# Twisted Gelfand pairs (Reference and examples)

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?

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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 '12 at 22:30
Usually one wants this algebra to be commutative to be a twisted Gelfand pair. – Benjamin Steinberg Jan 22 '12 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 '12 at 7:51
My understanding is that the term twisted Gelfand pair is only used when the algebra is commutative. – Benjamin Steinberg Jan 26 '12 at 14:26

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.

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And do you know references for the maximal compact subgroups in Lie groups? I would be mostly interested how twisted Gelfand pairs are related to Hecke eigenvalues. – Marc Palm Jan 23 '12 at 7:51
I'm not a specialist of real groups, unfortunately. – Paul Broussous Jan 23 '12 at 19:52
Unfortunately, they only deal with locally profinite groups, instead of discussing special properties of $GL(n)$. – Marc Palm Jan 25 '12 at 7:05

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.

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Compact groups are essentially finite groups;) – Marc Palm Jan 26 '12 at 15:22