On the structure of Borel stable abelian subalgebras in infinitesimal symmetric spaces

Let g=g_0+g_1 be a Z_2-graded Lie algebra. We study the posets of abelian subalgebras of g_1 which are stable w.r.t. a Borel subalgebra of g_0. In particular, we find out a natural parametrization of maximal elements and dimension formulas for them. We recover as special cases several results of Kostant, Panyushev, Suter.Comment: Latex file, 35 pages, minor corrections, some examples added. To appear in Selecta Mathematic

Affine Toric SL(2)-embeddings

In 1973 V.L.Popov classified affine SL(2)-embeddings. He proved that a locally transitive SL(2)-action on a normal affine three-dimensional variety X is uniquely determined by a pair (p/q, r), where 0<p/q<=1 is an uncancelled fraction and r is a positive integer. Here r is the order of the stabilizer of a generic point. In this paper we show that the variety X is toric, i.e. admits a locally transitive action of an algebraic torus, if and only if r is divisible by q-p. To do this we prove the following necessary and sufficient condition for an affine G/H-embedding to be toric. Suppose X is a normal affine variety, G is a simply connected semisimple algebraic group acting regularly on X, H is a closed subgroup of G such that the character group $\mathfrak{X}(H)$ is finite and G/H -> X is a dense open equivariant embedding. Then X is toric if and only if there exist a quasitorus T and a $(G\times T)$-module V such that $X\stackrel{G}{\cong} V//T$. The key role in the proof plays D. Cox's construction.Comment: 16 page

Harmonic analysis on spherical homogeneous spaces with solvable stabilizer

For all spherical homogeneous spaces G/H, where G is a simply connected semisimple algebraic group and H a connected solvable subgroup of G, we compute the spectra of the representations of G on spaces of regular sections of homogeneous line bundles over G/H.Comment: v2: 14 pages, minor correction

Affine spherical homogeneous spaces with good quotient by a maximal unipotent subgroup

For an affine spherical homogeneous space G/H of a connected semisimple algebraic group G, we consider the factorization morphism by the action on G/H of a maximal unipotent subgroup of G. We prove that this morphism is equidimensional if and only if the weight semigroup of G/H satisfies some simple condition.Comment: v2: title and abstract changed; v3: 16 pages, minor correction

Geodesic flows on Riemannian g.o. spaces

We prove the integrability of geodesic flows on the Riemannian g.o. spaces of compact Lie groups, as well as on a related class of Riemannian homogeneous spaces having an additional principal bundle structure.Comment: 12 pages, minor corrections, final versio

Parabolic subgroups with abelian unipotent radical as a testing site for invariant theory.

Abstract. Let L be a simple algebraic group and P a parabolic subgroup with Abelian unipotent radical Pu. Many familiar varieties (determinantal varieties, their symmetric and skew-symmetric analogues) arise as closures of P-orbits in Pu. We give a unified invariant-theoretic treatment of various properties of these orbit closures. We also describe the closures of the conormal bundles of these orbits as the irreducible components of some commuting variety and show that the polynomial algebra k[Pu] is a free module over the algebra of covariants