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

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.

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.