A Construction of the Symmetric Monoidal Structure of the Geometric Whittaker Model

Abstract

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 $U\subseteq G$ be a maximal unipotent subgroup, $T$ a maximal torus normalizing $U$ and $W$ the Weyl group of $G$. Let $\mathcal{L}$ be a non-degenerate multiplicative ${\overline{\mathbb{Q}}}_{\ell }$-local system on $U$. R. Bezrukavnikov and the second author have proved that the bi-Whittaker category, namely the triangulated monoidal category of $(U,\mathcal{L})$-biequivariant ${\overline{\mathbb{Q}}}_{\ell }$-complexes on $G$ is monoidally equivalent to an explicit thick triangulated monoidal subcategory ${\mathcal{D}}_W^{\circ }(T)\subseteq {\mathcal{D}}_W(T)$ of "central sheaves" on the torus. In particular it has the structure of a symmetric monoidal category coming from the symmetric monoidal structure on ${\mathcal{D}}_W(T)$. In this paper, we give another construction of a symmetric monoidal structure on the above category and prove that it agrees with the one coming from the above construction. For this, among other things, we generalize a proof by Gelfand for finite groups to the geometric setup.

Submitted Version, 25 pages