Let G be a semisimple and simply connected algebraic group, and let H^0 be the subgroup of points fixed by an involution of G. Let V be an irreducible representation of G with a nonzero vector v fixed by H 0. In this article, we prove a property of the normalization of the coordinate ring of the closure of G·[v] in P(V).

Let ρ be half the sum of the positive roots of a root system. We prove that if ⋋ is a dominant weight, ⋋ ≤ 2ρ with respect to the dominance order, and d is a saturation factor for the complex Lie algebra associated to the root system, then the irreducible representation V (d⋋) appears in the tensor product V (dρ) ⊗ V (dρ).

We present explicit formulas for a set of generators of the ideal of relations among the pfaffians of the principal minors of the antisymmetric matrices of fixed dimension. These formulas have an interpretation in terms of the standard monomial theory for the spin module of orthogonal groups.

A general framework for the reduction of the equations defining classes of spherical varieties to (possibly infinite-dimensional) grassmannians is proposed. This is applied to model varieties of types A, B and C; in particular, a standard monomial theory for these varieties is presented.

Given a classical semisimple complex algebraic group GG and a symmetric pair (G,K)(G,K) of non-Hermitian type, we study the closures of the spherical nilpotent KK-orbits in the isotropy representation of KK. For all such orbit closures, we study the normality, and we describe the KK-module structure of the ring of regular functions of the normalizations.

Given a crystallographic reduced root system and an element γ of the lattice generated by the roots, we study the minimum number |γ|, called the length of γ , of roots needed to express γ as sum of roots. This number is related to the linear functionals presenting the convex hull of the roots. The map γ → |γ| turns out to be the upper integral part of a piecewise-linear function with linearity domains the cones over the facets of this convex hull. In order to show this relation, we investigate the integral closure of the monoid generated by the roots in a facet. We study also the positive length, i.e., the minimum number of positive roots needed to write an element, and we prove that the two notions of length coincide only for the types A and C.

A general setting for a standard monomial theory on a multiset is introduced and applied to the Cox ring of a wonderful variety. This gives a degeneration result of the Cox ring to a multicone over a partial flag variety. Further, we deduce that the Cox ring has rational singularities.