Institute for Mathematical Physics Nearly Holomorphic Functions and Relative Discrete Series of Weighted L 2 -spaces on Bounded Symmetric Domains Nearly Holomorphic Functions and Relative Discrete Series of Weighted L 2 -spaces on Bounded Symmetric Domains

Let = 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) ?p. Let dd (z) = h(z; z) dm(z), > ?1, be the weighted measure on. The group G acts unitarily on the space L 2 ((;) 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 2-space. Let D = B(z; z)@ be the invariant Cauchy-Riemann operator. We realize the relative discrete series as the kernels of the power D m+1 of the invariant Cauchy-Riemann operator D and thus as nearly holomorphic functions in the sense of Shimura. We prove that, roughly speaking, the operators D m 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).