Sequences of Orthogonal Polynomials related to Isotropy Orbits of Symmetric Spaces

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 $D_m$.