Gelfand Triples and Their Hecke

Algebras Harmonic Analysis for

Multiplicity Free Induced
Representations of Finite Groups Tullio
Ceccherini-Silberstein
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Lecture Notes in Mathematics 2267

Tullio Ceccherini-Silberstein
Fabio Scarabotti
Filippo Tolli

Gelfand
Triples and
Their Hecke
Algebras
Harmonic Analysis
for Multiplicity-Free Induced
Representations of Finite Groups
Foreword by
Eiichi Bannai
Lecture Notes in Mathematics

Volume 2267

Editors-in-Chief
Jean-Michel Morel, CMLA, ENS, Cachan, France
Bernard Teissier, IMJ-PRG, Paris, France

Series Editors
Karin Baur, University of Leeds, Leeds, UK
Michel Brion, UGA, Grenoble, France
Camillo De Lellis, IAS, Princeton, NJ, USA
Alessio Figalli, ETH Zurich, Zurich, Switzerland
Annette Huber, Albert Ludwig University, Freiburg, Germany
Davar Khoshnevisan, The University of Utah, Salt Lake City, UT, USA
Ioannis Kontoyiannis, University of Cambridge, Cambridge, UK
Angela Kunoth, University of Cologne, Cologne, Germany
Ariane Mézard, IMJ-PRG, Paris, France
Mark Podolskij, University of Luxembourg, Esch-sur-Alzette, Luxembourg
Sylvia Serfaty, NYU Courant, New York, NY, USA
Gabriele Vezzosi, UniFI, Florence, Italy
Anna Wienhard, Ruprecht Karl University, Heidelberg, Germany
This series reports on new developments in all areas of mathematics and their
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Tullio Ceccherini-Silberstein • Fabio Scarabotti •
Filippo Tolli

Gelfand Triples and Their

Hecke Algebras
Harmonic Analysis for Multiplicity-Free
Induced Representations of Finite Groups

Foreword by Eiichi Bannai

Tullio Ceccherini-Silberstein Fabio Scarabotti
Dipartimento di Ingegneria Dipartimento SBAI
Università degli Studi del Sannio Università degli Studi di Roma
Benevento, Italy “La Sapienza”
Roma, Italy

Filippo Tolli
Dipartimento di Matematica e Fisica
Università degli Studi Roma Tre
Roma, Italy

ISSN 0075-8434 ISSN 1617-9692 (electronic)

Lecture Notes in Mathematics
ISBN 978-3-030-51606-2 ISBN 978-3-030-51607-9 (eBook)
https://doi.org/10.1007/978-3-030-51607-9

Mathematics Subject Classification: 20C15, 43A65, 20C08, 20G05, 43A90, 43A35

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To Francesca
To the memory of my father
To Valentina
Foreword

A (finite) Gelfand pair is a pair (G, H ) consisting of a (finite) group G and

a subgroup H such that G, with its action on the coset space G/H , yields a
multiplicity-free transitive permutation group, equivalently, the induced represen-
tation IndG
H (1H ) of the identity representation 1H of H to G is multiplicity-free,
i.e., decomposes into pairwise-inequivalent irreducible representations of G.
Gelfand pairs constitute an important mathematical concept that plays a central
role in several areas of mathematics. The purely combinatorial counterpart of
Gelfand pairs are the (commutative) association schemes. Gelfand pairs, indeed,
constitute the most important class of (commutative) association schemes. Both
Gelfand pairs and association schemes have already been extensively studied,
because of their importance in their own right. But there are many other objects and
theories that have been studied and developed under this framework, for example,
distance-transitive and/or distance-regular graphs, coding theory and design theory
(Delsarte theory), probability theory and statistics (Diaconis theory), witnessing,
once more, the strong connections of Gelfand pairs and association schemes with
many other areas of mathematics.
There have been many attempts to generalize the concept of a Gelfand pair. Some
authors have already successfully dealt with the case of a triple (G, H, η), where G
and H are a group and a subgroup as before, and η is a linear representation of H
such that IndGH (η) is multiplicity-free.
The next more general step would naturally be to study the case of a triple
(G, H, η), where now η is an arbitrary (not necessarily linear) irreducible repre-
sentation of H such that IndG H (η) is multiplicity-free. The authors of this memoir
call (G, H, η) a Gelfand triple. It seems that this situation has been regarded by
many experts to be a bit too general, and this may explain why not much general
theory has been developed so far.
The present memoir tackles this problem very seriously and bravely from the
very front side. Frankly speaking, in spite of some success achieved in this work,
many questions are still left unanswered and waiting to be studied. However, I
believe that this text provides us with very useful foundations and information to

vii
viii Foreword

start a serious research in this direction. This memoir is “the” pioneering work on
general Gelfand triples.
The three authors are very strong researchers working in representation theory
and discrete harmonic analysis, as well as in many related fields of mathematics.
They published already several excellent books on these topics. This memoir has a
mixed nature of both a research paper and a book. Indeed, on the one hand, all the
detailed proofs are carefully given as in a research paper, and, on the other hand, the
authors masterfully describe the mathematical philosophy of this research direction,
as witnessed in any excellent book.
I believe that this volume in the Springer LNM series will provide another good
addition to this general research direction, namely “harmonic analysis on finite
groups.” It will be read and very welcomed, not only by experts but also by a broad
range of mathematicians.

Tokyo, Japan Eiichi Bannai

May 2020
Preface

Finite Gelfand pairs play an important role in mathematics and have been studied
from several points of view: in algebra (we refer, for instance, to the work of Bump
and Ginzburg [7, 8] and Saxl [57]; see also [15]), in representation theory (as wit-
nessed by the new approach to the representation theory of the symmetric groups by
Okounkov and Vershik [54], see also [13]), in analysis (with relevant contributions
to the theory of special functions by Dunkl [30] and Stanton [67]), in number theory
(we refer to the book by Terras [69] for a comprehensive introduction; see also
[11, 17]), in combinatorics (in the language of association schemes as developed by
Bannai and Ito [1]), and in probability theory (with the remarkable applications to
the study of diffusion processes by Diaconis [24]; see also [10, 11]). Indeed, Gelfand
pairs arise in the study of algebraic, geometrical, or combinatorial structures with a
large group of symmetries such that the corresponding permutation representations
decompose without multiplicities: it is then possible to develop a useful theory of
spherical functions with an associated spherical Fourier transform.
In our preceding work, we have shown that the theory of spherical functions may
be studied in a more general setting, namely for permutation representations that
decompose with multiplicities [14, 58], for subgroup-conjugacy-invariant functions
[16, 59], and for general induced representations [61]. Indeed, a finite Gelfand
pair may be considered as the simplest example of a multiplicity-free induced
representation (the induction of the trivial representation of the subgroup), and this
is the motivation for the present monograph.
The most famous of these multiplicity-free representations is the Gelfand–Graev
representation of a reductive group over a finite field [34] (see also Bump [7]).
In this direction, we have started our investigations in Part IV of our monograph
[17], where we have developed a theory of spherical functions and spherical
representations for multiplicity-free induced representations of the form IndG K χ,
where χ is a one-dimensional representation of subgroup K. This case was
previously investigated by Stembridge [68], Macdonald [46, Exercise 10, Chapter
VII], and Mizukawa [49, 50]. We have applied this theory to the Gelfand–Graev
representation of GL(2, Fq ), following the beautiful expository paper of Piatetski-
Shapiro [53], where the author did not use the terminology/theory of spherical

