ON PRO-p-IWAHORI INVARIANTS OF R-REPRESENTATIONS OF REDUCTIVE p-ADIC GROUPS

Let F be a locally compact field with residue characteristic p, and let G be a connected reductive F-group. Let Li be a pro-p Iwahori subgroup of G = G(F). Fix a commutative ring R. If pi is a smooth R[G1-representation, the space of invariants pi(U) is a right module over the Hecke algebra H of U in G. Let P be a parabolic subgroup of G with a Levi decomposition P = MN adapted to U. We complement a previous investigation of Ollivier-Vigneras on the relation between taking U-invariants and various functor like Ind(q)(G), and right and left adjoints. More precisely the authors' previous work with Herzig introduced representations I-G(P, sigma, Q) where sigma is a smooth representation of M extending, trivially on N, to a larger parabolic subgroup P(sigma), and Q is a parabolic subgroup between P and P(sigma). Here we relate I-G(P,sigma,Q)(u) to an analogously defined H-module I-H(P, sigma(u)m,Q), where U-M = U boolean AND M and sigma(u)m is seen as a module over the Hecke algebra H-M of U-M in M. In the reverse direction, if V is a right Wm-module, we relate LH(P, V, Q) 0 c-Indi 1 to /G (P, V Mc-Indtilm 1, Q). As an application we prove that if R is an algebraically closed field of characteristic p, and it is an irreducible admissible representation of G, then the contragredient of it is 0 unless it has finite dimension.