SPHERICAL FUNCTIONS: THE SPHERES VS. THE PROJECTIVE SPACES

JUAN A. TIRAO; IGNACIO N. ZURRIÁN

JOURNAL OF LIE THEORY

HELDERMANN VERLAG

Lugar: Lemgo; Año: 2014 p. 147 - 157

0949-5932

In this paper we establish a close relationship between the spherical functions of the $n$-dimensional sphere $S^nsimeqSO(n+1)/SO(n)$ and those of the $n$-dimensional real projective space $P^n(mathbb{R})simeqSO(n+1)/mathrm{O}(n)$. In fact, for $n$ odd a function on $SO(n+1)$is an irreducible spherical function of some type $piinhatSO(n)$ if and only if it is an irreducible spherical function of some type $gammainhat {mathrm{O}}(n)$. 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.