ON LIE-ALGEBRA DECOMPOSITIONS, RELATED TO SPHERICAL HOMOGENEOUS SPACES

Let G be a connected, reductive, linear algebraic group over an algebr aically closed field k of characteristik zero. Let H-1 and H-2 be two spherical subgroups of G. It is shown that for all g in a Zariski open subset of G one has a Lie algebra decomposition g = h1 + a + Adg . h2 , where a is the Lie algebra of a torus and dim a less-than-or-equal-t o min (rank G/H-1, rank G/H-2). As an application one obtains an estim ate of the transcendence degree of the field k(G/H-1 x G/H-2)G for the diagonal action of G. If k = C and G(R) is a real form of G defined b y an antiholomorphic involution sigma : G --> G then for a spherical s ubgroup H subset-of G and for all g in a Hausdorff open subset of G on e has a decomposition g = g(R) + a + Adg . h, where a is the Lie algeb ra of a sigma-invariant torus and dim a less-than-or-equal-to rank G/H .