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Untwisted Gaiotto equivalence

Roman Travkin, Ruotao Yang

Abstract

This is a successive paper of arXiv:1909.11492. We prove an equivalence between the category of finite-dimensional representations of degenerate supergroup $\underline{GL}(M|N)$ and the category of $(GL_M(O) \ltimes U_{M, N}(F), χ_{M, N})$-equivariant D-modules on $Gr_{N}$. We also prove that we can realize the category of finite-dimensional representations of degenerate supergroup $\underline{GL}(M|N)$ as a category of D-modules on the mirabolic subgroup $Mir_L(F)$ with certain equivariant conditions for any $L$ bigger than $N$ and $M$.

Untwisted Gaiotto equivalence

Abstract

This is a successive paper of arXiv:1909.11492. We prove an equivalence between the category of finite-dimensional representations of degenerate supergroup and the category of -equivariant D-modules on . We also prove that we can realize the category of finite-dimensional representations of degenerate supergroup as a category of D-modules on the mirabolic subgroup with certain equivariant conditions for any bigger than and .
Paper Structure (38 sections, 34 theorems, 225 equations)

This paper contains 38 sections, 34 theorems, 225 equations.

Table of Contents

  1. Introduction
  2. Notations
  3. Reminder on mirabolic Satake equivalence
  4. Untwisted Gaiotto conjecture in the case $M=N-1$
  5. Untwisted Gaiotto conjecture in general case
  6. Strategy of the proof
  7. Symmetric definition of $D^{{\mathop{\operatorname{\rm GL}}}_M(\mathbf O) \ltimes U_{M, N}(\mathbf F), \chi_{M, N}}\left({\mathbf{Gr}}_{N}\right)$
  8. Iwahori version of the conjecture
  9. Coherent side
  10. Degenerate supergroup
  11. Symmetric algebra realization
  12. Nilpotent support
  13. Convolution
  14. Endofunctor of ${{D}}^{{\mathop{\operatorname{\rm GL}}}_M\times {\mathop{\operatorname{\rm GL}}}_N}(\mathfrak{B}_{M|N})$
  15. Tensor product of categories of representations
  16. ...and 23 more sections

Key Result

Theorem 1.2.1

There are monoidal equivalences of DG-categories, which commute with left and right convolutions with ${\operatorname{Perv}}_{{\mathop{\operatorname{\rm GL}}}_N({\mathbf{\mathbf O}})}({\mathbf{Gr}}_N)\simeq {\operatorname{Rep}}^{\operatorname{fin}}({\mathop{\operatorname{\rm GL}}}_N)^\heartsuit$.

Theorems & Definitions (83)

  • Theorem 1.2.1
  • Theorem 1.3.1
  • Theorem 1.4.1
  • Conjecture 1.7.1
  • Remark 2.2.1
  • Remark 2.2.2
  • Remark 2.2.3
  • Remark 2.3.1
  • Remark 2.4.1
  • Remark 2.4.2
  • ...and 73 more