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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

t.b.
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Mary
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    Try the polar decomposition and identify $SL(2,\mathbb{C})/SU(2)$ with the positive definite, skew-Hermitian matrices $P$ of determinant one. This gives you an obvious map $s:P \to SL(2,\mathbb{C})$. – t.b. Oct 11 '11 at 17:16
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    Well, I think you need to specify which properties your map needs to have. Otherwise one can probably construct many different maps, for example a trivial one: $s(g) = 1$ for $g\in SL(2,\mathbb C)/SU(2)$ and $1$ is the identity element of $SL(2,\mathbb C)$. – Heidar Oct 11 '11 at 17:17
  • tb: I guess my problem is that I don't see the 'obvious map'. It is just s: P -> SL(2,C) such that p -> sp for s in SU(2)? – Mary Oct 11 '11 at 19:11
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    You have too many $s$'es. The set $P$ of positive definite skew-Hermitian matrices is a subset of the group $G = SL(2,\mathbb{C})$. By uniqueness of the polar decomposition every $g$ is uniquely $g= pu$ with $p = \sqrt{a^\ast a}\in P$ and $u = p^{-1}a \in SU(2)$. This gives you a unique coset representative $p$ for each $SU(2)$-coset and hence the projection $\pi:SL(2,\mathbb{C}) \to SL(2)/SU(2)$ restricts to a bijection of the set $P$ with the quotient $SL(2,\mathbb{C})/SU(2)$. Then the map is just the inclusion $s: P \to SL(2,\mathbb{C})$ which maps $p$ to $p$, so $s(p) = p$. – t.b. Oct 11 '11 at 19:32
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    Which you can then compose with $(\pi|_P)^{-1}: SL(2,\mathbb{C})/SU(2) \to P$ if you wish. – t.b. Oct 11 '11 at 19:35

2 Answers2

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I hope it helps viewing SL2(C) as the complexification of SU(2), which is the compact real form, hence exhibiting a "doubling" construction allowing to look at a decomposition (Re & Im) and project etc.

Alternatively SL2(C) has two real forms, depending on the signature of the Killing form: SL2(R) (non-compact with signature (2,1)) and SU(2) (signature (0,2)); then it is clear the the complexification of the real non-compact form SL(2;R) is the non-compact SL(2;C) (not changing signature), helping understand why the same complexification of SU(2), with negative definite Killing form "turns into" the "same" Lie algebra and group (over complex numbers as coefficients there is only one signature).

Lucian Ionescu
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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.

Marc Palm
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