Sequences of Orthogonal Polynomials related to Isotropy Orbits of Symmetric Spaces

TL;DR

This paper identifies a family of polynomials related to symmetric space isotropy orbits as special Jacobi polynomials, confirming a conjecture about their roots and geometric properties.

Contribution

It proves that Reiswich's polynomials are specific Jacobi polynomials, confirming the conjecture on their roots and minimal isotropy orbits in symmetric spaces.

Findings

Reiswich's polynomials are special cases of Jacobi polynomials.

Confirmed the conjecture that these polynomials have distinct roots in [0,1].

Established a link between polynomial roots and geometric isotropy orbits.

Abstract

Studying the isotropy orbits of compact symmetric spaces Reiswich introduced a family of explicit polynomials in one variable in order to describe the unique minimal isotropy orbit of compact symmetric spaces with Dynkin diagram of type $D_m$. Based on this geometric interpretation he conjectured that these polynomials all have pairwise different real roots in the interval $[\,0,1\,]$. In this article the polynomials constructed by Reiswich will be identified as special cases of Jacobi polynomials thus proving the conjecture about minimal isotropy orbits of compact symmetric spaces with Dynkin diagram of type $D_m$.