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Let $G$ be a locally compact group and let $K$ be a compact group. Let $(\tau, V_\tau)$ be an irreducible representation of $K$.

We consider the space of $Endo_K(\tau)$-valued, compactly supported continuous functions $f$ on $G$
with $$ f(k_1 g k_2) = \tau(k_1) f(g) \tau(k_2), $$ which is an $*$ algebra under convolution.

What is a good reference for such algebras, especially in the context with reductive group over local fields and the connection to representation theory?

edited Jan 22 2012 at 16:13

asked Jan 22 2012 at 16:01

Marc Palm
6,956●7●20

	Since you have no assumption on $K$ (except compactness), I don't see the connection with (twisted) Gelfand pairs... In particular, taking for $K$ the identity subgroup, you get the convolution algebra of compactly supported functions on $G$. Is this really what you want? – Alain Valette Jan 22 2012 at 22:30
	Usually one wants this algebra to be commutative to be a twisted Gelfand pair. – Benjamin Steinberg Jan 22 2012 at 23:57
	Of course, I am mostly interested in some non trivial compact subgroups such as the maximal compact in the case of $GL(N)$. Commutativity doesn't hold in general, I guess? – Marc Palm Jan 23 2012 at 7:51
	My understanding is that the term twisted Gelfand pair is only used when the algebra is commutative. – Benjamin Steinberg Jan 26 2012 at 14:26

Paul Broussous · Accepted Answer · 2012-01-22 18:08:55Z UTC

These Hecke algebras are intensively studied in the field of "type theory" for reductive $p$-adic groups.

You have a nice summary of basic facts with proofs in chapter 4 of Bushnell and Kutzko's book "The admissible dual of ${\rm GL}(N)$ via compact open subgroups" (the chapter is entitled "Interlude with Hecke algebras").

You may also read the monography "The Langlands conjecture for ${\rm GL}(2)$", written by Bushnell and Henniart. You'll find there a nice introduction to these algebras.

There are many other references. But it depends on what exactly you're interested in.

Benjamin Steinberg · Answer 2 · 2012-01-22 17:20:04Z UTC

1

A classical reference for twisted Gelfand pairs is J.R. Stembridge, On Schur's Q-functions and the primitive idempotents of a commutative Hecke algebra, J. Algebr. Comb. 1 (1992) 71–95, but I believe this paper considers only finite groups.

answered Jan 22 2012 at 17:20

Benjamin Steinberg
10.7k●1●14●49

Compact groups are essentially finite groups;) – Marc Palm Jan 26 2012 at 15:22

Twisted Gelfand pairs (Reference and examples)

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