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Let $G$ be a complex reductive group. The spherical Hecke category of $G$ canbe presented as the category of $G_{\mathcal O}$-equivariant constructiblesheaves on the affine Grassmannian $\mathrm{Gr}_G$. This category admits aconvolution product, extending the convolution product in the Satake category.In this paper, we upgrade the mentioned convolution product to an $\mathbbE_3$-monoidal structure in $\infty$-categories. The construction is intrinsic,in the sense that it is not performed by pullback from the spectral side of theDerived Satake Theorem by Bezrukavnikov and Finkelberg. We use classicalproperties of the Beilinson-Drinfeld Grassmannian, Lurie's characterization of$\mathbb E_k$-algebras via the topological Ran space, Gaitsgory andRozenblyum's formalism of correspondences and the homotopy theory of stratifiedspaces.

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