On the derived category of the Iwahori-Hecke algebra

TL;DR

This paper explores a conjecture linking the derived category of smooth representations of p-adic groups with that of sheaves on L-parameter stacks, focusing on the principal block of GL_n and proposing a derived tensor product approach.

Contribution

It proposes a new conjectural equivalence between representation categories and sheaves on L-parameters, supported by a specific case analysis for GL_n.

Findings

The conjecture relates representation categories to sheaves on L-parameter stacks.

The functor is proposed to be a derived tensor product with interpolating representations.

Evidence is provided for the principal block of GL_n.

Abstract

We state a conjecture that relates the derived category of smooth representations of a p-adic split reductive group with the derived category of (quasi-)coherent sheaves on a stack of L-parameters. We investigate the conjecture in the case of the principal block of GL_n by showing that the functor should be given by the derived tensor product with the family of representations interpolating the modified Langlands correspondence over the stack of L-parameters that is suggested by the work of Helm and Emerton-Helm.