Equivalences of tame blocks for p-adic linear groups

TL;DR

This paper establishes equivalences between certain blocks of smooth representations of p-adic linear groups and unipotent blocks of product groups, using algebraic and cohomological techniques.

Contribution

It introduces a method to relate level 0 blocks of GL n over p-adic fields to unipotent blocks of product groups via Morita equivalences.

Findings

Level 0 blocks of GL n(F) are equivalent to modules over a specific algebra.

The algebra can be split using Deligne-Lusztig cohomology.

Morita equivalences are constructed to relate different blocks.

Abstract

Let p and $\ell$ be two distinct primes, F a p-adic field and n an integer. We show that any level 0 block of the category of smooth Z $\ell$-valued representations of GL n (F) is equivalent to the unipotent block of an appropriate product of GL n i (F i). To this end, we first show that the level 0 category of GL n (F) is equivalent to a category of " modules " over a certain Z $\ell$-algebra " with many objects " whose definition only involves n and the residue field of F. Then we use fine properties of Deligne-Lusztig cohomology to split this algebra and produce suitable Morita equivalences.