Orthogonal polynomials of compact simple Lie groups

TL;DR

This paper introduces a recursive algebraic method to construct orthogonal polynomials associated with all semisimple Lie groups, unifying several classical polynomial families and revealing their connections to orbit functions and Macdonald polynomials.

Contribution

It provides a uniform recursive construction of orthogonal polynomials for any semisimple Lie group, linking Chebyshev, Macdonald, and orbit function properties.

Findings

Constructed two families of polynomials for all semisimple Lie groups

Established orthogonality and discretization properties

Derived recurrence relations for specific Lie group types

Abstract

Recursive algebraic construction of two infinite families of polynomials in $n$ variables is proposed as a uniform method applicable to every semisimple Lie group of rank $n$. Its result recognizes Chebyshev polynomials of the first and second kind as the special case of the simple group of type $A_1$. The obtained not Laurent-type polynomials are proved to be equivalent to the partial cases of the Macdonald symmetric polynomials. Basic relation between the polynomials and their properties follow from the corresponding properties of the orbit functions, namely the orthogonality and discretization. Recurrence relations are shown for the Lie groups of types $A_1$, $A_2$, $A_3$, $C_2$, $C_3$, $G_2$, and $B_3$ together with lowest polynomials.