@MISC{Jablonski804goodrepresentations, author = {M. Jablonski}, title = {Good Representations and Homogeneous Spaces}, year = {804} }

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Abstract

Let G be a complex reductive affine algebraic group. Let F, H be algebraic reductive subgroups. The homogeneous space G/F has a natural, transitive left action of G on it. We will consider the induced action of H on G/F. Hereafter a property of a space will be called generic if it occurs on a nonempty Zariski open set. Our main result is the following. The following theorem and its corollaries are true for both real and complex algebraic groups. Moreover, the results actually hold for real semi-algebraic groups by passing to finite index subgroups and finite covers of manifolds. We omit the details of the proofs for semi-algebraic groups. Theorem 1. Consider the induced action of H on G/F, then generic H-orbits are closed in G/F; that is, there is a nonempty Zariski open set of G/F such that the H-orbit of any point in this open set is closed. Corollary 2. Let G, H, F be as above. If H is normal in G, then all orbits of H are closed in G/F. Consequently, if G acts on V and the orbit Gv is closed, then Hv is also closed. Corollary 3. Let G be a reductive algebraic group. If H, F are generic reductive subgroups, then H ∩ F is also reductive. More precisely, take any two reductive subgroups H, F of G. Then H ∩ gFg −1 is reductive for generic g ∈ G.