Logo
ScienceStack
Table of Contents
Fetching ...
Sign In

Geometric Satake equivalence in mixed characteristic and Springer correspondence

Katsuyuki Bando

Abstract

The geometric Satake equivalence and the Springer correspondence are closely related when restricting to small representations of the Langlands dual group. We prove this result for étale sheaves, including the case of the mixed characteristic affine Grassmannian, assuming a sufficient ramification. In this process, we construct a monoidal structure on the restriction functor of Satake categories. We construct also a canonical isomorphism between a mixed characteristic affine Grassmannian under a sufficient ramification and an equal characteristic one.

Geometric Satake equivalence in mixed characteristic and Springer correspondence

Abstract

The geometric Satake equivalence and the Springer correspondence are closely related when restricting to small representations of the Langlands dual group. We prove this result for étale sheaves, including the case of the mixed characteristic affine Grassmannian, assuming a sufficient ramification. In this process, we construct a monoidal structure on the restriction functor of Satake categories. We construct also a canonical isomorphism between a mixed characteristic affine Grassmannian under a sufficient ramification and an equal characteristic one.

Paper Structure

This paper contains 25 sections, 26 theorems, 139 equations.

Table of Contents

  1. Introduction
  2. Notations
  3. Preliminaries on affine Grassmannians and geometric Satake equivalences
  4. Affine Grassmannians
  5. Geometric Satake equivalence
  6. Partial isomorphism between affine Grassmannians
  7. Preliminaries on affine Grassmannian for $\mathrm{GL}_n$
  8. Isomorphism for $G=\mathrm{GL}_n$
  9. Isomorphism for general $G$
  10. Geometric Satake equivalence and Springer correspondence
  11. Preliminaries on Springer correspondence
  12. Functor $\mathscr{S}_G^{\mathrm{sm}}$
  13. Functor $\Phi_{\check{G}}$
  14. Functor $\Psi_{G}$
  15. Statement of main theorem
  16. ...and 10 more sections

Key Result

Theorem 1.1

Let $F_1,F_2$ be two complete discrete valuation fields with the same residue field $k$. Let $(G_{\mathbb{Z}},B_{\mathbb{Z}},T_{\mathbb{Z}})$ be a triple of a split reductive group over $\mathbb{Z}$, its Borel group and its maximal torus such that $T\subset B$. Put $(G_1,B_1,T_1)=(G_{\mathbb{Z}},B_{ Moreover, the Schubert cells $\mathrm{Gr}_{G_1,\lambda}$ and $\mathrm{Gr}_{G_2,\lambda}$ correspond

Theorems & Definitions (48)

  • Theorem 1.1
  • Theorem 1.2
  • Remark 3.1
  • Definition 3.2
  • Lemma 4.1
  • proof
  • Theorem 4.2
  • Lemma 4.3
  • Lemma 4.4
  • proof
  • ...and 38 more