ix
x Preface

functions but, actually, computed them. In such a way, we have shed light on the
results and the calculations in [53] by framing them in a more comprehensive theory.
In the present monograph, we face the more general case: we study multiplicity-
free induced representations of the form IndG K θ , where θ is an irreducible K-
representation, not necessarily one-dimensional. In this case, borrowing a termi-
nology used by Bump in [7, Section 47], we call (G, K, θ ) a multiplicity-free
triple. Since this constitutes a generalization of Gelfand pairs, we shall also refer
to (G, K, θ ) as to a Gelfand triple, although we are aware that such a terminology is
already widely used in another setting, namely in functional analysis and quantum
mechanics, as a synonym of a rigged Hilbert space [23, 35].
Our first target (cf. Sect. 2.1) is a deep analysis of Mackey’s formula for
invariants. We show that the commutant EndG (IndG K θ ) of an arbitrary induced
representation IndG K θ , with θ an irreducible K-representation, is isomorphic to both
a suitable convolution algebra of operator-valued functions defined on G and to a
subalgebra of the group algebra of G. We call it the Hecke algebra of the triple
(G, K, θ ) (cf. Bump [7, Section 47], Curtis and Reiner [22, Section 11D], and
Stembridge [68]; see also [17, Chapter 13] and [60]). Note that this study does
not assume multiplicity-freeness. In fact, we shall see (cf. Theorem 3.1) that the
triple (G, K, θ ) is multiplicity-free exactly when the associated Hecke algebra is
commutative.
We then focus on our main subject of study, namely multiplicity-free induced
representations (cf. Chap. 3); we extend to higher dimensions a criterion of Bump
and Ginzburg from [8]: this constitutes an analogue of the so-called weakly
symmetric Gelfand pairs (cf. [11, Example 4.3.2 and Exercise 4.3.3]); we develop
the theory of spherical functions in an intrinsic way, that is, by regarding them as
eigenfunctions of convolution operators (without using the decomposition of IndG Kθ
into irreducible representations) and obtain a characterization of spherical functions
by means of a functional equation. This approach is suitable to more general
settings, such as compact or locally compact groups: here we limit ourselves to the
finite case since the main examples that we have discovered (and that we have fully
analyzed) fall into this setting. Later (cf. Sect. 3.3), we express spherical functions as
matrix coefficients of irreducible (spherical) representations. In Sect. 3.6, we prove a
Frobenius–Schur type theorem for multiplicity-free triples (it provides a criterion for
determining the type of a given irreducible spherical representation, namely being
real, quaternionic, or complex).
As mentioned before, the case when θ is a one-dimensional representation and
the example of the Gelfand–Graev representation of GL(2, Fq ) were developed, in
full details, in [17, Chapters 13 and 14] (the last chapter is based on [53]; see also
the pioneering work by Green [38]). Here (cf. Sect. 3.4) we recover the analysis of
the one-dimensional case from the general theory we have developed so far and we
briefly sketch the Gelfand–Graev example (cf. Sect. 3.5) in order to provide some of
the necessary tools for our main new examples of multiplicity-free triples to which
the second part of the monograph (Chaps. 5 and 6) is entirely devoted.
Preface xi

A particular case of interest is when the subgroup K = N ≤ G is normal

(cf. Chap. 4). In the classical framework, (G, N) is a Gelfand pair if and only if
the quotient group G/N is Abelian and, in this case, the spherical Fourier analysis
simply reduces to the commutative harmonic analysis on G/N. In Chap. 4, we face
the corresponding analysis for multiplicity-free triples of the form (G, N, θ ), where
θ is an irreducible N-representation. Now, G acts by conjugation on the dual of
N and we denote by IG (θ ) the stabilizer of θ (this is the inertia group in Clifford
theory; cf. the monographs by Berkovich and Zhmud [5], Huppert [43], and Isaacs
[44]; see also [12, 14]). First of all, we study the commutant of IndG N θ —we show that
it is isomorphic to a modified convolution algebra on the quotient group IG (θ )/N—
and we describe the associated Hecke algebra: all of this theory is developed without
assuming multiplicity-freeness. We then prove that (G, N, θ ) is a multiplicity-
free triple if and only if IG (θ )/N is Abelian and the multiplicity of θ in each
irreducible representation of IG (θ ) is at most one. Moreover, if this is the case, the
associated Hecke algebra is isomorphic to L(IG (θ )/N), the (commutative) group
algebra of IG (θ )/N with its ordinary convolution product. Thus, as for Gelfand
pairs, normality of the subgroup somehow trivializes the analysis of multiplicity-
free triples.
As mentioned above, the last two chapters of the monographs are devoted to two
examples of multiplicity-free triples constructed by means of the group GL(2, Fq ).
Chapter 5 is devoted to the multiplicity-free triple (GL(2, Fq ), C, ν0 ), where C is
the Cartan subgroup of GL(2, Fq ), which is isomorphic to the quadratic extension
Fq 2 of Fq , and ν0 is an indecomposable multiplicative character of Fq 2 (that is, ν0
is a character of the multiplicative group F∗q 2 such that ν0 (z) is not of the form
ψ(zz), z ∈ F∗q 2 , where ψ is a multiplicative character of Fq and z is the conjugate
of z). Actually, C is a multiplicity-free subgroup, that is, (GL(2, Fq ), C, ν0 ) is
multiplicity-free for every multiplicative character ν0 . We remark that the case
ν0 = ιC (the trivial character of C) has been extensively studied by Terras under the
name of finite upper half plane [69, Chapters 19, 20] and corresponds to the Gelfand
pair (GL(2, Fq ), C). We have chosen to study, in full details, the indecomposable
case because it is quite different from the Gelfand pair case analyzed by Terras
and constitutes a new example, though much more difficult. We begin with a brief
description of the representation theory of GL(2, Fq ), including the Kloosterman
sums used for the cuspidal representations. We then compute the decomposition of
GL(2,Fq )
IndC ν0 into irreducible representations (cf. Sect. 5.4) and the corresponding
spherical functions (cf. Sects. 5.5 and 5.6). We have developed new methods: in
particular, in the study of the cuspidal representations, in order to circumvent some
technical difficulties, we use, in a smart way, a projection formula onto a one-
dimensional subspace.
In Chap. 6, we face the most important multiplicity-free triple of this monograph,
namely (GL(2, Fq 2 ), GL(2, Fq ), ρν ), where ρν is a cuspidal representation. Now the
representation that is induced is no more one-dimensional nor is itself an induced
representation (as in the parabolic case). We have found an intriguing phenomenon:
in the computations of the spherical functions associated with the corresponding
xii Preface

parabolic spherical representations, we must use the results of Chap. 5, in particular

GL(2,Fq )
the decomposition of an induced representation of the form IndC ξ , with ξ
a character of C. In other words, the methods developed in Chap. 5 (for the triple
(GL(2, Fq , C, ν0 )) turned out to be essential in the much more involved analysis of
the second triple (GL(2, Fq 2 ), GL(2, Fq ), ρν ).
We finally remark that it is not so difficult to find other examples of multiplicity-
free induced representations within the framework of finite classical groups: for
instance, as a consequence of the branching rule in the representation theory of the
symmetric groups (see, e.g., [13]), Sn is a multiplicity-free subgroup of Sn+1 for all
n ≥ 1. So, although the two examples that we have presented and fully analyzed
here are new and highly nontrivial, we believe that we have only scraped the surface
of the subject and that this deserves a wider investigation. For instance, in [3, 4]
several Gelfand pairs constructed by means of GL(n, Fq ) and other finite linear
groups are described. It would be interesting to analyze if some of these pairs give
rise to multiplicity free triples by induction of a nontrivial representation.
We also mention that, very recently, a similar theory for locally compact groups
has been developed by Ricci and Samanta in [56] (see also [47] and, for an earlier
reference, the seminal paper by Godement [36]). In particular, their condition (0.1)
(cf. also with (3) in [47]) corresponds exactly to our condition (2.1). In Sect. A.1,
we show that the Gelfand–Graev representation yields a solution to a problem raised
in the Introduction of their paper. For a classical account on the classification of
Gelfand pairs on Lie groups, we refer to the monograph by Wolf [71] and the survey
by Vinberg [70] (see also [72]).
The problem we addressed in the present monograph, namely the study of
multiplicity-free induced representations, is certainly too general for a comprehen-
sive and exhaustive study in its full generality. This is witnessed by the lack of
literature (at our knowledge), somehow leading to believe that researchers were
reluctant to attack such a general situation. So, although our research only barely
scraped the surface of the subject, we believe that our monograph constitutes a
serious attempt to obtain some reasonably meaningful new results in this direction.
We express our deep gratitude to Eiichi Bannai, Charles F. Dunkl, David
Ginzburg, Pierre de la Harpe, Hiroshi Mizukawa, Akihiro Munemasa, Jean-Pierre
Serre, Hajime Tanaka, Alain Valette, and the anonymous referees, for useful remarks
and suggestions. Finally, we warmly thank Elena Griniari from Springer Verlag
for her continuous encouragement and most precious help at various stages of the
preparation of the manuscript.

