Let $G$ be a reductive algebraic group (over some alg. closed field $k$ of char 0), and $H$ a subgroup such that $(G, H)$ is *spherical* (i.e., the Borel $B$ of $G$ has an open orbit on $G/H$). Then $k[G/H]$ is multiplicity-free as a $G$-module. I'm looking for a nice description of the irreducible $G$-representations it contains.

More precisely, let $\Lambda(G)$ be the lattice of integral weights for $G$ and $\Lambda_+(G)$ be the cone of dominant integral weights. For each $\lambda \in \Lambda_+(G)$ we have an irreducible representation $V_\lambda$. Let me write $\Lambda_+(G, H)$ for the set of $\lambda \in \Lambda_+(G)$ such that $V_\lambda$ appears in $k[G/H]$ (equivalently, such that $(V_\lambda)^H \ne 0$). The set $\Lambda_+(G, H)$ is a sub-monoid of $\Lambda_+(G)$ (see this question). Playing around with some examples suggests the following:

Claim: $\Lambda_+(G, H)$ is a lattice cone in $\Lambda(G)$, and its $\mathbb{Z}$-span is exactly the weights trivial on the $B$-stabiliser of a point in the open $B$-orbit on $G/H$.

Is this true? If so, is there a good reference that is easily digestible by non-experts like myself?

(Brion's paper "Spherical varieties" in the 1994 ICM proceedings, page 755, has a very general assertion about the sections of any $G$-equivariant line bundle on any spherical $G$-variety, which seems to be closely related; but I couldn't make head or tail of the proof, and surely there must be an easier approach for homogenous spherical varieties $G/H$ and trivial line bundles.)

unitaryrepresentations, but he somewhat cryptically notes that others can be discussed “by applying Weyl’s unitary trick” (p. 505). $\endgroup$ – Francois Ziegler Dec 15 '19 at 19:18