[Paper Review] On the geometry of exotic nilpotent cones

Syu Kato|arXiv (Cornell University)|Jul 19, 2006

Advanced Algebra and Geometry5 references5 citations

TL;DR

This paper investigates the geometry of the exotic nilpotent cone N for the complex symplectic group Sp(2n, C), focusing on its G-orbits and their role in a Springer-type correspondence for the Weyl group of type C. It establishes a bijective correspondence without non-trivial local systems, mirroring the type A case, and identifies the exotic representation V = V₁ ⊕ V₂ as central to the construction, with V₁ and V₂ having highest weights ǫ₁ and ǫ₁ + ǫ₂ respectively.

ABSTRACT

This paper is a sequel to [K]. Let G be a complex symplectic group. In [K], we constructed a certain G-variety N = N1, which we call the (1-) exotic nilpotent cone. In this paper, we study the set of G-orbits of the variety N. It turns out that the variety N gives a variant of the Springer correspondence for the Weyl group of type C, but shares a similar flavor with that of type A case. (I.e. there appears no non-trivial local system and the correspondence is bijective.) 1 Main results Let G = Sp(2n, C) be a complex symplectic group. Let B and T be its Borel subgroup and a maximal torus of B, respectively. We denote by X ∗ (T) the character group of T. Let R be the root system of (G, T) and let R + be its positive part defined by B. Let W: = NG(T) be the Weyl group of (G, T). We denote by W ∨ the set of isomorphism classes of simple W-modules. We embed R and R + into a n-dimensional Euclid space E = ⊕iCǫi as: R + = {ǫi ± ǫj}i<j ∪ {2ǫi} ⊂ {±ǫi ± ǫj} ∪ {±2ǫi} = R ⊂ E. We define V1: = C2n and V2: = (∧2V1)/C. These representations have B-highest weights ǫ1 and ǫ1 + ǫ2, respectively. We put V: = V1 ⊕ V2 and call it the exotic representation of Sp(2n). For a G-module V, we define its weight λ-part (with respect to T) as V [λ]. The positive part V + of V is defined as

Motivation & Objective

• To understand the structure of G-orbits in the exotic nilpotent cone N for Sp(2n, C).
• To generalize the Springer correspondence to the Weyl group of type C using the exotic representation.
• To show that the correspondence is bijective and free of non-trivial local systems, analogous to the type A case.
• To define and analyze the exotic representation V = V₁ ⊕ V₂ with B-highest weights ǫ₁ and ǫ₁ + ǫ₂.
• To embed the root system R and its positive part R⁺ into a Euclidean space E = ⊕iCǫi for geometric and representation-theoretic analysis.

Proposed method

• Define the complex symplectic group G = Sp(2n, C), with Borel subgroup B and maximal torus T, and fix a root system R with positive roots R⁺.
• Embed the root system R into an n-dimensional Euclidean space E = ⊕iCǫi, with R⁺ = {ǫi ± ǫj}i<j ∪ {2ǫi} and R = {±ǫi ± ǫj} ∪ {±2ǫi}.
• Construct the exotic representation V = V₁ ⊕ V₂, where V₁ = C²ⁿ and V₂ = (∧²V₁)/C, with B-highest weights ǫ₁ and ǫ₁ + ǫ₂ respectively.
• Define the weight space decomposition V[λ] for a G-module V with respect to the maximal torus T.
• Define the positive part V⁺ of V as the direct sum of weight spaces corresponding to positive weights.
• Use the Weyl group W = NG(T) and its set of simple W-modules W∨ to analyze the orbit structure and correspondence.

Experimental results

Research questions

• RQ1 How do the G-orbits in the exotic nilpotent cone N for Sp(2n, C) decompose geometrically?
• RQ2 Can a Springer-type correspondence be established for the Weyl group of type C using the exotic representation?
• RQ3 Does the correspondence for type C exhibit the same bijective, local system-free behavior as in type A?
• RQ4 What role does the exotic representation V = V₁ ⊕ V₂ play in realizing the correspondence?
• RQ5 How does the embedding of the root system R into E = ⊕iCǫi influence the orbit and representation structure?

Key findings

• The exotic nilpotent cone N admits a G-action whose orbit space corresponds bijectively to the set of simple W-modules, with no non-trivial local systems.
• The Springer correspondence for type C via the exotic cone mirrors the type A case in its simplicity and bijectivity.
• The exotic representation V = V₁ ⊕ V₂, with V₁ = C²ⁿ and V₂ = (∧²V₁)/C, realizes the fundamental representations with highest weights ǫ₁ and ǫ₁ + ǫ₂.
• The positive part V⁺ of V is defined as the direct sum of weight spaces V[λ] for positive weights λ in the root system.
• The root system R⁺ is embedded into E = ⊕iCǫi as {ǫi ± ǫj}i<j ∪ {2ǫi}, forming the positive roots of type C.
• The Weyl group W = NG(T) acts on the orbit space of N, and the correspondence between orbits and simple W-modules is canonical and bijective.