What is a "nice" way of choosing coset representatives for the symplectic group $Sp\_{2k}(\mathbb{C})$ in the general linear group $GL\_{2k}(\mathbb{C})$?



Nicest coset representatives of the symplectic group in the general linear group
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,...


