Spherical functions: The spheres vs the projective spaces

J. TIRAO, I. ZURRIÁN

JOURNAL OF LIE THEORY

HELDERMANN VERLAG

Lugar: Lemgo; Año: 2014 vol. 24 p. 147 - 157

0949-5932

In this paper we establish a close relationship between the spherical functions of the n-dimensional sphere S^n ≃ 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 ∈ 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),O(n)). Summarizing, to find all spherical functions of one pair is equivalent to do so for the other pair.