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

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 ${\mathrm{G}\mathrm{r}}_G$. 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 $\mathrm{\infty }$-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.