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