Finite dimensional representations and subgroup actions on homogeneous spaces (Q1269769)

Let \(H\) be an \(\mathbb R\)-subgroup of a \(\mathbb Q\)-algebraic group \(G\). \(H\) is said to be \textit{observable} in \(G\) if it is the stabilizer of a vector in a finite dimensional algebraic representation, and \textit{epimorphic} in \(G\) if any \(H\)-invariant vector is already \(G\)-invariant. Observable subgroups were introduced by Bialynicki-Birula, Hochschild and Mostow in 1963. In the present paper it is shown that if \(H\) is a \(\mathbb Q\)-subgroup then H is observable in \(G\) if and only if a certain \(H\) orbit is closed in \(G/G_{\mathbb Z}\). Moreover, if \(H\) is epimorphic in \(G\) then the action of \(H\) on \(G/G_{\mathbb Z}\) is minimal (all orbits are dense), and the converse holds when \(H\) is a \(\mathbb Q\)-subgroup of \(G\).