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K-motives, Springer Theory and the Local Langlands Correspondence

Jens Niklas Eberhardt

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.

K-motives, Springer Theory and the Local Langlands Correspondence

Abstract

We construct a geometric realization of categories of representations of affine Hecke algebras and split reductive -adic groups via a -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 -motives within the Langlands program. To achieve our results, we introduce a six functor formalism for reduced -motives applicable to linearly reductive stacks and establish formality for categories of Springer -motives. We work within a broader framework of Hecke algebras derived from Springer data. This makes the results applicable, for example, to the (-theoretic) quiver Hecke and Schur algebra. Moreover, we relate our constructions to prior geometric realizations for graded Hecke algebras.
Paper Structure (31 sections, 22 theorems, 58 equations)

This paper contains 31 sections, 22 theorems, 58 equations.

Table of Contents

  1. Introduction
  2. Motivation
  3. Affine Hecke algebra via $K$-motives
  4. The formalism of $K$-motives
  5. Formality for reduced $K$-motives
  6. Relation to coherent Springer theory
  7. $K$-motives and Langlands duality
  8. Acknowledgements
  9. Categories of $K$-motives on stacks
  10. Quotient stacks
  11. $K$-motives on linearly reductive stacks
  12. Reduced $K$-motives
  13. Realization and completion
  14. Cellularity and formality
  15. Properties $(R)$ and $(C)$
  16. ...and 16 more sections

Key Result

Theorem 1

There is an equivalence of categories

Theorems & Definitions (43)

  • Theorem : \ref{['thm:formalityaffinehecke']}
  • Theorem : \ref{['thm:padicrepsviakmotives']}
  • Theorem : \ref{['thm:propertyrgivesformality']}
  • Theorem : Global Koszul duality
  • Conjecture : Global Soergel's conjecture
  • Conjecture : Quantum geometric Satake
  • Remark 2.1
  • Remark 2.2
  • Definition 3.1
  • Proposition 3.2
  • ...and 33 more