- 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?

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.)

isthe complexification of $SU(2)$. You can find some information and references here: eom.springer.de/c/c024210.htm – Theo Buehler Dec 13 '10 at 6:33