Twisted Gelfand pairs (Reference and examples) - MathOverflow most recent 30 from http://mathoverflow.net 2013-05-22T08:49:38Z http://mathoverflow.net/feeds/question/86381 http://www.creativecommons.org/licenses/by-nc/2.5/rdf http://mathoverflow.net/questions/86381/twisted-gelfand-pairs-reference-and-examples Twisted Gelfand pairs (Reference and examples) Marc Palm 2012-01-22T16:01:26Z 2012-01-22T18:08:55Z <p>Let $G$ be a locally compact group and let $K$ be a compact group. Let $(\tau, V_\tau)$ be an irreducible representation of $K$.</p> <p>We consider the space of $Endo_K(\tau)$-valued, compactly supported continuous functions $f$ on $G$<br> with $$f(k_1 g k_2) = \tau(k_1) f(g) \tau(k_2),$$ which is an $*$ algebra under convolution.</p> <p>What is a good reference for such algebras, especially in the context with reductive group over local fields and the connection to representation theory?</p> http://mathoverflow.net/questions/86381/twisted-gelfand-pairs-reference-and-examples/86386#86386 Answer by Benjamin Steinberg for Twisted Gelfand pairs (Reference and examples) Benjamin Steinberg 2012-01-22T17:20:04Z 2012-01-22T17:20:04Z <p>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. </p> http://mathoverflow.net/questions/86381/twisted-gelfand-pairs-reference-and-examples/86392#86392 Answer by Paul Broussous for Twisted Gelfand pairs (Reference and examples) Paul Broussous 2012-01-22T18:08:55Z 2012-01-22T18:08:55Z <p>These Hecke algebras are intensively studied in the field of "type theory" for reductive $p$-adic groups.</p> <p>You have a nice summary of basic facts <em>with proofs</em> 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"). </p> <p>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.</p> <p>There are many other references. But it depends on what exactly you're interested in.</p>