On the Djrbashian kernel of a Siegel domain

We establish an inversion formula for the M. M. Djrbashian & A. H. Karapetyan integral transform (cf. [6]) on the Siegel domain ${\displaystyle {\Omega }_n=\{\zeta \in {ℂ}^n:\varrho (\zeta )>0\}}$, ${\displaystyle \varrho (\zeta )=Im\left({\zeta }_1\right)-{|\zeta \prime |}^2}$. We build a family of Kähler metrics of constant holomorphic curvature whose potentials are the ${\displaystyle {\varrho }^{\alpha }}$-Bergman kernels, α > -1, (in the sense of Z. Pasternak-Winiarski [20] of ${\displaystyle {\Omega }_n}$. We build an anti-holomorphic embedding of ${\displaystyle {\Omega }_n}$ in the complex projective Hilbert space ${\displaystyle ℂ\mathbb{P}(H^2\_\alpha \left({\Omega }_n\right))}$ and study (in connection with work by A. Odzijewicz [18] the corresponding transition probability amplitudes. The Genchev transform (cf. [9]) is shown to be well defined on ${\displaystyle L^2(\Omega ,{\varrho }^{\alpha })}$, for any strip Ω ⊂ ℂ, and applied in a problem of approximation by holomorphic functions. Building on work by T. Mazur (cf. [15]) we prove the existence of a complete orthonormal system in ${\displaystyle H^2\_\alpha \left({\Omega }_n\right)}$ consisting of eigenfunctions of a certain explicitly defined operator ${\displaystyle V_a}$, ${\displaystyle a\in B_n}$.