For α, β ∈ N, consider the space Hα,β of polynomials P (z, z) (z ∈ CN )such that:

1. P is homogeneous of degree α in z and of degree β in z;

2. ∆C(P ) = 0.

Each Hα,β is invariant under the left action of unitary matrices; theresulting representations of the unitary group SU(N) on Hα,β are irre-ducible and pairwise inequivalent (again, see [51], chapter 11).

Proposition 4.1. The dimension of Hα,β is given by the followingformula:

dim Hα,β = (α+ β +N − 1)(α+N − 2)!(β +N − 2)!(N − 1)!(N − 2)!α!β! .

Via coordinate identifications, one can consider spaces Hα,β as sub-spaces of Hα+β. Given k ∈ N, this leads to the natural isomorphism(see [51], Chapter 11):

Hk '⊕

(α,β)∈N2

α+β=k

Hα,β . (4.3)

We denote by Yα,β the space of restrictions of elements of Hα,β to theunit sphere S4N−1. From (4.1) and (4.3) we deduce:

L2(S2N−1

)=

⊕(α,β)∈N2

Yα,β .

57

Consequently, in the compact picture:

Viλ,δ =⊕

(α,β)∈N2

δ=β−α

Yα,β . (4.4)

We now discuss a basic yet important point: we can work with ho-mogeneous harmonic polynomials rather than spherical harmonics. Inorder to explain why, let us fix any non-negative integers α and β anddenote by R the map that restricts polynomials of Hα,β to the unitsphere S2N−1; the target space of R is precisely Yα,β. Because of ho-mogeneity, the map R is a bijection; it is in fact a linear isomorphism.Let us:

• denote by π the representation of K that corresponds to the leftaction on Yα,β.

• denote by π′ the representation of K that corresponds to the leftaction on Hα,β.

The restriction map R commutes with both left actions of K. Saidotherwise, the following diagram commutes (with k ∈ K):

Hα,β π′(k)−−−−→ Hα,βyR yRYα,β −−−−→

π(k)Yα,β .

Because the spaces Hα,β and Yα,β are finite-dimensional, π and π′ aredifferentiable. The differentials (at the identity) dπ and dπ′ are alsorelated by a commuting diagram (with X ∈ k):

Hα,β dπ′(X)−−−−→ Hα,βyR yRYα,β −−−−→

dπ(X)Yα,β .

In other words, the linear isomorphism R intertwines dπ and dπ′. Thisimplies that weights of π coincide with weights of π′, weight vectors of

58

π correspond to weight vectors of π′ and highest weight vectors of πcorrespond to highest weight vectors of π′.

This is why we now identify π and π′, going back to our notation Land denoting by dL the differential of L (at the identity).

Remark: we will often identify, without mentioning it, polynomials ofHα,β and the corresponding spherical harmonics of Yα,β.

We now work with polynomials, looking for highest weight vectors andthe corresponding highest weights.

Given (α, β) ∈ N2, the space Hα,β is stable under the action of K,but not necessarily irreducible: it can break up into (obviously) finite-dimensional irreducible invariant subspaces. Each of these subspaces isgenerated by some highest weight vector under the action of K. Also,because these subspaces are finite-dimensional, the restriction of L toany one of them can be differentiated. Remembering (4.2), ifX belongsto the Lie algebra k of K, if P belongs to the carrying space of L andif z belongs to CN , then by definition:

dL(X)(P )(z, z) = d

dt

∣∣∣∣t=0

(P(exp(−tX)z, t

(exp(tX)

)z) ).

The chain rule of differentiation applied to functions of real variablesand then rewritten in terms of complex variables gives

dL(X)(P )(z, z) =(

∂P∂z (z, z) ∂P

∂z (z, z)) d

dt

∣∣∣t=0

(exp(−tX)z

)ddt

∣∣∣t=0

(t(

exp(tX))z)

and so we have

dL(X)(P )(z, z) =(

∂P∂z (z, z) ∂P

∂z (z, z))( −Xz

tXz

). (4.5)

The infinitesimal action dL can be complexified, in other words ex-tended in the natural way to the complexification g = sp(n,C) ofk = sp(n). This complexification of dL is also defined by Formula(4.5), but this time with X ∈ g.

59

4.1.2 Highest weight vectors

We fix k ∈ N and (α, β) ∈ N2 such that α+ β = k.

To make polynomial calculations clearer, we change our system of no-tation for complex variables: we write (z, w) = (z1, ..., zn, w1, ..., wn) in-stead of our initial z = (z1, ..., zN ), with of course z = (z1, ..., zn) ∈ Cnand w = (w1, ..., wn) ∈ Cn. We now state and prove a theorem whichis the heart of this chapter:

Theorem 4.2. Denote by Iα,β the set of integers γ such that

0 ≤ γ ≤ min(α, β).

For γ ∈ Iα,β, consider the polynomial Pα,βγ defined by:

Pα,βγ (z, w, z, w) = wα−γ1 z1β−γ(w2z1 − w1z2)γ .

1. Pα,βγ belongs to Hα,β.

2. For any element H of the Cartan subalgebra h of g:

dL(H)Pα,βγ =((α+ β − γ)L1 + γL2

)(H)Pα,βγ .

This means that Pα,βγ is a weight vector associated to the weight

σα,βγ = (α+ β − γ, γ, 0, . . . , 0) = (k − γ, γ, 0, . . . , 0).

3. If i and j denote positive integers, if X denotes Ei,j − En+j,n+ior Ei,n+j +Ej,n+i when 1 ≤ i < j ≤ n, or Ei,n+i when 1 ≤ i ≤ n,then:

dL(X)Pα,βγ = 0.

This means that the highest weight condition is satisfied.

In other words, Pα,βγ is a highest weight vector of L. Under the actionof L, Pα,βγ generates an irreducible invariant subspace V α,β

γ of Hα,β.We will call V α,β

γ a component of L.

Proof:

60

For simplicity, let us just write P instead of Pα,βγ .

1. Multiplying out the brackets of P gives a list of monomials that areall homogeneous of degree γ in both w and z. Combining this with thepowers of the terms w1 and z1 that sit before the brackets and the factthat no variable appears at the same time as its conjugate, we see thatP belongs to Hα,β(CN ).

2. and 3. We use Formula (4.5) for the specific elements of g we needto consider:

• ifH belongs to h and if h1, ..., hn,−h1, ...,−hn denote the complexdiagonal terms of H:

dL(H)P (z, w, z, w) =n∑i=1

hi

(−zi

∂P

∂zi+ wi

∂P

∂wi+ zi

∂P

∂zi− wi

∂P

∂wi

).

So:

dL(H)P (z, w, z, w) = h1w1∂P

∂w1+ h2w2

∂P

∂w2

+ h1z1∂P

∂z1+ h2z2

∂P

∂z2. (4.6)

• if X = Ei,j − En+j,n+i (1 ≤ i < j ≤ n):

dL(X)P (z, w, z, w) = −zj∂P

∂zi+ wi

∂P

∂wj+ zi

∂P

∂zj− wj

∂P

∂wi.

So:

– for j ≥ 3:dL(X)P = 0.

– for i = 1 and j = 2 (only remaining case):

dL(X)P (z, w, z, w) = w1∂P

∂w2+ z1

∂P

∂z2. (4.7)

61

• if X = Ei,n+j + Ej,n+i (1 ≤ i < j ≤ n):

dL(X)P (z, w, z, w) = −wj∂P

∂zi− wi

∂P

∂zj+ zj

∂P

∂wi+ zi

∂P

∂wj.

So:dL(X)P = 0.

• If X = Ei,n+i (1 ≤ i ≤ n):

dL(X)P (z, w, z, w) = −wi∂P

∂zi+ zi

∂P

∂wi.

So:dL(X)P = 0.

All that remains to be done is compute the partial derivatives in for-mulas (4.6) and (4.7) to find:

• (i) ∀ H ∈ h: dL(H)P =((α+ β − γ)h1 + γh2

)P .

• (ii) dL(E1,2 − En+2,n+1)P = 0.

To make computations easier we set B = w2z1 − w1z2.

To establish (i), let us compute dL(H)P (z, w, z, w):

h1w1((α− γ)wα−γ−1

1 z1β−γBγ − γz2w

α−γ1 z1

β−γBγ−1)

+

h2w2(γz1w

α−γ1 z1

β−γBγ−1)

+

h1z1((β − γ)wα−γ1 z1

β−γ−1Bγ + γw2wα−γ1 z1

β−γBγ−1)

+

h2z2(−γw1w

α−γ1 z1

β−γBγ−1). (4.8)

Then we can organise the terms of (4.8) to get

Bγ(h1(α− γ)wα−γ1 z1

β−γ + h1(β − γ)wα−γ1 z1β−γ

)+

Bγ−1h1γwα−γ1 z1

β−γ (w2z1 − w1z2) +Bγ−1h2γw

α−γ1 z1

β−γ (w2z1 − w1z2) , (4.9)

62

which can be rewritten((α+ β − γ)h1 + γh2

)wα−γ1 z1

β−γBγ . (4.10)

We finally recognise the desired expression:((α+ β − γ)h1 + γh2

)P .

To establish (ii), let us compute dL(E1,2 − En+2,n+1)P (z, w, z, w):(w1

∂P

∂w2+ z1

∂P

∂z2

)(z, w, z, w) =

w1γz1wα−γ1 z1

β−γBγ−1 − z1γw1wα−γ1 z1

β−γBγ−1,

which obviously equals 0.

End of proof.

Each component corresponds to a specific eigenvalue of the Casimiroperator ΩL of L. Proposition 2.6 and Theorem 4.2 imply:

Corollary 4.3. Consider the irreducible representation (L, V α,βγ ). We

remind the reader that its highest weight is (α+ β− γ)L1 + γL2. Then(denoting by Id the identity map):

ΩL = −((α+ β − γ)2 + 2n(α+ β) + γ2 − 2γ

)Id.

4.1.3 Isotypic decomposition with respect to Sp(n)Again we fix k ∈ N and (α, β) ∈ N2 such that α+β = k. We now wantto establish that Hα,β is the direct sum of all subspaces V α,β

γ . We dothis by computing dimensions and showing that the dimensions of thevarious V α,β

γ add up to the dimension of Hα,β. We point out that inthe formulas to come, we use the standard convention 0! = 1.

Proposition 4.4. The dimension of V α,βγ is given by the following

formula:

dimV α,βγ = (k − γ +N − 2)! (γ +N − 3)! (k − 2γ + 1) (k +N − 1)

(k − γ + 1)! γ! (N − 1)! (N − 3)! .

We denote this dimension dkγ (omitting the values of α and β).

63

Proof:

We apply Weyl’s dimension formula (2.4), choosing the highest weightσ = σα,βγ (given by Theorem 4.2), ρK = (n, n − 1, ..., 1) and using thethree types of positive roots Li − Lj , Li + Lj and 2Li introduced inSection 2.3.1 of Chapter 2. With these choices, dimV α,β

γ is equal to:

∏i<j

< σ + ρK , Li − Lj >< ρK , Li − Lj >

∏i<j

< σ + ρK , Li + Lj >

< ρK , Li + Lj >

n∏i=1

< σ + ρK , 2Li >< ρK , 2Li >

. (4.11)

Looking into each product, we see that the terms for i ≥ 3 all cancelout so that (4.11) can be rewritten as:

n∏j=2

< σ + ρK , L1 − Lj >< ρK , L1 − Lj >

n∏j=3

< σ + ρK , L2 − Lj >< ρK , L2 − Lj >

n∏j=2

< σ + ρK , L1 + Lj >

< ρK , L1 + Lj >

n∏j=3

< σ + ρK , L2 + Lj >

< ρK , L2 + Lj >

< σ + ρK , 2L1 >

< ρK , 2L1 >

< σ + ρK , 2L2 >

< ρK , 2L2 >. (4.12)

With σ + ρK = (k − γ + n, γ + n− 1, n− 2, ..., 1), (4.12) becomes:

(k − γ + n)− (γ + n− 1)n− (n− 1)

n∏j=3

(k − γ + n)− (n− j + 1)n− (n− j + 1)

(k − γ + n) + (γ + n− 1)n+ (n− 1)

n∏j=3

(γ + n− 1)− (n− j + 1)(n− 1)− (n− j + 1)

n∏j=3

(k − γ + n) + (n− j + 1)n+ (n− j + 1)

n∏j=3

(γ + n− 1) + (n− j + 1)(n− 1) + (n− j + 1)

2(k − γ + n)2n

2(γ + n− 1)2(n− 1) . (4.13)

64

Formula (4.13) can be rewritten as:

k − 2γ + 11

n∏j=3

k − γ + j − 1j − 1

n∏j=3

γ + j − 2j − 2

k +N − 1N − 1

n∏j=3

k − γ +N − j + 1N − j + 1

n∏j=3

γ +N − jN − j

k − γ + n

n

γ + n− 1n− 1 . (4.14)

Formula (4.14) can be reorganised as:

n∏j=2

k − γ + j

j

n−1∏j=1

γ + j

j

n∏j=3

k − γ +N + 1− jN + 1− j

n∏j=3

γ +N − jN − j

(k − 2γ + 1)(k +N − 1)(N − 1) . (4.15)

Writing products in terms of factorials we get the following expression:

(k − γ + n)!(k − γ + 1)!n!

