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A Betti geometric Casselman-Shalika equivalence

Colton Sandvik

Abstract

Whittaker sheaves are ubiquitous in geometric representation theory; however, their definition requires one to restrict the sheaf-theoretic setting to either étale sheaves or $D$-modules. Gaitsgory and Lysenko proposed a solution to this problem called the Kirillov model, which is well-defined for many sheaf theories. In this paper, we advance the study of the Kirillov model in the setting of Betti sheaves with a particular emphasis on developing a theory of Iwahori-Whittaker sheaves on the affine Grassmannian. Using this framework, we prove a Betti geometric Casselman-Shalika equivalence, which relates Iwahori-Whittaker perverse sheaves on the affine Grassmannian with the Satake category.

A Betti geometric Casselman-Shalika equivalence

Abstract

Whittaker sheaves are ubiquitous in geometric representation theory; however, their definition requires one to restrict the sheaf-theoretic setting to either étale sheaves or -modules. Gaitsgory and Lysenko proposed a solution to this problem called the Kirillov model, which is well-defined for many sheaf theories. In this paper, we advance the study of the Kirillov model in the setting of Betti sheaves with a particular emphasis on developing a theory of Iwahori-Whittaker sheaves on the affine Grassmannian. Using this framework, we prove a Betti geometric Casselman-Shalika equivalence, which relates Iwahori-Whittaker perverse sheaves on the affine Grassmannian with the Satake category.

Paper Structure

This paper contains 24 sections, 35 theorems, 157 equations.

Table of Contents

  1. Introduction
  2. Geometric Satake
  3. The Geometric Casselman--Shalika Equivalence
  4. Acknowledgements
  5. Affine Grassmannians and Affine Flag Varieties
  6. Notation
  7. Geometry
  8. Constructible Sheaves
  9. Convolution
  10. The Kirillov Model
  11. Definition and First Properties
  12. The Prototypical Example
  13. Averaging and Translation Functors
  14. Sheaf Functors
  15. Riemann--Hilbert Correspondence
  16. ...and 9 more sections

Key Result

Theorem 1.2.1

There is an equivalence of categories, $\blacktriangleleft$$\blacktriangleleft$

Theorems & Definitions (72)

  • Theorem 1.2.1
  • Remark 1.2.2
  • Lemma 2.3.1: MV
  • Lemma 2.3.2: BGMRR
  • Lemma 2.4.1
  • proof
  • Lemma 2.4.2
  • proof
  • Remark 2.4.3
  • Lemma 3.1.1
  • ...and 62 more