After introducing L^2-cohomology in my first talk, the second talk will give a rigorous definition of the Hilbert modules of complex L^2-cohomology, for domains in C^n, and include the proof that one actually obtains genuine Hilbert modules with a bounded polynomial action. This requires some checking in the non-holomorphic case. In addition, we discuss the problem of constructing a reproducing kernel for this new type of Hilbert modules, which cannot be the standard sesqui-holomorphic series expansion, and solve this problem from a group theoretical perspective for any discrete series representation of a semi-simple Lie group.