(γ + n− 1)!γ!(n− 1)!

(k − γ +N + 1− 3)!(N + 1− n− 1)!(k − γ +N + 1− n− 1)!(N + 1− 3)!

(γ +N − 3)!(N − n− 1)!(γ +N − n− 1)!(N − 3)!

(k − 2γ + 1)(k +N − 1)(N − 1) . (4.16)

Formula (4.16) becomes:

(k − γ + n)!(k − γ + 1)!n!

(γ + n− 1)!γ!(n− 1)!

(k − γ +N − 2)!n!(k − γ + n)!(N − 2)!

(γ +N − 3)!(n− 1)!(γ + n− 1)!(N − 3)!

(k − 2γ + 1)(k +N − 1)(N − 1) .

One finishes the proof by noticing various cancellations.

65

End of proof.

Proposition 4.5. The dimension of Hα,β can be written as the follow-ing sum:

dim Hα,β =∑

γ∈Iα,βγ

dimV α,βγ .

Proof:

We start by supposing α ≤ β, so that γ ≤ α. Using Propositions 4.1and 4.4 and several cancelations, we just need to prove this equality:

α∑γ=0

(k − γ +N − 2)!(γ +N − 3)!(k − 2γ + 1)(k − γ + 1)!γ! =

(α+N − 2)!(β +N − 2)!(N − 2)α!β! . (4.17)

If β ≤ α, so that γ ≤ β, we need to prove:

β∑γ=0

(k − γ +N − 2)!(γ +N − 3)!(k − 2γ + 1)(k − γ + 1)!γ! =

(α+N − 2)!(β +N − 2)!(N − 2)α!β! .

These two formulas are in fact equivalent, because of the symmetricalroles of α and β. So one just has to prove Formula (4.17); in thisformula, let us set µ = k + 1 and A = N − 3, which changes Formula(4.17) into

α∑γ=0

(A+ µ− γ)!(A+ γ)!(µ− 2γ)(µ− γ)!γ! =

(α+A+ 1)!(µ− α+A)!(A+ 1)α!(µ− α− 1)! . (4.18)

Forgetting about the exact expression of µ, we shall prove Formula(4.18) by induction on α, proving the following statement:

∀α ∈ N, ∀µ ≥ 2α+ 1 : Formula (4.18) is true. (4.19)

66

One easily sees that Statement (4.19) is true for α = 0.

Let us now suppose that it is true for given α ∈ N and consider α + 1and some µ ≥ 2(α+ 1) + 1. Then

α+1∑γ=0

(A+ µ− γ)!(A+ γ)!(µ− 2γ)(µ− γ)!γ!

separates into α∑γ=0

(A+ µ− γ)!(A+ γ)!(µ− 2γ)(µ− γ)!γ!

+

(A+ µ− α− 1)!(A+ α+ 1)!(µ− 2α− 2)(µ− α− 1)!(α+ 1)! .

Because µ ≥ 2(α + 1) + 1 ≥ 2α + 1, we can apply the induction hy-pothesis to the sum and obtain:

(α+A+ 1)!(µ− α+A)!(A+ 1)α!(µ− α− 1)! +

(A+ µ− α− 1)!(A+ α+ 1)!(µ− 2α− 2)(µ− α− 1)!(α+ 1)! =

(α+A+ 1)!(µ− α+A− 1)!(µ− α− 1)!α!

(µ− α+A

A+ 1 + µ− 2α− 2α+ 1

). (4.20)

One can check that the brackets can be written as(µ− α− 1)(α+A+ 2)

(A+ 1)(α+ 1)so that (4.20) becomes:

(α+A+ 1)!(µ− α+A− 1)!(µ− α− 1)!α!

(µ− α− 1)(α+A+ 2)(A+ 1)(α+ 1) =

(α+A+ 2)!(A+ µ− α− 1)!(µ− α− 1)(A+ 1)(µ− α− 1)!(α+ 1)! =(

(α+ 1) +A+ 1)!(µ− (α+ 1) +A

)!

(A+ 1)(α+ 1)!(µ− (α+ 1)− 1

)!.

This finishes the induction step and thus the proof.

67

End of proof.

The isotypic decomposition of the restriction of L to Hα,β follows from:

Theorem 4.6. Hα,β =⊕

γ∈Iα,βV α,βγ .

Proof:

The Peter-Weyl theorem 2.2 implies that two inequivalent subrepre-sentations of a unitary representation of a compact Lie group must acton orthogonal subspaces. Therefore the spaces V α,β

γ are pairwise or-thogonal (the corresponding subrepresentations each have a differenthighest weight, so they are pairwise inequivalent). Proposition 4.5 saysthat the dimensions of all the V α,β

γ add up to the dimension of Hα,β;Theorem 4.6 follows.

End of proof.

To summarise the whole of Section 4.1, putting it back into contextwith regards to our induced representations πiλ,δ, we have proved:

Theorem 4.7 (Isotypic decomposition of πiλ,δ). Consider any λ ∈ Rand δ ∈ Z. Then:

πiλ,δ∣∣K∼=

∑⊕

(α,β)∈N2

δ=β−αγ∈Iα,β

L∣∣V α,βγ

.

In this isotypic decomposition, the multiplicity of each K-type is 1 (onesays that the K-types are multiplicity free).

4.2 Right action of Sp(1)

4.2.1 Isotypic decomposition with respect to Sp(1)

Denote by(z, w) = (z1, . . . , zn, w1, . . . , wn)

68

the coordinates of CN , by h = (h1, . . . , hn) the coordinates of Hn anduse the identification h = (z + jw) ∈ Hn ←→ (z, w) ∈ CN .

Right multiplication of a quaternionic vector h by a quaternion q con-sists simply in multiplying all quaternionic coordinates by q on theright, obtaining the quaternionic vector hq.

Consider any subset S of Hn and assume that it is stable under rightmultiplication by unit quaternions. Let F be any subset of the complexvector space f : S −→ C. Then the right action R of UH on F isdefined by:

R(q)f(x) = f(xq),

for all (q, f, x) ∈ UH×F × S; we write R(q)f instead of(R(q)

)(f).

Consider a unit quaternion q = a+ jb ∈ UH. The rules of quaternionicmultiplication imply (as explained in Section 2.4.3):

∀h = z + jw ∈ Hn, hq = (az − bw) + j(aw + bz). (4.21)

The subset S of Hn identifies with a subset of CN , that we also denoteby S. As explained in Chapter 2 (section 2.4.3) the quaternion q (seenas a 1× 1 matrix) identifies with the matrix(

a −bb a

)∈ Sp(1)

that we also denote by q. These considerations explain our choice ofaction of Sp(1) on functions of the complex variables (z, w):

Definition 4.8. The right action of Sp(1) on F , again denoted by R,is defined for all q ∈ Sp(1) and (z, w) ∈ S by:

R(q)f(z, w) = f (az − bw, aw + bz) .

An element f of F is right-invariant if it is invariant under the rightaction of Sp(1), meaning that for all q ∈ Sp(1): R(q)f = f .

Let us take S = S2N−1 and come back to L2(S2N−1

). One can show:

69

Proposition 4.9. The right action R of Sp(1) on L2(S2N−1

)is a

continuous (it satisfies the continuity property stated in Section 2.1)and unitary representation.

Let us fix k ∈ N. Using the usual identification CN ' R2N , denote byFk the subspace of L2

(S2N−1

)that consists of functions on S2N−1 (one

should really speak of their equivalence classes) that are restrictions ofhomogeneous polynomial functions on R2N of degree k; Fk is stableunder the right action R of Sp(1). Because Fk is finite-dimensional, allits elements are smooth vectors of R. As we have done previously, ho-mogeneity enables us to identify elements of Fk with the homogeneouspolynomials they come from. The right action of Sp(1) is compatiblewith these identifications. We also denote by R the right action ofSp(1) on homogeneous polynomials of degree k; they are all smoothvectors of R.

Ultimately, our aim is to focus on the subspace Yk of Fk and the corre-sponding space of harmonic polynomials Hk; these spaces are invariantunder R, as we shall see.

As we have already mentioned, to study irreducible invariant subspacesunder the action of a compact group, one complexifies its Lie algebra.The complexification of sp(1) = su(2) is:

su(2)⊕ i su(2) = sl(2,C).

A basis over the field R of su(2) is given by the three matrices :

A =(i 00 −i

); B =

(0 ii 0

); C =

(0 −11 0

).

A basis over the field C of sl(2,C) is given by the three matrices :

• h =(

1 00 −1

)= 0 + i(−A) ∈ su(2)⊕ isu(2);

• e =(

0 10 0

)= −C

2 + i(−B2

)∈ su(2)⊕ isu(2);

70

• f =(

0 01 0

)= C

2 + i(−B2

)∈ su(2)⊕ isu(2).

We have for t ∈ R the following exponentials:

• exp(−tA) =(e−it 0

0 eit

);

• exp(−tB

2

)=(

cos(t2)

−i sin(t2)

−i sin(t2)

cos(t2) )

;

• exp(−tC

2

)=(

cos(t2)

sin(t2)

− sin(t2)

cos(t2) ).

These exponentials belong to Sp(1) and therefore correspond to specificunit quaternions qt = at+jbt, with (at, bt) ∈ C×C (see Lemma 2.11 torecall how one identifies quaternionic matrices with complex matrices):

• exp(−tA) corresponds to qt = e−it + 0j;

• exp(−tB

2

)corresponds to qt = cos

(t2)− j i sin

(t2);

• exp(−tC

2

)corresponds to qt = cos

(t2)− j sin

(t2).

Consider any homogeneous polynomial P of degree k. We know thatP is a smooth vector of R, so, by definition, we have for all X ∈ sp(1)and (z, w) ∈ Cn×Cn:

dR(X)P (z, w) = d

dt

∣∣∣∣t=0

(R (exp(tX))P (z, w)

).

Extend dR to a complex representation dCR of sl(2,C) by defining forall Z = X + iY ∈ sl(2,C) (taking (X,Y ) ∈ sp(1)× sp(1)):

dCR(Z) = dR(X) + idR(Y ).

We will write Z instead of dCR(Z) and Z · P instead of dCR(Z)P (inother words, we see Z as a differential operator).

Using the formulas that link e, f, h to A,B,C, the exponentials of−tA, −tB2 , −tC2 and the definitions of dR and dCR, one can prove:

71

Proposition 4.10. Simply denote by e, f, g the respective differentialoperators dCR(e) , dCR(f) , dCR(h). Then:

• e = ∑nr=1

(wr

∂∂zr− zr ∂

∂wr

);

• f = ∑nr=1

(zr

∂∂wr− wr ∂

∂zr

);

• h = ∑nr=1

(zr

∂∂zr− zr ∂

∂zr+ wr

∂∂wr− wr ∂

∂wr

).

Let us now look into the direct sum Hk =k⊕s=0

Hk−s,s. We know that

each Hk−s,s breaks intomin(k−s,s)⊕

γ=0V k−s,sγ and that each V k−s,s

γ contains

a highest weight vector denoted by P k−s,sγ (see Theorem 4.2). Usingformulas of Proposition 4.10, one shows:

Theorem 4.11. Consider integers k, α, β, γ such that 0 ≤ γ ≤ min(α, β)and α+ β = k. Then:

• h · Pα,βγ = (α− β)Pα,βγ ;

• When γ ≤ β : e · Pα,βγ = (β − γ)Pα+1,β−1γ ;

• When γ ≤ α : f · Pα,βγ = (α− γ)Pα−1,β+1γ .

