info:eu-repo/semantics/article
Spherical Functions: The Spheres Vs. The Projective Spaces
Date
2014-01Registration in:
Tirao, Juan Alfredo; Zurrián, Ignacio Nahuel; Spherical Functions: The Spheres Vs. The Projective Spaces; Heldermann Verlag; Journal Of Lie Theory; 24; 1-2014; 147-157
0949-5932
CONICET Digital
CONICET
Author
Tirao, Juan Alfredo
Zurrián, Ignacio Nahuel
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 $\pi\in\hat\SO(n)$ if and only if it is an irreducible spherical function of some type γ∈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.