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Relative Langlands duality for $\mathfrak{osp}(2n + 1|2n)$

Alexander Braverman, Michael Finkelberg, David Kazhdan, Roman Travkin

Abstract

We establish an $S$-duality converse to the one studied by the 1st, 2nd and 4th authors; this is also a case of a twisted version of the relative Langlands duality of Ben Zvi, Sakellaridis and Venkatesh.. Namely, we prove that the $S$-dual of $\text{SO}(2n+1)\times \text{Sp}(2n)$ acting on the tensor product of their tautological representations is the symplectic mirabolic space $\text{Sp}(2n)\times\text{Sp}(2n)$ acting on the product $T^* \text{Sp}(2n)$ and the tautological representations of $\text{Sp}(2n)$. (Note that due to the anomaly, the dual of the second factor $\text{Sp}(2n)$ is the metaplectic dual, i.e. $\text{Sp}(2n)$). We also formulate the corresponding global conjecture, which describes explicitly the categorical theta-correspondence on the Langlands dual side.

Relative Langlands duality for $\mathfrak{osp}(2n + 1|2n)$

Abstract

We establish an -duality converse to the one studied by the 1st, 2nd and 4th authors; this is also a case of a twisted version of the relative Langlands duality of Ben Zvi, Sakellaridis and Venkatesh.. Namely, we prove that the -dual of acting on the tensor product of their tautological representations is the symplectic mirabolic space acting on the product and the tautological representations of . (Note that due to the anomaly, the dual of the second factor is the metaplectic dual, i.e. ). We also formulate the corresponding global conjecture, which describes explicitly the categorical theta-correspondence on the Langlands dual side.
Paper Structure (16 sections, 18 theorems, 13 equations)

This paper contains 16 sections, 18 theorems, 13 equations.

Table of Contents

  1. Introduction
  2. Ring objects in the Satake category and relative Langlands duality
  3. The non-cotangent case
  4. The subject of bft
  5. The subject of this paper
  6. The global conjecture
  7. Acknowledgments
  8. Setup
  9. Weyl algebra
  10. Main theorem
  11. The proof
  12. The two Hecke actions can be identified
  13. The Hecke action generates ${\mathcal{W}}\operatorname{-mod}^{{\mathop{\operatorname{\rm SO}}}(V')_{\mathcal{O}}\times{\mathop{\operatorname{\rm Sp}}}(V)_{\mathcal{O}}}$
  14. A deequivariantized Ext-algebra
  15. A pseudo-slice
  16. ...and 1 more sections

Key Result

Corollary 1.5.2

Let $\Psi$ denote the natural functor from $D({\operatorname{Bun}}_{{\mathop{\operatorname{\rm SO}}}(2n+1)})$ to $D_{-1/2}({\operatorname{Bun}}_{{\mathop{\operatorname{\rm Sp}}}(2n)}^\omega)$ defined by the kernel $\iota^!\Theta$ (here $D_{-1/2}({\operatorname{Bun}}_{{\mathop{\operatorname{\rm Sp}}}

Theorems & Definitions (27)

  • Conjecture 1.5.1
  • Corollary 1.5.2
  • Lemma 2.1.1
  • Theorem 2.2.1
  • Lemma 3.1.1
  • proof
  • Lemma 3.2.1
  • Lemma 3.2.2
  • proof
  • Corollary 3.2.3
  • ...and 17 more