One-step spherical functions of the pair (SU(n+1),U(n))

2012/09/20 by Pacharoni, Ines, Tirao, Juan A.
#22E45 - 33C45 - 33C47 #Classical Analysis and ODEs (math.CA) #FOS: Mathematics #Representation Theory (math.RT)

paper · doi:10.48550/arxiv.1209.4500

Abstract

The aim of this paper is to determine all irreducible spherical functions of the pair (G,K)=(SU(n+1),U(n)), where the highest weight of their K-types are of the form (m+l,...,m+l,m,...,m). Instead of looking at a spherical function Φof type πwe look at a matrix-valued function H defined on a section of the K-orbits in an affine subvariety of P_n(C). The function H diagonalizes, hence it can be identified with a column vector-valued function. The irreducible spherical functions of type πturn out to be parameterized by S=(w,r)∈ Z x Z : 0≤ w, 0 ≤ r ≤ l, 0≤ m+w+r. A key result to characterize the associated function H_w,r is the existence of a matrix-valued polynomial function Ψof degree l such that F_w,r(t)=Ψ(t)^-1H_w,r(t) becomes an eigenfunction of a matrix hypergeometric operator with eigenvalue λ(w,r), explicitly given. In the last section we assume that m≥ 0 and define the matrix polynomial P_w as the (l+1) x (l+1) matrix whose r-row is the polynomial F_w,r. This leads to interesting families of matrix-valued orthogonal Jacobi polynomials P_w^α,β for α,β>-1.

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