Quotients of commuting schemes associated to Symmetric Pairs

2022/03/10 by Santosh Nadimpalli, Nadimpalli, Santosh, Santosha Pattanayak +1
Mathematics · #14L24 #14L30 #20G05 #Advanced Algebra and Geometry #Advanced Topics in Algebra #Algebraic Geometry (math.AG) #Algebraic structures and combinatorial models #FOS: Mathematics #Representation Theory (math.RT)

paper · pdf · doi:10.48550/arxiv.2203.05341

openalex publication_date 2022/03/10 · openalex created_date 2025/10/10 · openalex updated_date 2026/07/28

Abstract

Let \mathfrakg=\mathfrakg₀⊕ \mathfrakg₁ be a \mathbb Z₂-grading of a classical Lie algebra such that (\mathfrakg, \mathfrakg₀) is a classical symmetric pair. Let G be a classical group with Lie algebra \mathfrakg and let G₀ be the connected subgroup of G with \rm Lie (G₀)=\mathfrak g₀. For d ≥ 2, let \mathfrakC^d(\mathfrakg₁) be the d-th commuting scheme associated with the symmetric pair (\mathfrak g, \mathfrak g₀). In this article, we study the categorical quotient \mathfrakC^d(\mathfrakg₁)//G₀ via the Chevalley restriction map. As a consequence we show that the categorical quotient scheme \mathfrak C^d(\mathfrak g₁)//G₀ is normal and reduced. As a part of the proof, we describe a generating set for the algebra k[\mathfrakg₁^d]^G₀, which are of independent interest.

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