Nearly holomorphic functions and relative discrete series of weighted $L^2$-spaces on bounded symmetric domains

Abstract

Let $\Omega = G/K$ 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 $d\mu _{\alpha}(z) = h(z, \bar{z})^{\alpha}dm(z)$, $\alpha > -1$, be the weighted measure on $\Omega$. The group $G$ acts unitarily on the space $L^{2}(\Omega , \mu_\alpha )$ 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 $\bar{D} = B(z, \bar{z})\partial$ be the invariant Cauchy-Riemann operator. We realize the relative discrete series as the kernels of the power $\bar{D}^{m+1}$ of the invariant Cauchy-Riemann operator $\bar{D}$ and thus as nearly holomorphic functions in the sense of Shimura. We prove that, roughly speaking, the operators $\bar{D}^{m}$ 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 $\Omega$).