Journal of Lie Theory 24 (2014), No. 1, 147--157
Copyright Heldermann Verlag 2014
Spherical Functions: The Spheres versus the Projective Spaces
Juan Tirao
CIEM-FaMAF, Universidad Nacional, Cordoba 5000, Argentina
tirao@famaf.unc.edu.ar
Ignacio Zurrián
CIEM-FaMAF, Universidad Nacional, Cordoba 5000, Argentina
zurrian@famaf.unc.edu.ar
[Abstract-pdf]
\def\R{{\Bbb R}} \def\Aut{\mathop{\rm Aut}\nolimits} \def\SO{{\rm SO}} We establish a close relationship between the spherical functions of the $n$-dimensional sphere $S^n\cong\SO(n+1)/\SO(n)$ and those of the $n$-dimensional real projective space $P^n(\R)\cong\SO(n+1)/{\rm O}(n)$. 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 $\gamma\in\hat{\rm O}(n)$. 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 $(\SO(n+1),{\rm O}(n))$. Summarizing, to find all spherical functions of one pair is equivalent to do so for the other pair.
Keywords: Spherical functions, orthogonal group, special orthogonal group, group representations.
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