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The Frobenius semiradical, generic stabilizers, and Poisson center for nilradicals
- Part of
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- Journal:
- Canadian Journal of Mathematics / Volume 76 / Issue 6 / December 2024
- Published online by Cambridge University Press:
- 27 October 2023, pp. 2019-2048
- Print publication:
- December 2024
-
- Article
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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 
${\mathfrak n}$.
of the symmetric algebra 
, 
as 
-module, and 
