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Perverse sheaves on twisted affine flag varieties and Langlands duality

Rızacan Çiloğlu

Abstract

We provide a description of Iwahori-Whittaker equivariant perverse sheaves on affine flag varieties associated to tamely ramified reductive groups, in terms of Langlands dual data. This extends the work of Arkhipov-Bezrukavnikov from the case of split reductive groups. To achieve this, we first extend the theory of Wakimoto sheaves to our context and prove convolution exact central objects admit a filtration by such. We then establish the tilting property of the Iwahori-Whittaker averaging of certain central objects arising from the geometric Satake equivalence, which enables us to address the absence of an appropriate analogue of Gaitsgory's central functor for non-split groups.

Perverse sheaves on twisted affine flag varieties and Langlands duality

Abstract

We provide a description of Iwahori-Whittaker equivariant perverse sheaves on affine flag varieties associated to tamely ramified reductive groups, in terms of Langlands dual data. This extends the work of Arkhipov-Bezrukavnikov from the case of split reductive groups. To achieve this, we first extend the theory of Wakimoto sheaves to our context and prove convolution exact central objects admit a filtration by such. We then establish the tilting property of the Iwahori-Whittaker averaging of certain central objects arising from the geometric Satake equivalence, which enables us to address the absence of an appropriate analogue of Gaitsgory's central functor for non-split groups.

Paper Structure

This paper contains 61 sections, 82 theorems, 230 equations.

Table of Contents

  1. Introduction
  2. Main result
  3. Strategy of the proof
  4. Other results
  5. Contents
  6. Twisted affine flag varieties
  7. Loop groups and affine flag varieties
  8. Iwahori-Weyl group and Bruhat order
  9. Coweights and some partial orders
  10. Orbit stratification
  11. Constructible sheaves on the Hecke stack
  12. Convolution product
  13. Affine Grassmannians
  14. Geometric Satake equivalence
  15. BD-Grassmannian and nearby cycles
  16. ...and 46 more sections

Key Result

Theorem 1.1

There is an equivalence of categories where $D^{\mathrm{b}}\mathsf{Coh}^{\check G^I}(\widetilde{\mathcal{N}}_{\check{G}^{I,\circ}})$ denotes the bounded derived category of $\mathsf{Coh}^{\check G^I}(\widetilde{\mathcal{N}}_{\check{G}^{I,\circ}})$.

Theorems & Definitions (184)

  • Theorem 1.1
  • Theorem 1.3
  • Theorem 1.4
  • Definition 2.1
  • Remark 2.2
  • Remark 2.3
  • Lemma 2.4
  • proof
  • Remark 2.5
  • Remark 2.6
  • ...and 174 more