Action of $SL(2,\mathbb{C})$ on representations of $SU(2)$  
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*I want to precisely understand in what sense is (if it is!) $SL(2,\mathbb{C})$ the "complexified" version of $SU(2)$? 


Can I think of it like choosing a natural matrix basis of the real three dimensional Lie algebra of $SU(2)$ (say the Pauli matrices) and looking at the vector space they would span over $\mathbb{C}$ (i.e look at the vector space of matrices spanned by linear sums of Pauli matrices with complex coefficients) and then exponentiate it down? 


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*Is there a natural action of $SL(2,\mathbb{C})$ on the $5$ dimensional irreducible representation of $SU(2)$? If yes then how does one best understand the quotient space and precisely in what way does this action respect the representation. 
(I would be interested about the general picture if there is any behind such actions, I chose the above particular example since it is most relevant to my current pursuits.) 
 A: I am not an expert in this field but I think that the general picture looks more or less like that:
1.
$\mathfrak{sl}(N,\mathbb{C}) = \mathfrak{su}_{\mathbb{C}}(N)$ ($\mathfrak{su}(N)$ is a compact real form of $\mathfrak{sl}(N,\mathbb{C} )$). Therefore, there is one to one correspondence between finite dimensional complex representations of Lie algebras:  $\mathfrak{sl}(N,\mathbb{C})$ and $\mathfrak{su}(N)$. Since $SU(N)$ is compact, all finite dimensional complex representations of $\mathfrak{su}(N)$ are unitary and therefore completely reducible (finite dimensional complex representations of $\mathfrak{sl}(N,\mathbb{C})$ share this property). Moreover - there is one to one correspondence between complex, irreducible, finite dimensional representations of $\mathfrak{su}(N)$ and $ \mathfrak{sl}(N,\mathbb{C})$.
2. Since $SL(N,\mathbb{C})$ is simply- connected (see http://en.wikipedia.org/wiki/General_linear_group ), each finite dimensional irreducible complex representation of $\mathfrak{sl}(N,\mathbb{C})$ gives irreducible representation of  $SL(N,\mathbb{C})$ (by matrix exponentiation of a Lie algebra representation) and vice-versa. The same thing can be said about $SU(N)$ and $\mathfrak{su}(N)$. As a result you have a natural action of $SL(N,\mathbb{C})$ on each irreducible complex representation of $SU(N)$ (note that given complex Lie group representation is irreducible iff corresponding representation of a Lie algebra is irreducible). 
