Coherent Springer theory and the categorical Deligne-Langlands correspondence

TL;DR

This paper uses derived algebraic geometry to connect affine Hecke algebras with coherent sheaves on stacks of Langlands parameters, advancing the categorical understanding of the local Langlands correspondence.

Contribution

It identifies the affine Hecke algebra with endomorphisms of a coherent sheaf on the unipotent Langlands parameters stack, confirming categorical aspects of the local Langlands correspondence.

Findings

Realized the derived category of -modules as a subcategory of coherent sheaves.

Constructed a full embedding of  representations into sheaves on Langlands parameters.

Confirmed conjectures relating the local Langlands correspondence to coherent sheaves.

Abstract

Kazhdan and Lusztig identified the affine Hecke algebra $\mathcal{H}$ with an equivariant $K$-group of the Steinberg variety, and applied this to prove the Deligne-Langlands conjecture, i.e., the local Langlands parametrization of irreducible representations of reductive groups over nonarchimedean local fields $F$ with an Iwahori-fixed vector. We apply techniques from derived algebraic geometry to pass from $K$-theory to Hochschild homology and thereby identify $\mathcal{H}$ with the endomorphisms of a coherent sheaf on the stack of unipotent Langlands parameters, the coherent Springer sheaf. As a result the derived category of $\mathcal{H}$-modules is realized as a full subcategory of coherent sheaves on this stack, confirming expectations from strong forms of the local Langlands correspondence (including recent conjectures of Fargues-Scholze, Hellmann and Zhu). In the case of the general linear group our result allows us to lift the local Langlands classification of irreducible representations to a categorical statement: we construct a full embedding of the derived category of smooth representations of $\mathrm{GL}_n(F)$ into coherent sheaves on the stack of Langlands parameters.