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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?

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  • $\begingroup$ 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? $\endgroup$
    – Alain Valette
    Commented Jan 22, 2012 at 22:30
  • $\begingroup$ Usually one wants this algebra to be commutative to be a twisted Gelfand pair. $\endgroup$
    – Benjamin Steinberg
    Commented Jan 22, 2012 at 23:57
  • $\begingroup$ 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? $\endgroup$
    – Marc Palm
    Commented Jan 23, 2012 at 7:51
  • $\begingroup$ My understanding is that the term twisted Gelfand pair is only used when the algebra is commutative. $\endgroup$
    – Benjamin Steinberg
    Commented Jan 26, 2012 at 14:26

2 Answers 2

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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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  • $\begingroup$ 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. $\endgroup$
    – Marc Palm
    Commented Jan 23, 2012 at 7:51
  • $\begingroup$ I'm not a specialist of real groups, unfortunately. $\endgroup$
    – Paul Broussous
    Commented Jan 23, 2012 at 19:52
  • $\begingroup$ Unfortunately, they only deal with locally profinite groups, instead of discussing special properties of $GL(n)$. $\endgroup$
    – Marc Palm
    Commented Jan 25, 2012 at 7:05
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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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  • $\begingroup$ Compact groups are essentially finite groups;) $\endgroup$
    – Marc Palm
    Commented Jan 26, 2012 at 15:22

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