Non-symmetric Jacobi polynomials of type BC₁ as vector-valued polynomials Part 1: spherical functions

2023/07/07 by van Horssen, Max, van Pruijssen, Maarten
#33C45 #33C52 #33E30 #Classical Analysis and ODEs (math.CA) #FOS: Mathematics #Representation Theory (math.RT)

paper · doi:10.48550/arxiv.2307.03857

Abstract

We study non-symmetric Jacobi polynomials of type BC₁ by means of vector-valued and matrix-valued orthogonal polynomials. The interpretation as matrix-valued orthogonal polynomials yields a new expression of the non-symmetric Jacobi polynomials of type BC₁ in terms of the symmetric Jacobi polynomials of type BC₁. In this interpretation, the Cherednik operator, that has the non-symmetric Jacobi polynomials as eigenfunctions, corresponds to two shift operators for the symmetric Jacobi polynomials of type BC₁. We show that the non-symmetric Jacobi polynomials of type BC₁ with so-called geometric root multiplicities, interpreted as vector-valued polynomials, can be identified with spherical functions on the sphere S^2m+1=Spin(2m+2)/Spin(2m+1) associated with the fundamental spin-representation of Spin(2m+1). The Cherednik operator corresponds to the Dirac operator for the spinors on S^2m+1 in this interpretation.

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