2023/10/23 by Bezrukavnikov, Roman, Deshpande, Tanmay
#Algebraic Geometry (math.AG) #FOS: Mathematics #Representation Theory (math.RT)
paper · doi:10.48550/arxiv.2310.14834
Let G be a connected reductive algebraic group over an algebraically closed field k of characteristic p>0 and let ℓ be a prime number different from p. Let U⊂ G be a maximal unipotent subgroup, and let T be a maximal torus normalizing U with normalizer N=NG(T). Let W=N/T be the Weyl group of G. Let L be a non-degenerate ℓ-adic multiplicative local system on U. In this paper we prove that the bi-Whittaker category, namely the triangulated monoidal category of (U,L)-bi-equivariant complexes on G, is monoidally equivalent to an explicit thick triangulated monoidal subcategory \mathscrD^∘W(T)⊂ \mathscrDW(T) of ''W-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 \mathscrD^∘G(G)⊂ \mathscrDG(G) of ''vanishing sheaves'' and prove that it is braided monoidally equivalent to an explicit thick triangulated monoidal subcategory \mathscrD^∘N(T)⊂ \mathscrDN(T) of ''N-equivariant central sheaves'' on the torus. The above equivalence is given by an enhancement of the parabolic restriction functor restricted to the subcategory \mathscrD^∘G(G).