[Paper Review] Equivariant Sheaves on Flag Varieties

Olaf M. Schnürer|arXiv (Cornell University)|Sep 27, 2008

Algebraic structures and combinatorial models17 references4 citations

TL;DR

This paper establishes a t-exact equivalence between the Borel-equivariant derived category of constructible sheaves on a flag variety $G/P$ and the perfect derived category of differential graded modules over the extension algebra of the simple equivariant perverse sheaves. The key result proves a conjecture of Soergel and Lunts by showing formality of the equivariant derived category via ABCD-approximations and inverse limits, providing an algebraic description of equivariant sheaf theory on flag varieties.

ABSTRACT

We show that the Borel-equivariant derived category of sheaves on the flag variety of a complex reductive group is equivalent to the perfect derived category of dg modules over the extension algebra of the direct sum of the simple equivariant perverse sheaves. This proves a conjecture of Soergel and Lunts in the case of flag varieties.

Motivation & Objective

• To establish an algebraic description of the Borel-equivariant derived category of sheaves on flag varieties.
• To prove the conjecture of Soergel and Lunts that the equivariant derived category is equivalent to the perfect derived category of dg modules over the extension algebra of simple equivariant perverse sheaves.
• To extend known formality results for toric and symmetric varieties to the case of flag varieties.
• To provide a t-exact equivalence between the category of equivariant perverse sheaves and a full subcategory of dg modules.
• To clarify the relationship between equivariant and non-equivariant derived categories via forgetful and extension of scalars functors.

Proposed method

• Constructs an ABCD-approximation of the $B$-stratified flag variety $(G/P, \mathcal{S})$ using a sequence of equivariant vector bundles and quotients.
• Applies inverse limits to realize the equivariant derived category as a limit of categories of sheaves on approximating spaces.
• Uses the formality of the extension algebra $\operatorname{Ext}(\mathcal{I}\mathcal{C}_B(\mathcal{S}))$ with trivial differential to establish equivalence to the perfect derived category of dg modules.
• Defines a t-structure on the dg module category via the subcategory $\operatorname{dgFlag}(\mathcal{E})$ of dg modules with cohomology concentrated in degree zero.
• Establishes a commutative diagram involving the forgetful functor and extension of scalars, linking equivariant and non-equivariant settings.
• Employs the induction equivalence to relate $\mathcal{D}^b_{B,c}(G/P)$ to $\mathcal{D}^b_{G,c}(G\times_B G/P)$, reducing to the case of a $G$-variety.

Experimental results

Research questions

• RQ1 Is the Borel-equivariant derived category of sheaves on a flag variety equivalent to the perfect derived category of dg modules over the extension algebra of simple equivariant perverse sheaves?
• RQ2 Does the formality of the extension algebra $\operatorname{Ext}(\mathcal{I}\mathcal{C}_B(\mathcal{S}))$ with trivial differential imply a t-exact equivalence to $\operatorname{dgPer}(\mathcal{E})$?
• RQ3 Can the conjecture of Soergel and Lunts on equivariant derived categories for projective varieties with finitely many orbits be verified in the case of flag varieties?
• RQ4 How do the equivariant and non-equivariant derived categories relate via the forgetful and extension of scalars functors?
• RQ5 Is the heart of the t-structure on $\operatorname{dgPer}(\mathcal{E})$ equivalent to the category of equivariant perverse sheaves?

Key findings

• There is a t-exact equivalence $\mathcal{D}^b_{B,c}(G/P) \cong \operatorname{dgPer}(\operatorname{Ext}(\mathcal{I}\mathcal{C}_B(\mathcal{S})))$, proving the main conjecture of Soergel and Lunts in the flag variety case.
• The equivalence restricts to an equivalence between the hearts: $\operatorname{Perv}_B(G/P) \cong \operatorname{dgFlag}(\operatorname{Ext}(\mathcal{I}\mathcal{C}_B(\mathcal{S})))$.
• The simple equivariant perverse sheaf $\mathcal{I}\mathcal{C}_B(S)$ corresponds to the indecomposable projective dg $\mathcal{E}$-module $e_S \mathcal{E}$.
• The forgetful functor $\operatorname{For}: \mathcal{D}^b_{B,c}(G/P) \to \mathcal{D}^b(G/P,\mathcal{S})$ induces a commutative diagram with the extension of scalars functor $?-\overset{L}{\otimes}_{\mathcal{E}} \mathcal{F}$, linking equivariant and non-equivariant settings.
• The non-equivariant analog holds: $\mathcal{D}^b(G/P,\mathcal{S}) \cong \operatorname{dgPer}(\operatorname{Ext}(\mathcal{I}\mathcal{C}(\mathcal{S})))$.
• The complexified version of the main equivalence is also valid, as shown by the commutativity of the diagram in Remark 72.