For a spherical pair $(G, H)$, which $G$-representations appear in $k[G/H]$?

Question

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.)

Answer 1

Let $X_0=B/B_{x_0}=Bx_0$ be the open $B$-orbit in $X=G/H$. Then every character of $B$ which is trivial on $B_{x_0}$ yields a $B$-semiinvariant $f_\lambda$ on $X_0$. In general, it does not extend to a regular function on $X$. If it does, this means that $V_\lambda$ occurs in $k[X]$. So your second problems asks whether every $B$-semiinvariant on $X_0$ is a quotient of two regular $B$-semiinvariants on $X$. This is true if $X$ is quasiaffine (e.g., if $H$ is reductive) but false in general. Take, e.g., $G=SL(3,k)$ and $H=ker(\chi)$ where $\chi:B\to G_m$ is a character which is neither dominant nor antidominant (e.g. $\chi=a_{22}$). Then $V_\chi^H=0$ for all $\lambda$.

I don't know what you mean by lattice cone but if it means a monoid which is the intersection of a convex cone by a lattice then your first claim is true. This is a consequence of the normality of $X$. More precisely, let $\overline\Lambda_+$ be the intersection of the convex cone and the lattice generated by $\Lambda_+(G,H)$. It is called a saturation. An element $\lambda$ is in $\overline \Lambda_+$ if there is $m\ge1$ such that $m\lambda\in\Lambda_+(G,H)$. Now, since $\lambda$ is in the lattice generated by $\Lambda_+$ it corresponds to a rational $B$-semiinvariant $f_\lambda$ on $X$. The power $f_\lambda^m$ is regular. So $f_\lambda$ is regular and $\lambda\in\Lambda_+$.