A Formula of Gauss-Kummer and the Trace of Certain Intertwining Operators

In our opinion one of the most interesting aspects of the orbit method is the occurrence of “independence of polarization”. From the perspective of geometric quantization, given two transverse polarizations, one may set up a formal kernel operator which then intertwines the two corresponding quantizations. This has been referred to as the BKS kernel. See e.g. [Sn], § 5. For the case of real polarizations of a coadjoint orbit of a nilpotent Lie group this more or less comes down to the Fourier transform. However for hyperbolic coadjoint orbits of a semisimple Lie group one may easily relax the condition of transversality and then the transforms become the well-known and well-studied intertwining operators associated to the spherical principal series. As one knows these operators are parameterized by elements of the Weyl group. The particular case when the polarizations are transverse corresponds to the long element of the Weyl group. In this paper we find that an analytic continuation of this operator is traceable and an expression for the trace relates to a known formula of Gauss-Kummer for hyper geometric series.