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I am trying to write a map $s$, such that $s: SL(2,\mathbb{C})/SU(2) \to SL(2,\mathbb C)$.

Using Iwasawa decomposition, I see that any element $g$ in $SL(2,\mathbb{C})$ can be written as $g = su$, where $u$ is an element of $SU(2)$ and $s$ is a matrix that is a product of two matrices $a$ and $k$, where $a$ is a member of the subgroup of diagonal matrices with positive diagonal components, and $k$ is a member of the subgroup of upper triangular nilpotent matrices.

However, I don't exactly see how to formulate this map; I see that it should come from this decomposition, but I am a bit stuck after that.

Thanks

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    Which you can then compose with $(\pi|_P)^{-1}: SL(2,\mathbb{C})/SU(2) \to P$ if you wish.2011-10-11

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I don't think that the Iwasawa decomposition can help you, because $B \cap SU(2) \neq \{1\}$, so you have to work with the subset of $B$, where the diagonal entries are strictly positive.