Speaker
Description
Let F be a non Archimedean local field and let p be its residue characteristic. Let R be an algebraically closed field with char(R) different from p. We consider a connected reductive group G over F, and look at irreducible smooth R-representations of the locally pro-p group G=G(F). Classifying all such representations up to isomorphism is very involved, but their behaviour under restriction to small enough open pro-p subgroups is expected to be more uniform. When R=C and char(F)=0, that can be obtained from Harish-Chandra's germ expansion. One can hope for a similar control for general R and char(F)>=0. This has consequences for the asymptotics of fixed points under congruence subgroups of G. We shall survey what is known for G=GL(N) and G=SL(2) (Joint work with Vignéras) and what it suggests for the general case.