Spherical Functions: The Spheres versus the Projective Spaces
Journal of Lie Theory, Volume 24 (2014) no. 1, pp. 147-157
\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.
DOI:
10.5802/jolt.779
Keywords:
Spherical functions, orthogonal group, special orthogonal group, group representations
@article{JOLT_2014_24_1_a6,
author = {J. Tirao and I. Zurri\'an},
title = {Spherical {Functions:} {The} {Spheres} versus the {Projective} {Spaces}},
journal = {Journal of Lie Theory},
pages = {147--157},
year = {2014},
volume = {24},
number = {1},
doi = {10.5802/jolt.779},
zbl = {1291.43010},
url = {https://jolt.centre-mersenne.org/articles/10.5802/jolt.779/}
}
J. Tirao; I. Zurrián. Spherical Functions: The Spheres versus the Projective Spaces. Journal of Lie Theory, Volume 24 (2014) no. 1, pp. 147-157. doi: 10.5802/jolt.779
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