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Determine the equivalence classes of irreducible unitary representations of a solvable lie group. $$\begin{bmatrix}ae^t & 0\\ 0 & be^{-t}\end{bmatrix}$$ for $a,b\in \mathbb{R}, t\in \mathbb{R}$

This is the sketch of how I would like to approach this problem.

  1. Suppose $\pi$ is an irreducible unitary representation the lie group.

  2. Find the center of the group and note that by Schur's lemma, $\pi$ is just a character on the center of the group.

  3. Then consider the map $g \rightarrow Tr(\pi(g))$ where $g \in$$\begin{bmatrix}ae^t & 0\\ 0 & be^{-t}\end{bmatrix}$ and Tr=trace of the matrix.

Then I don't know how to go further from here.

Does this approach reasonable?

Thanks.

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