A functoriality principle for blocks of p-adic linear groups

TL;DR

This paper proposes a functoriality principle involving dual groups to explain and predict equivalences among Bernstein blocks of p-adic reductive groups, extending known results for groups related to GL(n).

Contribution

It establishes a precise functoriality principle for groups related to GL(n) and conjectures its extension to more general reductive groups and integral l-adic representations.

Findings

Proves a functoriality principle for groups related to GL(n).

Predicts new equivalences among Bernstein blocks based on dual group relations.

Suggests broader conjectural framework for general reductive groups.

Abstract

Bernstein blocks of complex representations of p-adic reductive groups have been computed in a large amount of examples, in part thanks to the theory of types a la Bushnell and Kutzko. The output of these purely representation-theoretic computations is that many of these blocks are equivalent. The motto of this paper is that most of these coincidences are explained, and many more can be predicted, by a functoriality principle involving dual groups. We prove a precise statement for groups related to GL n , and then state conjectural generalizations in two directions : more general reductive groups and/or integral l-adic representations.