Principal series component of Gelfand-Graev representation

Abstract

Let G be a connected reductive group defined over a nonarchimedean local field F. Let B be a minimal F-parabolic subgroup with Levi factor T and unipotent radical U. Let. be a non-degenerate character of U(F) and lambda a character of T(F). Let (K,rho) be a Bushnell-Kutzko type associated to the Bernstein block of G(F) determined by the pair (T,lambda). We study the rho-isotypical component (c-ind(U(F))(G(F))). of the induced space c-ind(U(F))(G(F)) psi of functions compactly supported mod U(F). We show that (c-ind(U(F))(U(F))psi)(rho) is cyclic module for the Hecke algebra H(G,rho) associated to the pair (K,rho). When T is split, we describe it more explicitly in terms of H(G,rho). We make assumptions on the residue characteristic of F and later also on the characteristic of F and the center of G depending on the pair (T,lambda). Our results generalize the main result of Chan and Savin in [Math. Z. 288 (2018), pp. 125-133] who treated the case of lambda = 1 for T split.