The Bott-Borel-Weil Theorem for direct limit groups

We show how highest weight representations of certain infinite dimensional Lie groups can be realized on cohomology spaces of holomorphic vector bundles. This extends the classical Bott-Borel-Weil Theorem for finite-dimensional compact and complex Lie groups. Our approach is geometric in nature, in the spirit of Bott's original generalization of the Borel-Weil Theorem. The groups for which we prove this theorem are strict direct limits of compact Lie groups, or their complexifications. We previously proved that such groups have an analytic structure. Our result applies to most of the familiar examples of direct limits of classical groups. We also introduce new examples involving iterated embeddings of the classical groups and see exactly how our results hold in those cases. One of the technical problems here is that, in general, the limit Lie algebras will have root systems but need not have root spaces, so we need to develop machinery to handle this somewhat delicate situation.

     

Differentiable structure for direct limit groups

A direct limit of (finite-dimensional) Lie groups has Lie algebra and exponential map exp _G : gG. BothG and g carry natural topologies.G is a topological group, and g is a topological Lie algebra with a natural structure of real analytic manifold. In this Letter, we show how a special growth condition, natural in certain physical settings and satisfied by the usual direct limits of classical groups, ensures thatG carries an analytic group structure such that exp _G is a diffeomorphism from a certain open neighborhood of 0g onto an open neighborhood of 1 _G G. In the course of the argument, one sees that the structure sheaf for this analytic group structure coincides with the direct limit C (G ) of the sheaves of germs of analytic functions on theG .L.N. partially supported by a University of California Dissertation Year Fellowship.E.R.C. partially supported by N.S.F. Grant DMS 89 09432.J.A.W. partially supported by N.S.F. Grant DMS 88 05816.