Actions with a finite number of orbits
Let G be a connected reductive linear algebraic group, defined over \({\mathbb{C}}\), and \(U\subset G\) a closed subgroup of it. Given a rational character \(\rho: U\to {\mathbb{C}}^*\) denote by \(L^{\rho}\) the associated homogeneous algebraic line bundle over G/U. Theorem. The following conditions are equivalent: (i) for any equivariant open embedding \(G/U\hookrightarrow X\), where X is a G-variety, the number of G-orbits on X is finite; (ii) for any rational character \(\rho: U\to {\mathbb{C}}^*\) the induced representation of G in the space of regular sections of \(L^{\rho}\) is multiplicity-free. - If \(G/U\) is quasiaffine, then each of these conditions is equivalent to: (iii) the representation of G in the space of regular functions on \(G/U\) is multiplicity-free. It was known, that (ii) is equivalent to the fact, that a Borel subgroup \(B\subset G\) has an open orbit on \(G/U\). Thus the implication (ii)\(\Rightarrow\)(i) follows also from the proposition 7.5 of a paper by \textit{D. Luna} and \textit{Th. Vust} [Comment. Math. Helv. 58, 186-245 (1983; Zbl 0545.14010)].
scientific article; zbMATH DE number 3920672