\in \mathbb{A},$$ where $$\label{explicitKir}
 W_\phi(y)=\rho_\phi({\mathfrak{o}}){\varepsilon}_\pi(\operatorname{sgn}(y))\tilde{W}_{q/2,
\nu_{\pi}}(y), \qquad y\in K_\infty^\times,\qquad \phi\in V_{\pi,
q}({\mathfrak{c}}_{\pi}).$$ An intrinsic definition of $W_\phi$ becomes apparent upon choosing $y_\text{fin}=(1,1,\dots)$ and $x_\text{fin}=(0,0,\dots)$ in and picking by orthogonality and $\operatorname{vol}(K\backslash\mathbb{A})=1$ the term corresponding to $r=1$: $$\label{Kir}W_{\phi}(y) := \int_{K\backslash\mathbb{A}} \phi\left(\begin{pmatrix} y & x\\ 0 &
1\end{pmatrix}\right) \psi(-x) \,dx,\qquad y\in K_\infty^\times.$$ We have verified and for pure weight newforms $\phi\in V_{\pi, q}({\mathfrak{c}}_{\pi})$ but then, by linearity, it extends to all smooth vectors $\phi\in
V_\pi^\infty({\mathfrak{c}}_{\pi})$. Using also , we obtain a linear map from $V_\pi^\infty({\mathfrak{c}}_{\pi})$ to a dense subspace of $L^2(K_\infty^\times,d^\times y)$ given by $\phi\mapsto W_\phi$. We will prove in Section \[Lfunctionsection\] that $$\label{constant}
\langle\phi_1,\phi_2\rangle=C_\pi\langle
W_{\phi_1},W_{\phi_2}\rangle$$ for some positive constant $C_\pi$ depending only on $\pi$. It follows that the map $\phi\mapsto W_\phi$ extends to a vector space isomorphism $V_{\pi}({\mathfrak{c}}_{\pi})\to L^2(K_\infty^\times,d^\times y)$, called the (archimedean) Kirillov map of $\pi$, and Lemma \[lemma2\] below shows that it is essentially an isometry (i.e. $C_\pi\approx 1$). In particular, and hold for all $\phi\in V_\pi({\mathfrak{c}}_\pi)$.\
Now let ${\mathfrak{c}}\subseteq {\mathfrak{c}}_{\pi}$ be any ideal. It will be important to investigate in detail vectors in the larger space $V_{\pi}({\mathfrak{c}})$, classically called “oldforms". The proofs of the following facts depend partly on the theory of Eisenstein series that we will develop 