Local coherence for representations of amalgams

TL;DR

This paper explores how the structure of amalgamated groups influences the finiteness properties of their smooth representation categories, linking group colimits to categorical limits in the local Langlands context.

Contribution

It introduces a framework connecting amalgam structures of groups with the categorical limits of their representation categories, revealing new finiteness properties.

Findings

Representation categories of amalgams are limits of subgroups' categories.

Finiteness properties of Mod(G) can be deduced from subgroup categories.

Applicable to p-adic groups in the local Langlands program.

Abstract

In all forms of the local Langlands program the abelian category of smooth representations of p-adic groups G in vector spaces over a field k plays a central role. Of particular interest are its finiteness properties. If the field k has characteristic zero then, by work of Bernstein, this category is most of the time locally noetherian. But if the field has characteristic p then this remains the case only for very special groups. The basic idea of this paper is that if G is an amalgam, i.e., a colimit of certain subgroups then this is reflected by Mod(G) being the limit of the corresponding categories for these subgroups. This allows to deduce finiteness properties of Mod(G) from finite properties of the categories in the limit diagram.