On Lie algebra decompositions, related to spherical homogeneous spaces (Q1313581)

Let \(G\) be a complex connected reductive group. An algebraic subgroup \(H\subset G\) is a spherical subgroup iff a Borel subgroup of \(G\) has an open orbit on \(G/H\). The author proves the following two theorems: (1) Let \(H_ 1\) and \(H_ 2\) be spherical subgroups of \(G\). Then there exists a torus \(A\) in \(G\) such that \[ L(G)= L(H_ 1)+ L(A)+ Ad(g).L(H_ 2) \] for all \(g\) in a Zariski open subset of \(G\) (\(L(G)\) is the Lie algebra of \(G\)). (2) Let \(H\) be a spherical subgroup of \(G\) and let \(s: G\to G\) be an antiholomorphic involution defining a real form \(G_ r\) of \(G\). Then there exists an \(s\)-invariant torus \(A\) in \(G\) such that \[ L(G)= L(G_ r)+ L(A)+ Ad(g).L(H) \] for all \(g\) in a Hausdorff open subset of \(G\).