Let $B$ be the Borel subgroup of $G = \text{GL}_2\big( \Bbb{R}\big)$, let ${\bf \alpha}:B \longrightarrow \Bbb{C}^*$ be a character, and consider the induced representation $\text{Ind}_B^G ({\bf \alpha} )$ of $G$. Without getting into analytic technicalities, is it possible to interpret this induced representation as the space of solutions to a Dirichlet problem in the upper half-plane $\frak{H}$ whose boundary conditions (one for each solution) vary over some appropriate family of functions on the boundary $\Bbb{R} \cup \{\infty\}$ ? Specifically, given a vector $v$ in $\text{Ind}_B^G ({\bf \alpha} )$ is there a real-valued function $f_v:\Bbb{R} \longrightarrow \Bbb{R}$ with some regulation on its behaviour at $\infty$ such that the solution $u_v: \frak{H} \longrightarrow \Bbb{R}$ to the Dirichlet problem $\Delta_{\frak{H}} u_v = 0$ with $u_v {\big|}_{\Bbb{R}} = f_v$ lives in the standard upper-half plane model $V_{\bf \alpha} \subset L^2\big( \frak{H} \big)$ for the induced representation, and as $v$ varies (and $f_v$ varies respectively over this appropriate class of boundary conditions) the corresponding solutions $u_v$ fill up this standard model $V_{\bf \alpha}$ ?




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