Laguerre functions associated to Euclidean Jordan algebras

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,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 L²(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 L²(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 L²(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 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).

Name thesis:degree_name

Doctor of Philosophy (PhD)

Level thesis:degree_level

Dissertation

Discipline thesis:degree_discipline

Applied Mathematics

Grantor

Mathematics

Year dc:date.available

2005

Author dc:creator

• Aristidou, Michael

• laguerre functions
• laguerre polynomials
• lie groups
• recursion relations
• lie algebras
• jordan algebras
• symmetric cones
• highest weight representations

Statement dc:rights

• unrestricted
• Release the entire work immediately for access worldwide.

Identifier

etd-07132005-160351
https://repository.lsu.edu/gradschool_dissertations/1403

OAI identifier oai:identifier

oai:repository.lsu.edu:gradschool_dissertations-2402