in 2006, kostant and wallach constructed an integrable system on the n x n complex matrices $m_{n}( c)$ using gelfand-zeitlin theory. this system can be viewed as a complexified version of the one studied by guillemin and sternberg on the n x n hermitian matrices, which is related to the classical gelfand-zeitlin basis for irreducible representations of the unitary group via geometric quantization. in this talk, we discuss joint work with sam evens in which we develop a geometric description of the fibres of the moment map for the complexified gelfand-zeitlin system. our approach uses the theory of orbits of a symmetric subgroup k of the group g of all invertible n x n complex matrices on the flag variety of $m_{n}( c)$ . these orbits play a central role in the geometric construction of harish-chandra modules for the pair $(m_{n} (c ), k)$ using the beilinson-bernstein correspondence. we indicate how our work provides the foundation for the geometric construction of a category of generalized harish-chandra modules studied by drozd, futorny, and ovsienko.