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Drinfeld-Gaitsgory-Vinberg interpolation Grassmannian and geometric Satake equivalence (with appendix by Dennis Gaitsgory)

Michael Finkelberg, Vasily Krylov, Ivan Mirković

Abstract

Let $G$ be a reductive complex algebraic group. We fix a pair of opposite Borel subgroups and consider the corresponding semiinfinite orbits in the affine Grassmannian $Gr_G$. We prove Simon Schieder's conjecture identifying his bialgebra formed by the top compactly supported cohomology of the intersections of opposite semiinfinite orbits with $U(\check{\mathfrak n})$ (the universal enveloping algebra of the positive nilpotent subalgebra of the Langlands dual Lie algebra $\check{\mathfrak g}$). To this end we construct an action of Schieder bialgebra on the geometric Satake fiber functor. We propose a conjectural construction of Schieder bialgebra for an arbitrary symmetric Kac-Moody Lie algebra in terms of Coulomb branch of the corresponding quiver gauge theory.

Drinfeld-Gaitsgory-Vinberg interpolation Grassmannian and geometric Satake equivalence (with appendix by Dennis Gaitsgory)

Abstract

Let be a reductive complex algebraic group. We fix a pair of opposite Borel subgroups and consider the corresponding semiinfinite orbits in the affine Grassmannian . We prove Simon Schieder's conjecture identifying his bialgebra formed by the top compactly supported cohomology of the intersections of opposite semiinfinite orbits with (the universal enveloping algebra of the positive nilpotent subalgebra of the Langlands dual Lie algebra ). To this end we construct an action of Schieder bialgebra on the geometric Satake fiber functor. We propose a conjectural construction of Schieder bialgebra for an arbitrary symmetric Kac-Moody Lie algebra in terms of Coulomb branch of the corresponding quiver gauge theory.

Paper Structure

This paper contains 91 sections, 36 theorems, 194 equations.

Table of Contents

  1. Introduction
  2. Review of Schieder's work
  3. General recollections
  4. The affine Grassmannian bd, zh
  5. The Beilinson-Drinfeld Grassmannian bd, zh, mv
  6. Tensor structure on $\operatorname{Perv}_{G_{\mathcal{O}}}({\operatorname{Gr}}_G)$ via Beilinson-Drinfeld Grassmanian
  7. Rational morphisms
  8. The Tannakian approach
  9. Definition of $\overline{S}_\nu$ via Tannakian approach
  10. Definition of $\overline{T}_\nu$ via Tannakian approach
  11. Definition of $\overline{S}_{\theta_1,\theta_2}$
  12. Definition of $\overline{T}_{\theta_1,\theta_2}$
  13. The Vinberg semigroup
  14. Rees construction of the Vinberg semigroup v2
  15. Construction of the morphism $\Upsilon\colon \operatorname{Vin}_G \rightarrow T^+_{\operatorname{ad}}$
  16. ...and 76 more sections

Key Result

Proposition 2.3

For $n \in {\mathbb{N}}$, the functors ${\operatorname{Gr}}_{G,X^n}$ and ${\operatorname{Gr}}'_{G,X^n}$ are isomorphic.

Theorems & Definitions (92)

  • Definition 1.1
  • Remark 1.2
  • Remark 2.1
  • Remark 2.2
  • Proposition 2.3
  • proof
  • Remark 2.4
  • Remark 2.5
  • Proposition 2.6: Peter-Weyl theorem
  • Definition 2.7
  • ...and 82 more