Homogeneous spherical data of orbits in spherical embeddings

Abstract

Let $G$ be a connected reductive complex algebraic group. Luna assigned to any spherical homogeneous space $G/H$ a combinatorial object called a homogeneous spherical datum. By a theorem of Losev, this object uniquely determines $G/H$ up to $G$-equivariant isomorphism. In this paper, we determine the homogeneous spherical datum of a $G$-orbit $X_0$ in a spherical embedding $G/H \hookrightarrow X$. As an application, we obtain a description of the colored fan associated to the spherical embedding $X_0 \hookrightarrow \bar{X_0}$.