The Weighted Super Bergman Kernels Over the Supermatrix Spaces

Abstract

The purpose of this paper is threefold. Firstly, using Howe duality for \((\mathfrak {gl}(m_{1}|n_{1}),\mathfrak {gl}(m_{2}|n_{2}))\), we obtain integral formulas of the super Schur functions with respect to the super standard Gaussian distributions. Secondly, we give explicit expressions of the super Szego kernels and the weighted super Bergman kernels for the Cartan superdomains of type I. Thirdly, combining these results, we obtain duality relations of integrals over the unitary groups and the Cartan superdomains, and the marginal distributions of the weighted measure.