A Szegö Kernel for Discrete Series

where / i s the boundary function for jpand da is Lebesgue measure on the sphere. When m = 1, (1) easily transforms into the Cauchy integral formula. In dimension m9 the formula extends to be defined on all / in L , always yielding holomorphic functions. If we identify holomorphic functions with their boundary values, the extended operator can be regarded as the orthogonal projection from L to the holomorphic functions in L. This projection property characterizes the kernel. In terms of semisimple Lie groups, functions on the sphere suggest nonunitary principal series representations and holomorphic functions on the ball suggest discrete series representations, and the Szegö kernel should suggest a map from the one to the other. Actually formula (1) will not arise with discrete series but with socalled limits of discrete series. For ordinary discrete series representations, we shall use operators that are more analogous to the formula for the {n — l)st complex derivative in the disc,