The Derived l-modular unipotent block of p-adic GLnMPIM

by Rose Berry (MPIM)

Thursday Mar 19, 2026, 3:00 PM → 4:00 PM Europe/Berlin

MPIM, Vivatsgasse, 7 - Lecture Hall (Max Planck Institute for Mathematics)

MPI-Oberseminar

Smooth representations of p-adic groups are of interest as they appear on one side of the local Langlands correspondence. Over the complex numbers, they decompose into explicit blocks, which are in turn known in many cases to be equivalent to modules over explicit Hecke algebras. Over an algebraically closed field of characteristic l not equal to p, neither the decomposition nor the algebras are known. For GLn, the same block decomposition holds, but the Hecke algebra is now too small to describe the whole block. Instead, there is mild extension of the Hecke algebra, known as the Schur algebra, whose modules include every irreducible smooth unipotent l-modular representation. However, this still only captures a subcategory of the unipotent block, and the categorical Langlands correspondence uses the full derived category of the unipotent block. We thus give a description of a derived category of the unipotent block in terms of a dg-enhancement of the Schur algebra. Along the way, we give an analogous and independently interesting result for GLn(k) for a finite field k.