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 $\Omega =H/L$. The Laguerre functions ${\ell }_{\mathbf{n}}^{\nu }$, $\mathbf{n}\in \mathbf{\Lambda }$, form an orthogonal basis in $L^2(\Omega ,d{\mu }_{\nu }{)}^L$ and are related via the Laplace transform to an orthogonal set in the representation space of a highest weight representations $({\pi }_{\nu },{\mathcal{H}}_{\nu })$ of the automorphism group $G$ corresponding to a tube domain $T(\Omega )$. In this article we consider the case where $\Omega$ is the space of positive definite Hermitian matrices and $G=\mathrm{SU}(n,n)$. We describe the Lie algebraic realization of ${\pi }_{\nu }$ acting in $L^2(\Omega ,d{\mu }_{\nu })$ and use that to determine explicit differential equations and recurrence relations for the Laguerre functions.