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# Spherical Functions on $\SO_0(p,q)/\SO(p)\times \SO(q)$

Published:1999-12-01
Printed: Dec 1999
• P. Sawyer
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## Abstract

An integral formula is derived for the spherical functions on the symmetric space $G/K=\break \SO_0(p,q)/\SO(p)\times \SO(q)$. This formula allows us to state some results about the analytic continuation of the spherical functions to a tubular neighbourhood of the subalgebra $\a$ of the abelian part in the decomposition $G=KAK$. The corresponding result is then obtained for the heat kernel of the symmetric space $\SO_0(p,q)/\SO (p)\times\SO (q)$ using the Plancherel formula. In the Conclusion, we discuss how this analytic continuation can be a helpful tool to study the growth of the heat kernel.
 MSC Classifications: 33C55 - Spherical harmonics 17B20 - Simple, semisimple, reductive (super)algebras 53C35 - Symmetric spaces [See also 32M15, 57T15]

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