[Paper Review] Quasi-exceptional sets and equivariant coherent sheaves on the nilpotent cone

Roman Bezrukavnikov|ArXiv.org|Feb 6, 2001

Algebraic structures and combinatorial models10 citations

TL;DR

This paper establishes a bijection between dominant weights and pairs of nilpotent orbits with irreducible representations of their centralizers using a new t-structure on the derived category of G-equivariant coherent sheaves on the nilpotent cone. By showing that the perverse t-structure (from middle perversity) coincides with the t-structure defined by a quasi-exceptional set of standard objects, it constructs a canonical correspondence between irreducible objects (indexed by (O,L)) and dominant weights, resolving a conjecture of Lusztig and Vogan via geometric representation theory.

ABSTRACT

In math.AG/0005152 a certain $t$-structure on the derived category of equivariant coherent sheaves on the nil-cone of a simple complex algebraic group was introduced (the so-called perverse $t$-structure corresponding to the middle perversity). In the present note we show that the same $t$-structure can be obtained from a natural quasi-exceptional set generating this derived category. As a consequence we obtain a bijection between the sets of dominant weights and pairs consisting of a nilpotent orbit, and an irreducible representation of the centralizer of this element, conjectured by Lusztig and Vogan (and obtained by other means in math.RT/0010089).

Motivation & Objective

• To construct a canonical bijection between the set of dominant weights Λ⁺ and the set of pairs (O,L) consisting of a nilpotent orbit O and an irreducible representation L of its centralizer.
• To provide a direct, geometric construction of the perverse t-structure on Dᵇ(Cohᴳ(𝒩)) using a quasi-exceptional set, avoiding deep results from the geometric Langlands program.
• To show that the core of this t-structure is quasi-hereditary, with standard and costandard objects indexed by dominant weights.
• To establish that the irreducible objects in the core are precisely the intersection cohomology sheaves IC_{O,L}, indexed by (O,L) ∈ O.
• To provide a new, elementary proof of the Lusztig–Vogan bijection, independent of the methods in [B1] and [G].

Proposed method

• Define a new t-structure on Dᵇ(Cohᴳ(𝒩)) via a dualizable quasi-exceptional set {∇^λ = A_λ[d]} indexed by dominant weights λ ∈ Λ⁺.
• Show that the set {∇^λ} forms a quasi-exceptional set generating the derived category, with the associated t-structure being equivalent to the perverse t-structure of middle perversity.
• Prove that the core 𝒫 of this t-structure is quasi-hereditary, with costandard objects ∇^λ and irreducible objects IC_{O,L} indexed by (O,L) ∈ O.
• Use the fact that the Springer resolution π: 𝒩̃ → 𝒩 is semi-small and that π_* preserves cohomological dimension to verify that A_λ[d] lies in the heart of the t-structure.
• Leverage duality and proper base change to show that the dual of A_λ[d] is A_{-λ}[d], confirming the t-structure axioms.
• Establish the equivalence between the t-structure from the quasi-exceptional set and the perverse t-structure via stalkwise cohomological vanishing conditions on orbits.

Experimental results

Research questions

• RQ1 Can the perverse t-structure on Dᵇ(Cohᴳ(𝒩)) be constructed directly from a quasi-exceptional set, without relying on perverse sheaves on the affine flag variety?
• RQ2 Is there a canonical bijection between dominant weights and pairs (O,L) of nilpotent orbits and irreducible centralizer representations?
• RQ3 Does the core of the perverse t-structure on Dᵇ(Cohᴳ(𝒩)) admit a quasi-hereditary structure with standard and costandard objects indexed by Λ⁺?
• RQ4 Can the irreducible objects in the core be identified with intersection cohomology sheaves IC_{O,L}?
• RQ5 Is the transition matrix between the basis of irreducible objects [IC_{O,L}] and the basis of costandard objects [A_λ[d]] upper triangular with 1s on the diagonal?

Key findings

• The perverse t-structure on Dᵇ(Cohᴳ(𝒩)) corresponding to middle perversity coincides with the t-structure defined by the quasi-exceptional set {A_λ[d]} indexed by dominant weights λ ∈ Λ⁺.
• The core 𝒫 of this t-structure is a quasi-hereditary category with irreducible objects {IC_{O,L}} indexed by (O,L) ∈ O and costandard objects {A_λ[d]} indexed by λ ∈ Λ⁺.
• A canonical bijection between Λ⁺ and O is established via the condition that Hom(IC_{O_λ,L_λ}, A_λ[d]) ≠ 0.
• The transition matrix between the basis {[IC_{O,L}]} and {[A_λ]} is upper triangular with 1s on the diagonal, confirming the canonical nature of the bijection.
• The Grothendieck group K⁰(Cohᴳ(𝒩)) is freely generated by both {[IC_{O,L}]} and {[A_λ]}, and the change-of-basis matrix is upper triangular modulo v⁻¹.
• The result provides a new, direct proof of the Lusztig–Vogan bijection, independent of the geometric Langlands correspondence and the results of [B1] and [G].