The moduli scheme of affine spherical varieties with a given weight monoid

Given a connected reductive group G and a submonoid Gamma of its dominant weights, in 2005 Alexeev and Brion constructred an affine scheme M_Gamma of finite type and equipped it with an action of an adjoint torus T_ad of G in such a way that T_ad-orbits in M_Gamma are in bijection with affine spherical G-varieties whose weight monoid is Gamma. They also proved that M_Gamma contains only finitely many T_ad-orbits and exactly one T_ad-fixed point X_0. The goal of the talk is to give an idea of computing the T_ad-module structure of the tangent space of M_Gamma at X_0 and discuss some applications.

The talk is based on a joint work with S. Cupit-Foutou.