#### Identifier

etd-07132005-160351

#### Degree

Doctor of Philosophy (PhD)

#### Department

Mathematics

#### Document Type

Dissertation

#### 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 L^{2}(Ω,dμ_{v}). 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 L^{2}(Ω,dμ_{v}). To obtain these representations, we start from the highest weight representations of G on H_{v}(T(Ω)), the Hilbert space of holomorphic functions on T(Ω), and we transfer the representations to L^{2}(Ω,dμ_{v}) via the Laplace transform. The Laguerre functions correspond to an orthogonal set of functions in H_{v}(T(Ω)) and they form an orthogonal basis in L^{2}(Ω,dμ_{v})^{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 sl_{2}(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).

#### Date

2005

#### Document Availability at the Time of Submission

Release the entire work immediately for access worldwide.

#### Recommended Citation

Aristidou, Michael, "Laguerre functions associated to Euclidean Jordan algebras" (2005). *LSU Doctoral Dissertations*. 1403.

http://digitalcommons.lsu.edu/gradschool_dissertations/1403

#### Committee Chair

Gestur Olafsson