## Noncommutative Geometry Seminar

Date: November 13, 2019

Time: 4:00PM - 5:00PM

Location: BLOC 220

Speaker: Tullio Ceccherini-Silberstein, University of Sannio

Title: Hecke algebras of multiplicity-free induced representations

Abstract: Given a finite group G and a subgroup K, one says that (G,K) is a Gelfand pair provided the associated permutation representation (\lambda, L(G/K)) is multiplicity-free (that is, decomposes into pairwise non-equivalent irreducible subrepresentations). This condition is equivalent to the algebra End_G(L(G/K)) of interwining operators being commutative. Observe that \lambda is nothing but the induced representation Ind_K^G \iota_K of the trival representation \iota_K of K. In [CS-S-T] we consider triples (G,K,\theta), where \theta is, more generally, an irreducible K-representation and introduce a Hecke-type algebra H(G,K,\theta) - analogous to End_G(L(G/K)) - and show that that Ind_K^G\theta is multiplicity-free if and only if H (G,K,\theta) is commutative. We apply our results in the context of the representation theory of GL_2(q), the general linear group of a field with q elements. [CS-S-T] Harmonic analysis and spherical functions for multiplicity-free induced representations of finite groups. Springer (to appear) arXiv: 1811.09526.

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