Differential Recursion Relations for Laguerre Functions on Hermitian Matrices

Abstract

In our previous papers \cite{doz1,doz2} we studied Laguerre functions and polynomials on symmetric cones $Ω=H/L$. The Laguerre functions $\ell^ν_{\mathbf{n}}$, $\mathbf{n}\in\mathbfΛ$, form an orthogonal basis in $L^{2}(Ω,dμ_ν)^{L}$ and are related via the Laplace transform to an orthogonal set in the representation space of a highest weight representations $(π_ν, \mathcal{H}_ν)$ of the automorphism group $G$ corresponding to a tube domain $T(Ω)$. In this article we consider the case where $Ω$ is the space of positive definite Hermitian matrices and $G=\mathrm{SU}(n,n)$. We describe the Lie algebraic realization of $π_ν$ acting in $L^{2}(Ω,dμ_ν)$ and use that to determine explicit differential equations and recurrence relations for the Laguerre functions.