V. F. Molchanov: Discrete series and analyticity

Abstract: Let X be a hyperboloid in Rⁿ, it is the homogeneous space G/H, where G = SO₀(p,q), H = SO₀(p, q-1), n=p+q. The decomposition of the quasiregular representation of G on L²(X) contains irreducible unitary representations of two series: continuous and discrete. In the paper we give explicit expressions for the projection on the subspace of L²(X) where the discrete series acts.

If q=2, then X is a symmetric space of Hermitian type, it is a Shilov boundary of some complex manifolds, and the discrete series splits into two series: analytic and antianalytic. We give explicit expressions for the Cauchy-Szegö kernels and for the projections on the subspaces where the analytic and antianalytic series act.

This separation of series is the first step of the Gelfand-Gindikin program for homogeneous spaces G/H.

The formulae are obtained by a direct summation of a series of spherical functions of the discrete (and analytic and antianalytic for q=2) series.

The formulae for the projections as well as the explicit formulae for the spherical functions of the discrete series suggest that some analyticity is displayed here. At the end of the paper some problems are formulated which are connected with attempts to understand this analyticity.

{ last modified : Jan 20 1998 }