4.2.2 K-type diagram

Figure 4.1 captures the contents of Theorems 4.2 and 4.11. In this di-agram, the thick black dots represent the highest weight vectors Pα,βγ

(α+ β = k).

Let us take a closer look at Figure 4.1. For any fixed integer γ suchthat 0 ≤ γ ≤ E

(k2

)(the function E assigns to a real number its integer

part):

72

Figure 4.1: combining the left action of Sp(n) and the right action ofSp(1) within Hk, assuming here k ≥ 5.

73

• The arrow beneath γ points to a vertical set of componentswhose direct sum defines an invariant subspace V k

γ (obviously notirreducible) under the left action of Sp(n):

V kγ =

k−γ⊕α=γ

V α,k−αγ .

• The components in V kγ all correspond to the highest weight

(k−γ, γ, 0, · · · , 0) (given by Theorem 4.2) and are therefore equi-valent; in particular they all have the same dimension dkγ .

• A vertical set of thick black dots defines a basis of an irre-ducible invariant subspaceW k

γ under the right action of Sp(1) (ofdimension k − 2γ + 1):

W kγ = VectCPα,k−αγ α=γ,...,k−γ ⊂ Vγ .

The polynomial P k−γ,γγ is a highest weight vector of W kγ and

corresponds to the highest weight k − 2γ.

• Because the left action of Sp(n) and the right action of Sp(1)commute, applying L toW k

γ gives irreducible invariant subspacesof R contained in V k

γ ; these subspaces define subrepresentationsof R which are equivalent to the restriction of R toW k

γ ; therefore,they all have the same dimension k − 2γ + 1.

From the fact that the left action L of Sp(n) is transitive in each com-ponent, we deduce:

Theorem 4.12. The isotypic of the right action R of Sp(1) on Hk isgiven by:

Hk =E( k2 )⊕γ=0

dkγWkγ .

In particular, the multiplicity of R|Wkγis equal to dkγ.

Remark 4.13. The structure of Hk, which we have studied with respectto the left action of Sp(n) and the right action of Sp(1), appears in [22]

74

(see Proposition 5.1), where it is expressed as follows in terms of tensorproducts (using our system of notation):

Hk∣∣Sp(n)×Sp(1) =

E( k2 )∑γ=0

V k−γ,γγ ⊗W k

γ .

The credit one can give to our work is to explain this structure in apersonal and self-contained way.

4.2.3 Bi-invariant polynomials

This result follows from Theorem 4.12:

Corollary 4.14.

1. Consider k ∈ N and P ∈ Hk. If the subspace VectCP is stableunder the right action R of Sp(1), then k is even and P belongsto V α,α

α , where we set α = k2 .

2. Consider k ∈ N, suppose k is even and set α = k2 .

• The subspace VectCPα,αα is stable under the right actionR of Sp(1); in fact, Pα,αα is right-invariant.

• The component V α,αα decomposes into dkα one-dimensional

irreducible subspaces, along which R is just the identity re-presentation. This implies that all elements of V α,α

α areright-invariant.

Another consequence of the previous section is:

Corollary 4.15. Consider an element f of L2(S2N−1). Then f isright-invariant if and only if f belongs to the following Hilbert sum:⊕

α∈NV α,αα .

Let us now consider the subgroup 1× Sp(n− 1) ⊂ Sp(n): it consists of

block diagonal quaternionic matrices(

1 00 A

), where A ∈ Sp(n− 1).

75

Definition 4.16. Consider k ∈ N. A polynomial of Hk and its cor-responding spherical harmonic in Yk are said to be bi-invariant if:

• they are invariant under the left action of 1× Sp(n− 1);

• they are also invariant under the right action of Sp(1).

Theorem 4.17.

1. Consider any (α, β) ∈ N2 and any γ ∈ N such that γ ≤ min(α, β).Denote by Invα,βγ the complex vector space that consists of allpolynomials of V α,β

γ which are invariant under the left action of1× Sp(n− 1). Then:

dimC(Invα,βγ ) = α+ β − 2γ + 1.

We point out that this dimension is higher than or equal to 1 andthat it equals 1 if and only if α = β = γ.

2. Consider k ∈ N.

• If a polynomial P ∈ Hk is bi-invariant, then k is even andP belongs to V α,α

α , where we set α = k2 .

• Assume that k is even and set α = k2 . Then Hk contains a

unique, up to a constant, bi-invariant polynomial (and thispolynomial belongs to V α,α

α ).

Proof:

Item 1: our proof relies on Želobenko’s branching theorem with respectto the pair (Sp(n), Sp(n− 1)), namely Theorem 4 of Chapter XVIII in[53]. This theorem implies that the number of times an irreduciblerepresentation of 1 × Sp(n − 1) ' Sp(n − 1) with highest weight c =(c1, · · · , cn−1) occurs in an irreducible representation of Sp(n) withhighest weight a = (a1, · · · , an) is equal to the number of non-negativeintegral n-tuples b = (b1, · · · , bn) such that the following two rows ofinequalities are satisfied:

a1 ≥ b1 ≥ a2 ≥ b2 ≥ a3 ≥ b3 ≥ · · · ≥ an ≥ bn ≥ 0;b1 ≥ c1 ≥ b2 ≥ c2 ≥ b3 ≥ · · · ≥ cn−1 ≥ bn.

76

Remember that the highest weight of the left action of Sp(n) on V α,βγ

is the n-tuple a = (α+ β − γ, γ, 0, · · · , 0).

Also, the highest weight theorem (see for instance [27], Section 7 ofChapter IV) implies that a polynomial P ∈ V α,β

γ is invariant underthe left action of 1 × Sp(n − 1) if and only if the restriction of theleft action of 1 × Sp(n − 1) to the 1-dimensional space VectC(P ) de-fines an irreducible representation of Sp(n− 1) whose highest weight isc = (0, · · · , 0).

Taking a = (α+ β − γ, γ, 0, · · · , 0) and c = (0, · · · , 0), the two rows ofinequalities become:

α+ β − γ ≥ b1 ≥ γ ≥ b2 ≥ 0 = b3 = · · · = 0 = bn = 0;b1 ≥ 0 = b2 = 0 = b3 = · · · = 0 = bn.

Thus the only non-zero coordinate of the n-tuple b is b1 and we are leftwith counting the number of integers b1 between α+β−γ and γ. Thisnumber is α + β − 2γ + 1. Because γ ∈ [0,min(α, β)], this number ishigher than or equal to 1 and it equals 1 if and only if α = β = γ.

Item 2: it just follows from Item 1 and Corollary 4.14.

End of proof.

Let us fix any α ∈ N and write k = 2α. We now set to determine,explicitly, the unique (up to a constant) bi-invariant polynomial of Hk

(given by Theorem 4.17).

Define the following set:

Aα = (a, b) ∈ N2 / a+ b = α.

Let us consider the polynomials U, V, P defined by:

• U = z1z1 + w1w1;

• V = ∑nr=2 (zrzr + wrwr);

77

• P = ∑A∈Aα νA U

a V b, where the various νA denote complexscalars (undetermined at this stage).

Polynomial P is obviously:

• homogeneous, of homogeneous degree (α, α);

• bi-invariant.

We want P to be harmonic (in which case P actually belongs to thesubspace V α,α

α of Hα,α). To ensure that this be indeed the case, weneed to apply the complex Laplace operator and make P belong to itskernel. Remember that this operator is:

∆C = 4n∑r=1

(∂2

∂zr∂zr+ ∂2

∂wr∂wr

).

One easily checks that:

∆CP (z, w) = νA(a2 + a)Ua−1 V b + νA(b2 + (2n− 3)b)Ua V b−1.

Given A ∈ A, let us define:

• C1(A) = νA(a2 + a) and T1(A) = Ua−1 V b;

• C2(A) = νA(b2 + (2n− 3)b) and T2(A) = Ua V b−1.

We have:

∆CP (z;w) = 4∑A∈A

(C1(A) · T1(A) + C2(A) · T2(A)

). (4.22)

By multiplying the brackets out in powers of U and V , we can writedown the list of monomials that come from Ti(A), given A = (a, b) ∈ Aand i ∈ 1, 2; let us denote byMi(A) the set of such monomials.

Proposition 4.18. Consider two pairs A = (a, b) and A′ = (a′, b′)such that A 6= A′. Then:

1. M1(A) ∩M1(A′) = ∅;

2. M2(A) ∩M2(A′) = ∅;

78

3.(M1(A) ∩M2(A′) 6= ∅

)⇔(A′ = (a− 1, b+ 1)

);

4.(M2(A) ∩M1(A′) 6= ∅

)⇔(A′ = (a+ 1, b− 1)

).

Proof:

The monomials of Mi(A) (resp. Mi(A′)), again with i ∈ 1, 2, splitinto a first part which is a homegeneous polynomial of the variablesz1, w1, z1, w1 and of homogeneous degree (a, a) (resp. (a′, a′)) anda second part which is a homegeneous polynomial of the variablesz2, . . . , zn, w2, . . . , wn, z2, . . . zn, w2, . . . wn and of homogeneous degree(b, b) (resp. (b′, b′)). For monomials to coincide, their homogeneousdegrees must of course coincide, which is enough to establish (1), (2),(3) and (4).

End of proof.

Consider two pairs A = (a, b) ∈ A and A′ = (a′, b′) ∈ A such thatA 6= A′.

• If a monomial appears in both M1(A) and M2(A′) (implyingA′ = (a − 1, b + 1)), then the coefficients it appears with areC1(A) times some combinatorial coefficient and C2(A′) times thesame combinatorial coefficient. Thus we must have:

C1(A) + C2(A′) = 0.

• If a monomial appears in both M2(A) and M1(A′) (implyingA′ = (a+1, b−1)), then the coefficients it appears with are C2(A)times a combinatorial coefficient and C1(A′) times the same com-binatorial coefficient. Thus we must have:

C2(A) + C1(A′) = 0. (4.23)

Above considerations finally show us how to choose coefficients νA inthe definition of P so as to ensure that P be harmonic. Figure 4.2helps understand the following steps (assuming that α ≥ 1):

79

Figure 4.2: induction method to find bi-invariant polynomials.

• List all pairs of Aα from left to right:

(α, 0) , (α− 1, 1) , . . . , (1, α− 1) , (0, α).

• Start for instance with the pair A0 = (0, α) on the far right andassign to it any coefficient νA0 = ν0; it determines C1(A0) (whichis in fact 0) and C2(A0).

• Move on to the pair A1 = (1, α− 1) on the left and compute theonly suitable coefficient νA1 = ν1 by using Equation (4.23).

• And so on, until the pair Aα = (α, 0) has been reached and itscoefficient νAα = να computed.

Let us refer to this as the induction method.

Theorem 4.19. Given any even integer k ∈ N and setting α = k2 , the

induction method computes the unique (up to a constant) bi-invariantpolynomial of Hk (and this polynomial belongs to V α,α

α ).

Examples 4.2.1.

• For α = 1, one finds P = (1− n)U + V .

• For α = 2, one finds P = (2n−1)(n−1)3 U2 + (1− 2n)UV + V 2.

80

Chapter 5

Compact picture andhypergeometric equation

In this chapter, we work with the compact picture of our representa-tions πiλ,δ, using the setting and results of Chapter 4.

5.1 Quaternionic projective spaceIn this work, straight lines are defined with respect to right multiplica-tion: given x ∈ Hn, the quaternionic line through x is the vector space

xH = xh / h ∈ H.

The quaternionic projectice space Pn−1(H) is naturally the set of quater-nionic lines of Hn. Quaternionic matrices act naturally on quaternioniclines: given x ∈ Hn, an invertible matrix M ∈ GL(n,H) assigns toxH the quaternionic line through Mx. We call this action the natu-ral action of GL(n,H) on Pn−1(H). We now study the orbits underthe restriction of this action to the subgroup 1 × Sp(n − 1). GivenxH ∈ Pn−1(H), we denote by O(xH) the orbit of xH. We show in thenext proposition that the orbits can be parametrised by a single realvariable.

