One step spherical functions of the pair ( SU(n+1),U(n) ). ENVIADO A PUBLICAR

I. PACHARONI, J. TIRAO

"Lie Groups: Structure, Actions and Representations". (a volumen in Progress in Mathematics).

Birkhäuser

Año: 2011; p. 1 - 37

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 π are of the form (m+ℓ, . . . ,m+ℓ, m, . . . ,m). Instead of looking at a spherical function Φ of type π we look a function H defined on a section of the K-orbits in an affine subvariety of Pn(C). The function H diagonalize simultaneously, hence it can be identified with a column vector valued function. The irreducible spherical functions of type π turns out to be parameterized by the set S. A key result to characterize the associated function Hw;r is the existence of a matrix valued polynomial function Ψ of degree ℓ such that F_[w;r](t) = Ψ(t)^{-1}H_{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 is positive and define the matrix polynomial Pw as the (ℓ + 1)x(ℓ + 1) matrix whose r-row is the polynomial Fw;r. This yield to interesting families of matrix valued orthogonal Jacobi polynomials P _w for α, β >1.