# Approximation of special differential operator applied to special function on $\operatorname{SL}_2(\mathbb{R})$

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.