Proposition 5.1. Consider any x = (x1, ..., xn) ∈ Hn. There exists aunique θ ∈

[0, π2

]such that, denoting x(θ) = (cos θ, 0, ..., 0, sin θ) ∈ Hn,

81

the quaternionic line x(θ)H belongs to O(xH). Moreover, θ is explicitlygiven by the following formulas:

• if x1 6= 0, then θ = arctan (‖x′‖), taking x′ =(x2x1, ..., xnx1

);

• if x1 = 0, then θ = π2 .

Proof:

Existence:

Suppose first that x1 6= 0. Then, writing x′ =(x2

1x1, ..., xn

1x1

), we

have:

xH = (x1, ..., xn)H =(x1

1x1, x2

1x1, ..., xn

1x1

)H = (1, x′)H.

Because 1×Sp(n−1) acts transitively on spheres of Hn−1, there exists

a matrix(

1 00 k

), with k ∈ Sp(n − 1), that takes (1, x′)H to the

quaternionic line (1, 0, ..., 0, ‖x′‖)H (remember that ‖ · ‖ denotes thenorm of Hn−1).

This quaternionic line can be rewritten

(1, 0, ..., 0, ‖x′‖

)H =

(1√

1 + ‖x′‖2, 0, ..., 0, ‖x′‖√

1 + ‖x′‖2

)H

and then simply(1, 0, ..., 0, ‖x′‖

)H = (cos θ, 0, ..., 0, sin θ)H

for some θ ∈[0, π2

[(see Figure 5.1).

If now x1 = 0, thus ‖x′‖ = 1, then one can choose k so that(

1 00 k

)carries xH onto (0, ...0, 1)H, which gives θ = π

2 .

This establishes existence and formulas of θ.

82

Figure 5.1: a single variable θ to parametrise orbits.

83

Uniqueness:

Suppose θ and θ′ belong to[0, π2

]and that the quaternionic lines x(θ)H

and x(θ′)H belong to O(xH). Then there exists k ∈ 1×Sp(n−1) suchthat kx(θ′)H = x(θ)H, which implies that there exists q ∈ UH suchthat:

kx(θ′) = x(θ)q. (5.1)

Suppose that n ≥ 3 and write

k =

1 0 00 A B0 L b

with b ∈ H, A ∈ Mn−2,n−2(H), L ∈ M1,n−2(H) and B ∈ Mn−2,1(H).Equation (5.1) becomes: cos θ′

B sin θ′b sin θ′

=

q cos θ0

q sin θ

.We now split the rest of the proof into three cases.

• Case 1: θ, θ′ ∈[0, π2

[. Then q = cos θ′

cos θ , B sin θ′ = 0 and

b sin θ′ = sin θ cos θ′cos θ . (5.2)

Subcase 1: θ′ 6= 0. Then B must be 0 and Proposition 2.14 (seeChapter 2) implies L = 0 but more importantly for us |b| = 1.Thus from (5.2) it follows that | tan θ| = | tan θ′|, which thenimplies tan θ = tan θ′ and θ = θ′.Subcase 2: θ′ = 0. Then sin θ′ = 0 and cos θ′ = 1, so (5.2)becomes tan θ = 0 and finally θ = 0 = θ′.

• Case 2: θ = π2 . Then cos θ = 0 implies cos θ′ = 0 and finally

θ′ = π2 = θ.

• Case 3: θ′ = π2 . Then cos θ′ = 0 implies q cos θ = 0, thus cos θ = 0

and finally θ = π2 = θ′ (because q 6= 0).

84

There remains the case n = 2, for which we choose k as(

1 00 b

)with

b ∈ UH. Similarly to what is detailed above, we also obtain θ = θ′.

End of proof.

5.2 Reducing the number of variablesFrom now on, throughout the rest of this chapter, the setting is thefollowing:

• We fix an even integer k ∈ N and write α = k2 .

• We consider the space Hk and its unique, up to a constant, bi-invariant polynomial P , which in fact belongs to V α,α

α ⊂ Hα,α

(see Theorem 4.19).

• We denote by fP the spherical harmonic that corresponds to P ,that is:

fP = P |S2N−1 ∈ Yα,α .

We remind the reader that, by definition, the right action of Sp(1)corresponds, in the quaternionic setting, to right scalar multiplicationby unit quaternions.Invariance under the right action of Sp(1) enables fP to descend to afunction f on Pn−1(H):

f : Pn−1(H) −→ CxH 7→ fP (x),where x is assumed to belong to S2N−1.

Transferring, in the way one expects, the left action of K on sphericalharmonics to a left action, also denoted by L, of K on complex-valuedfunctions defined on Pn−1(H), we write

L(k)f(xH) = f((k−1x)H

)for all (k, x) ∈ K × S2N−1.Invariance of fP under the left action of 1×Sp(n− 1) implies that f isinvariant along the orbits of the left action of 1×Sp(n−1) on Pn−1(H).

85

Therefore, f descends to a function on the classifying set O of orbits,leading in turn to the following new function (via Proposition 5.1):

F :[0, π2

]−→ C

θ 7→ F (θ) = f (x(θ)H) = fP (x(θ)) .

Now, apply Formula (2.6) to write down the Casimir operator ΩL ofL. This is how this operator acts on fP , given an element x ∈ S4n−1:

ΩL(fP )(x) = ∑X∈Bk

∂2

∂t2

∣∣∣t=0

(π(exp(−tX))(fP )(x)

)=∑

X∈Bk

∂2

∂t2

∣∣∣∣t=0

(fP(

exp(−tX)x)),

as long as each expression ∂2

∂t2

∣∣∣t=0

(fP (exp(−tX)x)

)above is well

defined, which is the case because fP is smooth.

Corollary 4.3 says that the component V α,αα is associated to the eigen-

value Λ = −(2α2 + (4n− 2)α) of ΩL:

ΩL(fP )(x) = ΛfP (x).

Given θ ∈[0, π2

], one can apply this equation to x = x(θ) (see Section

5.1 for notation):

∑X∈Bk

∂2

∂t2

∣∣∣t=0

(fP (exp(−tX)x(θ))

)= ΛfP (x(θ)) . (5.3)

One can convert this into a differential equation satisfied by the func-tion F of the real variable θ. But this requires to know for each X ∈ Bkand each t ∈ R, which parameter ξX,θ(t) ∈

[0, π2

]labels the orbit of

exp(−tX)x(θ)H under the left action of 1 × Sp(n − 1). Proposition5.1 gives us the value of ξX,θ(t). Indeed, denoting by (y1, ..., yn) thecoordinates of exp(−tX)x(θ):

• when y1 6= 0, we have ξX,θ(t) = arctan ‖y′‖, where y′ = standsfor

(y2y1, ..., yny1

);

86

• when y1 = 0, we have ξX,θ(t) = π2 .

Then (5.3) can be written:

∑X∈Bk

∂2

∂t2

∣∣∣∣t=0

(F(ξX,θ(t)

))= ΛF (θ). (5.4)

We point out that in this equation, the expressions that appear in thesum are automatically well defined. Finally, all we have to do now isdetermine the coordinates of the various exp(−tX)x(θ) and use aboveformulas to compute ξX,θ(t); let us also point out that we obviouslyhave:

ξX,θ(0) = θ.

5.3 Hypergeometric equation

We continue to work with the setting of Section 5.2.

In Equation (5.3), all terms corresponding to elements of the Lie algebraof 1 × Sp(1) vanish, precisely because fP is invariant under the leftaction of 1× Sp(1). So the relevant elements of Bk are :

Zi = A1 , Zj = D1 , Zji = E1 , Xr = B1,r√2,

Yr,i = C1,r√2, Yr,j = F1,r√

2, Yr,ji = G1,r√

2,

where r denotes an integer such that 2 ≤ r ≤ n; let us denote by I theset of these integers.

Exponential formulas of Lemma 2.13 give, taking r ∈ I, t ∈ R andη ∈ i, j, ji:

exp(−tZη) =(

cos t− η sin t 00 Im

),

87

exp(−tXr) =

cos τ · · · − sin τ · · ·... . . . ...

sin τ · · · cos τ · · ·...

... . . .

,

exp(−tYr,η) =

cos τ · · · −η sin τ · · ·... . . . ...

−η sin τ · · · cos τ · · ·...

... . . .

,where:

• the non-explicited entries are 1 on the diagonal and 0 elsewhere;

• the dots show the diagonal, Rows 1 and r and Columns 1 and r;

• the variable τ denotes t√2 .

Dealing first with matrices Zη:

The following proposition is straightforward:

Proposition 5.2.

1. exp(−tZη)x(θ) = ((cos t− η sin t) cos θ, 0, ..., 0, sin θ).

2. Suppose θ ∈[0, π2

[. Then:

ξ′Zη ,θ(0) = 0 ; ξ′′Zη ,θ(0) = 0.

Dealing now with matrices Xr and Yr,η:

Case 1: r = n

This proposition is also straightforward:

Proposition 5.3.

88

1. exp(−tXn)x(θ) =

(cos τ cos θ − sin τ sin θ, 0, ..., 0, sin τ cos θ + cos τ sin θ) .

2. Suppose θ ∈]0, π2

[. Then for t small enough we have

cosτ cos θ − sin τ sin θ 6= 0

andξXn,θ(t) = τ + θ ∈

]0, π2

[.

Also:ξ′Xn,θ(0) = 1√

2; ξ′′Xn,θ(0) = 0.

Proposition 5.4. Consider η ∈ i, j, ji.1. exp(−tYn,η)x(θ) =

(cos τ cos θ − η sin τ sin θ, 0, ..., 0,−η sin τ cos θ + cos τ sin θ) .

2. Suppose θ ∈]0, π2

[. Then cos τ cos θ − η sin τ sin θ 6= 0 and

ξYn,η ,θ(t) = arctan

√cos2 τ sin2 θ + sin2 τ cos2 θ

cos2 τ cos2 θ + sin2 τ sin2 θ∈]0, π2

[.

Also:ξ′Yn,η ,θ(0) = 0 ; ξ′′Yn,η ,θ(0) = 1

tan(2θ) .

Proof:

A simple matrix multiplication proves Item 1. Let us now look at Item2. Because cos θ and sin θ are both non-zero and cos τ and sin τ cannotbe simultaneously equal to 0, one necessarily has

cosτ cos θ − η sin τ sin θ 6= 0.

Then the square root formula comes from Proposition 5.1 and the factthat, given any (a, b, η) ∈ R×R×i, j, ji:

|a+ ηb| =√a2 + b2.

The reason why ξYη ,θ(t) belongs to the open interval]0, π2

[is due to

the following facts:

89

• arctan maps [0,+∞[ onto[0, π2

[;

• the square root expression cannot take the value 0 because, again,neither cos θ nor sin θ are 0 and cos τ and sin τ cannot simulta-neously be equal to 0.

Now let us now compute ξ′Yη ,θ(0) and ξ′Yη ,θ(0). For this purpose, definefunctions T,U, V,W,R, S, φ by (t, x, y are real variables):

• T (t) = t√2 ;

• U(x) = cos2 x sin2 θ + sin2 x cos2 θ;

• V (x) = cos2 x cos2 θ + sin2 x sin2 θ;

• W = UV ;

• R(y) = √y;

• S = R W ;

• φ = arctan S.

One obviously has

U(0) = sin2 θ , V (0) = cos2 θ , W (0) = tan2(θ) , S(0) = tan θ , φ(0) = θ

and easily checks

U ′(0) = 0 , V ′(0) = 0 , U ′′(0) = 2 cos(2θ) and V ′′(0) = −2 cos(2θ).

By writing W ′ = U ′V−UV ′V 2 and S′ = W ′

2√W

and by differentiating W ′

and S′ one obtains:

W ′(0) = 0 , W ′′(0) = 2 cos(2θ)cos4 θ

, S′(0) = 0 and S′′(0) = cos(2θ)cos4 θ tan θ .

By writing φ′ = S′

1+S2 and by differentiating φ′ one gets:

φ′(0) = 0 and φ′′(0) = 2tan(2θ) .

