Matrix Valued Spherical Functions Associated to the Complex Projective Plane

The main purpose of this paper is to compute all irreducible spherical functions on G=\SU(3) of arbitrary type $\delta\in \hat K$, where $K={\mathrm{S}}(\mathrm{U}(2)\times\mathrm{U}(1))\simeq\mathrm{U}(2)$. This is accomplished by associating to a spherical function $\Phi$ on $G$ a matrix valued function $H$ on the complex projective plane $P_2(\mathbb{C})=G/K$. It is well known that there is a fruitful connection between the hypergeometric function of Euler and Gauss and the spherical functions of trivial type associated to a rank one symmetric pair $(G,K)$. But the relation of spherical functions of types of dimension bigger than one with classical analysis, has not been worked out even in the case of an example of a rank one pair. The entries of $H$ are solutions of two systems of ordinary differential equations. There is no ready made approach to such a pair of systems, or even to a single system of this kind. In our case the situation is very favorable and the solution to this pair of systems can be exhibited explicitely in terms of a special class of generalized hypergeometric functions ${}_{p+1}F_p$.Comment: 70 pages, 1 figur

Spherical Functions Associated With the Three Dimensional Sphere

In this paper, we determine all irreducible spherical functions \Phi of any K -type associated to the pair (G,K)=(\SO(4),\SO(3)). This is accomplished by associating to \Phi a vector valued function H=H(u) of a real variable u, which is analytic at u=0 and whose components are solutions of two coupled systems of ordinary differential equations. By an appropriate conjugation involving Hahn polynomials we uncouple one of the systems. Then this is taken to an uncoupled system of hypergeometric equations, leading to a vector valued solution P=P(u) whose entries are Gegenbauer's polynomials. Afterward, we identify those simultaneous solutions and use the representation theory of \SO(4) to characterize all irreducible spherical functions. The functions P=P(u) corresponding to the irreducible spherical functions of a fixed K-type \pi_\ell are appropriately packaged into a sequence of matrix valued polynomials (P_w)_{w\ge0} of size (\ell+1)\times(\ell+1). Finally we proved that \widetilde P_w={P_0}^{-1}P_w is a sequence of matrix orthogonal polynomials with respect to a weight matrix W. Moreover we showed that W admits a second order symmetric hypergeometric operator \widetilde D and a first order symmetric differential operator \widetilde E.Comment: 49 pages, 2 figure