2013/06/27 by Ignacio Zurrián, Zurrián, Ignacio N.
Mathematics · Physics and Astronomy · #22E45 - 33C45 - 33C47 #Advanced Algebra and Geometry #Advanced Mathematical Theories #Advanced Mathematical Theories and Applications #Advanced Topics in Algebra #FOS: Mathematics #Mathematical functions and polynomials #Representation Theory (math.RT)
paper · pdf · doi:10.48550/arxiv.1306.6581
openalex publication_date 2013/06/27 · openalex created_date 2022/09/01 · openalex updated_date 2026/07/28
In this work we start by determining all irreducible spherical functions\n\Φ of any K -type associated to the pair (G,K)=( SO(4), SO(3)). The\nfunctions P=P(u) corresponding to the irreducible spherical functions of a\nfixed K-type \π_\ℓ are appropriately packaged into a sequence of matrix\nvalued polynomials (Pw)w\≥0 of size (\ℓ+1)\×(\ℓ+1). Finally we\nprove that widetilde Pw=P0-1Pw is a sequence of matrix orthogonal\npolynomials with respect to a weight matrix W. Moreover, we show that W\nadmits a second order symmetric hypergeometric operator widetilde D and a\nfirst order symmetric differential operator widetilde E.\n Later, we establish a direct relationship between the spherical functions of\nthe n-dimensional sphere Sn\≃ SO(n+1)/ SO(n) and the spherical\nfunctions of the n-dimensional real projective space\nPn(\ℝ)\≃ SO(n+1)/\O(n). Concluding that to find all the\nspherical functions of one of these pairs is equivalent to do the same it with\nthe other.\n Finally, we study the spherical functions of certain types of the\nn-dimensional sphere Sn\≃ SO(n+1)/ SO(n), for any n. More\nprecisely, we give explicitly all the spherical functions whose associated\nfunctions H are scalar valued, including those of trivial type, and then we\nstudy the irreducible spherical functions of fundamental type, describing them\nin terms of matrix valued hypergeometric functions 2!F1. Thereafter, for\nevery fundamental type we build a sequence of ortogonal matrix valued\npolynomials with respect to a weight W, which are associated to the spherical\nfunctions. We also prove that, for any n, W admits a second order symmetric\ndifferential operator.\n