Nearly Holomorphic Functions and Relative Discrete Series of Weighted L-spaces on Bounded Symmetric Domains

Let Ω = G/K be a bounded symmetric domain in a complex vector space V with the Lebesgue measure dm(z) and the Bergman reproducing kernel h(z, w). Let dμα(z) = h(z, z̄) dm(z), α > −1, be the weighted measure on Ω. The group G acts unitarily on the space L(Ω, μα) via change of variables together with a multiplier. We consider the discrete parts, also called the relative discrete series, in the irreducible decomposition of the L-space. Let D̄ = B(z, z̄)∂ be the invariant Cauchy-Riemann operator. We realize the relative discrete series as the kernels of the power D̄ of the invariant Cauchy-Riemann operator D̄ and thus as nearly holomorphic functions in the sense of Shimura. We prove that, roughly speaking, the operators D̄ are intertwining operators from the relative discrete series into the standard modules of holomorphic discrete series (as Bergman spaces of vector-valued holomorphic functions on Ω).