<s>[INST] Non-unitary representation How to prove $\pi :\mathbb R\to \mathbb C^2$,   defined by $t\mapsto \begin{pmatrix} 1 & t\\  0 & 1\end{pmatrix}$ is a non-unitary representation? 
Is the following correct? $\pi$ is a representation because 
$$\pi (t+s)=\begin{pmatrix}1 & t+s \\ 0 & 1\end{pmatrix}=\begin{pmatrix}1 & t\\  0  &1\end{pmatrix}\begin{pmatrix}1 & s \\ 0 & 1\end{pmatrix}  = \pi(t)\pi(s)$$
For non-unitary, I need to prove $U:\mathbb C^2\to \mathbb C^2$ is not unitary. 
 [/INST] For $\pi$ to be a unitary representation, $\pi(x)$ must be a unitary matrix for every element $x$ of the group. Here, 
$$\begin{pmatrix}1 & t \\ 0 & 1\end{pmatrix}$$ 
is not a unitary matrix unless $t=0$. Indeed,
$$\begin{pmatrix}1 & t \\ 0 & 1\end{pmatrix}^* \begin{pmatrix}1 & t \\ 0 & 1\end{pmatrix} 
=\begin{pmatrix}1 & 0 \\ \bar t & 1\end{pmatrix} \begin{pmatrix}1 & t \\ 0 & 1\end{pmatrix}   = \begin{pmatrix}1 & t \\ \bar t & 1+|t|^2\end{pmatrix} $$   which is not the identity matrix.
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