Finally, writing ξ = φ T , we have:

90

• ξ′Yη ,θ(t) = (φ T )′(t) = φ′(T (t)) · T ′(t) = 1√2 · φ

′(T (t));

• ξ′′Yη,θ(t) = 1√2 · (φ

′ T )′(t) = 1√2 · φ

′′(T (t)) · T ′(t) = 12 · φ

′′(T (t));

• ξ′Yη,θ(0) = 0;

• ξ′′Yη,θ(0) = 1tan(2θ) .

Throughout the calculations, one can check that all expressions arewell defined.

End of proof.

Case 2: 2 ≤ r < n (when n ≥ 3 only)

Proposition 5.5.

1. exp(−tXr)x(θ) = (cos τ cos θ, 0, ..., 0, sin τ cos θ, 0, ..., 0, sin θ),where the term sin τ cos θ is the rth coordinate.

2. Take θ ∈]0, π2

[. If τ ∈ R is such that |τ | is small enough, then

ξXr,θ(t) = arctan( 1

cos τ√

sin2 τ + tan2 θ

).

Also:ξ′Xr,θ(0) = 0 ; ξ′′Xr,θ(0) = 1

2 tan θ .

Proof:

The proof is straightforward; here are the steps:

• Define U(τ) = sin2 τ + tan2 θ and check:

U(0) = tan2(θ) , U ′(0) = 0 and U ′′(0) = 2.

• Define V =√U and check:

V (0) = tan θ , V ′(0) = 0 and V ′′(0) = 1tan θ .

91

• Define W (τ) = cos τ and Z = VW ; then check:

Z(0) = tan θ , Z ′(0) = 0 and Z ′′(0) = 2sin(2θ) .

• Define A = arctan Z and check:

A(0) = θ , A′(0) = 0 and A′′(0) = 1tan θ .

These are the calculations with respect to the variable τ ; rememberingthat τ = t√

2 establishes Item 2.

End of proof.

Above calculations also establish:

Proposition 5.6. Consider η ∈ i, j, ji.

1. exp(−tYr,η)x(θ) = (cos τ cos θ, 0, ..., 0,−η sin τ cos θ, 0, ..., 0, sin θ),where the term −η sin τ cos θ is the rth coordinate.

2. Take θ ∈]0, π2

[. If τ ∈ R is such that |τ | is small enough, then

ξYr,θ(t) = arctan( 1

cos τ√

sin2 τ + tan2 θ

).

Also:ξ′Yr,θ(0) = 0 ; ξ′′Yr,θ(0) = 1

2 tan θ .

Assembling all terms in (5.4)

Equation (5.4) can be written:

∑η ∈ i, j, ji2 ≤ r ≤ n

M ∈ Xr, Yr,η, Zη

F ′′(ξM,θ(0)

) (ξ′M,θ(0)

)2+

F ′(ξM,θ(0)

)ξ′′M,θ(0) = ΛF (θ).

92

Combining this to Propositions 5.2, 5.3, 5.4, 5.5 and 5.6 finally givesfor θ ∈

]0, π2

[:

F ′′(θ) +( 6

tan(2θ) + 4n− 8tan θ

)F ′(θ)− 2ΛF (θ) = 0. (5.5)

If one considers the smooth diffeomorphism (onto)

ψ :]0, π2

[−→ ]0, 1[

θ 7−→ cos2 θ

and applies the change of variables u = ψ(θ), then one obtains thestandard hypergeometric equation given in the theorem below, whichis one of our main results and which summarises this chapter. Beforewe state it:

Definition 5.7. The reduced version of fP is the function

ϕP = F ψ−1.

Theorem 5.8 (Compact picture and hypergeometric equation). Con-sider α ∈ N and k = 2α. Consider the unique (up to a constant)bi-invariant spherical harmonic of Yk. Then the restriction ϕ of itsreduced version to the open interval ]0, 1[ satisfies the following hyper-geometric equation:

u(1− u)ϕ′′(u) + (2−Nu)ϕ′(u)− Λ2 ϕ(u) = 0,

where Λ = −(2α2 + (4n− 2)α

).

93

94

Chapter 6

Non-standard picture andBessel functions

We use the definition of the non-standard picture given in [8] (Section5.1, where it is actually called the non-standard model). This picturewas initiated by the authors of [35], who studied degenerate principalseries of the real symplectic group Sp(n,R).

We now fix (λ, δ) ∈ R×Z.

6.1 Non-standard picture

Let us denote by (s, u, v) the coordinates on C×Cm×Cm ' C2m+1

and define some partial Fourier transforms along specific variables ofC2m+1:

• Denote by L1τ the set of equivalence classes of functions

g : C2m+1 −→ C

such that for almost all (u, v) ∈ Cm×Cm:∫C|g(τ, u, v)| dτ < +∞.

95

For such functions g and such pairs (u, v) we set:

Fτ (g)(s, u, v) =∫Cg(τ, u, v) e−2iπRe(sτ) dτ.

• Denote by L1ξ the set of equivalence classes of functions

g : C2m+1 −→ C

such that for almost all (s, u) ∈ C×Cm:∫Cm|g(s, u, ξ)| dξ < +∞.

For such functions g and such pairs (s, u) we set:

Fξ(g)(s, u, v) =∫Cm

g(s, u, ξ) e−2iπRe<v,ξ> dξ.

Density of L1τ ∩ L2(C2m+1) and L1

ξ ∩ L2(C2m+1) in L2(C2m+1) impliesthat above formulas completely define two partial Fourier transforms,written respectively:

Fτ : L2(C2m+1) −→ L2(C2m+1);Fξ : L2(C2m+1) −→ L2(C2m+1).

We now define the partial Fourier transform F on which is based thenon-standard picture:

F = Fτ Fξ. (6.1)For functions f ∈ L1(C2m+1) ∩ L2(C2m+1), one can write:

F(f)(s, u, v) =∫C×Cm

f(τ, u, ξ) e−2iπRe(sτ+〈v,ξ〉) dτdξ. (6.2)

Finally:

Definition 6.1. The non-standard picture of πiλ,δ has L2(C2m+1) ascarrying space. The action of G is then the conjugate under F of theaction of G in the non-compact picture; in other words, F intertwinesthe action of G in the non-compact picture and the action of G in thenon-standard picture.

96

6.2 Aim of this chapter: specific highestweight vectors

Decomposition (4.4) and Theorem 4.6 tell us what the K-types of πiλ,δare: they are the isotypic components of the left action of K which sitin the spaces Hα,β such that δ = β−α. This point of view comes fromthe compact picture. Though this picture has helped us describe allK-types, it is interesting to see what they look like in other pictures,in particular the non-standard one. This is the point of the rest ofthis chapter. Recalling Theorem 4.2, for technical issues we restrict toK-types labelled by γ = 0, namely the components V α,β

0 whose highestweight vectors are

Pα,β0 (z, w, z, w) = wα1 z1β.

Because we intend to use the right action of Sp(1), we consider anentire space Hk of harmonic polynomials.

So we now fix (for the rest of this chapter) any k ∈ N and considervalues of α and β such that:

α+ β = k.

We denote by gα,β the restriction of Pα,β0 to the unit sphere S2N−1.

Let us call g the function in the induced picture that corresponds togα,β. It extends gα,β, meaning: g|

S2N−1 = gα,β. Remember that g mustsatisfy the covariance relation for all non-zero complex numbers c:

g(c ·) =(c

|c|

)−δ|c|−iλ−N g.

We definea(s, u) =

√1 + 4|s|2 + ‖u‖2

andr(s, u, v) =

√a2(s, u) + ‖v‖2.

97

By restricting g to the complex hyperplane 1 × C×Cm×Cm, weobtain the function Gα,β defined on C2m+1 by:

Gα,β(s, u, v) = g(1, u, 2s, v)

= (r(s, u, v))−iλ−N g( 1r(s, u, v) ,

u

r(s, u, v) ,2s

r(s, u, v) ,v

r(s, u, v)

)=( 1r(s, u, v)

)iλ+Ngα,β

( 1r(s, u, v) ,

u

r(s, u, v) ,2s

r(s, u, v) ,v

r(s, u, v)

).

Finally, due to the total homogeneity degree k of Pα,β0 :

Gα,β(s, u, v) = (2s)α

(a2(s, u) + ‖v‖2)iλ+k

2 +n. (6.3)

The function Gα,β is the non-compact form of gα,β. We point outthat we automatically know that Gα,β belongs to the Hilbert spaceL2(C2m+1): this follows from the identifications between the carryingspaces in the various pictures.

The aim of the rest of this chapter is to determine the non-standardform F(Gα,β) of Gα,β (using the composition formula F = Fτ Fξ).The explicit non-standard form we end up with is given in Theorem6.11 and also in Theorem 6.12, which is a recap that puts all these cal-culations back into context. Calculations will involve Bessel functionsand various technical results which, for convenience, we put togetherin the next section.

6.3 Bessel functions and useful formulas

We shall need to compute various integrals. Some will be expressed interms of Bessel functions. Let us recall definitions of these functions.In these definitions, following for instance [37] (Sections 5.3 and 5.7),we take ν to be any complex number (called the order) and the variablez ∈ C \0 to be such that −π < Arg(z) < π:

1. The Bessel function of the first kind is the function Jν defined

98

by:

Jν(z) =∞∑k=0

(−1)kΓ(k + 1)Γ(k + ν + 1)

(z

2

)ν+2k.

We point out that for ν ∈ Z, J−ν(z) = (−1)νJν(z).

2. The Bessel function of the second kind is the function Yν definedby:

Yν(z) = Jν(z) cos(νπ)− J−ν(z)sin(νπ)

when ν /∈ Z and, when ν ∈ Z, by

Yν(z) = limε→ν

0<|ε−ν|<1Yε(z).

We point out that for ν ∈ Z, Y−ν(z) = (−1)νYν(z).

3. The modified Bessel function of the first kind is the function Iνdefined by:

Iν(z) =∞∑k=0

1Γ(k + 1)Γ(k + ν + 1)

(z

2

)ν+2k.

We point out that for ν ∈ Z, I−ν(z) = Iν(z).

4. The modified Bessel function of the third kind is the function Kν

defined by

Kν(z) = π

2I−ν(z)− Iν(z)

sin(νπ)

when ν /∈ Z and, when ν ∈ Z, by

Kν(z) = limε→ν

0<|ε−ν|<1Kε(z).

We point out that K−ν(z) = Kν(z).

Remarks:

99

• The Gamma function is a meromorphic function that has no zerosand whose poles are 0,−1,−2 · · · (they are all simple poles). Inthe series above, some coefficients may involve terms 1

Γ(x) for cer-tain poles x (in which case there are only finitely many coefficientsof this sort); but this is not a problem, since such coefficients aresimply 0 (because |Γ(x)| = +∞).

• At the end of this work, we recall the differential equations thatare satisfied by the various Bessel functions.

When the order is a nonnegative integer, one can define Bessel functionsas integrals (see [17], Chapter 8, Section 8.411, Formula 1.11, or [10],Chapter VII, Section 7.3.1, Formula (2)):

Proposition 6.2 (Bessel’s integral representation). When ν ∈ N, onecan define Jν as follows:

Jν(z) = 12π

∫ 2π

0eiz sin θ e−iνθdθ.

As a consequence, by writing

cos a cos θ + sin a sin θ = cos(a− θ) = sin(π

2 − a+ θ

)and using a change of variables one can prove:

Corollary 6.3. Given any ν ∈ N, ρ > 0 and a > 0:∫ 2π

0eiνθ e−iρ(cos a cos θ+ sin a sin θ) dθ = 2πeiν(a−

π2 ) Jν(ρ).

The following equalities are stated in [10] (Chapter VII, Section 7.11,items (25) and (26)); one can also find them in [37] (Chapter 5, Section5.7):

Proposition 6.4.

• Kν−1(z)−Kν+1(z) = −2νz Kν(z);

• Kν−1(z) +Kν+1(z) = −2K ′ν(z).

100

As a consequence: ∂Kν∂z (z) = ν

zKν(z)−Kν+1(z).Formulas in this proposition are stated in [11] (Chapter VIII: Formula(20) of Section 8.5 and Formula (35) of Section 8.14); one can also findthem in [17] (Section 6.565, Formula 4., page 678 and Section 6.596,Formula 7.8, page 693):Proposition 6.5 (Two integral formulas involving Bessel functions).

