The Epstein zeta functions define certain automorphic forms on \(\text{SL}(n,\mathbb Z)\backslash\text{SL}(n,\mathbb R)\). The representations of \(G=\text{SL}(n,\mathbb R)\) generated by right translations of these automorphic forms are very degenerate principal series representations.
The authors consider the case when \(n=3\). Then there are three types of parabolic subgroups in \(G\). The Epstein zeta function \(Z(g,s)\) on \(G\) associated with the maximal parabolic subgroup \[ P_1:=\left\{\left(\begin{matrix} *&*&*\\ *&*&*\\ 0&0&*\end{matrix}\right)\in G\right\}, \] an Eisenstein series of class \(1\), is defined by \[ Z(g,s):=\sum_{\mathbf m\in\mathbb Z^3\backslash\{0\}} (\mathbf m\, g \;{}^t{g}\;{}^t{\mathbf m})^{-s}, \] for \(g\in G\) and \(s\in\mathbb C\) such that \(\operatorname{Re} s>3/2\).
The authors obtain multiplicity-free results on (degenerate) Whittaker models and generalized Whittaker models on \(\text{SL}(3,\mathbb R)\) belonging to the degenerate principal series and find an explicit formula for them. They also discuss the three types of Fourier expansions of the automorphic forms belonging to the spherical principal series representations.