Differential recursion relations for Laguerre functions on Hermitian matrices

In our previous papers [1,2] we studied Laguerre functions and polynomials on symmetric cones Omega=H/L. The Laguerre functions l(n)(nu), n is an element of Lambda, form an orthogonal basis in L-2(Omega, dmu(nu))(L) and are related via the Laplace transform to an orthogonal set in the representation space of a highest weight representations (pi(nu), H-nu) of the autormorphism 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=SU(n,n). We describe the Lie algebraic realization of pi(nu) acting in L-2(Omega, dmu(nu)) and use that to determine explicit differential equations and recurrence relations for the Laguerre functions.