Number Theory Seminar

Let G be a reductive p-adic group. In the case where G splits over a tamely ramified extension, we describe a class of irreducible admissible representations of G, called positive regular representations. The equivalence classes of positive regular representations of G correspond to pairs (T,φ) where T is a tamely ramified maximal torus of G and φ is a quasicharacter of T whose restriction to the maximal pro-p-subgroup of T satisfies a specific regularity condition. Let H be the group of fixed points of an involution of G. A representation of G is said to be H-distinguished if there exists a nonzero H-invariant linear form on the space of the representation. The H-distinguished irreducible admissible representations of G play a key role in harmonic analysis on the p-adic symmetric variety G/H. We will describe various results (valid under mild restrictions on the residual characteristic of F) about distinction of positive regular representations. We describe necessary conditions for H-distinction of the positive regular representation corresponding to (T,φ), expressed in terms of properties of T and φ relative to the involution. We will discuss some examples. An H-distinguished representation of G is said to be H-relatively supercuspidal if its generalized matrix coefficients have compact support modulo HZ, where Z is the centre of G. (This is a p-adic symmetric space analogue of the notion of supercuspidal representation.) We will discuss conditions under which a distinguished positive regular representation is relatively supercuspidal.