Endoscopy for metaplectic affine Hecke categories

TL;DR

This paper establishes a deep connection between affine Hecke categories associated with twisted loop groups and Soergel bimodules, advancing the understanding of metaplectic groups and quantum geometric Langlands.

Contribution

It identifies affine Hecke categories with Soergel bimodules for twisted loop groups, including central extensions, and proves applications like endoscopic equivalences and the derived Satake equivalence.

Findings

Affine Hecke categories are equivalent to combinatorial categories of Soergel bimodules.

Results hold for mod $\\ell$ or integral $\\ell$-adic coefficients.

Provides evidence for conjectures in quantum geometric Langlands.

Abstract

For a possibly twisted loop group $LG$, and any character sheaf of its Iwahori subgroup, we identify the associated affine Hecke category with a combinatorial category of Soergel bimodules. In fact, we prove such results for affine Hecke categories arising from central extensions of the loop group $LG$. Our results work for mod $\ell$ or integral $\ell$-adic coefficients. As applications, we obtain endoscopic equivalences between affine Hecke categories, including the derived Satake equivalence for metaplectic groups, and a series of conjectures by Gaitsgory in quantum geometric Langlands.