Roma, Italy Tullio Ceccherini-Silberstein

Roma, Italy Fabio Scarabotti
Roma, Italy Filippo Tolli
May 2020
Contents

1 Preliminaries . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .. . . . . . . . . . . . . . . . . . . . 1
1.1 Representations of Finite Groups .. . . . . . . . . . . . . . . . .. . . . . . . . . . . . . . . . . . . . 1
1.2 The Group Algebra, the Left-Regular and the Permutation
Representations, and Gelfand Pairs . . . . . . . . . . . . . . . .. . . . . . . . . . . . . . . . . . . . 3
1.3 The Commutant of the Left-Regular and Permutation
Representations .. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .. . . . . . . . . . . . . . . . . . . . 5
1.4 Induced Representations . . . . . . . . . . . . . . . . . . . . . . . . . . .. . . . . . . . . . . . . . . . . . . . 7
2 Hecke Algebras .. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .. . . . . . . . . . . . . . . . . . . . 11
2.1 Mackey’s Formula for Invariants Revisited . . . . . . .. . . . . . . . . . . . . . . . . . . . 11
2.2 The Hecke Algebra Revisited .. . . . . . . . . . . . . . . . . . . . .. . . . . . . . . . . . . . . . . . . . 21
3 Multiplicity-Free Triples . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .. . . . . . . . . . . . . . . . . . . . 31
3.1 A Generalized Bump–Ginzburg Criterion.. . . . . . . .. . . . . . . . . . . . . . . . . . . . 34
3.2 Spherical Functions: Intrinsic Theory . . . . . . . . . . . . .. . . . . . . . . . . . . . . . . . . . 35
3.3 Spherical Functions as Matrix Coefficients . . . . . . .. . . . . . . . . . . . . . . . . . . . 41
3.4 The Case dim θ = 1 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .. . . . . . . . . . . . . . . . . . . . 47
3.5 An Example: The Gelfand–Graev Representation of GL(2, Fq ). . . . . 48
3.6 A Frobenius–Schur Theorem for Multiplicity-Free Triples . . . . . . . . . . 50
4 The Case of a Normal Subgroup . . . . . . . . . . . . . . . . . . . . . . .. . . . . . . . . . . . . . . . . . . . 53
4.1 Unitary Cocycles . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .. . . . . . . . . . . . . . . . . . . . 54
4.2 Cocycle Convolution . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .. . . . . . . . . . . . . . . . . . . . 55
4.3 The Inertia Group and Unitary Cocycle Representations.. . . . . . . . . . . . 56
4.4 A Description of the Hecke Algebra H˜ (G, N, θ ) . . . . . . . . . . . . . . . . . . . 61
4.5 The Hecke Algebra H (G, N, ψ) . . . . . . . . . . . . . . . . .. . . . . . . . . . . . . . . . . . . . 63
4.6 The Multiplicity-Free Case and the Spherical Functions .. . . . . . . . . . . . 65
5 Harmonic Analysis of the Multiplicity-Free Triple
(GL(2, Fq ), C, ν).. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .. . . . . . . . . . . . . . . . . . . . 71
5.1 The Multiplicity-Free Triple (GL(2, Fq ), C, ν). . .. . . . . . . . . . . . . . . . . . . . 71
5.2 Representation Theory of GL(2, Fq ): Parabolic Representations .. . . 74

xiii
xiv Contents

5.3 Representation Theory of GL(2, Fq ): Cuspidal Representations . . . . 75

5.4 The Decomposition of IndG C ν0 . . . . . . . . . . . . . . . . . . . . .. . . . . . . . . . . . . . . . . . . . 77
5.5 Spherical Functions for (GL(2, Fq ), C, ν0 ): the Parabolic Case . . . . . 79
5.6 Spherical Functions for (GL(2, Fq ), C, ν0 ): the Cuspidal Case . . . . . 83
6 Harmonic Analysis of the Multiplicity-Free Triple
(GL(2, Fq 2 ), GL(2, Fq ), ρν ) . . . . . . . . . . . . . . . . . . . . . . . . . . . .. . . . . . . . . . . . . . . . . . . . 95
6.1 Spherical Functions for (GL(2, Fq 2 ), GL(2, Fq ), ρν ): the
Parabolic Case . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .. . . . . . . . . . . . . . . . . . . . 96
6.2 Spherical Functions for (GL(2, Fq 2 ), GL(2, Fq ), ρν ): the
Cuspidal Case. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .. . . . . . . . . . . . . . . . . . . . 104
6.3 A Non-normalized L̃ ∈ HomG1 (ρν , ResG 2
G1 ρμ ) . . .. . . . . . . . . . . . . . . . . . . . 116

Appendix A .. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .. . . . . . . . . . . . . . . . . . . . 121
A.1 On a Question of Ricci and Samanta . . . . . . . . . . . . . .. . . . . . . . . . . . . . . . . . . . 121
A.2 The Gelfand Pair (GL(2, Fq 2 ), GL(2, Fq )) . . . . . . .. . . . . . . . . . . . . . . . . . . . 123
A.3 On some Questions of Dunkl . . . . . . . . . . . . . . . . . . . . . .. . . . . . . . . . . . . . . . . . . . 125
References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .. . . . . . . . . . . . . . . . . . . . 130
Index . . . . . . . . .. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .. . . . . . . . . . . . . . . . . . . . 135
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·, ·V The scalar product on a vector space V 1
·, ·Hom(W,U ) The Hilbert–Schmidt scalar product on Hom(W, U ) 1
·V The norm on a vector space V 1
·End(W ) The Hilbert–Schmidt norm on End(W ) 2
δg The Dirac function supported by g ∈ G 4
η A generator of the (cyclic) group F∗q 49
(gθ, V ) The g-conjugate of the irreducible N-representation (θ, V ), g ∈ G, 56
NG
(θ s , Vs ) The Ks -representation associated with the K-representation (θ, V ) 13
and s ∈ S
Θ(q) The unitary in End(V ) conjugating (gθ, V ) and (θ, V ), q ∈ Q 57
(ιG , C) The trivial representation of a group G 2
(λG , L(G)) The left regular representation of a group G 4
(λ, L(G)K ) The permutation representation of G with respect to the subgroup K 4
μ A multiplicative character of Fq 2 77
μ A generic indecomposable character of Fq 4 95
ν A multiplicative character of Fq 2 73
ν A fixed indecomposable multiplicative character of Fq 2 95
ν0 An indecomposable multiplicative character of Fq 2 77
ξ The ∗-isomorphism between H (G, K, θ) and EndG (IndG V ) 15
K
ξ1 , ξ2 Multiplicative characters of Fq 2 95
Ξ A right-inverse of the ∗-isomorphism ξ 18
(ρG , L(G)) The right regular representation of a group G 4
ρν The (q − 1)-dimensional cuspidal representation of GL(2, Fq ), 77
ν∈F  ∗
2
q
(σ, W ) A group representation σ : G → GL(W ) of a group G 2
(σ , W ) The conjugate of the group representation (σ, W ) 50
σ ⊕ρ The direct sum of two representations σ and ρ 3

