De Concini and Procesi have defined the wonderful compactification $\overline{X}$ of a symmetric space $X=G/G^{\sigma }$ where $G$ is a complex semisimple adjoint group and $G^{\sigma }$ the subgroup of fixed points of $G$ by an involution $\sigma$. It is a closed subvariety of a Grassmannian of the Lie algebra $𝔤$ of $G$. In this paper we prove that, when the rank of $X$ is equal to the rank of $G$, the variety is defined by linear equations. The set of equations expresses the fact that the invariant alternate trilinear form $w$ on $𝔤$ vanishes on the $(-1)$-eigenspace...

V. Alexeev and M. Brion introduced, for a given a complex reductive group, a moduli scheme of affine spherical varieties with prescribed weight monoid. We provide new examples of this moduli scheme by proving that it is an affine space when the given group is of type $\mathsf{A}$ and the prescribed weight monoid is that of a spherical module.

We obtain several several results on the multiplicative structure constants of the T-equivariant Grothendieck ring $K_T(G/B)$ of the flag variety G/B. We do this by lifting the classes of the structure sheaves of Schubert varieties in $K_T(G/B)$ to R(T) ⊗ R(T), where R(T) denotes the representation ring of the torus T. We further apply our results to describe the multiplicative structure constants of $K{\left(X\right)}_{\mathbb{Q}}$ where X denotes the wonderful compactification of the adjoint group of G, in terms of the structure constants of...