Spherical Functions: The Spheres Vs. The Projective Spaces
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...
| Autores: | , |
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| Tipo de recurso: | artículo |
| Estado: | Versión publicada |
| Fecha de publicación: | 2014 |
| País: | Argentina |
| Institución: | Consejo Nacional de Investigaciones Científicas y Técnicas |
| Repositorio: | CONICET Digital (CONICET) |
| Idioma: | inglés |
| OAI Identifier: | oai:ri.conicet.gov.ar:11336/32176 |
| Acceso en línea: | http://hdl.handle.net/11336/32176 |
| Access Level: | acceso abierto |
| Palabra clave: | Spherical Functions Orthogonal Group Special Orthogonal Group Group Representations. https://purl.org/becyt/ford/1.1 https://purl.org/becyt/ford/1 |
| Sumario: | 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. |
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