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A modular ramified geometric Satake equivalence

Pramod N. Achar, João Lourenço, Timo Richarz, Simon Riche

Abstract

We extend the ramified geometric Satake equivalence due to Zhu (for tamely ramified groups) and the third named author (in full generality) from rational coefficients to include modular and integral coefficients.

A modular ramified geometric Satake equivalence

Abstract

We extend the ramified geometric Satake equivalence due to Zhu (for tamely ramified groups) and the third named author (in full generality) from rational coefficients to include modular and integral coefficients.
Paper Structure (71 sections, 62 theorems, 413 equations, 1 table)

This paper contains 71 sections, 62 theorems, 413 equations, 1 table.

Table of Contents

  1. Introduction
  2. Ramified geometric Satake equivalence
  3. Ramified affine Grassmannians
  4. Fixed points of groups of pinning-preserving automorphisms of reductive group schemes
  5. Nearby cycles and the relation with the dual group
  6. Comments on the proof
  7. Motivation
  8. Contents
  9. Acknowledgements
  10. Affine Grassmannians associated with special facets
  11. Loop groups and affine Grassmannians
  12. Reductive groups and special facets
  13. Tori, (co)weights, and (co)roots
  14. Iwahori--Weyl group and Schubert varieties
  15. Convolution schemes
  16. ...and 56 more sections

Key Result

Lemma 2.1

If $\mathcal{H}$ is a smooth affine group scheme over $k [\space[ {x} ]\space]$, then we have $\mathrm{Gr}_{\mathcal{H}} = [\mathrm{L} H/\mathrm{L}^+ \mathcal{H}]_{\mathrm{\acute{e}t}}$, i.e. the functor $\mathrm{Gr}_\mathcal{H}$ is also the étale sheafification of the presheaf eq:Gr-presheaf.

Theorems & Definitions (132)

  • Remark 1.1
  • Remark 1.2
  • Remark 1.3
  • Remark 1.4
  • Lemma 2.1
  • Remark 2.2
  • Lemma 2.3
  • Lemma 2.4
  • proof
  • Lemma 2.5
  • ...and 122 more