Block decomposition of the category of l-modular smooth representations of finite length of GL(m,D)

TL;DR

This paper analyzes the structure of the category of finite length l-modular smooth representations of inner forms of GL(n,F), revealing a complex block decomposition and establishing equivalences with principal blocks of division algebra groups.

Contribution

It provides a detailed description of the block decomposition for l-modular representations of inner forms of GL(n,F), including supercuspidal support classification and block equivalences.

Findings

Multiple supercuspidal supports can correspond to the same block.

Supercuspidal blocks are equivalent to principal blocks of division algebra groups.

Identifies irreducible representations with nontrivial extensions to supercuspidal ones.

Abstract

Let F be a non-Archimedean locally compact field of residue characteristic p, let G be an inner form of GL(n,F) with n>0, and let l be a prime number different from p. We describe the block decomposition of the category of finite length smooth representations of G with coefficients in an algebraically closed field of characteristic l. Unlike the case of complex representations of an arbitrary p-adic reductive group and that of l-modular representations of GL(n,F), several non-isomorphic supercuspidal supports may correspond to the same block. We describe the (finitely many) supercuspidal supports corresponding to a given block. We also prove that a supercuspidal block is equivalent to the principal (that is, the one which contains the trivial character) block of the multiplicative group of a suitable division algebra, and we determine those irreducible representations having a nontrivial extension with a given supercuspidal representation of G.