xv
xvi Symbols

Symbol Definition Page

σ ρ The representation (σ, W ) is contained in the representation (ρ, V ) 3
τ An antiautomortphism of a group G 32
τ A unitary cocycle H × H → T 54
τρ A unitary cocycle which is a coboundary (ρ : H → T) 54
φ A spherical function in the Hecke algebra H (G, K, ψ) 35
φσ The spherical function associated with σ ∈ J 44
φν The spherical function associated with the cuspidal 93
representation ρ ν
Φ A linear functional on the Hecke algebra H (G, K, ψ) 38
χν The character of the cuspidal representation ρ ν 107
χσ The character of the representation σ 6
χψ1 ,ψ2 The one-dimensional representation of B associated with 75
∗q
ψ1 , ψ2 ∈ F
ψ0
χ The one-dimensional representation of GL(2, Fq ) associated with 75
ψ∈F ∗q
ψ1
χ ∗q
The q-dimensional parabolic representation of GL(2, Fq ), ψ ∈ F 75
ψ1 ,ψ2
χ The (q + 1)-dimensional parabolic representation of GL(2, Fq ), 75
ψ1 = ψ2 ∈ F ∗q
ψ The element in L(K) associated with (θ, V ) and v ∈ V , v = 1 21
ψ A multiplicative character of Fq 77
Ψ The decomposable character of F∗ 2 associated with ψ ∈ F∗q 76
q
Aff(Fq ) The affine group over Fq 71
B The Borel subgroup of GL(2, Fq ) 48
B1 The Borel subgroup of GL(2, Fq ) 95
B2 The Borel subgroup of GL(2, Fq 2 ) 95
B(T, H ) The subgroup of all coboundaries of H 54
C The Cartan subgroup of GL(2, Fq ) 48
C1 The Cartan subgroup of GL(2, Fq ) 95
C2 The Cartan subgroup of GL(2, Fq 2 ) 95
C (T, H ) The group of all unitary cocycles H → T 54
dσ The dimension dim(W ) of a group representation (σ, W ) 2
D The diagonal subgroup of GL(2, Fq ) 49
Eρ The orthogonal projection onto the ρ-isotypic component of (σ, W ) 7
Eσ The orthogonal projection from I (G, K, ψ) onto Sσ Wσ 46
End(W ) The algebra of all linear endomorphism T : W → W 1
EndG (W ) The commutant of (σ, W ) 2
f∗ The function f ∗ (g) = f (g −1 ) for all f ∈ L(G) and g ∈ G 4
f1 ∗ f2 The convolution product of two functions f1 , f2 ∈ L(G) 3
f1 ∗ η f2 The cocycle convolution product of f1 , f2 ∈ L(H ) (η ∈ C (T, H )) 55
F∗ The adjoint of an element F ∈ H (G, K, θ) 12
F0 (x, y) The function defined in (5.22) 84
F0 (x, y) The function defined in (5.27) 87
F1 ∗ F2 The convolution product of elements F1 , F2 ∈ H (G, K, θ) 12
Symbols xvii

Symbol Definition Page

Fν The element in Vχψ1 ,ψ2 associated with ν, ν  = ψ1 ψ2 , 79
ψ1 = ψ2 ∈ F ∗q
F The spherical Fourier transform 45
Fq The field with q := pn elements 48
F∗q The multiplicative subgroup of the field Fq 49
Fq 2 The quadratic extension of Fq 49

G The dual of a group G, i.e., a (fixed, once and for all) complete set 2
of pairwise-inequivalent irreducible representations of G
G (G, H, K) The Greenhalgebra associated with K ≤ H ≤ G 125
G0 (x, y) The function defined as F0 (x, y) with μ0 in place of ν0 92
0 (x, y)
G The function defined as F00 (x, y) with μ0 in place of ν0 92
G1 The general linear group GL(2, Fq ) 95
G2 The general linear group GL(2, Fq 2 ) 95
GL(W ) The general linear group of the vector space W 1
GL(2, Fq ) The general linear group of rank 2 over Fq 48
Hom(W, U ) The space of all linear operators T : W → U 1
HomG (W, U ) The space of all intertwiners of the G-representations (σ, W ) and 2
(ρ, U )
H(G, K, θ) The Hecke algebra associated with the triple (G, K, θ) 11
H (G, K, ψ) The Hecke algebra associated with the triple (G, K, θ) and ψ 22
H 2 (T, H ) The second cohomology group C (T, H )/B(T, H ) of H 54
±i The square roots (in Fq 2 ) of η 49
IG (θ) The inertia group of the irreducible N-representation (θ, V ), 57
NG
IV The identity operator V → V 42
I (G, K, ψ) The range Tv (IndGK V ) ⊂ L(G) of the operator Tv 22
(IndG G
K θ, Ind K V ) The induction to the group G ≥ K of the K-representation (θ, V ) 7
j = jχ ,ν The generalized Kloosterman sum associated with nontrivial 76
χ ∈F q and indecomposable ν ∈ F  ∗
q2
J A complete set of pairwise-inequivalent irreducible 9
G-representations contained in IndGKθ
J = Jχ,μ The generalized Kloosterman sum associated with nontrivial 104
χ ∈Fq , cf. (6.19), and indecomposable μ ∈ F∗
q4
Ks The subgroup K ∩ sKs −1 , where s ∈ S 12
L The isometric immersion L(F∗q ) → L(F∗q 2 ) 107
L(H )η The twisted group algebra of H , η ∈ C (T, H ) 55
L(G) The group algebra of G 3
L(G)K The L(G)-subalgebra of K-right-invariant functions on G 4
L(G, H, K) The Hirschman subalgebra of H − K-invariant functions in L(G) 125
Lθ The operator L(F∗q ) → L(F∗q 2 ), θ ∈ Fq 105
Lσ The isometric map V → W such that HomK (V , ResG K W ) = CLσ 41
LT (G, K, θ) associated with
The element in H 13
T ∈ HomKs (ResK s
Ks θ, θ )
KL(G)K The L(G)-subalgebra of bi-K-invariant functions on G 4
xviii Symbols

Symbol Definition Page

ρ
mσ The multiplicity of the representation σ in the representation ρ 3
P The orthogonal projection of L(G) onto I (G, K, ψ) 22
P The projection of L(F∗q 2 ) onto the subspace of ρν 107
Q A complete set of representatives for the cosets of N in IG (θ), 56
1G ∈ Q
Q1 The operator L(F∗q 2 ) → L(F∗q 2 ) 107
(ResG
K σ, W ) The restriction of the G-representation (σ, W ) to the subgroup 7
K≤G
S A complete set of representatives for the double K-cosets in G 12
S0 The set of all s ∈ S such that HomKs (ResK s
Ks θ, θ ) is nontrivial 14
S (G, K, ψ) The set of all spherical functions in the Hecke algebra H (G, K, ψ) 35
Sσ The isometric immersion of Wσ into I (G, K, ψ) 43
Sv The ∗-antiisomorphism from H (G, K, θ) onto H (G, K, ψ) 22
tr(·) The trace of linear operators 1
T → T  An antiautomorphism of the algebra End(V ) 32
T A transversal for the left-cosets of K in G 8
Tσ The isometric immersion of W into IndG KV 41
Tv The isometry in HomG (IndG K V , L(G)) 22
T The group R/Z 54
T∗ The adjoint ∈ Hom(U, W ) of the linear operators T ∈ Hom(W, U ) 1
Tf The convolution operator in EndG (L(G)) with kernel f ∈ L(G) 5
uσi,j A matrix coefficient for the representation σ 6
U The unipotent subgroup of GL(2, Fq ) 49
U1 The unipotent subgroup of GL(2, Fq ) 95
U2 The unipotent subgroup of GL(2, Fq 2 ) 95
Uφ The spherical representation associated with the spherical 40
function φ  
01
w The matrix ∈ GL(2, Fq ) 49
10
wσ σ v ∈
The vector L  Wσ , σ ∈ J 43
i 1
W The matrix ∈ GL(2, Fq 2 ) 96
10
WK The subspace of K-invariant vectors of W 4
Wσ The representation space of the spherical representation σ ∈ J 43
W⊥ The orthogonal complement of the subspace W ⊂ U 3
Z The center of GL(2, Fq ) 49
Chapter 1
Preliminaries

In this chapter, we fix notation and recall some basic facts on linear algebra and
representation theory of finite groups that will be used in the proofs of several results
in the sequel.

