Matrix Valued Spherical Functions Associated to the Complex Projective Plane

By F. Alberto Grünbaum, Inés Pacharoni, J. Tirao. 2001

163 citations recorded by the source corpus.

Abstract The main purpose of this paper is to compute all irreducible spherical functions on G=SU(3) of arbitrary type δ∈K, where K=S(U(2)×U(1))≃U(2). This is accomplished by associating to a spherical function Φ on G a matrix valued function H on the complex projective plane P2( 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 can be described by 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 explicitly in terms of a special class of generalized hypergeometric functions p+1Fp.

Subfields: Algebra and Number Theory, Geometry and Topology, Mathematical Physics

View Paper: https://www.semanticscholar.org/paper/93a0b6084cc2ffb5fd3597a2ee6ce339ebe1f2d6

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Citations in the corpus

Orthogonal matrix polynomials satisfying first order differential equations: a collection of instructive examples (2005, methodology): “for all matrix valued polynomial functions P and Q. For a number of recent results along this line see [7, 8, 14 , 15].”
Matrix valued Jacobi polynomials (2003, background): “The results in [12–14] yield, apparently for the first time, examples of matrix valued Jacobi polynomials where the parameters α,β take the values α ∈Z 0, β = 1.”
A survey on orthogonal matrix polynomials satisfying second order differential equations (2005, methodology): “The choice of this example allows for fairly explicit computations carried out in [32].”
The Analytic Theory of Matrix Orthogonal Polynomials (2007, methodology): “The corresponding real line theory (MOPRL) is still more recent: Following early work of Krein [133] and Berezan’ski [9] on block Jacobi matrices, mainly as applied to self-adjoint extensions, there was a seminal paper of Aptekarev–Nikishin [4] and a flurry of papers since the 1990s [10, 11, 12, 14,”
The Algebra of Differential Operators Associated to a Weight Matrix (2007, methodology): “The set of those D satisfying (1.1) will be denoted by D(W ). Starting with [ 15 ], [16] and [10, 8] one has a growing collection of weight matrices W (x )f or which the algebra D(W ) is not trivial, i.e.”
Second-Order Differential Operators Having Several Families of Orthogonal Matrix Polynomials as Eigenfunctions (2007, background): “this phenomenon have been recently found by F. A. Gru¨nbaum and M. M. Castro [CG2] and other authors contributed more later. Some of the examples arise from group representation theory. For instance, [GPT1] discusses two second order diﬀerential operators acting on matrix spherical functions which w”