# Proof, Via Smooth Homology, of the Existence of Rational Families of H-Invariant Linear Forms on G-Induced Representations, When G/H is a Symmetric, Reductive, p-Adic Space, Via Smooth Homology

 JOINT IAS/PU NUMBER THEORY SEMINAR Topic: Proof, Via Smooth Homology, of the Existence of Rational Families of H-Invariant Linear Forms on G-Induced Representations, When G/H is a Symmetric, Reductive, p-Adic Space, Via Smooth Homology Speaker: Philippe Blanc Affiliation: Institut de Mathematiques de Luminy, Marseille, France Date: Thursday, April 8 Time/Room: 4:30pm - 5:30pm/S-101

We fix F a local non archmedean field of characteristic zero. Let G the points over F of an algebraic reductive group defined over F and s a rational involution of G defined over F. We denote by H the group of fixed points of G under the action of s and by X(G,s) the identity component of the set of complex characters of G antiinvariant under the action of s. Let P be a s-parabolic subgroup of G, which means that the intersection M of P with s(P) is a s-stable Levi-subgroup of P. We construct for each irreducible, smooth, representation r of M, a rational family of H-invariant linear forms on the smooth induced representation ind(P,G, r ) above the algebraic variety X(G,s). Our main trick is the use of homology of groups.