Square integrable harmonic forms and representation theory (Q1974807)

The authors are interested in obtaining a construction of the irreducible unitary representations of a semisimple Lie group \(G\) in terms of the geometry of the orbits. The indefinite invariant Hermitian metric on a semisimple Lie group is used to define a global invariant form on \({\mathfrak L}_X\)-valued type-\((0,s)\) differential forms. The authors indicate how to choose representatives for each \(K\)-finite cohomology class for which the integral defining the global form converges. The main tool for picking out cohomology classes is an intertwining operator \({\mathcal S}\) from a principal series representation into the space of closed forms of type \((0,s)\). In the case where \(G/L\) is an indefinite Kähler symmetric space, the authors prove square integrability. Finally, they obtain a continuous Hilbert space representation.