Hecke Algebras

Abstract

In this work, we study multiplicity-free induced representations of finite groups. We analyze in great detail the structure of the Hecke algebra corresponding to the commutant of an induced representation and then specialize to the multiplicityfree case. We then develop a suitable theory of spherical functions that, in the case of induction of the trivial representation of the subgroup, reduces to the classical theory of spherical functions for finite Gelfand pairs.We also examine in detail the case when we induce from a normal subgroup, showing that the corresponding harmonic analysis can be reduced to that on a suitable Abelian group.The second part of the work is devoted to a comprehensive study of two examples constructed by means of the general linear group GL(2, F q ), where F q is the finite field on q elements. In the first example we induce an indecomposable character of the Cartan subgroup. In the second example we induce to GL(2, F q 2 ) a cuspidal representation of GL(2, F q ). Contents 1. Introduction 2. Preliminaries 2.1. Representations of finite groups 2.2. The group algebra, the left-regular and the permutation representations, and Gelfand pairs 2.3. The commutant of the left-regular and permutation representations 2.4. Induced representations 3. Hecke algebras 3.1. Mackey's formula for invariants revisited 3.2. The Hecke algebra revisited 4. Multiplicity-free triples 4.1. A generalized Bump-Ginzburg criterion 4.2. Spherical functions: intrinsic theory 4.3. Spherical functions as matrix coefficients