1.1 Representations of Finite Groups

All vector spaces considered here are complex. Moreover, we shall equip every finite
dimensional vector space V with a scalar product denoted by ·, ·V and associated
norm ·V ; we usually omit the subscript if the vector space we are referring to is
clear from the context. Given two finite dimensional vector spaces W and U , we
denote by Hom(W, U ) the vector space of all linear maps from W to U . When
U = W we write End(W ) = Hom(W, W ) and denote by GL(W ) ⊆ End(W ) the
general linear group of W consisting of all bijective linear self-maps of W . Also, for
T ∈ Hom(W, U ) we denote by T ∗ ∈ Hom(U, W ) the adjoint of T .
We define a (normalized) Hilbert-Schmidt scalar product on Hom(W, U ) by
setting

1
T1 , T2 Hom(W,U ) = tr(T2∗ T1 ) (1.1)
dim W
for all T1 , T2 ∈ Hom(W, U ), where tr(·) denotes the trace of linear operators; note
that this scalar product (as well as all other scalar products which we shall introduce
thereafter) is conjugate-linear in the second argument. Moreover, by centrality of
the trace (so that tr(T2∗ T1 ) = tr(T1 T2∗ )), we have

dim U ∗ ∗
T1 , T2 Hom(W,U ) = T , T Hom(U,W ) . (1.2)
dim W 2 1

© Springer Nature Switzerland AG 2020 1

T. Ceccherini-Silberstein et al., Gelfand Triples and Their Hecke Algebras,
Lecture Notes in Mathematics 2267, https://doi.org/10.1007/978-3-030-51607-9_1
2 1 Preliminaries

√
In particular, the map T → dim U/ dim W T ∗ is an isometry from Hom(W, U )
onto Hom(U, W ). Finally, note that denoting by IW : W → W the identity operator,
we have IW End(W ) = 1.
We now recall some basic facts on the representation theory of finite groups. For
more details we refer to our monographs [11, 13, 17]. Let G be a finite group. A
unitary representation of G is a pair (σ, W ) where W is a finite dimensional vector
space and σ : G → GL(W ) is a group homomorphism such that σ (g) is unitary
(that is, σ (g)∗ σ (g) = IW ) for all g ∈ G. In the sequel, the term “unitary” will be
omitted. We denote by dσ = dim(W ) the dimension of the representation (σ, W ).
We denote by (ιG , C) the trivial representation of G, that is, the one-dimensional
G-representation defined by ιG (g) = IdC for all g ∈ G.
Let (σ, W ) be a G-representation. A subspace V ≤ W is said to be G-invariant
provided σ (g)V ⊆ V for all g ∈ G. Writing σ |V (g) = σ (g)|V for all g ∈ G,
we have that (σ |V , V ) is a G-representation, called a subrepresentation of σ . We
then write σ |V ≤ σ . One says that σ is irreducible provided the only G-invariant
subspaces are trivial (equivalently, σ admits no proper subrepresentations).
Let (σ, W ) and (ρ, U ) be two G-representations. We denote by

HomG (W, U ) = {T ∈ Hom(W, U ) : T σ (g) = ρ(g)T , for all g ∈ G},

the space of all intertwining operators. When U = W we write EndG (W ) =

HomG (W, W ). We equip HomG (W, U ) with a scalar product by restricting the
Hilbert-Schmidt scalar product (1.1).
Observe that if T ∈ HomG (W, U ) then T ∗ ∈ HomG (U, W ). Indeed, for all
g ∈ G,

T ∗ ρ(g) = T ∗ ρ(g −1 )∗ = (ρ(g −1 )T )∗ = (T σ (g −1 ))∗ = σ (g −1 )∗ T ∗ = σ (g)T ∗ .

(1.3)
One says that (σ, W ) and (ρ, U ) are equivalent, and we shall write (σ, W ) ∼
(ρ, U ) (or simply σ ∼ ρ), if there exists a bijective intertwining operator T ∈
HomG (W, U ).
The vector space EndG (W ) of all intertwining operators of (σ, W ) with itself,
when equipped with the multiplication given by the composition of maps and the
adjoint operation is a ∗-algebra (see [13, Chapter 7], [17, Sections 10.3 and 10.6]),
called the commutant of (σ, W ). We can thus express the well known Schur’s lemma
as follows: (σ, W ) is irreducible if and only if its commutant is one-dimensional (as
a vector space), that is, it reduces to the scalars (the scalar multiples of the identity
IW ).
We denote by G  a (fixed, once and for all) complete set of pairwise-inequivalent
irreducible representations of G and we refer to it as to the dual of G. It is well
known (cf. [11, Theorem 3.9.10] or [17, Theorem 10.3.13.(ii)]) that the cardinality
of G equals the number of conjugacy classes in G so that, in particular, G  is finite.
Moreover, if σ, ρ ∈ G  we set δσ,ρ = 1 (resp. = 0) if σ = ρ (resp. otherwise).
Let (σ, W ) and (ρ, U ) be two G-representations.
1.2 The Group Algebra, the Left-Regular and the Permutation. . . 3

The direct sum of σ and ρ is the representation (σ ⊕ ρ, W ⊕ U ) defined by

[(σ ⊕ ρ)(g)](w, u) = (σ (g)w, ρ(g)u) for all g ∈ G, w ∈ W and u ∈ U .
Moreover, if σ is a subrepresentation of ρ, then denoting by W ⊥ = {u ∈ U :
u, wU = 0 for all w ∈ W } the orthogonal complement of W in U , we have
that W ⊥ is a G-invariant subspace and ρ = σ ⊕ ρ|W ⊥ . From this, one deduces
that every representation ρ decomposes as a (finite) direct sum of irreducible
subrepresentations. More generally, when σ is equivalent to a subrepresentation of
ρ, we say that σ is contained in ρ and we write σ ρ (clearly, if σ ≤ ρ then
σ ρ).
ρ
Suppose that (σ, W ) is irreducible. Then the number m = mσ =
dim HomG (W, U ) denotes the multiplicity of σ in ρ. This means that one may
decompose U = U1 ⊕ U2 ⊕ · · · ⊕ Um ⊕ Um+1 with (ρ|Ui , Ui ) ∼ (σ, W ) for
all i = 1, 2, . . . , m and σ is not contained in ρ|Um+1 . The G-invariant subspace
U1 ⊕U2 ⊕· · ·⊕Um ≤ U is called the W -isotypic component of U and is denoted by
mW . One also says that ρ (or, equivalently, U ) contains m copies of σ (resp. of W ).
If this is the case, we say that T1 , T2 , . . . , Tm ∈ HomG (W, U ) yield an isometric
orthogonal decomposition of mW if Ti ∈ HomG (W, U ), Ti W ≤ U  Um+1 , and,
in addition,

Ti w1 , Tj w2 U = w1 , w2 W δi,j (1.4)

for all w1 , w2 ∈ W and i, j = 1, 2, . . . , m. This implies that the subrepresentation

mW = U1 ⊕ U2 ⊕ · · · ⊕ Um is equal to the orthogonal direct sum T1 W ⊕ T2 W ⊕
· · · ⊕ Tm W , and each operator Tj is a isometry from W onto Uj ≡ T Wj . For a quite
detailed analysis of this decomposition, we refer to [17, Section 10.6].
Finally, a representation (ρ, U ) is multiplicity-free if every (σ, W ) ∈ G  has
multiplicity at most one in ρ, that is, dim HomG (W, U ) ≤ 1. In other words, given
a decomposition of ρ = ρ1 ⊕ ρ2 ⊕ · · · ⊕ ρn into irreducible subrepresentations, the
ρi ’s are pairwise inequivalent. Alternatively, as suggested by de la Harpe [40], one
has that (ρ, U ) is multiplicity-free if for any nontrivial decomposition ρ = ρ1 ⊕ ρ2
(with (ρ1 , U1 ) and (ρ2 , U2 ) not necessarily irreducible) there is no (σ, W ) ∈ G 
such that σ ρi (i.e., dim HomG (W, Ui ) ≥ 1) for i = 1, 2. The equivalence
between the two definitions is an immediate consequence of the isomorphism
HomG (W, U1 ⊕ U2 ) ∼ = HomG (W, U1 ) ⊕ HomG (W, U1 ).

1.2 The Group Algebra, the Left-Regular and the

Permutation Representations, and Gelfand Pairs

We denote by L(G) the group algebra of G.

