Laguerre functions associated to Euclidean Jordan algebras

Abstract

Certain differential recursion relations for the Laguerre functions, defined on a symmetric cone Ω, can be derived from the representations of a specific Lie algebra on L2(Ω,dμv). This Lie algebra is the corresponding Lie algebra of the Lie group G that acts on the tube domain T(Ω)=Ω+iV, where V is the associated Euclidean Jordan algebra of Ω. The representations involved are the highest weight representations of G on L2(Ω,dμv). To obtain these representations, we start from the highest weight representations of G on Hv(T(Ω)), the Hilbert space of holomorphic functions on T(Ω), and we transfer the representations to L2(Ω,dμv) via the Laplace transform. The Laguerre functions correspond to an orthogonal set of functions in Hv(T(Ω)) and they form an orthogonal basis in L2(Ω,dμv)L, where L is a specific subgroup of G. The recursion relations result by restricting the representation to a distinguished 3-dimensional subalgebra which is isomorphic to sl2(C). First, we construct the differential recursion relations for Laguerre functions defined on Ω = Sym+(n,R), the cone of positive definite real symmetric matrices, from the highest weight representations of Sp(2n,R). These relations generalize the 'classical' relations for Laguerre functions on R+. Then, we consider highest weight representations of any simple Lie group G to construct general differential recursion relations, for Laguerre functions defined on any symmetric cone, that generalize both the 'classical' recursion relations for Laguerre functions on Ω = R+ and the ones for Laguerre functions on Ω = Sym+(n,R).