Base change fundamental lemma for Bernstein centers of principal series blocks

Abstract: Let G be an unramified group over a p-adic field F, and F_r/F an unramified extension of degree r. Let H(G) (resp. H(G(F_r)) denote the Hecke algebra of G(F) (resp. G(F_r)). Roughly speaking, we say two functions \phi\in H(G(F_r)) and f\in H(G) are associated (or matching functions) if they have the same stable orbital integrals. One main question is: how can we construct matching functions? In 1986, Kottwitz proved the unit elements of some Hecke algebras are associated. In 1990, Clozel defined a base change map between spherical Hecke algebras and proved the two functions corresponded by the base change map are associated. Later in 2009 and 2012, Haines generalized Clozel's result to centers of parahoric Hecke algebras and Bernstein centers of depth zero principal series block. In this talk, we will briefly introduce the history and set up of base change fundamental lemma, and focus on how we can generalize the result to general principal series blocks. This requires the concrete constructions of types for principal series blocks of unramified groups, and some concrete computations of root groups, which might give some inspirations on future study on deeper level structures.

number theoryrepresentation theory

Audience: researchers in the topic

University of Utah Representation Theory / Number Theory Seminar