Nicest coset representatives of the symplectic group in the general linear group What is a "nice" way of choosing coset representatives for the symplectic group $Sp_{2k}(\mathbb{C})$ in the general linear group $GL_{2k}(\mathbb{C})$?
 A: There's a slight issue, even when $k=2$. In this case, $Sp_2=SL_2$, so you want to find coset reps. of $SL_2$ in $GL_2$. The natural choice is the diagonal matrices, which is $\mathbb{C}^{\ast} I$,  but there is an ambiguity, since $-I \in SL_2$. Essentially, you need to take a "branch cut" of $\sqrt{}$ to get a unique representative. 
In general, I think one can get an analogue of the QR matrix decomposition by performing a version of Gram-Schmidt orthogonalization to make a symplectic form standard, but I'm not sure how canonical this decomposition may be chosen. It would be interesting if there is an analogue of the Polar decomposition. 
A: I suppose that "nice" is very much application-dependent, but let me give it a try.
The first thing to notice is that, of course, you will not be able to find a global coset representative, since the principal bundle
$$\mathrm{Sp}(2k,\mathbb{C}) \to \mathrm{GL}(2k,\mathbb{C}) \to M = \mathrm{GL}(2k,\mathbb{C})/\mathrm{Sp}(2k,\mathbb{C})$$
is not trivial and hence it has no global (continous) section.  So the best you can do is find a section over some $U \subset M$.
The subgroup $\mathrm{Sp}(2k,\mathbb{C})$ is the stabilizer of a symplectic structure $\Omega$ on $\mathbb{C}^{2k}$.  A convenient choice is
$$\Omega = \pmatrix{ 0 & -\mathbf{1} \cr \mathbf{1} & 0}$$
where $\mathbf{1}$ is the $k\times k$ identity matrix.
The Lie algebra $\mathfrak{sp}(2k,\mathbb{C})$ consists of those $2k \times 2k$ complex matrices $X$ such that $\Omega X$ is symmetric.  A complementary vector subspace of $\mathfrak{sp}(2k,\mathbb{C})$ in $\mathfrak{gl}(2k,\mathbb{C})$ is given by
$$\mathfrak{sp}(2k,\mathbb{C})^\perp := \big\lbrace \Omega X \mid X \in \mathfrak{so}(2k,\mathbb{C})\big\rbrace$$
Explicitly and for our choice of $\Omega$ above, this subspace consists of the matrices of the form
$$\pmatrix{B^t & C \cr A & B}$$
for $k\times k$ matrices $A,B,C$ with $A$ and $C$ skewsymmetric.
You can now exponentiate these matrices to find a coset representative.  Depending on the calculation,  though, you might it easier to write the coset representative as a product of exponentials,...
A: There is a one to one correspondence between 2n*2n complex matrices and n*n quaternionic matrices. In the n*n quaternionic representation the coset representatives of GL(2n,C)/Sp(2n,C) are self dual n*n quaternionic matrices. This can be seen from the 2n*2n complex matrix representation A of a self dual quaternionic matrix which satisfies the constraint:
A = - J A^t J
Where J is the block skew symmetric symplectic form. 
From this expression, one can see that the isotropy group of a fixed A in GL(2n, C) is Sp(2n,C).
