Geodesic
Spherical Functions on SO0(p, q)/ SO(p) × SO(q)
Canadian mathematical bulletin, Tome 42 (1999) no. 4, pp. 486-498

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An integral formula is derived for the spherical functions on the symmetric space ${G}/{K\,=\,{\text{S}{{\text{O}}_{0}}\left( p,\,q \right)}/{\text{SO}\left( p \right)\,\times \,\text{SO}\left( q \right)}\;}\;$ . 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 ${\text{S}{{\text{O}}_{0}}\left( p,\,q \right)}/{\text{SO}\left( p \right)\,\times \,\text{SO}\left( q \right)}\;$ 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.
DOI : 10.4153/CMB-1999-056-5
Mots-clés : 33C55, 17B20, 53C35 
Sawyer, P. Spherical Functions on SO0(p, q)/ SO(p) × SO(q). Canadian mathematical bulletin, Tome 42 (1999) no. 4, pp. 486-498. doi: 10.4153/CMB-1999-056-5
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     author = {Sawyer, P.},
     title = {Spherical {Functions} on {SO0(p,} q)/ {SO(p)} {\texttimes} {SO(q)}},
     journal = {Canadian mathematical bulletin},
     pages = {486--498},
     year = {1999},
     volume = {42},
     number = {4},
     doi = {10.4153/CMB-1999-056-5},
     url = {http://geodesic.mathdoc.fr/articles/10.4153/CMB-1999-056-5/}
}
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