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

Abstract

In this paper we provide a detailed construction of an associative and braided convolution product on the category of equivariant constructible sheaves on the affine Grassmannian through derived geometry. This product extends the convolution product on equivariant perverse sheaves and is constructed as an $E_3$-algebra object in $\infty$-categories. The main tools amount to a formulation of the convolution and fusion procedures over the Ran space involving the formalism of 2-Segal objects and correspondences from Dyckerhoff and Kapranov, "Higher Segal Spaces I", and Gaitsgory and Rozenblyum, "A Study in Derived Algebraic Geometry I"; of factorising constructible cosheaves over the Ran space from Lurie, "Higher Algebra" Chapter 5; and of constructible sheaves via $\infty$-categorical exit paths