Matrix-valued orthogonal polynomials related to \(\mathrm{SU}(N+1)\), their algebras of differential operators, and the corresponding curves. (English) Zbl 1149.33002
The authors start with the Casimir operator, i.e., more precisely, a differential operator acting on matrix-valued functions on the group \(\text{SU}(N+1)\), and derive certain ordinary differential operators on the interval \([0,1]\) acting on matrix-valued functions. In this way, the authors present two examples of algebras of differential operators associated with orthogonal polynomials arising from representations of \(\text{SU}(N+1)\). The one-step example here is commutative, while the second example with two steps is noncommutative.
Reviewer: Michael Voit (Dortmund)
MSC:
33C80 | Connections of hypergeometric functions with groups and algebras, and related topics |
33C45 | Orthogonal polynomials and functions of hypergeometric type (Jacobi, Laguerre, Hermite, Askey scheme, etc.) |
42C05 | Orthogonal functions and polynomials, general theory of nontrigonometric harmonic analysis |
22E45 | Representations of Lie and linear algebraic groups over real fields: analytic methods |