[Paper Review] On the action of the dual group on the cohomology of perverse sheaves on the affine grassmannian

Éric Vasserot|arXiv (Cornell University)|May 2, 2000

Advanced Algebra and Geometry3 citations

TL;DR

This paper investigates the action of the dual group $G^{lat}$ on the cohomology of perverse sheaves on the affine Grassmannian $\mathrm{Gr}^G$, establishing that the operators $\mathbb{e}_i$ and $\mathbb{f}_i$ associated to simple roots satisfy the Serre relations, thereby realizing the action of the universal enveloping algebra $U(\mathfrak{g}^\vee)$ on the cohomology. The key result is that the cohomology of intersection cohomology complexes on $\mathrm{Gr}^G$ carries a representation of the Langlands dual Lie algebra $\mathfrak{g}^\vee$, with explicit verification of the Serre relations in the case of quasi-minuscule coweights.

ABSTRACT

It was proved by Ginzburg and Mirkovic-Vilonen that the $G(O)$-equivariant perverse sheaves on the affine grassmannian of a connected reductive group $G$ form a tensor category equivalent to the tensor category of finite dimensional representations of the dual group $G^\vee$. The proof use the Tannakian formalism. The purpose of this paper is to construct explicitely the action of $G^\vee$ on the global cohomology of a perverse sheaf.

Motivation & Objective

• To understand the action of the Langlands dual group $G^\vee$ on the cohomology of $G(O)$-equivariant perverse sheaves on the affine Grassmannian $\mathrm{Gr}^G$.
• To establish that the cohomology of intersection cohomology complexes on $\mathrm{Gr}^G$ carries a representation of the universal enveloping algebra $U(\mathfrak{g}^\vee)$.
• To verify the Serre relations for the operators $\mathbb{e}_i$ and $\mathbb{f}_i$ arising from the convolution action on cohomology.
• To analyze the structure of the cohomology of $\overline{\mathrm{Gr}}_{\lambda^\vee}$ for quasi-minuscule coweights $\lambda^\vee$ via restriction to Levi subgroups and fixed-point loci.

Proposed method

• Uses the Mirković–Vilonen convolution product on the category $\mathcal{P}_G$ of $G(O)$-equivariant perverse sheaves on $\mathrm{Gr}^G$.
• Applies the restriction functor $\mathrm{res}^{GM}$ to relate $\mathcal{P}_G$ to categories of sheaves on $\mathrm{Gr}^M$ for Levi subgroups $M\subset G$.
• Analyzes the cohomology of strata $S_{\mu^\vee} \subset \overline{\mathrm{Gr}}_{\lambda^\vee}$ via the action of operators $\mathbb{e}_i$ and $\mathbb{f}_i$ induced by convolution and restriction.
• Studies the geometry of $\overline{\mathrm{Gr}}_{\lambda^\vee} \cap \mathrm{Gr}^{M_i}$ for $\lambda^\vee$ quasi-minuscule, identifying fixed-point sets under torus actions and using line bundles $\mathcal{L}(\lambda)$.
• Verifies the Serre relations $[\mathbb{e}_i, \mathbb{f}_j] = \delta_{ij} \mathbb{h}_i$ by analyzing non-vanishing of cohomology and using the pairing $\langle \alpha_i, \mu^\vee \rangle$.
• Uses the fact that $H^*_{c}(S_{\mu^\vee}, \mathcal{IC}_{\lambda^\vee}) \neq 0$ only for specific $\mu^\vee$ to constrain the support of the operators.

Experimental results

Research questions

• RQ1 How does the dual group $G^\vee$ act on the cohomology of $G(O)$-equivariant perverse sheaves on the affine Grassmannian?
• RQ2 What is the structure of the cohomology of the intersection cohomology complex $\mathcal{IC}_{\lambda^\vee}$ for a quasi-minuscule dominant coweight $\lambda^\vee$?
• RQ3 Do the operators $\mathbb{e}_i$ and $\mathbb{f}_i$ associated to simple roots satisfy the Serre relations in the cohomology of $\mathrm{Gr}^G$?
• RQ4 How do the fixed-point loci of torus actions on $\overline{\mathrm{Gr}}_{\lambda^\vee}$ relate to the support of the cohomology of strata?

Key findings

• The cohomology of the intersection cohomology complex $\mathcal{IC}_{\lambda^\vee}$ on $\mathrm{Gr}^G$ carries a representation of the universal enveloping algebra $U(\mathfrak{g}^\vee)$.
• For a quasi-minuscule dominant coweight $\lambda^\vee$, the cohomology $H^*_{c}(S_{\mu^\vee}, \mathcal{IC}_{\lambda^\vee})$ is non-zero only if $\langle \alpha_1, \mu^\vee \rangle = -1$, $\mu^\vee = 0$, or $\mu^\vee = -\alpha_1^\vee$.
• The operator $\mathbb{e}_1$ acts non-trivially only when $\langle \alpha_1, \mu^\vee \rangle = -1$, and similarly for $\mathbb{f}_2$ when $\langle \alpha_2, \mu^\vee \rangle = 1$.
• The relation $[\mathbb{e}_1, \mathbb{f}_2] = 0$ holds because the condition $\langle \alpha_1, \mu^\vee \rangle = -1$ and $\langle \alpha_2, \mu^\vee + \alpha_1^\vee \rangle = 1$ leads to $\langle \alpha_2, \mu^\vee \rangle = 4$, which is impossible for $\mu^\vee \in \Omega(\lambda^\vee)$.
• The relation $[\mathbb{e}_2, \mathbb{f}_1] = 0$ is similarly verified by contradiction, as $\langle \alpha_1, \mu^\vee \rangle = -2$ is not realizable under the constraints of $\Omega(\lambda^\vee)$.