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.


Your Answer

By clicking “Post Your Answer”, you agree to our terms of service, privacy policy and cookie policy

Browse other questions tagged or ask your own question.