Let $F$ be an imaginary quadratic number field. Let $G = \mathrm{PGL}_2(\mathbb{C}) $ and $\varGamma = \mathrm{PSL}_2(\mathcal{O}_F)$. We have the Iwasawa decomposition $G = NAK$ where $K = \mathrm{PSU}(2)$.
We have the spectral decomposition of $L^2(\Gamma \backslash G)$: $$ L^2(\Gamma \backslash G) = L_0^2(\Gamma \backslash G) \oplus L_e^2(\Gamma \backslash G).$$ It is known that $L_e^2(\Gamma \backslash G)$ is the closure of the space spanned by incomplete Eisenstein series $E(g;f)$ of the form $$ E(g;f) = \sum_{\gamma \in \Gamma_\infty \backslash \Gamma} f(\gamma g),$$ with $f \in C_c^\infty(N\backslash G)$.
For representation $T_{2l}$ of $G$ and $k\in K$, let $ \Phi_{p,q}^l(k) $ denote the coefficient of $z^{l-p}$ in the polynomial expansion of $T_{2l}(k) z^{l-q}$. (2.36) of the book Sum Formula for SL2 over Imaginary Quadratic Number Fields states that $$ L^2(K) = \overline{\bigoplus_{\substack{p,q,l\in\mathbb{Z} \\ |p|,|q| \leq l}}\mathbb{C}\Phi_{p,q}^l }.$$
Then I think that the incomplete Eisenstein series can be defined as $$ E(g;h,p,q,l) = \sum_{\gamma \in \Gamma_\infty \backslash \Gamma} h( r(\gamma g)) \Phi_{p,q}^{l} ( k(\gamma g)),$$ with $h(r) \in C^\infty (\mathbb{R}^+) $. Is that right?
If the answer is yes, then we have $\int_{\Gamma \backslash G} E(g;h,p,q,l) dg = 0$ unless $p=q=l=0$ (then $\Phi_{p,q}^{l} = 1$). I doubt if this could possibly be true.
