Holomorphic Discrete Series for Hyperboloids of Hermitian Type

Abstract

In the present paper we study the holomorphic discrete series (h.d.s.) for hyperboloids of Hermitian type. They are the spaces GÂH where G=SO 0 ( p, 2), H=SO 0 ( p, 1). We find some complex hulls Y \ of GÂH (they correspond to minimal G-invariant cones in the Lie algebra of G ), consider the Hardy spaces H 2 (Y \ ), and give explicit expressions for the corresponding Cauchy Szego kernels. Earlier these expressions were known for p=2 (Gelfand Gindikin) and p=1 (Molchanov). We compute the projections E \ of the L 2 (GÂH ) onto the images of the Hardy spaces. The difference Q=E + &E & is most interesting, it is an analogue of the Hilbert transform. We study the relations between the h.d.s. for G and the h.d.s. for GÂH. It turns out that for p>2 there is a finite number of representations of the h.d.s. for GÂH which are not representations of the h.d.s. of G. We give explicit expressions for the kernels of the projections on the subspaces of the Hardy spaces where the h.d.s. for G acts.

Hardy spaces and analytic continuation of Bergman spaces

“…Other data lead to the cases considered by Koufany-0rsted [K097], Molchanov [Mo97], and Betten-Olafsson [B098].…”

Separation of series for hyperboloids

“…Let Q be the corresponding distribution. In this case [12], if n is odd, then For odd n, the support of Q is the set x,, = 1, and, for even n, it is [x,[ < 1. The operator Q~ looks especially nice for n = 3 (p = 1, q = 2), see [8]:…”

“…As can be seen from the present paper (see also [12]), in this circle of problems, the case in which n is odd is simpler than that in which n is even; the method of [10] fits for odd n only.…”

“…For the case q = 2, the hyperboloid X is the Shilov boundary of some complex manifolds, and the discrete series splits into two series, namely, the analytic and antianalytic series. In [12] we obtained the projections related to each of these series, together with the corresponding Cauchy-Szeg6 kernels. For completeness we write out the formulas for these projections in the present paper.…”

Causal compactification and Hardy spaces

“…We shall call these causal compactifications. The idea of using such compactifications has been studied before in special cases, see for example [1], [9] and [10], where Cayley type spaces are treated as well as the case of the metaplectic group, and also recently by V. Molchanov for the case of the orthogonal group of Hermitian type and the corresponding hyperboloid [12]. The causal compactification is especially convenient for studying the noncommutative analogue of the Hardy space, since it is a Shilov boundary of a classical bounded domain.…”