## Spherical Functions with Respect to the Semisimple Symmetric Pair $(Sp(2,\Bbb R),SL(2,\Bbb R) Ã SL(2,\Bbb R))$

J. Math. Sci. Univ. Tokyo
Vol. 6 (1999), No. 1, Page 127--179.

Moriyama, Tomonori
Spherical Functions with Respect to the Semisimple Symmetric Pair $(Sp(2,\Bbb R),SL(2,\Bbb R) Ã SL(2,\Bbb R))$
Let $Ï$ be a generalized principal series representation with respect to the Jacobi parabolic subgroup or a large discrete series representation of $G=Sp(2,\Bbb R)$. A spherical function is the image of a $K$-finite vector by the intertwining operator from $Ï$ to the {\rep } induced from an irreducible unitary representation of $SL(2,\Bbb R)^2$ in $G$. We obtain differential equations for the spherical functions except for a few cases. We write down the solutions of these differential equations by means of the Gaussian hypergeometric functions.