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The proof is given below:

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But I do not understand why " we first determine which of the spaces of $V_{n}$ are invariant under T " as he said in the second sentence. could anyone explain this for me please?

$$A(z) = \begin{bmatrix} z & 0 \\ 0 & z^{-1} \end{bmatrix}$$

And $z \in \mathbb{C}.$

This link describes what is $\Phi_{n}$:

A discrepancy in a paragraph in the construction of a series of irreducible complex representations of $SU_{2}$

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