Matrix spherical functions and orthogonal polynomials: An instructive example.

PACHARONI, I.

REVISTA DE LA UNIóN MATEMáTICA ARGENTINA

Unión Matemática Argentina

Lugar: Bahia Blanca; Año: 2008 vol. 49 p. 1 - 15

0041-6932

In the scalar case, it is well known that the zonal spherical functionsof any compact Riemannian symmetric space of rank one can be expressedin terms of the Jacobi polynomials. The main purpose of this paperis to revisit the matrix valued spherical functions associated to the complexprojective plane to exhibit the interplay among these functions, the matrixhypergeometric functions and the matrix orthogonal polynomials. We alsoobtain very explicit expressions for the entries of the spherical functions inthe case of 2×2 matrices and exhibit a natural sequence of matrix orthogonalpolynomials, beyond the group parameters.