Toeplitz Operators, K\"ahler Manifolds, and Line Bundles

This is a survey paper. We discuss Toeplitz operators in K\"ahler geometry, with applications to geometric quantization, and review some recent developments.Comment: This is a contribution to the Proceedings of the 2007 Midwest Geometry Conference in honor of Thomas P. Branson, published in SIGMA (Symmetry, Integrability and Geometry: Methods and Applications) at http://www.emis.de/journals/SIGMA

A note on Iwasawa-type decomposition

We study the Iwasawa-type decomposition of an open subset of SL(n,C) as SU(p,q)AN. We show that the dressing action of SU(p,q) is globally defined on the space of admissible elements in AN. We also show that the space of admissible elements is a multiplicative subset of AN. We establish a geometric criterion: the symmetrization of an admissible element maps the positive cone in C^n into itself.Comment: 10 page

A Poisson structure on compact symmetric spaces

We present some basic results on a natural Poisson structure on any compact symmetric space. The symplectic leaves of this structure are related to the orbits of the corresponding real semisimple group on the complex flag manifold.Comment: 11 pages, exposition streamline