Connected components of a spherical subgroup from spherical data?

Question

This question is in a similar spirit to this one by Mikhail Borovoi.

Let $G$ be a reductive group over $\mathbb{C}$ and let $X=G/H$ be a homogeneous spherical variety. Losev proved that the spherical $G$-variety $X$ is uniquely determined by its spherical datum $(M,\Sigma,\mathcal D,\rho,\varsigma)$, which I will recall below. For example, in the cited question one sees how the automorphism group of $X$ may be read off of this data.

My question regards the group of connected components $\pi_0(H)$. I am in a situation where it would be great to be able to compute this group abstractly (without knowing $H$, for example) from the datum.

Question. Can one compute $\pi_0(H)$ from the spherical datum of $X$?

Frankly, partial results would be helpful (eg: can one at least detect if the component group is non-trivial?).

Let me now recall the description of the spherical datum of $X$ as stated in the above linked question:

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 $\alpha\in S$.

Answer 1

Let $H^\circ\subseteq H$ be the connected component of unity and let $M^\circ$ be weight lattice of $X^\circ:=G/H^\circ$. Since $\pi_0(H)=H/H^\circ$ acts as automorphisms on $X^\circ$ it follows that $\pi_0(H)$ is the dual group of, hence isomorphic to, the finite abelian group $M^\circ/M$. Thus, everything boils down to compute $M^\circ$. Now, this has been done by Hofscheier in his paper Containment Relations among Spherical Subgroups. Corollary 1.5 states, in particular, that $$M^\circ=\{\chi\in M_{\mathbb Q}\cap\mathcal X(B)\mid \langle\rho(\mathcal D),\chi\rangle\subseteq\mathbb Z\}.$$