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](https://mathoverflow.net/questions/327654)). Playing around with some examples suggests the following:



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> **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 $G/H$.
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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](https://www.mathunion.org/fileadmin/ICM/Proceedings/ICM1994.2/ICM1994.2.ocr.pdf), 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.)



For a spherical pair $(G, H)$, which $G$-representations appear in $k[G/H]$?
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 $\lambda\in\Lambda\_+$.


