Restrictions and Expansions of Holomorphic Representations

Let D, be a homogeneous bounded domain in Cl’ containing the origin 0. Let G, be the group of holomorphic transformations of D, and let us consider a representation T, of G inside a Hilbert space H, of holomorphic functions on D, . As a simple example, suppose that D,-, = {D, n z, = 0} is a homogeneous bounded domain in P-l for a subgroup G,_, of G, . We consider the restriction of the representation T, to G,_, . Clearly, as the Hilbert space H, consists of holomorphic functions, the restriction map R,:f(z, , z2 ,..., .z,~-~ , z,) -+ f(q , 22 ,..., X,-l 9 0) intertwines the representation T, with a representation T n-1 of G,,-l inside a space of holomorphic functions on D,-, . The kernel of R, is the subspace of holomorphic functions in H,, which vanish where z, = 0. Similarly, the maps lip = ((a/ik,)p .f) (2,zo are well defined, therefore it is natural, in order to calculate T,, jCnel , to expand the functions f in H, in Taylor series with respect to the variable z, . Let D = G/K be a Hermitian symmetric space. We will consider a representation T of G of the holomorphic discrete series. The preceding simple idea of taking normal derivatives gives us the decomposition of the restriction of T to any