Spherical Functions: The Spheres Vs. The Projective Spaces

Abstract

In this paper we establish a close 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 $P^n(\mathbb{R})\simeq\SO(n+1)/\mathrm{O}(n)$. In fact, for $n$ odd a function on $\SO(n+1)$ is an irreducible spherical function of some type $π\in\hat\SO(n)$ if and only if it is an irreducible spherical function of some type $\gamma \in \hat{\mathrm{O}}(n)$. When $n$ is even this is also true for certain types, and in the other cases we exhibit a clear correspondence between the irreducible spherical functions of both pairs $(\SO(n+1),\SO(n))$ and $(\SO(n+1),\mathrm{O}(n))$. Summarizing, to find all spherical functions of one pair is equivalent to do so for the other pair.

References in corpus (1)

• Spherical Functions Associated With the Three Dimensional Sphere

Cited by in corpus (2)

• Spherical Functions of Fundamental $K$-Types Associated with the $n$-Dimensional Sphere
• Funciones Esféricas Matriciales Asociadas a las Esferas y a los Espacios Proyectivos Reales