Representations of certain solvable Lie groups on Hilbert spaces of holomorphic functions and the application to the holomorphic discrete series of a Semisimple Lie group

If ($\text{b}$,$\text{j,?}{}_0$) is a normal j-algebra and b^? = {X + ijX;X∈b} then $\text{b}$^? is a positive polarization at /tf₀ in the sense of Auslander and Kostant. The representation $\text{ρ(?}{}_0\text{,b}{}^{\mathrm{?}}\text{)}$ defined as in the orbit theory is shown to be given on a space of holomorphic functions on a Siegel II domain. The Fourier transformation is shown to be an intertwining operator between $\text{ρ(?}{}_0\text{,b}{}^{\mathrm{?}}\text{)}$ and $\text{ρ(?}{}_0\text{,)}$ for a suitable real polarization $\text{h}$. These explicit results are used to deduce the Harish-Chandra conditions defining the holomorphic discrete series of a Semisimple Lie Group.