The Bernstein Center of a p-adic Unipotent Group

TL;DR

This paper explores the structure of the Bernstein center for p-adic unipotent groups by characterizing the dual space topology and establishing formal similarities with sheaves, extending known results from abelian cases.

Contribution

It provides three different characterizations of the dual space topology for p-adic unipotent groups and describes the Bernstein center in this context.

Findings

Three characterizations of the dual space topology for p-adic unipotent groups

Formal similarities established between smooth representations and sheaves

Concrete description of the Bernstein center for these groups

Abstract

Francois Rodier proved that it is possible to view smooth representations of certain totally disconnected abelian groups (the underlying additive group of a finite-dimensional p-adic vector space, for example) as sheaves on the Pontryagin dual group. For nonabelian totally disconnected groups, the appropriate dual space necessarily includes representations which are not one-dimensional, and does not carry a group structure. The general definition of the topology on the dual space is technically unwieldy, so we provide three different characterizations of this topology for a large class of totally disconnected groups (which includes, for example, p-adic unipotent groups), each with a somewhat different flavor. We then use these results to demonstrate some formal similarities between smooth representations and sheaves on the dual space, including a concrete description of the Bernstein center of the category of smooth representations.