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 $\mathrm{Gr}_G$. 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.