Holomorphic Discrete Series for Hyperboloids of Hermitian Type

In the present paper we study the holomorphic discrete series (h.d.s.) for hyperboloids of Hermitian type. They are the spacesG/HwhereG=SO0(p, 2),H=SO0(p, 1). We find some complex hullsY±ofG/H(they correspond to minimalG-invariant cones in the Lie algebra ofG), consider the Hardy spacesH2(Y±), and give explicit expressions for the corresponding Cauchy–Szegö kernels. Earlier these expressions were known forp=2 (Gelfand–Gindikin) andp=1 (Molchanov). We compute the projectionsE±of theL2(G/H) onto the images of the Hardy spaces. The differenceQ=E+−E−is most interesting, it is an analogue of the Hilbert transform. We study the relations between the h.d.s. forGand the h.d.s. forG/H. It turns out that forp>2 there is a finite number of representations of the h.d.s. forG/Hwhich are not representations of the h.d.s. ofG. We give explicit expressions for the kernels of the projections on the subspaces of the Hardy spaces where the h.d.s. forGacts.