Invariant measures on polarized submanifolds in group quantization

We provide an explicit construction of quasi-invariant measures on polarized coadjoint orbits of a Lie group G. The use of specific (trivial) central extensions of G by the multiplicative group R+ allows us to restore the strict invariance of the measures and, accordingly, the unitarity of the quantization of coadjoint orbits. As an example, the representations of SL(2,R) are recovered. (C) 2000 American Institute of Physics. [S0022-2488(00)00410-2].