Regular Functions on the K-Nilpotent Cone

2022/04/21 by Lucas Mason‐Brown, Mason-Brown, Lucas
Mathematics · #22E46 #Advanced Algebra and Geometry #Algebraic Geometry and Number Theory #Algebraic structures and combinatorial models #FOS: Mathematics #Representation Theory (math.RT)

paper · pdf · doi:10.48550/arxiv.2204.10118

openalex publication_date 2022/04/21 · openalex created_date 2025/10/10 · openalex updated_date 2026/07/28

Abstract

Let G be a complex reductive algebraic group with Lie algebra \mathfrakg and let G_ℝ be a real form of G with maximal compact subgroup K_ℝ. Associated to G_ℝ is a K × ℂ^×-invariant subvariety N_θ of the (usual) nilpotent cone N ⊂ \mathfrakg^*. In this article, we will derive a formula for the ring of regular functions ℂ[N_θ] as a representation of K × ℂ^×. Some motivation comes from Hodge theory. In arXiv:1206.5547, Schmid and Vilonen use ideas from Saito's theory of mixed Hodge modules to define canonical good filtrations on many Harish-Chandra modules (including all standard and irreducible Harish-Chandra modules). Using these filtrations, they formulate a conjectural description of the unitary dual. If G_ℝ is split, and X is the spherical principal series representation of infinitesimal character 0, then conjecturally gr(X) ≃ ℂ[N_θ] as representations of K × ℂ^×. So a formula for ℂ[N_θ] is an essential ingredient for computing Hodge filtrations.

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