Prehomogeneous spaces for parabolic group actions in classical groups. (Q1883065)

Let \(G\) be a reductive linear algebraic group, \(P\) a parabolic subgroup of \(G\), and \({\mathfrak p} _u\) the Lie algebra of the unipotent radical for \(P\). The authors are mainly concerned with the question of when terms \({\mathfrak p }_u^{(l)}\) of the descending central series for \({\mathfrak p }_u\) contain a dense \(P\)-orbit under the adjoint action. In the classical case, the authors prove that if \(B\) is a Borel subgroup of \(G\) and the characteristic of the base field is either zero or good for \(G\), then each \({\mathfrak b}_u^{(l)}\) contains a dense \(B\)-orbit. In the case \(G=\text{GL}_n\), the authors also address more general parabolics. Specifically, the authors prove that if \(P=P(d_1,\dots,d_t)\) with \(d_1\leq\cdots\leq d_s\geq d_{s+1}\geq\cdots\geq d_t\) for some \(s\), then each \({\mathfrak p}_u^{(l)}\) contains a dense \(P\)-orbit.