Cubature formulae for orthogonal polynomials in terms of elements of finite order of compact simple Lie groups

TL;DR

This paper generalizes Chebyshev polynomial properties to multivariable polynomials related to compact simple Lie groups, deriving Gaussian cubature formulas based on finite order elements and Lie theory methods.

Contribution

It introduces a unified Lie theoretical approach to construct cubature formulas for multivariable polynomials associated with all simple Lie groups, extending previous A-type lattice results.

Findings

Derived Gaussian cubature formulas for all simple Lie groups

Connected cubature points to regular elements of finite order in Lie groups

Unified Lie theoretical framework simplifies multivariable polynomial analysis

Abstract

The paper contains a generalization of known properties of Chebyshev polynomials of the second kind in one variable to polynomials of $n$ variables based on the root lattices of compact simple Lie groups $G$ of any type and of rank $n$. The results, inspired by work of H. Li and Y. Xu where they derived cubature formulae from $A$-type lattices, yield Gaussian cubature formulae for each simple Lie group $G$ based on interpolation points that arise from regular elements of finite order in $G$. The polynomials arise from the irreducible characters of $G$ and the interpolation points as common zeros of certain finite subsets of these characters. The consistent use of Lie theoretical methods reveals the central ideas clearly and allows for a simple uniform development of the subject. Furthermore it points to genuine and perhaps far reaching Lie theoretical connections.