Let $\Gamma$ be the $2n \times 2n$ complex matrix
$$
\Gamma =
\begin{pmatrix}
U & V^* \\
V & U^*
\end{pmatrix}
$$
where $U$ and $V$ are $n \times n$ complex matrices. Now suppose $\Gamma$ satisfies
$$
\Gamma^\dagger \sigma_z \Gamma = \sigma_z
$$
where $\dagger$ is the hermitian conjugate and $\sigma_z = \text{diag}(I_n, - I_n)$. This is equivalent to saying that $U^\dagger U-V^\dagger V = I_n$ and $UV^T = VU^T$ where $T$ denotes the matrix transpose. It is also easy to see that $\Gamma$ also then satisfies
$$
\Gamma^T \Omega \Gamma = \Omega
$$
where
$$
\Omega =
\begin{pmatrix}
0 & I_n\\
-I_n & 0
\end{pmatrix}
$$
With this in mind, we can view $\Gamma$ as an element of the group U(n,n) $\cap$Sp(2n,$\mathbb{C}$), where $U(n,n)$ is the group of matrices which preserve the quadratic form $\sigma_z$. Furthermore, consider the subgroup of $U(n,n)\cap Sp(2n, \mathbb{C})$
$$
\Omega =
\begin{pmatrix}
A & 0\\
0 & A^*
\end{pmatrix}
$$
where $A \in U(n)$ is an arbitrary $n \times n$ unitary matrix. Consider the quotient group $G(n)$ defined as $G(n) = (U(n,n)\cap Sp(2n, \mathbb{C}))/U(n)$.
My question is: is there a stable homotopy theory for this group? In the sense that, for large enough $n$, can we compute $\pi_k(G(n)) for a given $k$?$ Is this related to the regular Boot periodicity of the the unitary, orthogonal and symplectic group? Thanks!