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On deformation method in invariant theory

Dmitri Panyushev — 1997

Annales de l'institut Fourier

In this paper we relate the deformation method in invariant theory to spherical subgroups. Let G be a reductive group, Z an affine G -variety and H G a spherical subgroup. We show that whenever G / H is affine and its semigroup of weights is saturated, the algebra of H -invariant regular functions on Z has a G -invariant filtration such that the associated graded algebra is the algebra of regular functions of some explicit horospherical subgroup of G . The deformation method in its usual form, as developed...

Reductive group actions on affine varieties and their doubling

Dmitri I. Panyushev — 1995

Annales de l'institut Fourier

We study G -actions of the form ( G : X × X * ) , where X * is the dual (to X ) G -variety. These actions are called the doubled ones. A geometric interpretation of the complexity of the action ( G : X ) is given. It is shown that the doubled actions have a number of nice properties, if X is spherical or of complexity one.

On spherical nilpotent orbits and beyond

Dmitri I. Panyushev — 1999

Annales de l'institut Fourier

We continue investigations that are concerned with the complexity of nilpotent orbits in semisimple Lie algebras. We give a characterization of the spherical nilpotent orbits in terms of minimal Levi subalgebras intersecting them. This provides a kind of canonical form for such orbits. A description minimal non-spherical orbits in all simple Lie algebras is obtained. The theory developed for the adjoint representation is then extended to Vinberg’s θ -groups. This yields a description of spherical...

An extension of Rais' theorem and seaweed subalgebras of simple Lie algebras

Dmitri I. Panyushev — 2005

Annales de l’institut Fourier

We prove an extension of Rais' theorem on the coadjoint representation of certain graded Lie algebras. As an application, we prove that, for the coadjoint representation of any seaweed subalgebra in a general linear or symplectic Lie algebra, there is a generic stabiliser and the field of invariants is rational. It is also shown that if the highest root of a simple Lie algerba is not fundamental, then there is a parabolic subalgebra whose coadjoint representation do not...

Abelian ideals of a Borel subalgebra and root systems

Dmitri I. Panyushev — 2014

Journal of the European Mathematical Society

Let 𝔤 be a simple Lie algebra and 𝔄𝔟 o the poset of non-trivial abelian ideals of a fixed Borel subalgebra of 𝔤 . In [8], we constructed a partition 𝔄𝔟 o = μ 𝔄𝔟 μ parameterised by the long positive roots of 𝔤 and studied the subposets 𝔄𝔟 μ . In this note, we show that this partition is compatible with intersections, relate it to the Kostant-Peterson parameterisation and to the centralisers of abelian ideals. We also prove that the poset of positive roots of 𝔤 is a join-semilattice.

A remarkable contraction of semisimple Lie algebras

Dmitri I. PanyushevOksana S. Yakimova — 2012

Annales de l’institut Fourier

Recently, E.Feigin introduced a very interesting contraction 𝔮 of a semisimple Lie algebra 𝔤 (see arXiv:1007.0646 and arXiv:1101.1898). We prove that these non-reductive Lie algebras retain good invariant-theoretic properties of 𝔤 . For instance, the algebras of invariants of both adjoint and coadjoint representations of 𝔮 are free, and also the enveloping algebra of 𝔮 is a free module over its centre.

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