Block decomposition of the category of l-modular smooth representations of GL(n,F) and its inner forms

TL;DR

This paper establishes a decomposition of the category of smooth representations of GL(m,D) over an algebraically closed field, classifying representations via inertial classes of supercuspidal pairs, and proves each component is indecomposable.

Contribution

It introduces a block decomposition of the category of smooth representations of GL(m,D) based on inertial classes, showing each block is indecomposable, extending understanding of representation categories.

Findings

Category decomposes into blocks indexed by inertial classes

Each block corresponds to a unique inertial class of supercuspidal pairs

Blocks are indecomposable components of the category

Abstract

Let F be a non-Archimedean locally compact field of residue characteristic p, let D be a finite dimensional central division F-algebra and let R be an algebraically closed field of characteristic different from p. To any irreducible smooth representation of G=GL(m,D) with coefficients in R, we can attach a uniquely determined inertial class of supercuspidal pairs of G. This provides us with a partition of the set of all isomorphism classes of irreducible representations of G. We write R(G) for the category of all smooth representations of G with coefficients in R. To any inertial class O of supercuspidal pairs of G, we can attach the subcategory R(O) made of smooth representations all of whose irreducible subquotients are in the subset determined by this inertial class. We prove that R(G) decomposes into the product of the R(O), where O ranges over all possible inertial class of supercuspidal pairs of G, and that each summand R(O) is indecomposable.