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Twisted Whittaker category on affine flags and the category of representations of the mixed quantum group

Ruotao Yang

Abstract

We prove the twisted Whittaker category on the affine flag variety and the category of representations of the mixed quantum group are equivalent. In particular, we prove that the quantum category O is equivalent to the category of twisted Whittaker D-modules on affine flags in the generic case.

Twisted Whittaker category on affine flags and the category of representations of the mixed quantum group

Abstract

We prove the twisted Whittaker category on the affine flag variety and the category of representations of the mixed quantum group are equivalent. In particular, we prove that the quantum category O is equivalent to the category of twisted Whittaker D-modules on affine flags in the generic case.

Paper Structure

This paper contains 82 sections, 53 theorems, 289 equations.

Table of Contents

  1. Introduction
  2. Reminder on work of [AB]
  3. Main theorem of this paper
  4. Other Motivations
  5. Fundamental local equivalence
  6. Relation with Kazhdan-Lusztig
  7. BGG Category $\mathsf{O}$
  8. Casselman-Shalika theorem
  9. Small quantum groups
  10. Strategy of proof
  11. Organization of the paper
  12. Generality of our results
  13. Sheaf theories
  14. Deformation parameters
  15. Conventions and notations
  16. ...and 67 more sections

Key Result

Lemma 2.2.3

Evaluation on fundamental weights gives rise to a morphism It induces an isomorphism of the sheafifications in the topology generated by finite surjective maps.

Theorems & Definitions (148)

  • Conjecture 1
  • Conjecture 2
  • Definition 2.1.1
  • Definition 2.1.2
  • Remark 2.1.3
  • Definition 2.1.4
  • Definition 2.1.6
  • Definition 2.1.7
  • Definition 2.2.1
  • Lemma 2.2.3
  • ...and 138 more