Consider the subgroup of the Heisenberg matrices given by:
$$ \mathcal{S} = \Big\{ \begin{bmatrix} 1 & 0 & z \\ 0 & 1 & 0 \\ 0 & 0 & 1 \end{bmatrix} : z \in \mathbb{R}/\mathbb{N} \Big\} $$ If we take a group rotation given by $$T(g) = \begin{bmatrix} 1 & 0 & \pi/4 \\ 0 & 1 & 0 \\ 0 & 0 & 1 \end{bmatrix} \cdot g $$ then I wish to find the eigenfunctions of $T$. According to theorem 3.5 of Walters' "Introduction to Ergodic theory" every eigenfunction is a multiple of a character. But I'm unsure how to use this theorem to back out the characters and eigenfunctions.
Any help appreciated.
If $T=\begin{bmatrix} 1 & 0 & \pi/4 \\ 0 & 1 & 0 \\ 0 & 0 & 1 \end{bmatrix}$, then it is easy to see that $T$ has only one eigenvalue: $1$.
Can you proceed ?