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Let $G = \operatorname{SL}_2 (\mathbb{R})$ and $G=ANK$ be the Iwasawa decomposition, so for $g \in G$ we write $g = a\hat{n}k$ where $$a=\begin{pmatrix}\hat{a} & \\ & \hat{a}^{-1}\end{pmatrix}\text{ for }\hat{a} >0, ~ \hat{n} = \begin{pmatrix}1&x\\&1\end{pmatrix},\text{ and } k=k(\theta)=\begin{pmatrix} \cos \theta & - \sin \theta \\ \sin \theta & \cos \theta \end{pmatrix}.$$ For $n \in \mathbb{N}$ and $t >0$ define a function on $G$ as follows:$$\phi_n^{(t)} (a\hat{n}k (\theta)) := \hat{a}^{1+it} \frac{1}{\sqrt{2 n + 1}} \frac{\sin \big( (n+\frac{1}{2}) \theta \big)}{\sin \frac{\theta}{2}} .$$

So $\phi_n^{(t)}$ restricted to $K=\operatorname{SO}_2$ is a normalized Dirichlet kernel.

Also it is an element of a principal series representation of $\operatorname{SL}_2 (\mathbb{R})$, if that helps.

Furthermore we define the differential operator $$H f (g) := \frac{\partial}{\partial s} f\left( g \begin{pmatrix} e^s & \\ & e^{-s} \end{pmatrix} \right).$$

I want to show that, for large $t$, the funtions $\phi_n^{(t)}$ become close to Eigenfuntions for the operator $H$ with Eigenvalue $\lambda = 1+it$.

By become close I mean for the $L^2$-norm on $K$, i.e. $||f||^2=\int_K \vert f(k) \vert^2 ~ \mathrm{d}k$.

This is not a problem for $g \in AN$ but because the $\phi_n^{(t)}$ depend on the Iwasawa decomposition of the input matrix one has to compute the Iwasawa decomposition of $g\begin{pmatrix}e^s & \\ & e^{-s}\end{pmatrix}=ank\begin{pmatrix}e^s & \\ & e^{-s}\end{pmatrix}$ where one has to "pass the $e^s , e^{-s}$-matrix around the $k$", which gives huge terms when simply derived. So I'm looking for a more theoretical approach to why this holds.

TL;DR: I have a special function coming from the representation space of a principal series representation of $\operatorname{SL}_2 (\mathbb{R})$ and want to show, that in some limit it becomes an Eigenfunction of some special differential operator.

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