Title: The socle filtrations of principal series representations of $SL(3,\mathbb{R})$ and $Sp(2,\mathbb{R})$,
Abstract: We study the structure of the $(\mathfrak{g},K)$-modules of the principal
series representations of $SL(3,\mathbb{R})$ and $Sp(2,\mathbb{R})$ induced
from minimal parabolic subgroups, in the case when the infinitesimal character
is nonsingular. The composition factors of these modules are known by
Kazhdan-Lusztig-Vogan conjecture. In this paper, we give complete descriptions
of the socle filtrations of these modules.