4/15/2010

Philippe Blanc
Institut de Mathématiques de Luminy

Proof, via smooth homology, of the existence of rational families of H-invariant linear forms on G-induced representations, when G/H is a symetric, reductive, p-adic space, via smooth homology

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