Holomorphic vector bundles on homogeneous spaces

Abstract

LET G be a connected, compact, semisimple, Lie group, and V a Lie subgroup of maximal rank. We shall assume that the coset space G/V is provided with a holomorphic structure which is invariant under the action of G. Any representation p of I’ gives rise to a vector bundle on G/V, viz. the fibre bundle associated to the fibration G + G/V. Moreover, this bundle can be provided with a natural holomorphic structure. (There are, in fact, several ways of doing this, but if p is irreducible, this procedure is unique.) We shall denote this holomorphic vector bundle by E,, and refer to this as a homogeneous vector bundle. Using the results of Bott on homogeneous vector bundles, we prove: Assume that V does not contain any simple component of G. If p is irreducible, then any holomorphic endomorphism of Ee is a scalar multiple of the identity. Moreover, tfp, and pz are two irreducible representations of V, then E,,, and EPI are holomorphically equivalent tfand only zfp, and pz are equivalent. This in particular implies that EP is indecomposable. The latter was proved by Griffiths [7, theorem 81. However, the fact that the only endomorphism of EP are scalars, seems likely to be useful in the study of deformation of such bundles. The equivalence problem was first raised by Ise [9, $141. We then study the homogeneous vector bundles over an irreducible Hermitian symmetric space. In this case, the second Betti number of G/V is 1 and since G/V is an algebraic manifold, there is a canonical generator for H’(G/V, Z). This enables one to talk of positive and negative elements of H2(G/V, Z). Following the definition of Mumford [13] of stable vector bundles over projective algebraic curves, we define a stable vector bundle E over G/V to be a vector bundle such that any proper subbundle F has the property