• For any real number y > 0 and any complex numbers a, ν, µ suchthat Re(a) > 0 and −1 < Re(ν) < 2Re(µ) + 3

2 , one has:∫ ∞0

xν+ 12(x2 + a2

)−µ−1Jν(xy)√xy dx =

aν−µyµ+ 12Kν−µ(ay)

2µΓ(µ+ 1) .

• For any real number y > 0 and any complex numbers a, β, ν, µsuch that Re(a) > 0, Re(β) > 0 and Re(ν) > −1, one has:∫ ∞

0xν+ 1

2(x2 + β2

)−µ2 Kµ

(a(x2 + β2)

12)Jν(xy)√xy dx =

a−µβν+1−µyν+ 12 (a2 + y2)

µ2−

ν2−

12Kµ−ν−1

(β(a2 + y2)

12).

This next formula is stated in [10] (Section 7.4.1, Item (4), page 24):Proposition 6.6 (Asymptotic expansion for Modified Bessel func-tions). For any fixed P ∈ N \0 and ν ∈ C:

Kν(z) =(π

2z

) 12e−z

P−1∑p=0

Γ(

12 + ν + p

)p! Γ

(12 + ν − p

) (2z)−p+ O

(|z|−P

) .The following proposition can be derived from Section 9.6 of [12]:Proposition 6.7 (The Bochner formula). Consider any integer p ≥ 2.For ξ′ ∈ Sp−1 and s > 0:∫

Sp−1e−2iπsξ·ξ′dσ(ξ) = 2πs1− p2J p

2−1(2πs),

where ξ · ξ′ denotes the Euclidean scalar product of Rp applied to theelements of the sphere ξ and ξ′ seen as elements of Rp.

101

We end this section by recalling two standard results: the first gives suf-ficient conditions to differentiate integrals with respect to parametersand the second relates the standard polar change of coordinates.

Proposition 6.8 (Integrals and parameters). Given an integer p ≥ 1and a pair (a, b) ∈ R2 such that a < b, consider a function

f : Rp×]a, b[−→ C .

Denote by (x, λ) the coordinates on Rp×]a, b[ (λ is the parameter).Assume that:

• for all fixed λ ∈]a, b|, the function x 7−→ f(x, λ) is integrable onRp;

• the function λ 7−→ f(x, λ) is differentiable for all fixed x ∈ Rp;

• there exists an integrable function g : Rp −→ [0,∞[ such that forall (x, λ) ∈ Rp×]a, b[: ∣∣∣∣∂f∂λ(x, λ)

∣∣∣∣ ≤ g(x).

Then, for all λ ∈]a, b[, the function x 7−→ ∂f∂λ(x, λ) is integrable on Rp,

the functionφ : ]a, b[ −→ C

λ 7−→∫Rp f(x, λ) dx

is differentiable and

φ′(λ) =∫Rp∂f

∂λ(x, λ) dx.

Proposition 6.9 (Integrals and polar coordinates). Consider any in-teger p ≥ 2 and a measurable function f : Rp −→ C (with respect tothe Lebesgue measure of Rp). Write elements x ∈ Rp \0 as rξ withr > 0 and ξ ∈ Sp−1.Then f is integrable on Rp if and only if the function

(r, ξ) ∈]0,+∞[×Sp−1 7−→ f(rξ)

102

is integrable on ]0,+∞[×Sp−1 with respect to the measure rp−1dr dσ(ξ).When integrability is satisfied, we have:∫

Rpf(x)dx =

∫ ∞0

(∫Sp−1

f(rξ)dσ(ξ))rp−1dr.

We now set to prove the main result of this chapter. The calculationsare long and technical. We spread them over two sections: Section 6.4(which applies Fξ) and Section 6.5 (which applies Fτ ). In Section 6.6,we summarise in a single and self-contained statement what we haveachieved; this is our main theorem, namely Theorem 6.12.

6.4 First transform

By definition:

Fξ(Gα,β)(s, u, v) =∫Cm

(2s)α

(a2(s, u) + ‖ξ‖2)iλ+k

2 +ne−2iπRe<v,ξ> dξ. (6.4)

In real coordinates, writing ξ = x+ iy and v = a+ ib (elements x, y, a, beach belong to Rm) and identifying ξ and v with the elements (x, y)and (a, b) of Rm×Rm, formula (6.4) reads:

Fξ(Gα,β)(s, u, v) =∫Rm×Rm

(2s)α

(a2(s, u) + ‖x‖2 + ‖y‖2)iλ+k

2 +ne−2iπ(a·x−b·y) dxdy. (6.5)

We remind the reader that the dots in the exponential term refer tothe usual Euclidian scalar product of Rm. Proposition 6.9 allows usto switch to polar coordinates in Formula (6.5): given non-zero pairs(x, y) and (a,−b), we write (x, y) = rM and (a,−b) = r′M ′, where wetake M and M ′ to belong to the sphere S2m−1 of R2m and where weset:

r =√‖x‖2 + ‖y‖2 = ‖ξ‖ > 0;

r′ =√a2 + (−b)2 = ‖v‖ > 0.

103

This is possible because the function

(r,M) ∈]0,+∞[×S2m−1 7−→ G(s, u, rM) e−2iπM ·M ′

is integrable with respect to the measure r2m−1dr dσ(ξ). This followsfrom the inequality ∣∣∣∣∣∣ r2m−1

(a2(s, u) + r2)iλ+k

2 +n

∣∣∣∣∣∣ ≤ 1rk+3

and Riemann’s usual criterion for integrability.

Here, we denote byM ·M ′ the Euclidean scalar product of R2m appliedto the points M and M ′ of the sphere S2m−1 seen as vectors of R2m.Polar coordinates change the integral of Formula (6.5) into∫ ∞

0

∫S2m−1

(2s)α

(a2(s, u) + r2)iλ+k

2 +ne−2iπrr′M ·M ′dσ(M)

r2m−1dr.

(6.6)Integral (6.6) can be written:∫ ∞

0

(2s)α

(a2(s, u) + r2)iλ+k

2 +n

(∫S2m−1

e−2iπrr′M ·M ′dσ(M))r2m−1dr.

(6.7)Proposition 6.7 then changes (6.7) into∫ ∞

0

(2s)α

(a2(s, u) + r2)iλ+k

2 +n2π(rr′)1−mJm−1(2πrr′)r2m−1dr. (6.8)

Because r′ = ‖v‖, (6.8) becomes

2α+1πsα‖v‖1−m∫ ∞

0

rm

(a2(s, u) + r2)iλ+k

2 +nJm−1(2π‖v‖r)dr. (6.9)

We now want to apply Proposition 6.5. But it uses another notationsystem than ours. To understand how to switch from one to the other,let us define new variables x, y, µ by:

x = r ; y = 2π‖v‖ ; µ = iλ+ k

2 + n− 1 ; ν = m− 1.

104

Then (6.9) becomes:

2α+1πsα‖v‖1−my−12

∫ ∞0

xν+ 12

(a2(s, u) + r2)µ+1Jm−1(xy)√xydx. (6.10)

Proposition 6.5 (first formula) now gives (as long as ‖v‖ > 0)

2α+1πsα‖v‖1−my−12aν−µyµ+ 1

2Kν−µ(ay)2µΓ(µ+ 1) ,

which, back to our own notation choices, is equal to

2α+1sαπiλ+k

2 +n

Γ(iλ+k

2 + n) ( ‖v‖

a(s, u)

) iλ+k2 +1

K−( iλ+k2 +1)(2πa(s, u)‖v‖).

Because Kν = K−ν whatever the value of ν, we have proved so far:

Proposition 6.10. Given any λ ∈ R and any (α, β) ∈ N2, considerthe non-compact version Gα,β of the highest weight vector gα,β. Then,for all (s, u, v) in C×Cm×Cm such that v 6= 0:

Fξ(Gα,β)(s, u, v) = 2α+1sαπΛ+n

Γ (Λ + n)

( ‖v‖a(s, u)

)Λ+1KΛ+1(2πa(s, u)‖v‖),

where k = α+ β and Λ = iλ+k2 .

6.5 Second transformGiven any (s, u, v) ∈ C×Cm×Cm (we assume v 6= 0), we want toapply Fτ to Fξ(Gα,β) by writing

Fτ(Fξ(Gα,β)

)(s, u, v) =

∫CFξ(Gα,β)(τ, u, v) e−2iπRe(sτ) dτ. (6.11)

Let us first check that the function τ 7−→ Fξ(Gα,β)(τ, u, v) is indeedintegrable. In other words, we wish to know whether the integral∫

C

∣∣Fξ(Gα,β)(τ, u, v)∣∣ dτ

105

or, more explicitly,∫C

∣∣∣2α+1ταπΛ+n

Γ (Λ + n)

( ‖v‖a(τ, u)

)Λ+1KΛ+1(2πa(τ, u)‖v‖)

∣∣∣ dτ (6.12)

is finite or not. Let us split (6.12) into two integrals:

• one integral, denoted by I1, over the unit ball B1 of C;

• one integral, denoted by I2, over C \B1.

Because ‖v‖ 6= 0 and a(τ, u) ≥ 1, the function that sits under Inte-gral (6.12) is continuous on B1 and therefore I1 is finite. Proposition6.6 (using the integer P = 1) and Proposition 6.10 allow us to write|Fξ(Gα,β)(τ, u, v)| in the following way (A,B,C denote positive con-stants and B ≥ 1):

A |τ |α

(B + 4|τ |2) k+34

e−C(B+4|τ |2)12

(1 + O

(1

C(B + 4|τ |2) 12

)). (6.13)

Then for |τ | large enough and a suitable constant D > 0,A

|τ |k+3

2 −αe−C|τ |

(1 + D

|τ |

)is an upper-bound of (6.13). The exponential term forces convergenceof I2 and this finally establishes the desired integrability assumption,which will allow us to switch to polar coordinates.

Let us again use the letters a, b, x, y, but this time taking them tosimply be real numbers such that s = a + ib and τ = x + iy. ThenRe(sτ) = ax− by and (6.11) becomes∫

R2

2α+1ταπΛ+n

Γ (Λ + n)

( ‖v‖a(τ, u)

)Λ+1KΛ+1(2πa(τ, u)‖v‖) e−2iπ(ax−by) dxdy,

which can be re-organised as

2α+1πΛ+n‖v‖Λ+1

Γ (Λ + n)

∫R2

ταKΛ+1(2πa(τ, u)‖v‖)(a(τ, u)

)Λ+1 e−2iπ(ax−by) dxdy.

(6.14)Let us again use polar coordinates (outside the origin):

106

• (x, y) = rvθ with r > 0, θ ∈ R and vθ = (cos θ, sin θ); accordingly,τ = reiθ.

• (a,−b) = r′vθ′ with r′ > 0, θ′ ∈ R and vθ′ = (cos θ′, sin θ′).

Let us point out that we obviously have r = |τ | and r′ = |s|. Let usalso write a(r, u) instead of a(τ, u):

a(r, u) =√

1 + 4r2 + ‖u‖2.

Integral 6.14 can now be written:

2α+1πΛ+n‖v‖Λ+1

Γ (Λ + n)

∫ ∞0

rαKΛ+1(2πa(r, u)‖v‖)(a(r, u)

)Λ+1(∫ 2π

0eiαθ e−2iπrr′(cos θ cos θ′+sin θ sin θ′)dθ

)rdr. (6.15)

Following Corollary 6.3, the inner integral∫ 2π

0eiαθ e−2iπrr′(cos θ cos θ′+ sin θ sin θ′) dθ

is equal to2πeiα(θ′−π2 ) Jα(2πrr′). (6.16)

Remembering that r′ = |s| and θ′ = Arg(s), (6.16) is equal to:

2πeiα(Arg(s)−π2 ) Jα(2πr|s|).

This turns (6.15) into:

2α+2πΛ+n+1‖v‖Λ+1eiα(Arg(s)−π2 )Γ (Λ + n)∫ ∞

0

rα+1KΛ+1(2πa(r, u)‖v‖)(a(r, u)

)Λ+1 Jα(2πr|s|) dr. (6.17)

We can now apply the second formula of Proposition 6.5. To help follownotation choices made in this proposition, we set:

107

• x = 2r and dx = 2dr;

• β =√

1 + ‖u‖2 > 0;

• a = 2π‖v‖ > 0 (careful: this variable a is not what we havedenoted a(r, u));

• y = π|s| > 0;

• ν = α;

• µ = Λ + 1.

