A model for the E3 fusion-convolution product of constructible sheaves on the affine Grassmannian

TL;DR

This paper constructs an intrinsic $ ext{E}_3$-monoidal convolution product on the category of constructible sheaves on the affine Grassmannian, advancing the understanding of automorphic categories in geometric representation theory.

Contribution

It introduces a novel $ ext{E}_3$-monoidal structure on the convolution product of constructible sheaves, utilizing advanced tools like the Beilinson--Drinfeld Grassmannian and Lurie's $ ext{E}_k$-algebra formalism.

Findings

Established a left t-exact $ ext{E}_3$-monoidal structure

Connected the convolution product to automorphic side via intrinsic construction

Applied homotopy theory of stratified spaces and correspondences

Abstract

Let $G$ be a complex reductive group. The spherical Hecke category of $G$ can be presented as the category of $G_{\mathcal O}$-equivariant constructible sheaves on the affine Grassmannian $\mathrm{Gr}_G$. This category admits a convolution product, extending the convolution product of equivariant perverse sheaves. In this paper, we upgrade the mentioned convolution product to a left t-exact $\mathbb E_3$-monoidal structure in $\infty$-categories. The construction is intrinsic to the automorphic side. Our main tools are the Beilinson--Drinfeld Grassmannian, Lurie's characterization of $\mathbb E_k$-algebras via the topological Ran space, the homotopy theory of stratified spaces, and the formalism of correspondences.