Geometry of Quotient Hilbert Modules
by
Ronald G. Douglas
Texas A&M University

Let M be a Hilbert module of holomorphic functions defined on a bounded domain in Cⁿ and M_o be the submodule of functions vanishing to order k on a hypersurface Z in the domain. Properties of the quotient module M_q can be described in terms of M, k and Z. The modules M and M_o together with the inclusion map yield a module resolution of M_q which is related to the model theory of Sz-Nagy and Foias when M is the Hardy module over the disk algebra.

In the talk, we will describe a generalized functional Hilbert space realization of M_q, involving matrix-valued functions. Alternatively, one can describe M_q using an extension of the Cowen-Douglas class in which the bundles involved are pullbacks from a flag-manifold and not just the Grassmanian. The latter structure arises by considering a partial jet bundle of order k determined by the normal direction to Z. Characterizing such classes is considered in work of Martin and Salinas. Finally, relations between geometric invariants for the three modules will be given.

Date received: April 22, 1999