Differential Recursion Relations For Laguerre Functions On Hermitian Matrices

In our previous papers [1, 2] we studied Laguerre functions and polynomials on symmetric cones = H=L. The Laguerre functions ` n , n 2 , form an orthogonal basis in L ; d ) and are related via the Laplace transform to an orthogonal set in the representation space of a highest weight representations ( ; H ) of the automorphism group G corresponding to a tube domain T( 9 In this article we consider the case where is the space of positive de nite Hermitian matrices and G = SU(n; n). We describe the Lie algebraic realization of acting in L ; d ) and use that to determine explicit dierential equations and recurrence relations for the Laguerre functions