This is the vector space of all functions f : G → C equipped with the
convolution product ∗ defined by setting [f1 ∗ f2 ](g) = h∈G f1 (h)f2 (h−1 g) =
−1
h∈G f1 (gh)f2 (h ), for all f1 , f2 ∈ L(G) and g ∈ G. We shall endow L(G)
4 1 Preliminaries

with the scalar product ·, ·L(G) defined by setting

f1 , f2 L(G) = f1 (g)f2 (g) (1.5)

g∈G

for all f1 , f2 ∈ L(G). The Dirac functions δg , defined by δg (g) = 1 and δg (h) = 0
if h = g, for all g, h ∈ G, constitute a natural orthonormal basis for L(G). We
shall also equip L(G) with the involution f → f ∗ , where f ∗ (g) = f (g −1 ), for all
f ∈ L(G) and g ∈ G. It is straightforward to check that (f1 ∗ f2 )∗ = f2∗ ∗ f1∗ , for
all f1 , f2 ∈ L(G). We shall thus regard L(G) as a ∗-algebra.
The left-regular representation of G is the G-representation (λG , L(G)) defined
by setting [λG (h)f ](g) = f (h−1 g), for all f ∈ L(G) and h, g ∈ G. Similarly,
the right-regular representation of G is the G-representation (ρG , L(G)) defined
by setting [ρG (h)f ](g) = f (gh), for all f ∈ L(G) and h, g ∈ G. Note that the
left-regular and right-regular representations commute, that is,

λG (g1 )ρG (g2 ) = ρG (g2 )λG (g1 ) (1.6)

for all g1 , g2 ∈ G.
Given a subgroup K ≤ G we denote by

L(G)K = {f ∈ L(G) : f (gk) = f (g), for all g ∈ G, k ∈ K}

and
K
L(G)K = {f ∈ L(G) : f (k1 gk2 ) = f (g), for all g ∈ G, k1 , k2 ∈ K}

the L(G)-subalgebras of K-right-invariant and bi-K-invariant functions on G,

respectively. Note that the subspace L(G)K ≤ L(G) is G-invariant with respect to
the left-regular representation. The G-representation (λ, L(G)K ), where λ(g)f =
λG (g)f for all g ∈ G and f ∈ L(G)K (equivalently, λ = λG |L(G)K ) is called the
permutation representation of G with respect to the subgroup K.
More generally, given a representation (σ, W ) we denote by

W K = {w ∈ W : σ (k)w = w, for all k ∈ K} ≤ W

the subspace of K-invariant vectors of W . This way, if (σ, W ) = (ρG , L(G)) we

K
have (L(G))K = L(G)K while, if (σ, W ) = (λ, L(G)K ) we have L(G)K =
KL(G)K .

For the following result we refer to [10] and/or to the monographs [11, Chapter
4] and [24].
1.3 The Commutant of the Left-Regular and Permutation Representations 5

Theorem 1.1 The following conditions are equivalent:

(a) The algebra KL(G)K is commutative;
(b) the permutation representation (λ, L(G)K ) is multiplicity-free;
(c) the algebra EndG (L(G)K ) is commutative;
(d)  one has dim(W K ) ≤ 1;
for every (σ, W ) ∈ G
(e)  one has dim HomG (W, L(G)K ) ≤ 1.
for every (σ, W ) ∈ G
Note that the equivalence (a) ⇔ (c) follows from the anti-isomorphism (1.11)
below.
Definition 1.1 If one of the equivalent conditions in Theorem 1.1 is satisfied, one
says that (G, K) is a Gelfand pair.

1.3 The Commutant of the Left-Regular and Permutation

Representations

Given f ∈ L(G), the (right) convolution operator with kernel f is the linear map
Tf : L(G) → L(G) defined by

Tf f = f ∗ f (1.7)

for all f ∈ L(G). We have

Tf1 ∗f2 = Tf2 Tf1 (1.8)

and

Tf ∗ = (Tf )∗ , (1.9)

for all f1 , f2 and f in L(G). Moreover, Tf ∈ EndG (L(G)) (this is a consequence

of (1.6)) and the map

L(G) −→ EndG (L(G))

(1.10)
f −→ Tf

is a ∗-anti-isomorphism of ∗-algebras (see [13, Proposition 1.5.2] or [17, Proposi-

tion 10.3.5]). Note that the restriction of the map (1.10) to the subalgebra KL(G)K
of bi-K-invariant functions on G yields a ∗-anti-isomorphism
K
L(G)K → EndG (L(G)K ). (1.11)
6 1 Preliminaries

It is easy to check that Tf δg = λG (g)f and tr(Tf ) = |G|f (1G ). We deduce that

1  1  
Tf1 , Tf2 End(L(G)) = tr (Tf2 )∗ Tf1 = tr Tf1 ∗f2∗
|G| |G|
∗

= f1 ∗ f2 (1G ) = f1 , f2 L(G) (1.12)

for all f1 , f2 ∈ L(G). This shows that the map (1.10) is an isometry.
Let (σ, W ) be a representation of G and let {w1 , w2 , . . . , wdσ } be an orthonormal
basis of W . The corresponding matrix coefficients uσj,i ∈ L(G) are defined by setting

uσj,i (g) = σ (g)wi , wj  (1.13)

for all i, j = 1, 2, . . . , dσ and g ∈ G.

 Then
Proposition 1.1 Let σ, ρ ∈ G.

ρ |G|
uσi,j , uh,k  = δσ,ρ δi,h δj,k (orthogonality relations), (1.14)
dσ

ρ |G|
uσi,j ∗ uh,k = δσ,ρ δj,h uσi,k (convolution properties), (1.15)
dσ

and
dσ
uσi,j (g1 g2 ) = uσi, (g1 )uσ,j (g2 ) (1.16)
=1

for all i, j = 1, 2, . . . , dσ , h, k = 1, 2, . . . , dρ , and g1 , g2 ∈ G.

Proof See [11, Lemma 3.6.3 and Lemma 3.9.14] or [17, Lemma 10.2.10,
Lemma 10.2.13, and Proposition 10.3.6]. 

dσ
The sum χ σ = i=1 ui,i ∈ L(G) of the diagonal entries of the matrix
σ

coefficients is called the character of σ . Note that χ σ (g) = tr(σ (g)) for all g ∈ G.
The following elementary formula is a generalization of [11, Exercise 9.5.8.(2)] (see
also [17, Proposition 10.2.26)]).
Proposition 1.2 Suppose (σ, W ) is irreducible and let w ∈ W be a vector of
norm 1. Consider the associated diagonal matrix coefficient φw ∈ L(G) defined
by φw (g) = σ (g)w, w for all g ∈ G. Then

dσ
χ σ (g) = φw (h−1 gh) (1.17)
|G|
h∈G

for all g ∈ G.
1.4 Induced Representations 7

Proof Let {w1 , w2 , . . . , wdσ } be an orthonormal basis of W with w1 = w and let

uσj,i as in (1.13). Then, for all g ∈ G and i = 1, 2, . . . , dσ , we have

dσ
σ (g)wi = uσj,i (g)wj
j =1

so that

φw (h−1 gh) = σ (g)σ (h)w1 , σ (h)w1 

h∈G h∈G
dσ
= uσj,1 (h)uσ,1 (h)σ (g)wj , w 
j,=1 h∈G

dσ
= uσj,1 uσ,1 σ (g)wj , w 
j,=1

|G| σ
(by (1.14)) = χ (g),
dσ

and (1.17) follows. 


From [13, Corollary 1.3.15] we recall the following fact. Let (σ, W ) and (ρ, V )
be two G-representations and suppose that ρ is irreducible and contained in σ . Then

dρ
Eρ = χ ρ (g)σ (g) (1.18)
|G|
g∈G

is the orthogonal projection onto the ρ-isotypic component of W .

1.4 Induced Representations

Let now K ≤ G be a subgroup. We denote by (ResG K σ, W ) the restriction of the G-

representation (σ, W ) to K, that is, the K-representation defined by [ResG K σ ](k) =
σ (k) for all k ∈ K.
Given a K-representation (θ, V ) of K, denote by λ = IndG K θ the induced
representation (see, for instance, [7, 9, 13, 14, 17, 32, 52, 65, 66, 69]). We recall
that the representation space of λ is given by
−1
K V = {f : G → V such that f (gk) = θ (k )f (g), for all g ∈ G, k ∈ K}
IndG
(1.19)
8 1 Preliminaries

and that

[λ(g)f ](g ) = f (g −1 g ), (1.20)

for all f ∈ IndG K V and g, g ∈ G.

