Let \(G\) be a reductive connected algebraic group over an algebraically closed field \(k\) of characteristic \(0\). Suppose \(H\) is an algebraic subgroup of \(G\). The homogeneous space \(G/H\) of the pair \((G,H)\) is said to be spherical if (among other equivalent conditions) there exists a Borel subgroup \(B\) of \(G\) such that \(BH\) is open in \(G\). M. Krämer [Compos. Math. 38, 129–153 (1979; Zbl 0402.22006)] described all spherical pairs \((G,H)\) where \(G\) is simple and simply connected and \(H\) is connected and reductive.
The author determines those pairs \((G,H)\) with \(G\) semisimple and simply connected, \(H\) connected and reductive. He remarks that I. V. Mikityuk [Math. Sb., Nov. Ser. 129 (171), 514–534 (1986; Zbl 0621.70005)] determined simultaneously these spherical pairs. If \((G',H')\), \((G'',H'')\) are spherical pairs, then \((G'\times G'', H'\times H'')\) is a spherical pair. So one may suppose that the pair is indecomposable. Furthermore, Krämer [loc. cit.] showed that one may suppose that \(G\) is semisimple. Then the author reduces the study to the case where \(H\) is reductive, handles the case where \(H\) is connected and each subgroup of \(G\) containing \(H\) is reductive, and then finishes the classification. The examination is often a case-by-case study.