Vanishing sheaves and the geometric Whittaker model

Abstract Let be a connected reductive algebraic group over an algebraically closed field of characteristic and let be a prime number different from . Let be a maximal unipotent subgroup, and let be a maximal torus normalizing with normalizer . Let be the Weyl group of . Let be a nondegenerate multiplicative ‐local system on . In this paper, we prove that the bi‐Whittaker category, namely, the triangulated monoidal category of ‐biequivariant ‐complexes on , is monoidally equivalent to an explicit thick triangulated monoidal subcategory of “‐equivariant central sheaves” on the torus, answering a question raised by Drinfeld. In particular, the bi‐Whittaker category has the structure of a symmetric monoidal category. We also study a certain thick triangulated monoidal subcategory of “vanishing sheaves” and prove that it is braided monoidally equivalent to an explicit thick triangulated monoidal subcategory of “‐equivariant central sheaves” on the torus. The above equivalence is given by an enhancement of the parabolic restriction functor restricted to the subcategory .