Spherical Functions on $\SO_0(p,q)/\SO(p)\times \SO(q)$
Printed: Dec 1999
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
In the Conclusion, we discuss how this analytic continuation can be
a helpful tool to study the growth of the heat kernel.
33C55 - Spherical harmonics
17B20 - Simple, semisimple, reductive (super)algebras
53C35 - Symmetric spaces [See also 32M15, 57T15]