Vanishing sheaves and the geometric Whittaker model

Let $G$ be a connected reductive algebraic group over an algebraically closed field $k$ of characteristic $p>0$ and let $\ell$ be a prime number different from $p$. Let $U\subset G$ be a maximal unipotent subgroup, and let $T$ be a maximal torus normalizing $U$ with normalizer $N=N_G(T)$. Let $W=N/T$ be the Weyl group of $G$. Let $\mathcal{L}$ be a non-degenerate $\ell$-adic multiplicative local system on $U$. In this paper we prove that the bi-Whittaker category, namely the triangulated monoidal category of $(U,\mathcal{L})$-bi-equivariant complexes on $G$, is monoidally equivalent to an explicit thick triangulated monoidal subcategory $\mathscr{D}^\circ_W(T)\subset \mathscr{D}_W(T)$ 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 $\mathscr{D}^\circ_G(G)\subset \mathscr{D}_G(G)$ of ''vanishing sheaves'' and prove that it is braided monoidally equivalent to an explicit thick triangulated monoidal subcategory $\mathscr{D}^\circ_N(T)\subset \mathscr{D}_N(T)$ 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 $\mathscr{D}^\circ_G(G)$.