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Semiinfinite sheaves on affine flag varieties

Pramod N. Achar, Gurbir Dhillon, Simon Riche

TL;DR

This work develops a geometric framework for semiinfinite constructible sheaves on the affine flag variety, defined over fields of arbitrary characteristic, and links them to Langlands dual representation theory. Central to the approach are the $$-categories $ extsf{Shv}(I^{ rac{ extinfty}{2}}ackslash ext{Fl})$ and $ extsf{Shv}(I^{ rac{ extinfty}{2}}ackslash ext{Gr})$, their standard/costandard objects indexed by $W_{ ext{ext}}$ and $X_*(T)$, and a perverse $t$-structure whose hearts capture simple objects parametrized by these groups; Raskin’s equivalences connect these semiinfinite categories to more familiar equivariant sheaf categories via Wakimoto objects. The paper also ties these categories to positive-characteristic representation theory of the Langlands dual group, with an anticipated t-exact equivalence relating $(g^ abla,^ abla)$-modules to semiinfinite sheaves, and provides a geometric realization of periodic Kazhdan--Lusztig polynomials through stalk/costalk data of IC objects. A major technical achievement is the construction and analysis of the Gaitsgory sheaf $ ext{Ga}^{ rac{ extinfty}{2}}$, whose stalks/costalks encode Kostant’s partition data and which plays a pivotal role in understanding both standard versus costandard phenomena and duality in this semiinfinite setting. Overall, the results extend Gaitsgory’s framework to arbitrary characteristics, establish robust links to representation theory, and lay groundwork for further development of modular geometric Langlands phenomena via semiinfinite sheaves.

Abstract

We study a category of semiinfinite sheaves on the affine flag variety of a connected reductive algebraic group, with coefficients in a field of arbitrary characteristic, generalizing some results of Gaitsgory and showing that this category behaves like familiar categories of sheaves on flag varieties in many respects. Our interest is motivated by expected relations with representation theory of the Lie algebra of the Langlands dual reductive group.

Semiinfinite sheaves on affine flag varieties

TL;DR

This work develops a geometric framework for semiinfinite constructible sheaves on the affine flag variety, defined over fields of arbitrary characteristic, and links them to Langlands dual representation theory. Central to the approach are the -categories and , their standard/costandard objects indexed by and , and a perverse -structure whose hearts capture simple objects parametrized by these groups; Raskin’s equivalences connect these semiinfinite categories to more familiar equivariant sheaf categories via Wakimoto objects. The paper also ties these categories to positive-characteristic representation theory of the Langlands dual group, with an anticipated t-exact equivalence relating -modules to semiinfinite sheaves, and provides a geometric realization of periodic Kazhdan--Lusztig polynomials through stalk/costalk data of IC objects. A major technical achievement is the construction and analysis of the Gaitsgory sheaf , whose stalks/costalks encode Kostant’s partition data and which plays a pivotal role in understanding both standard versus costandard phenomena and duality in this semiinfinite setting. Overall, the results extend Gaitsgory’s framework to arbitrary characteristics, establish robust links to representation theory, and lay groundwork for further development of modular geometric Langlands phenomena via semiinfinite sheaves.

Abstract

We study a category of semiinfinite sheaves on the affine flag variety of a connected reductive algebraic group, with coefficients in a field of arbitrary characteristic, generalizing some results of Gaitsgory and showing that this category behaves like familiar categories of sheaves on flag varieties in many respects. Our interest is motivated by expected relations with representation theory of the Lie algebra of the Langlands dual reductive group.

Paper Structure

This paper contains 51 sections, 47 theorems, 321 equations.

Table of Contents

  1. Introduction
  2. Sheaves on semiinfinite affine flag varieties vs. semiinfinite sheaves on affine flag varieties
  3. Structure of infinity-categories of semiinfinite sheaves
  4. Raskin's equivalences
  5. Relation with representation theory in positive characteristic
  6. A geometric realization of periodic Kazhdan--Lusztig polynomials
  7. Contents
  8. Acknowledgements
  9. Sheaves on affine flag varieties
  10. Sheaves
  11. Functorialities
  12. Affine flag varieties
  13. Sheaves on affine flag varieties
  14. Iwahori-equivariant sheaves on affine flag varieties
  15. Semiinfinite Iwahori-equivariant sheaves on affine flag varieties
  16. ...and 36 more sections

Key Result

Lemma 3.2

Let $\mu \in \mathbf{Y}_{+}$ and $w_{\mathrm{f}} \in W_{\mathrm{f}}$, and set $w=t_\mu w_{\mathrm{f}}$. Then multiplication and action on $z^\mu w_{\mathrm{f}} \mathrm{I}$ induce an isomorphism of $\mathbb{F}$-schemes

Theorems & Definitions (114)

  • Remark 1.1
  • Remark 1.2
  • Example 1.3
  • Remark 2.1
  • Remark 2.2
  • Remark 2.3
  • Remark 2.4
  • Remark 3.1
  • Lemma 3.2
  • Lemma 3.3
  • ...and 104 more