Introduction to Mathematical Logic, Sixth Edition
《Introduction to Mathematical Logic, Sixth Edition》 · 第 609 页 · Chapter 12，Section 12.5「Notes」
The spherical pairs $(G, K)$ with $G$ connected and $K$ reductive have been classified in Krämer [93]. The term spherical subgroup is also applied to any algebraic subgroup $L$ (not necessarily reductive) such that $L$ has an open orbit on $G/B$. Vinberg and Kimelfeld [147, Theorem 1] show that this orbit condition is necessary and sufficient for the representation $\operatorname{Ind}_L^G(\chi)$ to be multiplicity-free for all regular characters $\chi$ of $L$.