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!


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