Representations of noncompact symplectic groups

Representations of noncompact symplectic groups are constructed in the space of functions on their maximal compact subgroups. In the cases when an invariant bilinear form exists in the representation space a system of differential equations is obtained for the kernel of this form and some solutions of this system are found. The characters of the representations constructed are calculated. Orthogonality relations are also obtained for the matrix elements of the unitary representations of the continuous series.