in Section \[spectralsection\] that $V_{\chi_1, \chi_2}$ and $H(\chi_1, \chi_2)$ are isomorphic representations, in particular there is a decomposition $$\label{Htensor}
 H(\chi_1, \chi_2) = \bigotimes_v H_v(\chi_1, \chi_2).$$ In addition, $$\label{Vparam}
 V_{\chi_1, \chi_2}({\mathfrak{c}}) = \{E(\varphi(iy), \cdot) \in V_{\chi_1, \chi_2} \mid \varphi \in H(\chi_1, \chi_2, {\mathfrak{c}})\},$$ and by , $$H(\chi_1, \chi_2, {\mathfrak{c}}) = \bigoplus_{{\mathfrak{t}}\mid{\mathfrak{c}}{\mathfrak{c}}_{\pi}^{-1}} R_{\mathfrak{t}}H(\chi_1, \chi_2, {\mathfrak{c}}_{\pi}).$$ Here it is known (e.g. [@Ca p. 306]) that ${\mathfrak{c}}_\pi={\mathfrak{c}}_{\chi_1, \chi_2}={\mathfrak{c}}_{\chi_1}{\mathfrak{c}}_{\chi_2}$. In Section \[explicit\] we shall give a detailed proof of this fact for trivial central character.

Finally for ${\mathfrak{c}}\subseteq {\mathfrak{c}}_\omega$ we define, in harmony with the notation of the previous section, $$\label{defCE}
\begin{split}
\mathcal{C}_{\omega}({\mathfrak{c}}) & := \bigl\{\pi\in\mathcal{C}_\omega\mid {\mathfrak{c}}\subseteq {\mathfrak{c}}_{\pi} \subseteq {\mathfrak{c}}_{\omega} \bigr\},\\
\mathcal{E}_{\omega}({\mathfrak{c}}) & := \bigl\{\{\chi_1,
\chi_2\}\in\mathcal{E}_\omega\mid {\mathfrak{c}}\subseteq {\mathfrak{c}}_{\chi_1, \chi_2}
\subseteq {\mathfrak{c}}_{\omega} \bigr\},
\end{split}$$ and we shall drop the subscript $\omega$ in case $\omega$ is trivial.

Normalized Whittaker functions
------------------------------

Let $q\in{\mathbb{Z}}$. For $q$ even, let $\nu\in (\frac{1}{2}+{\mathbb{Z}})\cup
i{\mathbb{R}}\cup(-\frac{1}{2},\frac{1}{2})$. For $q$ odd, let $\nu\in{\mathbb{Z}}\cup
i{\mathbb{R}}$. For these parameters we define the normalized Whittaker function $$\label{normwhit}
 \tilde{W}_{\frac{q}{2}, \nu}(y) :=
 \frac{i^{\operatorname{sgn}(y)\frac{q}{2}}
 W_{\operatorname{sgn}(y)\frac{q}{2}, \nu}(4\pi|y|)}
 {\left\{\Gamma(\frac{1}{2}-\nu+\operatorname{sgn}(y)\frac{q}{2})\Gamma(\frac{1}{2}+\nu+\operatorname{sgn}(y)\frac{q}{2})\right\}^{1/2}},
\qquad y\in{\mathbb{R}}^\times,$$ where $W_{\alpha, \beta}$ is the standard Whittaker function, see [@WW Chapter XVI]. The right hand side is understood as $0$ if one of $\frac{1}{2} \pm \nu + \operatorname{sgn}(y)\frac{q}{2}$ is a nonpositive integer, 