Plugging these expressions in (6.17) and using Proposition 6.5, we fi-nally achieve what we announced at the end of Section 6.2: computethe non-standard form Fτ

(Fξ(Gα,β)

)of Gα,β (therefore of gα,β). The

formula we obtain is:

Theorem 6.11.

F(Gα,β)(s, u, v) =∫C×Cm

(2τ)α

(1 + 4|τ |2 + ‖u‖2 + ‖ξ‖2)iλ+k

2 +ne−2iπRe(sτ+〈v,ξ〉) dτdξ =

R(s, u, v) K iλ+δ2

(π√

1 + ‖u‖2√|s|2 + 4‖v‖2

),

where

R(s, u, v) = (−i s)α πiλ+β+n

2 iλ+k2 +1 Γ( iλ+k

2 + n)

(√|s|2 + 4‖v‖2π√

1 + ‖u‖2

) iλ+δ2

and:

• (k, α, n) ∈ N3, 0 ≤ α ≤ k and n ≥ 2 ;

• (s, u, v) ∈ C×Cm×Cm and (τ, ξ) ∈ C×Cm ;

• s 6= 0 and v 6= 0.

108

6.6 Statement of main result

The following result, which serves as a recap, binds together theformula of Theorem 6.11 and all the ingredients that are needed tounderstand the meaning of this formula.

In this theorem, we consider three groups M , A and N , respectivelyisomorphic to the groups U(1)× Sp(m,C), R+,∗

× (multiplicative groupof positive reals) and H2m+1

C (complex Heisenberg group), embeddedin Sp(n,C) as follows (denoting elements of M,A,N respectively bym, a, n):

• m =

eiθ(m) 0 0 0

0 A 0 C

0 0 e−iθ(m) 00 B 0 D

,with θ(m) ∈ R and

(A C

B D

)∈ Sp(m,C);

• a =

α(a) 0 0 0

0 Im 0 00 0

(α(a)

)−1 00 0 0 Im

,with α(a) ∈]0,∞[;

• n =

1 tu 2s tv0 Im v 00 0 1 00 0 −u Im

,with (s, u, v) ∈ C×Cm×Cm.

We consider the group Q = MAN .

Theorem 6.12 (Non-standard picture and Bessel functions).Consider n ∈ N such that n ≥ 2 and set N = 2n, m = n− 1.Let G be the group Sp(n,C) and Q = MAN its parabolic subgroupintroduced above; let K be the subgroup Sp(n). Consider (λ, δ) ∈ R×Z

109

and the character χiλ,δ defined on Q by

χiλ,δ(man) = eiδθ(m) (α(a))iλ+N

.

Consider the degenerate principal series representations of G (see Sec-tion 3.2.1 of Chapter 3 for details):

πiλ,δ = IndGQ χiλ,δ.

Consider k ∈ N and any (α, β) ∈ N2 such that α+β = k; set δ = β−α.Consider the irreducible (finite-dimensional) subrepresentation ofπiλ,δ

∣∣K

whose highest weight is (k, 0, · · · , 0). Then the correspondinghighest weight vector (up to a constant) is given:

• in the compact picture by the function

gα,β : S2N−1 ⊂ Cn×Cn −→ C(z, w) 7−→ wα1 z1

β;

• in the non-compact picture by the function

Gα,β : C×Cm×Cm −→ C(s, u, v) 7−→ (2s)α

(1+4|s|2+‖u‖2+‖v‖2)iλ+k

2 +n;

• in the non-standard picture by the function F(Gα,β), which isdefined for all (s, u, v) in C×Cm×Cm such that v 6= 0 and s 6= 0by

F(Gα,β)(s, u, v) = R(s, u, v) K iλ+δ2

(π√

1 + ‖u‖2√|s|2 + 4‖v‖2

),

with

R(s, u, v) = (−i s)α πiλ+β+n

2 iλ+k2 +1 Γ( iλ+k

2 + n)

(√|s|2 + 4‖v‖2π√

1 + ‖u‖2

) iλ+δ2

.

Remark: as we have pointed out before, the symbol N refers to a group,Symbols n,m to elements of groups, while the three symbols also referto dimensions; context makes intended meanings of these symbols clear.

110

6.7 Two interesting properties

In this section, we make two observations that seem relevant to us: wefeel they may prove useful in further investigation of the K-types ofthe various representations πiλ,δ in the non-standard picture.

6.7.1 A differential operator that connects certain K-types

If we look back at Figure 4.1 of Chapter 4, we see that the highestweight vectors we have studied in this present chapter are those thatcorrespond to the components of the column on the far left. Wesaw in Theorem 4.11 that the operator e applies a highest weightvector onto the one immediately above (up to a constant). Whenrestricting to the highest weight vectors of the left column, the actionof e can be interpreted, with respect to their non-standard versionsgiven by Theorem 6.12, as a differential operator that acts on Fouriertransforms; this present section makes this clear.

In our study of properties of F with respect to differentiablity andmultiplication by coordinates:

• (τ, u, ξ) will denote the coordinates on C×Cm×Cm with respectto initial functions f ;

• (s, u, v) will then denote the coordinates on C×Cm×Cm withrespect to F(f).

Theorem 6.13 (Differential induction formula). Suppose k ≥ 2 andβ ≥ 2. Then:

F(Gα+1,β−1)(s, u, v) = 2−iπ

∂

∂s

(F(Gα,β)(s, u, v)

).

Proof:

111

For 0 ≤ α ≤ k − 1 and (τ, u, ξ) ∈ C×Cm×Cm:

Gα+1,β−1(τ, u, ξ) =(r(τ, u, ξ)

)−iλ−N−kgα+1,β−1(1, u; 2τ, ξ)

=(r(τ, u, ξ)

)−iλ−N−k(2τ)α+1

= 2τ(r(τ, u, ξ)

)−iλ−N−k(2τ)α

= 2τ Gα,β(τ, u, ξ).

Theorem 6.13 then follows from Lemmas 6.14 and 6.15 given below.

End of proof.

Remark: as we see in the proof, one switches from Gα,β to Gα+1,β−1by multiplying by 2τ ; this corresponds to the action of the operator estudied in section 4.2.1.

Lemma 6.14. If β ≥ 2, then, given any u ∈ Cm, the maps

(τ, ξ) ∈ C×Cm 7−→ Gα,β(τ, u, ξ)

and(τ, ξ) ∈ C×Cm 7−→ τ Gα,β(τ, u, ξ)

are both integrable.

Proof:

Let us call ψ the map (τ, ξ) 7−→ τ Gα,β(τ, u, ξ) (we point out thatintegrability of this map forces integrability of the other map).

There is no problem as to the integrability of ψ on the unit ball B1(0)whose center is the origin. So one just needs to check integrability onthe set R2n \B1(0).

Given a non-zero element (τ, ξ) of C×Cm ' Cn, there exists a uniquer ∈]0,+∞[ and a unique M ∈ S2n−1 such that if (τM , ξM ) ∈ C×Cmdenote the coordinates of M seen as an element of C×Cm, then:

(τ, ξ) = r(τM , ξM ).

112

Identifying complex coordinates on C×Cm with real coordinates onR2×R2m ' R2n and applying Proposition 6.9, we see that ψ is inte-grable if the function

(r,M) ∈]0,+∞[×S2n−1 7−→ r Gα,β(rτM , u, rξM )

is itself integrable on ]0,+∞[×S2n−1 with respect to the measurer2n−1dr dσ(M). We have:

|r Gα,β(rτM , u, rξM )| r2n−1 =

∣∣∣∣∣∣ 2α(rτM )α r2n

(1 + 4|rτM |2 + ‖u‖2 + ‖rξM‖2)iλ+k

2 +n

∣∣∣∣∣∣≤ 2αrα+2n

(r2) k2 +n

= 2αrβ.

This finishes the proof, because β ≥ 2 and r ≥ 1.

End of proof.

Lemma 6.15. Consider a function f : C×Cm×Cm −→ C. Assumethat the maps

(τ, ξ) 7−→ f(τ, u, ξ)

and(τ, ξ) 7−→ τ f(τ, u, ξ)

are both integrable on C×Cm for all u ∈ Cm. Consider the map ψ :(τ, u, ξ) 7−→ τ f(τ, u, ξ). Then:

F(ψ)(s, u, v) = 1−iπ

∂

∂s

(F(f)(s, u, v)

).

Proof:

Switch to real variables:

• s = xs + iys, with (xs, ys) ∈ R2;

• u = xu + iyu, with (xu, yu) ∈ Rm×Rm;

113

• v = xv + iyv, with (xv, yv) ∈ Rm×Rm.

Similarly for the integration variables τ and ξ that enter the definitionof the non-standard Fourier transform:

• τ = xτ + iyτ , with (xτ , yτ ) ∈ R2;

• ξ = xξ + ixξ with (xξ, xξ) ∈ Rm×Rm.

To make expressions in the integral shorter, define:

• w = (xτ , xu, xξ, yτ , yu, yξ);

• dw = dxτ dxu dxξ dyτ dyu dyξ.

We remind the reader that the dot simply refers to the standard eu-cliean product (of Rm here). With these notation choices, Definition6.2 reads:

F(s, u, v) =∫R2n

fR(w) exp−2iπ(xsxτ−ysyτ+xv ·xξ−yv ·yξ) dw.

Coordinates xs and ys are just parameters that appear in the integral.Lemma 6.14 and Proposition 6.8 enable us to differentiate this integralwith respect to both parameters and obtain:

∂

∂xs

(F(s, u, v)

)=∫R2n−2iπxτfR(w) exp−2iπ(xsxτ−ysyτ+xv ·xξ−yv ·yξ) dw;

∂

∂ys

(F(s, u, v)

)=∫R2n

2iπyτfR(w) exp−2iπ(xsxτ−ysyτ+xv ·xξ−yv ·yξ) dw.

This finishes the proof because, by definition, the operator ∂∂s is equal

to 12

(∂∂xs− i ∂

∂ys

).

End of proof.

Corollary 6.16. Consider integers k, α, β such that k ≥ 2, α+ β = kand β ≥ 1. Then:

F(Gα,β)(s, u, v) =( 2−iπ

)α ∂α

∂sα

(F(G0,k)(s, u, v)

).

114

The following diagram shows how the operators of this section relateto one another, connecting functions and their transforms (assumingthat β ≥ 2):

gα+1,β−1 −−−−→ Gα+1,β−1 −−−−→ F(Gα+1,β−1)x 1βe

x2τIdx 2−iπ

∂∂s

gα,β −−−−→ Gα,β −−−−→ F(Gα,β).

6.7.2 Underlying Emden-Fowler equations

Theorem 6.12 gives the following expression for the highest weight vec-tors gα,β in the non standard picture:

(−i s)α πiλ+β+n

2 iλ+k2 +1 Γ( iλ+k

2 + n)

(√|s|2 + 4‖v‖2π√

1 + ‖u‖2

) iλ+δ2

K iλ+δ2

(π√

1 + ‖u‖2√|s|2 + 4‖v‖2

).

One inevitably notices a sort of two variable structure in this expres-sion. Indeed, the particular value α = 0 and the square root terms leadone to study the functions

ψν : ]0,+∞[×]0,+∞[ −→ C(x, y) 7−→

(xy

)νKν(xy),

where the parameter ν is taken to be any complex number.

For any given ν ∈ C, x0 > 0 and y0 > 0, define the functions:

ϕx0,ν : ]0,+∞[ −→ Cy 7−→ ψν(x0, y);

φy0,ν : ]0,+∞[ −→ Cx 7−→ ψν(x, y0).

Because the function Kν is a modified Bessel function of the third kind,we have:

K ′′ν (ξ) + 1ξK ′ν(ξ) −

(1 + ν2

ξ2

)Kν(ξ) = 0.

115

In this equation, ξ can be taken as a real or a complex variable; here,we choose ξ ∈ R. Combining this equation with the first and secondderivatives of the functions ϕx0,ν and φy0,ν , one can check the followingproposition.

