A Construction of the Symmetric Monoidal Structure of the Geometric Whittaker Model
Ashutosh Roy Choudhury, Tanmay Deshpande
TL;DR
The paper constructs a canonical symmetric monoidal structure on the geometric bi-Whittaker category $e_{\mathcal{L}}\mathscr{D}(G)e_{\mathcal{L}}$ by geometrizing Gelfand’s trick. It introduces a canonical anti-involution $\\bm{\\Psi}:G\to G$ preserving the non-degenerate local system $\mathcal{L}$ and proves a natural isomorphism $\\bm{\\Psi}^{*}\cong \mathrm{Id}$ on the bi-Whittaker subcategory, yielding a braiding via $\\mathcal{F}\\ast\mathcal{G}$. Beilinson gluing of perverse sheaves is adapted to glue local isomorphisms from Bruhat cells to the whole category, ensuring coherence; the construction is then compared with the established symmetric monoidal structure arising from the Beilinson–Bezrukavnikov–Drinfeld approach, via the equivalence $\\zeta: e_{\mathcal{L}}\mathscr{D}(G)e_{\mathcal{L}} \simeq \mathscr{D}_{W}^{\circ}(T)$. The result shows the two structures agree, giving a canonical symmetric monoidal framework that mirrors the Gelfand–Graev model in a geometric setting, with the Mellin transform and central sheaves playing a key role in the comparison.
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 $\mathscr{D}_{W}^{\circ}(T) \subseteq \mathscr{D}_{W}(T)$ of "central sheaves" on the torus. In particular it has the structure of a symmetric monoidal category coming from the symmetric monoidal structure on $\mathscr{D}_W(T)$. 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.