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Derived Satake category and Affine Hecke category in mixed characteristics

Katsuyuki Bando

Abstract

We construct a new affine Grassmannian which connects an equal characteristic affine Grassmannian and Zhu's Witt vector affine Grassmannian. As a result, we deduce the mixed characteristic version of the Bezrukavnikov-Finkelberg's derived Satake equivalence. By the same argument, we also obtain the mixed characteristic version of the Bezrukavnikov's equivalence between two categorifications of an affine Hecke algebra.

Derived Satake category and Affine Hecke category in mixed characteristics

Abstract

We construct a new affine Grassmannian which connects an equal characteristic affine Grassmannian and Zhu's Witt vector affine Grassmannian. As a result, we deduce the mixed characteristic version of the Bezrukavnikov-Finkelberg's derived Satake equivalence. By the same argument, we also obtain the mixed characteristic version of the Bezrukavnikov's equivalence between two categorifications of an affine Hecke algebra.
Paper Structure (20 sections, 50 theorems, 192 equations)

This paper contains 20 sections, 50 theorems, 192 equations.

Table of Contents

  1. Introduction
  2. Affine Grassmannians and Affine flag varieties
  3. Review on equal characteristic and Witt vector affine Grassmannians
  4. Affine Grassmannian over $\widetilde{\mathrm{Div}}_0^{\perp}$
  5. Definiton of affine Grassmannian
  6. Schubert varieties
  7. Affine flag variety over $\widetilde{\mathrm{Div}}_0^{\perp}$
  8. Affine Grassmannian over $\widetilde{\mathrm{Div}}_0^{\perp}$ is an ind-perfectly projective ind-scheme
  9. Preliminary on line bundles
  10. Affine Grassmannain of $\mathrm{GL}_n$
  11. Affine Grassmannian for general $G$
  12. Affine flag variety
  13. Derived category for affine Grassmannian and affine flag variety
  14. Derived category for affine Grassmannian
  15. Convolution products
  16. ...and 5 more sections

Key Result

Theorem 1.1

(Theorem thm:maineqmixedtorsion) Let $\Lambda$ be either Let $E$ be a finite extension of $\mathbb{Q}_p$, and $G$ a reductive group over $\mathcal{O}_E[[t]]$. There is a canonical equivalence of monoidal triangulated categories In particular, when $\Lambda=\overline{\mathbb{Q}}_\ell$, we have

Theorems & Definitions (108)

  • Theorem 1.1
  • Theorem 1.2: Derived geometric Satake correspondence in mixed characteristics
  • Theorem 1.3
  • Theorem 1.4
  • Theorem 1.5
  • Theorem 1.6
  • Definition 2.1
  • Remark 2.2
  • Remark 2.3
  • Lemma 2.4
  • ...and 98 more