# Eigenfunctions of subgroup of Heisenberg matrices

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: $$1$$.
• Ok so from that then we have that the eigenvalues of $T$ are given by $1 = \gamma(a)$ where $a$ is the matrix above and $\gamma$ is the character right? I'm unsure how to back out what the character function actually is from that. Is it a determinant of some kind? I guess I'm just confused about the character stuff itself as my background is more in analysis. – CAPM May 6 at 11:57