On the Self-adjoint properties of the standard Whittaker $(\mathfrak{g}, K)$-modules

The structure of the standard Whittaker $(\mathfrak{g}, K)$-module is examined in the case when the group in question is a real split reductive linear Lie group. This module is an injective object in the category of Harish-Chandra $(\mathfrak{g}, K)$-modules which admit a fixed infinitesimal character. The global character of this module is determined. The main theorem of this paper is that it has a self-adjoint structure. Also obtained are the explicit socle filtrations of the standard Whittaker $(\mathfrak{g}, K)$-modules for the rank two split groups $SL(3,\mathbb{R})$, $Sp(2,\mathbb{R})$ and $G_{2}$(split).