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2020 | Book

Preliminaries Read chapter Read first chapter

Gelfand Triples and Their Hecke Algebras

Harmonic Analysis for Multiplicity-Free Induced Representations of Finite Groups

Authors: Prof. Tullio Ceccherini-Silberstein, Prof. Fabio Scarabotti, Prof. Filippo Tolli

Publisher: Springer International Publishing

Book Series : Lecture Notes in Mathematics

Part of: Springer Professional "Wirtschaft+Technik" , Springer Professional "Technik" , Springer Professional "Wirtschaft"

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About this book

This monograph is the first comprehensive treatment of multiplicity-free induced representations of finite groups as a generalization of finite Gelfand pairs. Up to now, researchers have been somehow reluctant to face such a problem in a general situation, and only partial results were obtained in the one-dimensional case. Here, for the first time, new interesting and important results are proved. In particular, after developing a general theory (including the study of the associated Hecke algebras and the harmonic analysis of the corresponding spherical functions), two completely new highly nontrivial and significant examples (in the setting of linear groups over finite fields) are examined in full detail. The readership ranges from graduate students to experienced researchers in Representation Theory and Harmonic Analysis.

Table of Contents

Frontmatter
Chapter 1. Preliminaries
Abstract
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.
Tullio Ceccherini-Silberstein, Fabio Scarabotti, Filippo Tolli
Chapter 2. Hecke Algebras
Abstract
Let G be a finite group and K ≤ G a subgroup. Recalling the equality between the induced representation \((\mathrm {Ind}^G_K\iota _K,\mathrm {Ind}^G_K{\mathbb {C}})\) and the permutation representation (λ, L(G)K), (1.​11) yields a ∗-algebra isomorphism between the algebra of bi-K-invariant functions on G and the commutant of the representation obtained by inducing to G the trivial representation of K.
Tullio Ceccherini-Silberstein, Fabio Scarabotti, Filippo Tolli
Chapter 3. Multiplicity-Free Triples
Abstract
This chapter is devoted to the study of multiplicity-free triples and their associated spherical functions. After the characterization of multiplicity-freenes in terms of commutativity of the associated Hecke algebra (Theorem 3.1), in Sect. 3.1 we present a generalization of a criterion due to Bump and Ginzburg (J Algebra 278(1):294–313, 2004). In the subsequent section, we develop the intrinsic part of the theory of spherical functions, that is, we determine all their properties (e.g. the Functional Equation in Theorem 3.4) that may be deduced without their explicit form as matrix coefficients, as examined in Sect. 3.3. In Sect. 3.4 we consider the case when the K-representation (V, θ) is one-dimensional. This case is treated in full details in Chapter 13 of our monograph. We refer to the CIMPA lecture notes by Faraut (Analyse harmonique sur les paires de Guelfand et les espaces hyperboliques, CIMPA Lecture Notes, 1980) for an excellent classical reference in the case of Gelfand pairs.
Tullio Ceccherini-Silberstein, Fabio Scarabotti, Filippo Tolli
Chapter 4. The Case of a Normal Subgroup
Abstract
In this section we consider triples of the form (G, N, θ) in the particular case when the subgroup N ≤ G is normal.
Tullio Ceccherini-Silberstein, Fabio Scarabotti, Filippo Tolli
Chapter 5. Harmonic Analysis of the Multiplicity-Free Triple
Abstract
In this chapter we study a family of multiplicity-free triples on \(\mathrm {GL}(2, \mathbb F_q)\) that generalize the well known Gelfand pair associated with the finite hyperbolic plane (see [Terras, Fourier analysis on finite groups and applications. London mathematical society student texts, vol 43. Cambridge University Press, Cambridge, 1999, Chapters 19, 20, 21, and 23]). We suppose that q is an odd prime power (cf. Sect. 3.​5) and we denote by \(\widehat {\mathbb F_q^*}\) (respectively \(\widehat {\mathbb F_{q^2}^*}\)) the multiplicative characters of \(\mathbb F_q\) (respectively \(\mathbb F_{q^2}\)).
Tullio Ceccherini-Silberstein, Fabio Scarabotti, Filippo Tolli
Chapter 6. Harmonic Analysis of the Multiplicity-Free Triple
Abstract
In this chapter we study an example of a multiplicity-free triple where the representation that we induce has dimension greater than one.
Let q = p h with p an odd prime and h ≥ 1. Set
$$\displaystyle G_1 = \mathrm {GL}(2,\mathbb F_{q}) \ \mbox{ and } \ G_2 = \mathrm {GL}(2,\mathbb F_{q^2}). $$
Moreover (cf. Sect. 3.​5), we denote by B j (resp. U j, resp. C j) the Borel (resp. the unipotent, resp. the Cartan) subgroup of G j, for j = 1, 2. Throughout this chapter, with the notation as in Sect. 5.​3, we let \(\nu \in \widehat {\mathbb F_{q^2}^*}\) be a fixed indecomposable character. We assume that \(\nu ^\sharp = \mathrm {Res}^{\mathbb F_{q^2}^*}_{\mathbb F_{q}^*} \nu \) is not a square: this slightly simplifies the decomposition into irreducibles. Finally, ρ ν denotes the cuspidal representation of G 1 associated with ν.
Tullio Ceccherini-Silberstein, Fabio Scarabotti, Filippo Tolli
Backmatter
Metadata
Title
Gelfand Triples and Their Hecke Algebras
Authors
Prof. Tullio Ceccherini-Silberstein
Prof. Fabio Scarabotti
Prof. Filippo Tolli
Copyright Year
2020
Electronic ISBN
978-3-030-51607-9
Print ISBN
978-3-030-51606-2
DOI
https://doi.org/10.1007/978-3-030-51607-9

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