On Lie algebra decompositions related to spherical homogeneous spaces |
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Authors: | Dmitri N. Akhiezer |
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Affiliation: | 1. Moscow Construction Engineering Institute, 129337, Moscow, Russia
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Abstract: | LetG be a connected, reductive, linear algebraic group over an algebraically closed fieldk of characteristik zero. LetH 1 andH 2 be two spherical subgroups ofG. It is shown that for allg in a Zariski open subset ofG one has a Lie algebra decomposition g = h1 + Adg ? h2, where a is the Lie algebra of a torus and dim a ≤ min (rankG/H 1,rankG/H 2). As an application one obtains an estimate of the transcendence degree of the fieldk(G/H 1 xG/H 2) G for the diagonal action ofG. Ifk = ? andG a is a real form ofG defined by an antiholomorphic involution σ :G→G then for a spherical subgroup H ? G and for allg in a Hausdorff open subset ofG one has a decomposition g = ga + a Adg ? h, where a is the Lie algebra of σ-invariant torus and dim a ≤ rankG/H. |
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