K-motives, Springer Theory and the Local Langlands Correspondence

TL;DR

This paper develops a geometric framework using $K$-motivic Springer theory to realize categories of representations of affine Hecke algebras and $p$-adic groups, connecting to the Langlands program.

Contribution

It introduces a $K$-motivic Springer theory approach, establishing a categorical Chern character and formality results, linking geometric and algebraic representation theories.

Findings

Constructed a geometric realization of affine Hecke algebra representations.

Established a six functor formalism for reduced $K$-motives.

Connected $K$-motivic Springer theory to the Langlands program.

Abstract

We construct a geometric realization of categories of representations of affine Hecke algebras and split reductive $p$-adic groups via a $K$-motivic Springer theory. We suggest a connection to the coherent Springer theory of Ben-Zvi, Chen, Helm, and Nadler through a categorical Chern character and outline results and conjectures on $K$-motives within the Langlands program. To achieve our results, we introduce a six functor formalism for reduced $K$-motives applicable to linearly reductive stacks and establish formality for categories of Springer $K$-motives. We work within a broader framework of Hecke algebras derived from Springer data. This makes the results applicable, for example, to the ($K$-theoretic) quiver Hecke and Schur algebra. Moreover, we relate our constructions to prior geometric realizations for graded Hecke algebras.