Proposition 6.17. Given any ν ∈ C, x0 > 0, y0 > 0 and taking y > 0:

ϕ′′x0,ν(y) + (1 + 2ν)y

ϕ′x0,ν(y) − x20 ϕx0,ν(y) = 0;

φ′′y0,ν(x) + (1− 2ν)x

φ′y0,ν(x) − y20 φy0,ν(x) = 0.

Both equations can be written as (taking a ∈ C, b > 0 and t > 0):

u′′(t) + a

tu′(t) − b u(t) = 0. (6.18)

Such equations belong to the family of Emden-Fowler equations (orLane-Emden equations), which appear in various forms and have beenstudied in many works (see for instance [6], [2], [3], [43] and [45]).

The general solutions of (6.18) can be written as the following combi-nations (with t-dependent coefficients) of Bessel functions of the firstand second kind:

u(t) = C1 t1−a

2 Ja−12

(−it√b)

+ C2 t1−a

2 Ya−12

(−it√b).

This implies that ϕx0,ν and ϕx0,ν are such combinations, which givesnew formulas for the function ψν , thus other formulas for the non-standard version of the highest weight vector of Theorem 6.12 (whenα = 0).

116

Bibliography

[1] V. Bargmann, Irreducible unitary representations of the Lorentzgroup, Ann. of Math. (2) 48 (1947) 568–640.

[2] R. Bellman, Stability theory of differential equations, Dover Publi-cations, Inc., Mineola, New York, 2008. Reprint of the edition origi-nally published by the McGraw-Hill Book Company, Inc., in 1953.

[3] O. P. Bhutani and K. Vijayakumar, On certain new and exact so-lutions of the Emden-Fowler equation and Emden equation via in-variant variational principles and group invariance, The ANZIAMJournal 32 (4) (1991) 457–468.

[4] A. Borel, Semisimple groups and Riemannian symmetric spaces,Texts and Readings in Mathematics, Vol. 16, Hindustan BookAgency, New Delhi, 1998.

[5] W. Casselman and D. Miličić, Asymptotic behavior of matrix coef-ficients of admissible representations, Duke Math. J. 49 (4) (1982)869–930.

[6] S. Chandrasekhar, An introduction to the study of stellar structure,Dover Publications, Inc., New York, 1967. Unabridged and correctedrepublication of the work originally published by the University ofChicago Press in 1939.

[7] C. Chevalley, Theory of Lie groups—I, Princeton Mathematical Se-ries, Vol. 8, Princeton University Press, Princeton, N. J., 1946.

[8] P. Clare, On the degenerate principal series of complex symplecticgroups, J. Funct. Anal. 262 (9) (2012) 4160–4180.

117

[9] G. van Dijk and A. Pasquale, Canonical representations of Sp(1, n)associated with representations of Sp(1), Comm. Math. Phys. 202 (3)(1999) 651–667.

[10] A. Erdélyi, W. Magnus, F. Oberhettinger and F. G. Tricomi,Higher transcendental functions, Vol. II, McGraw-Hill Book Com-pany, Inc., New York-Toronto-London, 1953.

[11] A. Erdélyi, W. Magnus, F. Oberhettinger and F. G. Tricomi, Ta-bles of integral transforms, Vol. II, McGraw-Hill Book Company,Inc., New York-Toronto-London, 1954.

[12] J. Faraut, Analysis on Lie groups—An introduction, CambridgeStudies in Advanced Mathematics, Vol. 110, Cambridge UniversityPress, 2008.

[13] G. B. Folland, Harmonic analysis in phase space, Ann. Math.Studies, Vol. 122, Princeton University Press, Princeton, NJ, 1989.

[14] J. Fourier, Théorie analytique de la chaleur, Paris: Chez FirminDidot, Père et Fils, 1822.

[15] I. M. Gel’fand and M. A. Naımark, Unitary representations ofthe Lorentz group, Izvestiya Akad. Nauk SSSR. Ser. Mat. 11 (1947)411–504.

[16] I. M. Gel’fand and D. A. Raıkov, Irreducible unitary representa-tions of locally bicompact groups, Rec. Math. [Mat. Sbornik] N.S. 13(55) (1943) 301–316.

[17] I. S. Gradshteyn and I. M. Ryzhik, Table of integrals, series,and products, Seventh edition, Elsevier/Academic Press, Amster-dam, 2007.

[18] K. I. Gross, The dual of a parabolic subgroup and a degenerateprincipal series of Sp(n,C), Amer. J. Math. 93 (2) (1971) 398–428.

[19] W. R. Hamilton, On quaternions; or on a new system of imagi-naries in algebra, The London, Edinburgh and Dublin Philosoph-ical Magazine and Journal of Science (3rd Series) XXV-XXXVI(1844–1850).

118

[20] J. Hilgert, T. Kobayashi, G. Mano and J. Möllers, Orthogonalpolynomials associated to a certain fourth order differential equation,The Ramanujan Journal 26 (3) (2011) 295–310.

[21] J. Hilgert, T. Kobayashi, G. Mano and J. Möllers, Special func-tions associated with a certain fourth-order differential equation, TheRamanujan Journal 26 (1) (2011) 1–34.

[22] R. E. Howe and E.-C. Tan, Homogeneous functions on light cones:the infinitesimal structure of some degenerate principal series repre-sentations, Bull. Amer. Math. Soc. (N.S.) 28 (1) (1993) 1–74.

[23] J.-S. Huang, Lectures on representation theory, World ScientificPublishing Co., Inc., River Edge, NJ, 1999.

[24] H. Kalf, On the expansion of a function in terms of spherical har-monics in arbitrary dimensions, Bull. Belg. Math. Soc. Simon Stevin2 (4) (1995), 361–380.

[25] A. A. Kirillov, Lectures on the orbit method, American Mathemat-ical Society Providence, RI, Vol. 64, 2004.

[26] F. Klein, Le programme d’Erlangen, Jacques Gabay, 1991.

[27] A. W. Knapp, Representation theory of semisimple groups—Anoverview based on examples, Princeton Landmarks in Mathematics,Princeton University Press, Princeton, NJ, 2001, Reprint of the 1986original.

[28] A. W. Knapp, Lie groups beyond an introduction, Second edition,Progress in Mathematics, Vol. 140, Birkhäuser Boston, Inc., Boston,MA, 2002.

[29] T. Kobayashi, T. Kubo and M. Pevzner, Conformal symmetrybreaking operators for differential forms on spheres, Lecture Notesin Mathematics, Vol. 2170, Springer, Singapore, 2016.

[30] T. Kobayashi and G. Mano, The Schrödinger model for the mini-mal representation of the indefinite orthogonal group O(p, q), Mem-oirs of the AMS, Vol. 213, 2011.

119

[31] T. Kobayashi and J. Möllers, An integral formula for L2-eigenfunctions of a fourth-order Bessel-type differential operator, In-tegral Transforms and Special Functions 22 (7) (2011) 521–531.

[32] T. Kobayashi and B. Ørsted, Analysis on the minimal representa-tion of O(p, q)—I. Realization via conformal geometry, Adv. Math.180 (2) (2003) 486–512.

[33] T. Kobayashi and B. Ørsted, Analysis on the minimal represen-tation of O(p, q)—II. Branching laws, Adv. Math. 180 (2) (2003)513–550.

[34] T. Kobayashi and B. Ørsted, Analysis on the minimal represen-tation of O(p, q)—III. Ultrahyperbolic equations on Rp−1,q−1, Adv.Math. 180 (2) (2003) 551–595.

[35] T. Kobayashi, B. Ørsted and M. Pevzner, Geometric analysis onsmall unitary representations of GL(n,R), J. Funct. Anal. 260 (6)(2011) 1682–1720.

[36] T. Kobayashi and M. Pevzner, Differential symmetry breakingoperators—I. General theory and F-method, Selecta Math. 22 (2)(2016) 801–845.

[37] N. N. Lebedev, Special functions & their applications, Dover Pub-lications, Inc., New York, 1972. Unabridged and corrected repub-lication of the work originally published by Prentice-Hall, Inc., in1965.

[38] G. W. Mackey, On induced representations of groups, Amer. J.Math. 73 (1951) 576–592.

[39] G. W. Mackey, Induced representations of locally compact groupsI, Ann. of Math. (2) 55 (1952) 101–139.

[40] G. W. Mackey, Induced representations of locally compact groupsII. The Frobenius reciprocity theorem, Ann. of Math. (2) 58 (1953)193–221.

120

[41] G. W. Mackey, Harmonic analysis as the exploitation ofsymmetry—a historical survey, Bull. Amer. Math. Soc. (N.S.) 3 (1)(1980) 543–698.

[42] R. Mneimné and F. Testard, Introduction à la théorie des groupesde Lie classiques, Collection Méthodes, Hermann, Paris, 1986.

[43] B. Muatjetjeja and C. M. Khalique, Exact solutions of the gener-alized Lane-Emden equations of the first and second kind, PramanaJ. Phys. 77 (3) (2011) 545–554.

[44] P. Pajas and R. Rączka, Degenerate representations of the sym-plectic groups. I. The compact group Sp(n), J. Math. Phys. 9 (8)(1968) 1188–1201.

[45] K. Parand, M. Nikarya and J. A. Rad, Solving non-linear Lane-Emden type equations using Bessel orthogonal functions collocationmethod, Celestial Mech. Dynam. Astronom. 116 (1) (2013) 97–107.

[46] A. Pasquale, Maximal degenerate representations of SL(n+ 1,H),J. Lie Theory 9 (2) (1999) 369–382.

[47] F. Peter and H. Weyl, Die Vollständigkeit der primitiven Darstel-lungen einer geschlossenen kontinuierlichen Gruppe, Math. Annalen97 (1) (1927) 737–755.

[48] H. Samelson, Notes on Lie algebras, Second edition, Universitext,Springer-Verlag, New York, 1990.

[49] M. Tadić, GL(n,C) and GL(n,R), in Automorphic forms and L-functions II. Local aspects, 285–313, Contemp. Math., Vol. 489, IsraelMath. Conf. Proc., Amer. Math. Soc., Providence, RI, 2009.

[50] A. U. Klimyk and N. Ja. Vilenkin, Representations of Lie groupsand special functions—Volume 1: simplest Lie groups, special func-tions and integral transforms, Mathematics and Its Applications (So-viet Series), Vol. 72, Springer-Science+Business Media, B.V., 1993.

[51] A. U. Klimyk and N. Ja. Vilenkin, Representations of Lie groupsand special functions—Volume 2: class I representations, special

121

functions and integral transforms, Mathematics and Its Applications(Soviet Series), Vol. 74, Springer-Science+Business Media, B.V.,1993.

[52] D. A. Vogan, Jr., Unitary representations of reductive Lie groups,Ann. Math. Studies, Vol. 118, Princeton University Press, Princeton,NJ, 1987.

[53] D. P. Želobenko, Compact Lie groups and their representations,Translations of Mathematical Monographs, Vol. 40, American Math-ematical Society, Providence, R.I., 1973.

122

Index

Action, 15Adjoint representation, 17

Bessel functions, 98Bi-invariant, 75

Carrying space, 15Cartan subalgebra, 22Casimir operator, 28Compact picture, 47Complexification, 25Component, 60Continuous representation, 17Covariant, 46

Degenerate principal series, 46

Harmonic, 56Heisenberg group, 42Highest weight, 26Highest weight theorem, 26Highest weight vector, 26Hypergeometric equation, 93

Induced picture, 46Induced representation, 46Invariant subspace, 15Irreducible representation, 15Isotypic decomposition, 20

K-type, 20

Langlands classification, 3Langlands decomposition, 43Laplace operator, 55Left action, 16Left-invariant, 16Lie algebra, 17Lie bracket, 17Lie group, 16Linear Lie group, 17

Multiplicity, 20

Natural action, 15Non-compact picture, 47Non-standard picture, 96

Parabolic subgroup, 43Peter-Weyl Theorem, 19

Quaternion, 32Quaternionic projective space, 81

Representation, 15Right action, 69Right-invariant, 69Root, 23Root element, 23Root space, 23

Semisimple Lie algebra, 22

123

Smooth vector, 18Spherical harmonics, 56Standard symplectic form, 21Subrepresentation, 15Symplectic algebra, 21Symplectic group, 21Symplectic matrix, 21

Unit quaternion, 33Unitary dual, 18

Weight, 25Weight space, 25Weyl dimension formula, 26

124