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Apologies in advance for the vague question. Let $X$ be a spherical variety with the action of some reductive group $G$. I have been told in conversation several times that such spherical varieties ...

Setup Let $G$ be a (connected) reductive group over an algebraically closed field $k$, and fix a Borel subgroup $B \subset G$ and a maximal torus $T \subset B$. Let $\lambda: \mathbb{G}_m \to T$ be a ...