Let $ \alpha $ be the action of $ \Bbb{R} $ on the group $ \Bbb{R}^{2} $ defined by $ \alpha_{t} \! \left( \begin{bmatrix} a \\ b \end{bmatrix} \right) = \exp \! \left( \begin{bmatrix} t & 0 \\ 0 & - t \end{bmatrix} \right) \begin{bmatrix} a \\ b \end{bmatrix} $. Let $ G = \Bbb{R}^{2} \rtimes_{\alpha} \Bbb{R} $. Determine the equivalence classes of irreducible unitary representations of $ G $.
I came up with two approaches.
The first one is to actually work with the matrices.
The other one is to use Schur’s Lemma and use the fact that an irreducible representation $ \pi $ on the center of $ G $ is just a character.
But with those approaches, I don’t know how to go further and I guess it’s because I’m not really sure whether my approach is reasonable.
Thanks.