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Is there a book that finds all the irreducibile finite dimensional representations of $SL(2,\mathbb C)$ without considering the Lie algebra $sl(2,\mathbb C)$?

For example, how can I show "directly" that, for $G \in SL(2,\mathbb C)$, for $m,n \in \mathbb N \cup \{0\}$, the application $D_G^{(m/2,n/2)} : P_{m,n} \to P_{m,n}$ where the vector space $P_{n,m}$ is $P_{m,n} = \{ p : \mathbb C^2 \to \mathbb C : p(z,w) = \sum_{i=0}^m \sum_{j=0}^n p_{ij} z^i w^j, p_{ij} \in \mathbb C\}$

defined by $$ (D_G^{(m/2,n/2)}p)(z,w) := (bz+d)^m(\overline b w + \overline d)^n p \left( \frac{az+c}{bz+d},\frac{\overline a w+\overline c}{\overline b w + \overline d} \right) $$ for $G = \left( \begin{matrix} a & b \\ c & d \end{matrix} \right)$ gives an irreducible representation $D^{(m/2,n/2)} : SL(2,\mathbb C) \to Aut(P_{m,n})$, $G \mapsto D^{(m/2,n/2)}(G) := D_G^{(m/2,n/2)}$ ?

Also, how can I show that every finite dimensional irreducible representation of $SL(2,\mathbb C)$ is equivalent to one of this type?


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I'm pretty sure that Gel'fand-Graev-Vilenkin, Generalized functions, Vol. 5: Integral geometry and representation theory comes close. If I remember correctly, the word Lie algebra is not even mentioned in that book. – t.b. Jul 13 '11 at 23:09
Thanks for your answer! I'll search in the library for the book and check if I find the result in which I'm interested. – Robert Jul 15 '11 at 0:20

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