Parabolic Subgroups of Real Direct Limit Groups

Elizabeth Dan-Cohen; Ivan Penkov; Joseph A. Wolf

paper

Parabolic Subgroups of Real Direct Limit Groups

arXiv:0901.0295

Abstract

Let $G_{R}$ be a classical real direct limit Lie group and $g_{R}$ its Lie algebra. The parabolic subalgebras of the complexification $g_{C}$ were described by the first two authors. In the present paper we extend these results to $g_{R}$ . This also gives a description of the parabolic subgroups of $G_{R}$ . Furthermore, we give a geometric criterion for a parabolic subgroup $P_{C}$ of $G_{C}$ to intersect $G_{R}$ in a parabolic subgroup. This criterion involves the $G_{R}$ -orbit structure of the flag ind-manifold $G_{C} / P_{C}$ .

13 pages