Quantization and an invariant for unitary representations of nilpotent Lie groups (Q1089443)

In a previous paper the authors introduced an invariant \(\iota\) (\(\rho)\) for the irreducible unitary representations \(\rho\) of a simply connected nilpotent Lie group G. The invariant lives in the Lie algebra cohomology group \(H^{2q+1}({\mathfrak G})\) where 2q is the dimension of the coadjoint orbit corresponding to \(\rho\). The present paper provides an interpretation of the invariant using the machinery of geometric quantization. The invariant arises from a degree \(2q+1\) class in the G- invariant cohomology of the prequantization circle bundle over the orbit. The relevant cohomology group is computed using a model for this bundle developed in the paper. This approach to \(\iota\) (\(\rho)\) yields vanishing conditions for the invariant and applications to representations which are square integrable modulo their centers.