Commutative polarisations and the Kostant cascade

2021/08/17 by Dmitri I. Panyushev, Panyushev, Dmitri I.
Mathematics · #17B20 #17B22 #17B30 #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.2108.07750

openalex publication_date 2021/08/17 · openalex created_date 2025/10/10 · openalex updated_date 2026/07/28

Abstract

Let \mathfrak g be a complex simple Lie algebra. We classify the parabolic subalgebras \mathfrak p of \mathfrak g such that the nilradical of \mathfrak p has a commutative polarisation. The answer is given in terms of the Kostant cascade. It requires also the notion of an optimal nilradical and some properties of abelian ideals in a Borel subalgebra of \mathfrak g. Some invariant-theoretic consequences of the existence of a commutative polarisation are also discussed.

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