[Paper Review] Perverse $\mathbb{F}_p$-sheaves on the affine Grassmannian

Robert Cass|arXiv (Cornell University)|Oct 8, 2019

Advanced Algebra and Geometry33 references4 citations

TL;DR

This paper constructs the category of $L^+G$-equivariant perverse $\mathbb{F}_p$-sheaves on the affine Grassmannian for a reductive group $G$ in characteristic $p>0$, proves it is a symmetric monoidal category, and establishes an equivalence with the category of representations of an affine monoid scheme via Tannakian formalism. The key result is a geometrization of the inverse of the mod $p$ Satake isomorphism, with convolution of IC sheaves remaining simple and the cohomology of each IC sheaf being one-dimensional over $\mathbb{F}_p$. The work relies on Frobenius splitting and global $F$-regularity of affine Schubert varieties.

ABSTRACT

For a reductive group over an algebraically closed field of characteristic $p > 0$ we construct the abelian category of perverse $\mathbb{F}_p$-sheaves on the affine Grassmannian that are equivariant with respect to the action of the positive loop group. We show this is a symmetric monoidal category, and then we apply a Tannakian formalism to show this category is equivalent to the category of representations of a certain affine monoid scheme. We also show that our work provides a geometrization of the inverse of the mod $p$ Satake isomorphism. Along the way we prove that affine Schubert varieties are globally $F$-regular and we apply Frobenius splitting techniques to the theory of perverse $\mathbb{F}_p$-sheaves.

Motivation & Objective

• To define and study the category of $L^+G$-equivariant perverse $\mathbb{F}_p$-sheaves on the affine Grassmannian $\operatorname{Gr}$ in positive characteristic.
• To establish that this category is symmetric monoidal under convolution and admits a Tannakian reconstruction as representations of an affine monoid scheme $M_G$.
• To show that the cohomology functor $H = \bigoplus_i R^i\Gamma(-)$ is exact, faithful, and symmetric monoidal, leading to an equivalence with $\operatorname{Rep}_{\mathbb{F}_p}(M_G)$.
• To provide a geometric realization of the inverse of the mod $p$ Satake isomorphism via the Tannakian functor and IC sheaf cohomology.
• To prove that affine Schubert varieties are globally $F$-regular and apply Frobenius splitting techniques to analyze $\mathbb{F}_p$-perverse sheaves.

Proposed method

• Define perverse $\mathbb{F}_p$-sheaves on $\operatorname{Gr}$ using the middle perversity condition and the $\mathbb{F}_p$-structure of $\ell$-adic sheaves.
• Construct a convolution product on $P_{L^+G}(\operatorname{Gr}, \mathbb{F}_p)$ using the groupoid structure of $LG$ and the Hecke algebra action.
• Prove that the cohomology functor $H = \bigoplus_i R^i\Gamma(-)$ is exact and faithful, and upgrade it to a symmetric monoidal functor using the monoidal structure of the category.
• Apply Tannakian formalism to the fiber functor $H$ to show that the automorphism group of $H$ is represented by an affine monoid scheme $M_G$ over $\mathbb{F}_p$.
• Use Frobenius splitting techniques to establish that affine Schubert varieties are globally $F$-regular, which ensures the purity and vanishing of higher cohomology for IC sheaves.
• Establish the mod $p$ Satake isomorphism by comparing the Tannakian functor $\mathcal{T}$ on $K_0$ with the inverse of the Satake map $\mathcal{S}^{-1}$, using Herzig's formula for $\mathcal{S}^{-1}$.

Experimental results

Research questions

• RQ1 How can one define a well-behaved category of $L^+G$-equivariant perverse $\mathbb{F}_p$-sheaves on the affine Grassmannian in positive characteristic?
• RQ2 Is the category of such sheaves symmetric monoidal under convolution, and can it be reconstructed via Tannakian formalism?
• RQ3 Does the cohomology of the IC sheaves on affine Schubert varieties remain one-dimensional over $\mathbb{F}_p$, and what does this imply for the structure of the category?
• RQ4 Can the inverse of the mod $p$ Satake isomorphism be realized geometrically via a Tannakian functor on perverse $\mathbb{F}_p$-sheaves?
• RQ5 What singularities do affine Schubert varieties have, and how do Frobenius splitting techniques help in analyzing $\mathbb{F}_p$-perverse sheaves?

Key findings

• The category $P_{L^+G}(\operatorname{Gr}, \mathbb{F}_p)$ of $L^+G$-equivariant perverse $\mathbb{F}_p$-sheaves is a symmetric monoidal category under convolution.
• The cohomology functor $H = \bigoplus_i R^i\Gamma(-)$ is exact, faithful, and symmetric monoidal, leading to an equivalence $P_{L^+G}(\operatorname{Gr}, \mathbb{F}_p) \xrightarrow{\sim} \operatorname{Rep}_{\mathbb{F}_p}(M_G)$ for an affine monoid scheme $M_G$ over $\mathbb{F}_p$.
• The convolution of IC sheaves satisfies $\operatorname{IC}_{\mu_1} * \operatorname{IC}_{\mu_2} = \operatorname{IC}_{\mu_1 + \mu_2}$, showing that the product of simple objects remains simple.
• For every dominant cocharacter $\mu$, the $\mathbb{F}_p$-dimension of $H(\operatorname{IC}_\mu)$ is exactly 1, i.e., $\dim_{\mathbb{F}_p} H(\operatorname{IC}_\mu) = 1$.
• Affine Schubert varieties $\operatorname{Gr}_{\leq \mu}$ are globally $F$-regular, a key property used to control the cohomology and purity of $\mathbb{F}_p$-perverse sheaves.
• The Tannakian functor $\mathcal{T}$ on $K_0(P_{L^+G}(\operatorname{Gr}, \mathbb{F}_p)) \otimes \mathbb{F}_p$ coincides with the inverse of the mod $p$ Satake isomorphism $\mathcal{S}^{-1}$, thus geometrizing the inverse Satake map.