The automorphism group of a homogeneous almost complex manifold

Abstract

1. Introduction. Let M be a compact simply connected manifold of nonzero Euler characteristic that carries a homogeneous almost complex structure. We determine the largest connected group A0(M) of almost analytic automorphisms of M. Our hypotheses represent M as a coset space G/K where G is a maximal compact subgroup of the Lie group A0(M) and Ais a closed connected subgroup of maximal rank in G. In §2 we collect some information, decomposing M=MX x ■ ■ ■ x Mt as a product of irreducible factors along the decomposition of G as a product of simple groups; then every invariant almost complex structure or riemannian metric decomposes and every invariant riemannian metric is hermitian relative to any invariant almost complex structure; furthermore the decomposition is independent of G in a certain sense. In §3 we choose an invariant riemannian metric and determine the largest connected groups Af0(Mi) of almost hermitian isometries of the Mi, Then A0(M) is determined in §4. There it is shown that A0(M) = A0(M1) x ■ ■ ■ xA0(Mt), that A0(Mi)=H0(Mi) if the almost complex structure on Mi is not integrable, and that AQ(Mi) = H0(Mi)c if the almost complex structure on M{ is induced by a complex structure. A0(M) thus is a centerless semisimple Lie group whose simple normal analytic subgroups are just the A0(Mi). 2. Decomposition. Let M be an effective coset space of a compact connected Lie group G by a connected subgroup A of maximal rank. In other words M=G/K is compact, simply connected and of nonzero Euler characteristic; G is a compact centerless semisimple Lie group, rank AJ=rank G, and K contains no simple factor of G. Then