Let $G$ be a semisimple group over $\mathbb{C}$ and let $X=G/H$ be a homogeneous spherical variety. By Losev's theorem, the spherical $G$-variety $X$ is uniquely determined by its spherical datum, see below. In particular, the group ${\rm Aut}_G(X)=\mathcal{N}_G(H)/H$ is uniquely determined by the spherical datum.

Question.How can one compute ${\rm Aut}_G(X)$ from the spherical datum of $X$?

We specify a version of the spherical datum of $X$. Let $B\subset G$ be a Borel subgroup, and let $T\subset B\subset G$ be a maximal torus. Let $S=S(G,T,B)$ denote the corresponding system of simple roots. Let ${\mathcal{P}}(S)$ denote the set of subsets of $S$.

Let ${\mathcal{X}}(B)$ denote the character group of $B$, and let $M\subset {\mathcal{X}}(B)$ denote the weight lattice of $X$. Set $N={\rm Hom}(M,{\mathbb{Z}})$, $N_{\mathbb{Q}}={\rm Hom}(M,{\mathbb{Q}})$. Let $\mathcal{V}\subset N_{\mathbb{Q}}$ denote the valuation cone of $X$, and let $\Sigma\subset M$ denote the corresponding set of spherical roots.

Let ${\mathcal{D}}$ denote the set of colors of $X$ (the set of $B$-invariant prime divisors of $X$). We have two maps: $$\rho\colon {\mathcal{D}}\to N,\qquad \varsigma\colon {\mathcal{D}}\to {\mathcal{P}}(S). $$ Here, for $D\in{\mathcal{D}}$, $$ \varsigma(D)=\{\alpha\in S\ | \ P_\alpha\cdot D\neq D\}, $$ where $P_\alpha\supset B$ denotes the parabolic subgroup corresponding to $\alpha\in S$.

I wish to express ${\rm Aut}_G(X)$ in terms of the spherical datum $(M,\Sigma,\mathcal D,\rho,\varsigma)$ of $X$.