As an example, one checks that if (ιK , C) is the trivial representation of K,
then (IndGK ιK , IndK C) equals the permutation representation (λ, L(G) ) of G with
G K

respect to the subgroup K (see [14, Proposition 1.1.7] or [17, Example 11.1.6]).
Let T ⊆ G be a left-transversal for K, that is, a complete set of representatives
for the left-cosets gK of K in G. Then we have the decomposition

G= tK, (1.21)
t ∈T


where, from now on, denotes a disjoint union. For v ∈ V we define fv ∈ IndG
KV
by setting

θ (g −1 )v if g ∈ K
fv (g) = (1.22)
0 otherwise.

Then, for every f ∈ IndG

K V , we have

f = λ(t)fvt (1.23)
t ∈T

where vt = f (t) for all t ∈ T . The induced representation IndG

K θ is unitary with
respect to the scalar product ·, ·IndG V defined by
K

1
f1 , f2 IndG V = f1 (g), f2 (g)V = f1 (t), f2 (t)V (1.24)
K |K|
g∈G t ∈T

for all f1 , f2 ∈ IndG

K V . Moreover, if {vj : j = 1, 2, . . . , dθ } is an orthonormal basis
in V then the set

{λ(t)fvj : t ∈ T , j = 1, 2, . . . , dθ } (1.25)

is an orthonormal basis in IndG

K V (see [9, Theorem 2.1] and [17, Theorem 11.1.11]).
A well known relation between the induction of a K-representation (θ, V ) and
a G-representation (σ, W ) is expressed by the so called Frobenius reciprocity (cf.
[13, Theorem 1.6.11], [14, Theorem 1.1.19], or [17, Theorem 11.2.1]):

HomG (W, IndG ∼ K

K V ) = HomK (ResG W, V ). (1.26)
1.4 Induced Representations 9

Let J = {σ ∈ G:σ
K θ } denote a complete set of pairwise inequivalent
IndG
irreducible G-representations contained in IndGK θ . For σ ∈ J we denote by Wσ its
representation space and by mσ = dim HomG (σ, IndG K θ ) ≥ 1 its multiplicity in
IndGK θ . Then

IndG ∼
KV = mσ Wσ (1.27)
σ ∈J

is the decomposition of IndG

K V into irreducible G-representations and we have the
∗-isomorphism of ∗-algebras

EndG (IndG ∼
KV ) = Mmσ (C), (1.28)
σ ∈J

where Mm (C) denotes the ∗-algebra of all m × m complex matrices (cf. [17,
Theorem 10.6.3]). In particular:
Proposition 1.3 The following conditions are equivalent:

K θ is multiplicity-free (that is, mσ = 1 for all σ ∈ J );

(a) IndG
(b) the algebra EndG (IndG K
K V ) (i.e. the commutant of IndG V ) is commutative;
(c) EndG (IndK V ) is isomorphic to the ∗-algebra C = {f : J → C} equipped
G J

with pointwise multiplication and complex conjugation.

Remark 1.1 In (1.27) and (1.28) we have used the symbol ∼ = to denote an
isomorphism (with respect to the corresponding algebraic structure). We will use
the equality symbol = to denote an explicit decomposition. For instance, in the
multiplicity-free case this corresponds to a choice of an isometric immersion of Wσ
K V , that is, to a map Tσ ∈ HomG (Wσ , IndK V ) which is also an isometry.
into IndG G

K V ) = CTσ and
Clearly, in this case, HomG (Wσ , IndG

KV =
IndG Tσ Wσ (1.29)
σ ∈J

is the explicit decomposition. If multiplicities arise, then we decompose explicitly

each isotypic component as in Sect. 1.1 (cf. (1.4)).
Chapter 2
Hecke Algebras

Let G be a finite group and K ≤ G a subgroup. Recalling the equality between

the induced representation (IndG K ιK , IndK C) and the permutation representation
G

(λ, L(G) ), (1.11) yields a ∗-algebra isomorphism between the algebra of bi-K-
K

invariant functions on G and the commutant of the representation obtained by

inducing to G the trivial representation of K.
In Sect. 2.1, expanding the ideas in [7, Theorem 34.1] and in [53, Section 3],
we generalize this fact by showing that for a generic representation (θ, V ) of
K, the commutant of IndK G
G V , that is, EndG (IndK V ), is isomorphic to a suitable
convolution algebra of operator-valued maps on G. This may be considered as a
detailed formulation of Mackey’s formula for invariants (see [9, Section 6] or [17,
Corollary 11.4.4]).
Later, in Sect. 2.2, we show that, when θ is irreducible, the algebra EndG (IndG
KV )
is isomorphic to a suitable subalgebra of the group algebra L(G) of G.

2.1 Mackey’s Formula for Invariants Revisited

In this section, we study isomorphisms (or antiisomorphisms) between three ∗-

algebras. We explicitly use the terminology of a Hecke algebra only for the first
one (cf. Definition 2.1), although one may carry it also for the other two.
Let (θ, V ) be a K-representation. We denote by H (G, K, θ ) the set of all maps
F : G → End(V ) such that

F (k1 gk2 ) = θ (k2−1 )F (g)θ (k1−1 ), for all g ∈ G and k1 , k2 ∈ K. (2.1)

© Springer Nature Switzerland AG 2020 11

T. Ceccherini-Silberstein et al., Gelfand Triples and Their Hecke Algebras,
Lecture Notes in Mathematics 2267, https://doi.org/10.1007/978-3-030-51607-9_2
12 2 Hecke Algebras

Given F1 , F2 ∈ H (G, K, θ ) we define their convolution product F1 ∗ F2 : G →

End(V ) by setting

[F1 ∗ F2 ](g) = F1 (h−1 g)F2 (h) (2.2)

h∈G

for all g ∈ G, and their scalar product as

F1 , F2 H(G,K,θ) = F1 (g), F2 (g)End(V ) . (2.3)

g∈G

(G, K, θ ) we define the adjoint F ∗ : G → End(V ) by setting

Finally, for F ∈ H

F ∗ (g) = [F (g −1 )]∗ (2.4)

for all g ∈ G, where [F (g −1 )]∗ is the adjoint of the operator F (g −1 ) ∈ End(V ).

(G, K, θ ) is an associative unital algebra with respect
It is easy to check that H
to this convolution. The identity is the function F0 defined by setting F0 (k) =
−1
|K| θ (k ) for all k ∈ K and F0 (g) = 0 for g ∈ G not in K; see also (2.9) below.
1

Moreover, F ∗ still belongs to H (G, K, θ ), the map F → F ∗ is an involution, that

is, (F ) = F , and (F1 ∗ F2 ) = F2∗ ∗ F1∗ , for all F, F1 , F2 ∈ H
∗ ∗ ∗ (G, K, θ ).

