The moduli space of affine spherical varieties with a fixed weight monoid (based on a joint work with S. Cupit-Foutou)

Let $G$ be a connected reductive group. In a famous 2005 paper Alexeev and Brion constructed (among other things) the so-called moduli space of affine spherical $G$-varieties with prescribed weight monoid $\Gamma$. This space, denoted by $\mathrm M_\Gamma$, is an affine scheme of finite type over the base field equipped with a regular action of an adjoint torus $T_{\mathrm{ad}}$ in such a way that the orbits are in one-to-one correspondence with (considered up to $G$-equivariant isomorphism) affine spherical $G$-varieties with weight monoid $\Gamma$. Moreover, $\mathrm M_\Gamma$ contains a unique $T_{\mathrm{ad}}$-fixed closed point $X_0$, corresponding to the so-called $S$-variety with weight monoid $\Gamma$.
In the talk, we shall discuss computing the $T_{\mathrm{ad}}$-module structure on the tangent space $T_{X_0} \mathrm M_\Gamma$ to $\mathrm M_\Gamma$ at the point $X_0$. The main result is a complete description of this structure in terms of the given monoid $\Gamma$. In particular, the space $T_{X_0} \mathrm M_\Gamma$ is always a multiplicity-free $T_{\mathrm{ad}}$-module and its weights (up to a sign) belong to a fixed finite set depending only on $G$ (elements of this set are called the spherical roots of $G$).
Using the result on the $T_{\mathrm{ad}}$-module structure in $T_{X_0} \mathrm M_\Gamma$ we shall prove that the so-called root monoid of an affine spherical $G$-variety is free. Besides, we shall obtain new proofs of several known results on spherical $G$-varieties, among which are uniqueness theorems for affine spherical $G$-varieties and spherical homogeneous spaces first proved by I. V. Losev in 2009. Remarkably, the proofs of the above-mentioned results easily reduce to checking certain properties of spherical roots, which is easily done case by case.