Whittaker models and the integral Bernstein center for GLn

Abstract

We establish integral analogues of results of Bushnell and Henniart for spaces of Whittaker functions arising from the groups ${\mathrm{GL}}_n\left(F\right)$ for $F$ a $p$-adic field. We apply the resulting theory to the existence of representations arising from the conjectural “local Langlands correspondence in families” and reduce the question of the existence of such representations to a natural conjecture relating the integral Bernstein center of ${\mathrm{GL}}_n\left(F\right)$ to the deformation theory of Galois representations.