Equivalence of Hecke Categories with Deeper Level Structures

Abstract

Let G be a reductive group of classical type. We study representations of the loop group G((t)) and geometrizations of Hecke algebras. The representations are generalizations of epipelagic representations. They have positive depth and appear in the representation induced from some level group (J, ψ). The associated Hecke category is the category of mixed Ql-sheaves on G((t)) with equivariant conditions, and we prove that this monoidal category is equivalent to an affine Hecke category of a smaller group H (not necessarily split). This equivalence relates certain positive depth representations of G((t)) and tamely ramified representations of H. Therefore, it has potential applications to local geometric Langlands program in a wildly ramified setting. The proof relies on the theory of Soergel bimodules and a reduction step using hyperbolic localization. It can be potentially generalized to (J, ψ) defined using Yu’s data.