Whittaker models and the integral Bernstein center for GL_n

TL;DR

This paper develops integral analogues of Whittaker models for GL_n over p-adic fields, linking the integral Bernstein center to Galois deformation theory and advancing the local Langlands correspondence in families.

Contribution

It introduces integral versions of Whittaker models for GL_n and connects the integral Bernstein center to Galois deformation theory, facilitating progress on the local Langlands conjecture.

Findings

Established integral analogues of Whittaker models for GL_n

Linked the integral Bernstein center to Galois deformation theory

Reduced the existence of certain representations to a conjecture relating Bernstein center and Galois deformations

Abstract

We establish integral analogues of results of Bushnell and Henniart for spaces of Whittaker functions arising from the groups GL_n(F) 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" of Emerton-Helm, and reduce the question of the existence of such representations to a natural conjecture relating the integral Bernstein center of GL_n(F) to the deformation theory of Galois representations.