On extended weight monoids of spherical homogeneous spaces

Given a connected reductive complex algebraic group $G$ and a spherical subgroup $H\subset G$, the extended weight monoid ${\widehat{\mathrm{\Lambda }}}_G^{+}(G\mathrm{/}H)$ encodes the $G$-module structures on spaces of global sections of all $G$-linearized line bundles on $G\mathrm{/}H$. Assuming that $G$ is semisimple and simply connected and $H$ is specified by a regular embedding in a parabolic subgroup $P\subset G$, in this paper we obtain a description of $\widehat \Lambda^+_G(G/H)$viathesetofsimplesphericalrootsof$G/H$ together with certain combinatorial data explicitly computed from the pair $(P,H)$. As an application, we deduce a new proof of a result of Avdeev and Gorfinkel describing ${\widehat{\mathrm{\Lambda }}}_G^{+}(G\mathrm{/}H)$ in the case where $H$ is strongly solvable.