Orbits of strongly solvable spherical subgroups on the flag variety

Let G be a connected reductive complex algebraic group and B a Borel subgroup of G. We consider a subgroup H of B acting with finitely many orbits on the flag variety G/B, and we classify the H-orbits in G/B in terms of suitable combinatorial invariants. As well, we study the Weyl group action defined by Knop on the set of H-orbits in G/B, and we give a combinatorial model for this action in terms of weight polytopes.Comment: v4: final version, to appear on Journal of Algebraic Combinatorics. Apported some minor corrections to the previous versio

Projective normality of model varieties and related results

We prove that the multiplication of sections of globally generated line bundles on a model wonderful variety M of simply connected type is always surjective. This follows by a general argument which works for every wonderful variety and reduces the study of the surjectivity for every couple of globally generated line bundles to a finite number of cases. As a consequence, the cone defined by a complete linear system over M or over a closed G-stable subvariety of M is normal. We apply these results to the study of the normality of the compactifications of model varieties in simple projective spaces and of the closures of the spherical nilpotent orbits. Then we focus on a particular case proving two specific conjectures of Adams, Huang and Vogan on an analogue of the model orbit of the group of type E8.Comment: v2: 54 pages, new introduction and several minor changes, added Proposition 9.2. To appear on Representation Theor

Simple linear compactifications of spherical homogeneous spaces

Let G be a simply connected semisimple algebraic group over an algebraically closed field k of characteristic 0 and let V be a rational simple G-module. If G/H is a spherical orbit in P(V) and if X is its closure, then we describe the orbits of X and those of its normalization X. If moreover the wonderful completion of G/H is strict, then we give necessary and sufficient combinatorial conditions so that the normalization morphism X → X is a homeomorphism. Such conditions are trivially fulfilled if G is simply laced or if H is a symmetric subgroup. In the special case of an odd orthogonal group G regarded as a GxG variety, we give an explicit classification of its simple linear compactifications, namely those equivariant compactifications with a unique closed orbit which are obtained by taking the closure of the GxG-orbit of the identity in a projective space P(End(V)), where V is a finite dimensional rational G-module

Spherical nilpotent orbits and abelian subalgebras in isotropy representations

Let $G$ be a simply connected semisimple algebraic group with Lie algebra $\mathfrak g$, let $G_0 \subset G$ be the symmetric subgroup defined by an algebraic involution $\sigma$ and let $\mathfrak g_1 \subset \mathfrak g$ be the isotropy representation of $G_0$. Given an abelian subalgebra $\mathfrak a$ of $\mathfrak g$ contained in $\mathfrak g_1$ and stable under the action of some Borel subgroup $B_0 \subset G_0$, we classify the $B_0$-orbits in $\mathfrak a$ and we characterize the sphericity of $G_0 \mathfrak a$. Our main tool is the combinatorics of $\sigma$-minuscule elements in the affine Weyl group of $\mathfrak g$ and that of strongly orthogonal roots in Hermitian symmetric spaces.Comment: Latex file, 29 pages, minor revision, to appear in Journal of the London Mathematical Societ

Nilpotent orbits of height 2 and involutions in the affine Weyl group

Let G be an almost simple group over an algebraically closed field k of characteristic zero, let g be its Lie algebra and let B be a Borel subgroup of G. Then B acts with finitely many orbits on the variety N_2 of the nilpotent elements in g whose height is at most 2. We provide a parametrization of the B-orbits in N_2 in terms of subsets of pairwise orthogonal roots, and we provide a complete description of the inclusion order among the B-orbit closures in terms of the Bruhat order on certain involutions in the affine Weyl group of g.Comment: v2: 28 pages, 1 table. Minor revision. To appear in Indag. Mat

The Bruhat order on Hermitian symmetric varieties and on abelian nilradicals

Let G be a simple algebraic group and P a parabolic subgroup of G with abelian unipotent radical Pu, and let B be a Borel subgroup of G contained in P. Let pu be the Lie algebra of Pu and L a Levi factor of P, then L is a Hermitian symmetric subgroup of G and B acts with finitely many orbits both on pu and on G/L. In this paper we study the Bruhat order of the B-orbits in pu and in G/L, proving respectively a conjecture of Panyushev and a conjecture of Richardson and Ryan

Some combinatorial properties of skew Jack symmetric functions

Motivated by Stanley's conjecture on the multiplication of Jack symmetric functions, we prove a couple of identities showing that skew Jack symmetric functions are semi-invariant up to translation and rotation of a $\pi$ angle of the skew diagram. It follows that, in some special cases, the coefficients of the skew Jack symmetric functions with respect to the basis of the monomial symmetric functions are polynomials with nonnegative integer coefficients

Simple linear compactifications of odd orthogonal groups

We classify the simple linear compactifications of SO(2r+1), namely those compactifications with a unique closed orbit which are obtained by taking the closure of the SO(2r+1)xSO(2r+1)-orbit of the identity in a projective space P(End(V)), where V is a finite dimensional rational SO(2r+1)-module.Comment: v2: several simplifications, final version. To appear in J. Algebr