Is this subset with conjugate columns of $GL_n(\mathbb C)$ connected?

Question

Let us assume $n=2k$ to be even and for a matrix $A \in M_n(\mathbb C)$, we let $(a_1, \dots, a_n)$ denotes the columns. We define a set \begin{align*} \mathcal E =\{M \in GL_n(\mathbb C): m_i = m_{i+1}^* \text{ for } i = 1, 3, \dots, 2k-1\} . \end{align*} This says, the columns of every $M \in \mathcal E$ come in conjugate pairs. I am wondering whether this set is connected. If not, how many connected components will there be.

Answer 1

Let $D$ be the direct sum of $n/2$ copies of $\pmatrix{1&1\\ i&-i}$. Then $f:GL_n(\mathbb R)\ni X\mapsto XD\in\mathcal E$ is a homeomorphism. Hence $\mathcal E$ has the same number of connected components as $GL_n(\mathbb R)$, i.e. two.