A partial generalization of Lie’s triangular form theorem and applications.

Let $G = GL(n,{\bf C})$ and let $H$ denote the subgroup of upper triangular matrices. Lie’s triangular form Theorem states: any connected solvable subgroup of $G$ acting on $G/H$ leaves some point fixed. We give a partial generalization applying to any real Lie group $G$ and any closed subgroup $H$. As a consequence (together with previous results) the Euler characters of $G/H$ is $\geq 0$.

Additional consequence: $G/H$ has an iterated fibration with base space of all successive terms except the top term expressible as $G_i/\Gamma_i$, with $\Gamma_i$ discrete in the Lie group $G_i$; the top quotient has this form or else the form $K\times_{K’} E$ with $K$ compact and $E$ euclidean.