## Matrix Coefficients with Minimal {\large $K$}-Types of the Spherical and Non-Spherical Principal Series Representations of {\large $SL(3,\mathbf{R})$}

J. Math. Sci. Univ. Tokyo
Vol. 19 (2012), No. 1, Page 1--55.

Sono, Keiju
Matrix Coefficients with Minimal {\large $K$}-Types of the Spherical and Non-Spherical Principal Series Representations of {\large $SL(3,\mathbf{R})$}
We compute the holonomic system of rank 6 for the radial part of the matrix coefficients of spherical and non-spherical principal series representations of $SL(3,\mathbf{R})$. We obtain six power series solutions corresponding to the set of six characteristic roots, and express the matrix coefficients by linear combinations of these power series. Among others, the $c$-functions of non-spherical principal series are obtained.