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

http://dx.doi.org/10.4153/CMB-1999-056-5
Canad. Math. Bull. 42(1999), 486-498

  Published:1999-12-01
 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 formula. In the Conclusion, we discuss how this analytic continuation can be a helpful tool to study the growth of the heat kernel.