[Paper Review] $Z/m$-graded Lie algebras and perverse sheaves, II

G. Lusztig, Zhiwei Yun|arXiv (Cornell University)|Apr 3, 2016

Algebraic structures and combinatorial models17 references3 citations

TL;DR

This paper establishes a combinatorial framework for parametrizing simple $G_{ar{0}}$-equivariant perverse sheaves on the nilpotent cone of a $\mathbb{Z}/m$-graded Lie algebra, using a $\mathbb{Q}(v)$-vector space $\mathbb{V}$ with a sesquilinear form derived from Ext-groups. The key result is that the set of such sheaves in a fixed block is in natural bijection with the orbits of $\pm$ on a signed basis $\mathbb{B}'$, generalizing the Springer correspondence to the graded setting.

ABSTRACT

We consider a fixed block for the equivariant perverse sheaves with nilpotent support in the $1$-graded ccomponent of a semisimple cyclically graded Lie algebra. We give a combinatorial parametrization of the simple objects in that block.

Motivation & Objective

• To provide a combinatorial parametrization of simple $G_{\underline{0}}$-equivariant perverse sheaves on $\mathfrak{g}_{\delta}^{\text{nil}}$ in the $\mathbb{Z}/m$-graded setting.
• To generalize the Springer correspondence for $\mathbb{Z}$-graded Lie algebras to the $\mathbb{Z}/m$-graded case using geometric and combinatorial data.
• To establish purity and vanishing results for odd cohomology sheaves of intersection cohomology complexes in this setting.
• To define a $\mathbb{Q}(v)$-vector space $\mathbb{V}$ with a sesquilinear form and bar involution, and to construct a signed basis $\mathbb{B}'$ whose $\pm$-orbits index the blocks.

Proposed method

• Construct a $\mathbb{Q}$-vector space $\mathbb{E}$ with a hyperplane arrangement associated to a block $^\xi\mathfrak{B}$, whose complement forms chambers indexing spiral inductions.
• Define a $\mathbb{Q}(v)$-vector space $\mathbb{V}'$ from these chambers, equipped with a sesquilinear form $(:)$ defined via dimensions of $G_{\underline{0}}$-equivariant Ext-groups.
• Form the quotient $\mathbb{V} = \mathbb{V}' / \text{rad}(\,(:\,)$, where the radical is both left and right, ensuring a well-defined form on $\mathbb{V}$.
• Introduce a bar involution $\bar{\cdot}$ and an $\mathcal{A}$-lattice $\mathbb{V}_{\mathcal{A}}$ in $\mathbb{V}$, both defined combinatorially.
• Define the signed basis $\mathbb{B}' \subset \mathbb{V}_{\mathcal{A}}$ as elements fixed under the bar map with $ (b:b) \in 1 + v\mathbb{Z}[[v]] $, and show it is well-defined via geometric arguments.
• Establish a natural bijection between $\mathbb{B}' / \pm$ and the block $^\xi\mathfrak{B}$, providing a combinatorial index set for the perverse sheaves.

Experimental results

Research questions

• RQ1 How can the simple $G_{\underline{0}}$-equivariant perverse sheaves in a fixed block of $\mathfrak{g}_{\delta}^{\text{nil}}$ be parametrized combinatorially in the $\mathbb{Z}/m$-graded setting?
• RQ2 What is the structure of the $\mathbb{Q}(v)$-vector space $\mathbb{V}$ associated to a block, and how does its sesquilinear form relate to Ext-groups between spiral inductions?
• RQ3 Does the odd cohomology of intersection cohomology complexes vanish in the $\mathbb{Z}/m$-graded case, as in the ungraded case?
• RQ4 Can a signed basis $\mathbb{B}'$ of $\mathbb{V}$ be constructed such that its $\pm$-orbits index the perverse sheaves in a block, generalizing the ungraded case?
• RQ5 How do purity and vanishing properties of cohomology sheaves extend from the $\mathbb{Z}$-graded to the $\mathbb{Z}/m$-graded setting?

Key findings

• The number of simple $G_{\underline{0}}$-equivariant perverse sheaves in a block $^\xi\mathfrak{B}$ equals $\dim_{\mathbb{Q}(v)} \mathbb{V}$, where $\mathbb{V}$ is the quotient of $\mathbb{V}'$ by the radical of the sesquilinear form.
• The left and right radicals of the form $(:)$ on $\mathbb{V}'$ coincide, ensuring that $\mathbb{V}$ inherits a well-defined sesquilinear form.
• The vector space $\mathbb{V}$ carries a natural bar involution and an $\mathcal{A}$-lattice $\mathbb{V}_{\mathcal{A}}$, both defined combinatorially.
• The signed basis $\mathbb{B}'$ of $\mathbb{V}$ consists of elements $b \in \mathbb{V}_{\mathcal{A}}$ with $\bar{b} = b$ and $ (b:b) \in 1 + v\mathbb{Z}[[v]] $, and its $\pm$-orbits bijectively parametrize the block $^\xi\mathfrak{B}$.
• The cohomology sheaves $\mathcal{H}^a(A)$ of any simple $G_{\underline{0}}$-equivariant perverse sheaf $A$ vanish for odd $a$, generalizing a result from the ungraded case.
• Spiral induction complexes $I_\varpi$ and induced sheaves ${}^\epsilon\text{Ind}(A')$ have only even-degree cohomology, and are direct sums of shifts of perverse sheaves with even shifts.