Differential Recursion Relations for Laguerre Functions on Symmetric Cones

Abstract

Let $\Omega$ be a symmetric cone and $V$ the corresponding simple Euclidean Jordan algebra. In \cite{ado,do,do04,doz2} we considered the family of generalized Laguerre functions on $\Omega$ that generalize the classical Laguerre functions on $\mathbb{R}^+$. This family forms an orthogonal basis for the subspace of $L$-invariant functions in $L^2(\Omega,d\mu_\nu)$, where $d\mu_\nu$ is a certain measure on the cone and where $L$ is the group of linear transformations on $V$ that leave the cone $\Omega$ invariant and fix the identity in $\Omega$. The space $L^2(\Omega,d\mu_\nu)$ supports a highest weight representation of the group $G$ of holomorphic diffeomorphisms that act on the tube domain $T(\Omega)=\Omega + iV.$ In this article we give an explicit formula for the action of the Lie algebra of $G$ and via this action determine second order differential operators which give differential recursion relations for the generalized Laguerre functions generalizing the classical creation, preservation, and annihilation relations for the Laguerre functions on $\mathbb{R}^+$.