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Coherent Springer theory and the categorical Deligne-Langlands correspondence

David Ben-Zvi, Harrison Chen, David Helm, David Nadler

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.

Coherent Springer theory and the categorical Deligne-Langlands correspondence

Abstract

Kazhdan and Lusztig identified the affine Hecke algebra with an equivariant -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 with an Iwahori-fixed vector. We apply techniques from derived algebraic geometry to pass from -theory to Hochschild homology and thereby identify 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 -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 into coherent sheaves on the stack of Langlands parameters.

Paper Structure

This paper contains 47 sections, 56 theorems, 179 equations.

Table of Contents

  1. Introduction
  2. Springer theory and Hecke algebras
  3. Affine Hecke algebras
  4. Coherent Springer Theory
  5. Applications to the local Langlands correspondence
  6. Discussion: Categorical Langlands Correspondence
  7. Methods
  8. Bezrukavnikov's theorem
  9. Trace Decategorifications
  10. Trace of the affine Hecke category
  11. Assumptions and notation
  12. Categories
  13. Algebraic geometry
  14. Representation theory
  15. Acknowledgments
  16. ...and 32 more sections

Key Result

Theorem 1.2

KLCGlusztig bases Suppose that $G$ has simply connected derived subgroup. There is an isomorphism of algebras $\mathcal{H}\simeq K_0({\mathcal{Z}}/G_{\mathrm{gr}}) \otimes_{{\mathbb Z}} {\mathbb C}$, compatible with the Bernstein isomorphism $Z(\mathcal{H})\simeq {\mathbb C}[G_{\mathrm{gr}}]^{G_{\ma

Theorems & Definitions (171)

  • Definition 1.1
  • Theorem 1.2
  • Theorem 1.3
  • Theorem 1.4: Theorem \ref{['ThmHH']}, Corollary \ref{['thm no sc']}
  • Remark 1.5
  • Definition 1.6
  • Theorem 1.7: Theorem \ref{['thm endomorph']}
  • Definition 1.8
  • Theorem 1.9: Theorem \ref{['thm endomorph']}
  • Remark 1.10
  • ...and 161 more