Multiplicity free induced representations and orthogonal polynomials

TL;DR

This paper classifies multiplicity free systems arising from reductive spherical pairs and parabolic subgroups, explores their spectra, and introduces new families of multivariable orthogonal polynomials generalizing Jacobi polynomials.

Contribution

It provides a classification of multiplicity free systems and constructs new multivariable orthogonal polynomials from their spectra, extending classical polynomial families.

Findings

Explicit spectra for three multiplicity free systems

Families of multivariable orthogonal polynomials constructed

Polynomials are eigenfunctions of differential operators, satisfy recurrence, and orthogonality

Abstract

Let $(G,H)$ be a reductive spherical pair and $P\subset H$ a parabolic subgroup such that $(G,P)$ is spherical. The triples $(G,H,P)$ with this property are called multiplicity free systems and they are classified in this paper. Denote by $\pi^{H}_{\mu}=\mathrm{ind}_{P}^{H}\mu$ the Borel-Weil realization of the irreducible $H$-representation of highest weight $\mu\in P^{+}_{H}$ and consider the induced representation $\mathrm{ind}_{P}^{G}\chi_{\mu}=\mathrm{ind}_{H}^{G}\pi^{H}_{\mu}$, a multiplicity free induced representation. Some properties of the spectrum of the multiplicity free induced representations are discussed. For three multiplicity free systems the spectra are calculated explicitly. The spectra give rise to families of multi-variable orthogonal polynomials which generalize families of Jacobi polynomials: they are simultaneous eigenfunctions of a commutative algebra of differential operators, they satisfy recurrence relations and they are orthogonal with respect to integrating against a matrix weight on a compact subset. We discuss some difficulties in describing the theory for these families of polynomials in the generality of the classification.