### Good properties of algebras of invariants and defect of linear representations.

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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 $H\subset G$ 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...

We study $G$-actions of the form $(G:X\times {X}^{*})$, 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.

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 $\theta $-groups. This yields a description of spherical...

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...

Let $\U0001d524$ be a simple Lie algebra and ${\mathrm{\U0001d504\U0001d51f}}^{o}$ the poset of non-trivial abelian ideals of a fixed Borel subalgebra of $\U0001d524$. In [8], we constructed a partition ${\mathrm{\U0001d504\U0001d51f}}^{o}={\bigsqcup}_{\mu}{\mathrm{\U0001d504\U0001d51f}}_{\mu}$ parameterised by the long positive roots of $\U0001d524$ and studied the subposets ${\mathrm{\U0001d504\U0001d51f}}_{\mu}$. 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 $\U0001d524$ is a join-semilattice.

Recently, E.Feigin introduced a very interesting contraction $\U0001d52e$ of a semisimple Lie algebra $\U0001d524$ (see arXiv:1007.0646 and arXiv:1101.1898). We prove that these non-reductive Lie algebras retain good invariant-theoretic properties of $\U0001d524$. For instance, the algebras of invariants of both adjoint and coadjoint representations of $\U0001d52e$ are free, and also the enveloping algebra of $\U0001d52e$ is a free module over its centre.

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