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The Frobenius semiradical, generic stabilizers, and Poisson center for nilradicals

Part of: Algebraic groups Lie algebras and Lie superalgebras

Published online by Cambridge University Press:  27 October 2023

Dmitri I. Panyushev
Dmitri I. Panyushev*
Affiliation:
Laboratory of Algebra and Number Theory, Institute for Information Transmission Problems, Moscow, Russia
*
e-mail: [email protected]
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Abstract

Let ${\mathfrak g}$ be a complex simple Lie algebra and ${\mathfrak n}$ the nilradical of a parabolic subalgebra of ${\mathfrak g}$. We consider some properties of the coadjoint representation of ${\mathfrak n}$ and related algebras of invariants. This includes (i) the problem of existence of generic stabilizers, (ii) a description of the Frobenius semiradical of ${\mathfrak n}$ and the Poisson center of the symmetric algebra , (iii) the structure of as -module, and (iv) the description of square integrable (= quasi-reductive) nilradicals. Our main technical tools are the Kostant cascade in the set of positive roots of ${\mathfrak g}$ and the notion of optimization of ${\mathfrak n}$.

Keywords

Coadjoint actionKostant cascadeFrobenius semiradicalPoisson center

MSC classification

Primary: 14L30: Group actions on varieties or schemes (quotients) 17B30: Solvable, nilpotent (super)algebras
Secondary: 17B20: Simple, semisimple, reductive (super)algebras 17B22: Root systems
Type
Article
Information
Canadian Journal of Mathematics , Volume 76 , Issue 6 , December 2024 , pp. 2019 - 2048
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Copyright
© The Author(s), 2023. Published by Cambridge University Press on behalf of The Canadian Mathematical Society

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