Definition 2.1 The unital ∗-algebra H (G, K, θ ) is called the Hecke algebra
associated with the group G and the K-representation (θ, V ).
Let S ⊆ G be a complete set of representatives for the double K-cosets in G so
that

G= KsK. (2.5)
s∈S

We assume that 1G ∈ S , that is, 1G is the representative of K. For s ∈ S we

set

Ks = K ∩ sKs −1 (2.6)

and observe that given g ∈ KsK we have |{(k1, k2 ) ∈ K 2 : k1 sk2 = g}| = |Ks |.
Indeed, suppose that k1 sk2 = g = h1 sh2 , where k1 , k2 , h1 , h2 ∈ K. Then we have
h1 −1 k1 = sh2 k2 −1 s −1 which gives, in particular, h1 −1 k1 ∈ Ks . Thus there are
|Ks | = |k1 Ks | different choices for h1 , and since h2 = s −1 h1 −1 g is determined by
h1 , the observation follows.
As a consequence, given an Abelian group A (e.g. C, a vector space, etc.), for
any map Φ : G → A and s ∈ S we have

1
Φ(g) = Φ(k1 sk2 ). (2.7)
|Ks |
g∈KsK k1 ,k2 ∈K
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 Fig. 5 Fig. 6

On the other hand, let both the reference points “a” and “b”
(fig. 6) be situated short of the objective O, “a” being nearer the
objective and “b” nearer to the balloon. An error aa′ in spotting “a”
leads to an error OO′ in the objective greater than aa′. Notice that
this error diminishes as “A” approaches O, thus “a” being as a₁, the
error a₁a′₁ equal to aa′ leads to an error OO′₁, in the objective, less
than OO′. We would thus obtain an analogous result if we would
move the point “b” farther away.
 Therefore, when you are obliged to take the two reference
points between you and the objective, choose one as near the
objective as possible and the other as near as possible to the
balloon.
3. Investigation of range.
Identify details of the terrain situated over and short of the
objective on the alignment. Narrow this bracket down step by step;
situate the objective on the map according to its relative distance
from the two nearest identifiable reference points, taking into account
the deformations due to the laws of perspective and the relief of the
ground.
If you have a vertical photograph of the region, trace the
alignment on this photograph and make the investigation in range by
the same means.
The dangers against which I warned you before in connection
with the investigation of range apply in this case also, so it is
unnecessary to repeat them.
When the two last identifiable reference points are some
distance from each other, the situation of the objective has a
possible error, of which you know the size according to the distance
between the two reference points; it might be interesting to
remember this in case different information is obtained on this
objective from that obtained in the balloon.
This error can be considerably diminished if you use a vertical
photograph; the investigation can then be carried on by the same
method as on the map, with greater precision. In the case of a
battery, particularly, it is for the observer to find the position of each
piece.
In case, on account of dead ground or of a mask before the
battery, the observer sees the flashes or the smoke without seeing
the battery itself, he should mark the exact alignment in which the
flashes or smoke are seen, and determine the bracket in range—that
is, the reference points nearest the objective which are clearly over
and short. This document compared with other information can
facilitate the identification of the battery.
 Chapter III.
OBSERVATION OF FIRE.

The observation of fire is essentially the following operation,

repeated for each shot or salvo: Locating on the ground the position
of one point, which is the point of burst, and announcing its situation
in reference to another point, which is the target.
But it has been demonstrated that it is impossible, without using
the map, to determine the error in range of one point relatively to
another point not on the same alignment.
The operation must consist in:
1. Spotting on the map the point of burst.
2. Reading its position in reference to the target.
The observation of the burst—that is, the spotting of the point of
impact—is the same whether the observation is direct or lateral.

1. PREPARATION OF THE FIRE.

Draw on the map, and copy if possible on a vertical

photograph, the line balloon target (alignment of the center of
the objective) and draw through this point a perpendicular to
the alignment. In case the observation is lateral, draw also the
line battery target and its perpendicular.
To draw the line balloon target, it is not necessary to know the
horizontal projection of the balloon. It is enough to find on the ground
a point situated directly on the alignment of the center of the
objective.
 2. OBSERVATION OF A SHOT.

When the shell bursts, take quickly an alignment and

reference points in range of the point of burst; spot this point
on the map or on the photograph; give its error in reference to
the line battery-target, measuring it by the scale on the map or
on the photograph. (It is well to put the graphic scale on the
photos.)
The delicate part of the operation consists in seeing the shot at
the moment of burst. One must try to spot the apex of the inverted
cone formed by the burst, without paying any attention to the more or
less considerable cloud of smoke which follows and which will cause
mistakes if the burst was not seen immediately. The method of
situating the point is the same as that described above.
For the direction, one reference point is enough, because one
can consider the alignment of the point as practically parallel to the
balloon-target. For the range, a rapid study of the terrain before the
fire is sufficient to allow the observer to know the reference points by
which he can guide himself. His eyes must never leave the point of
burst until he has fixed well in his mind its situation in reference to
appropriate reference points. Not to do this would lead to errors and
to loss of time while he has to find the point again with his glasses to
study it some more.
When the observation is direct, the direction of the burst is, of
course, known as soon as it is seen. When the observation is lateral,
it is well to remember that the line battery-target can not be
materialized on the ground because it is in reality an oblique
alignment, leading to the same errors which we have discussed. It
follows all irregularities of the ground and, on account of this, can not
be followed exactly in oblique vision.
Particularly around batteries, the ground is often very irregular.
There may even be little spaces of dead ground, caused by hollows
which the map does not always show. The above method, applied
with the help of photographs, allows you to avoid errors resulting
from the existence of these hidden parts.
 PRACTICAL ADVICE.
1. Work sitting down, with the map on your knees and leaning
against the edge of the basket.
This position is preferable to all others, because it allows the
observer—
(a) To correct with his head and shoulders the movements of the
basket.
(b) To have the map always before him. He can consult it at any
moment, mark the necessary alignments without loss of time, use it
as a desk for drawing or taking notes, or as a wind shield when
looking at photographs.
The observer who works standing up must either pick up his
map from the bottom of the basket every time he needs it, which is
out of the question, or fasten it outside the basket; the latter solution
is inadvisable, it necessitates working in the wind when the map is
referred to, and every time the observer turns to look at his map he
disturbs the equilibrium of the basket.
It is advisable to work standing up only on days when there is
practically no wind, and the balloon is continually turning and never
becomes oriented.
2. Have always within reach a flat rule, a pencil, and a
duodecimeter rule.
To be able to trace an alignment on the map with precision, the
rule must rest on a firm surface. This happens when the map is
mounted on a drawing board; when it is mounted on a frame with
rollers, the frame should have, between the two thicknesses of the
map, a board level with the edges of the frame on which the rule can
slide with its whole length on the map.
With a hard pencil, well sharpened, precise and neat alignments
can be drawn.
The duodecimeter rule is for measuring distances on
photographs and on the map; chiefly in observations of fire.
 3. Hold the field glasses with both hands.
This advice, sometimes ignored by observers without
expedience, has a great influence on the accuracy of information.
When an observer holds the glasses in one hand, it is much more
difficult for him to correct the movements caused by the balloon and
to concentrate on a point. It is very important, therefore, to hold the
glasses firmly with both hands, especially when you are making a
delicate observation or when you wish to study an “objective” in
detail.
 PART III.

APPENDIX.
Note 1.—All observations of rounds refer to the line battery
target (b-t) and a line perpendicular to same passing through the
target. Observations are given in meters right and left and whether
the round is over or short. Indications as to deflection are given
before those of range. Indications as to the amount of error precede
those as to the sense of the error. Example, 50 meters “Right,”
“over.” Owing to the dispersion of fire when adjusting fire for field
artillery or howitzer, it is unnecessary and of little value to the battery
to give the amount of the error in range except when asked for by
the battery commander or when the error in range is abnormal (over
200 meters).
When the target is clearly visible and the effect of a round hitting
a target is evident, the observation “Target” is reported. Unless the
observer is certain of having seen the bursts “Unobserved” will be
sent. If, however, after a few seconds smoke can be seen rising from
trees, houses, etc., in proximity to the target, the observation
“Unobserved, but smoke seen rising left and over,” may be given.
Note 2.—Observers must beware of being over-confident in
their own powers of observation. True confidence only comes with
experience, and this is best attained by making ascents with a
trained observer when ranging a battery and checking one’s own
observations with those given by him. An observation must never be
given unless the observer is quite certain as to its correctness. It is
essential to good results that the artillery may be able to rely
absolutely on the observations sent down. The observer must watch
the target but must avoid straining his eyes by putting up his glasses
as soon as a round is fired. He should arrange for the chart room to
inform him when a shell is about to fall. The latter must know the
time of flight. Observers must learn to distinguish readily the bursts
of different kinds of shells.
Note 3.—If the balloon-target line makes an angle with the
battery-target line of more than 30° with field artillery and 20° with
heavy, the balloon position will be given to the battery, and all
observations will be given with reference to the balloon-target line
and the battery will replot accordingly.
 *** END OF THE PROJECT GUTENBERG EBOOK BALLOON
OBSERVATION, AND INSTRUCTIONS ON THE SUBJECT OF
WORK IN THE BASKET ***

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