Category decomposition of Rep_k(SL_n(F))

TL;DR

This paper provides a detailed category decomposition of the representation category of SL_n(F) over an algebraically closed field, based on supercuspidal classes, highlighting differences from the characteristic zero case.

Contribution

It establishes a new category and block decomposition of Rep_k(SL_n(F)) according to inertially equivalent supercuspidal classes, and illustrates the non-uniqueness of blocks in positive characteristic.

Findings

Category decomposition based on supercuspidal classes

Block decomposition of supercuspidal sub-category

Counterexample showing non-uniqueness of blocks in positive characteristic

Abstract

Let F be a non-archimedean local field with residual characteristic p, and k an algebraically closed field of characteristic l different from p. We establish a category decomposition of Rep_k(SL_n(F) according to the GL_n(F)-inertially equivalent supercuspidal classes of SL_n(F), and we give a block decomposition of the supercuspidal sub-category of Rep_k(SL_n(F)). Finally we give an example to show that in general a block of SL_n(F) is not defined according to a unique inertially equivalent supercuspidal classes of SL_n(F), which is different from the case when l=0.