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 Sn ≃ SO(n + 1)/SO(n) and the spherical functions of the n-dimensional real projective space P n(R) ≃ SO(n + 1)/O(n). In fact, for n odd a function on SO(n + 1) is an irreducible spherical function of some type π ∈ ˆSO(n) if and only if it is an irreducible spherical function of some type γ ∈ ˆO(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), O(n)). Summarizing, to find all spherical functions of one pair is equivalent to do so for the other pair.