## Exact power series in the asymptotic expansion of the matrix coefficients with the corner $K$-type of $P_J$-principal series representations of $Sp(2,\R)$

J. Math. Sci. Univ. Tokyo
Vol. 15 (2008), No. 4, Page 521--543.

IIDA, Masatoshi ; ODA, Takayuki
Exact power series in the asymptotic expansion of the matrix coefficients with the corner $K$-type of $P_J$-principal series representations of $Sp(2,\R)$
Let $G$ be a symplectic Lie group of rank $2$, $Sp(2,\mathbb{R})$ and $P_J$ be its maximal parabolic subgroup called the Jacobi parabolic subgroup with non-abelian unipotent radica. The radial part of matrix coefficients of the $P_J$-principal series representations of $G$ were studied in relation to the Appell's hypergeometric function. The leading terms of the expansion of the function around the infinity were well investigated in general cases (semisimple Lie groups and representations). In this paper, we determine the power series expansion other than leading terms for the above special case.