Homogeneous spaces defined by Lie group automorphisms. I

Abstract

This paper is a study of the structure and geometry of coset spaces X = G/K where G is a reductive Lie group and K is an open subgroup of the fixed point set G of a semisimple automorphism of G. The symmetric spaces are the case 2 = 1. There the structure and classification theory for is well known, and the geometry of X basically comes down to a knowledge of the linear isotropy representation of K and the "Cartan decomposition" of the Lie algebra of G into eigenspaces of . We follow this general outline, starting with a structure theory for , obtaining full classifications (including the linear isotropy representations) in the cases which we know to have significant geometric interest, and then turning to geometric applications utilizing the eigenspace decomposition of the Lie algebra. The geometric applications which we pursue are concerned with G-invariant almost complex structures and almost hermitian metrics on X. The almost complex structures themselves are used as a technical tool in passing from compact G to reductive G in the structure theory for .