The moduli space of affine spherical varieties with a fixed weight monoid (continuation)

In the talk, we shall present a sketch of the proof of the theorem on the $T_{\mathrm{ad}}$-module structure in the tangent space to $\mathrm M_\Gamma$ at the point $X_0$. Besides, we plan to discuss a combinatorial description of all affine spherical varieties with a given weight monoid based on the known classification of spherical homogeneous spaces and their affine embeddings.