Irreducibility of unitary group representations and reproducing kernels Hilbert spaces

We discuss unitary representations of groups in Hilbert spaces of functions given together with reproducing kernels, and in particular irreducibility. Our main focus is on examples, including spherical representations of orthogonal groups, distance-~ransitive finite graphs, irreducibility of induced representations, the discrete series of SL(2, R), and some representations of PGL(2, Qp). The appendix contains examples involving groups acting on rooted trees