Funciones Esf\'ericas Matriciales Asociadas a las Esferas y a los Espacios Proyectivos Reales

Abstract

In this work we start by determining all irreducible spherical functions $\Phi$ of any $K $-type associated to the pair $(G,K)=(\SO(4),\SO(3))$. 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 prove 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 show that $W$ admits a second order symmetric hypergeometric operator $\widetilde D$ and a first order symmetric differential operator $\widetilde E$. Later, we establish a direct relationship between the spherical functions of the $n$-dimensional sphere $S^n\simeq\SO(n+1)/\SO(n)$ and the spherical functions of the $n$-dimensional real projective space $P^n(\mathbb{R})\simeq\SO(n+1)/\mathrm{O}(n)$. Concluding that to find all the spherical functions of one of these pairs is equivalent to do the same it with the other. Finally, we study the spherical functions of certain types of the $n$-dimensional sphere $S^{n}\simeq \SO(n+1)/\SO(n)$, for any $n$. More precisely, we give explicitly all the spherical functions whose associated functions $H$ are scalar valued, including those of trivial type, and then we study the irreducible spherical functions of fundamental type, describing them in terms of matrix valued hypergeometric functions $_2!F_1$. Thereafter, for every fundamental type we build a sequence of ortogonal matrix valued polynomials with respect to a weight $W$, which are associated to the spherical functions. We also prove that, for any $n$, $W$ admits a second order symmetric differential operator.