described in the previous section. It is obviously connected. If we show that $\dim
\pi^{-1}(JNF\setminus \tilde{U})<2n$ then Corollary \[sm2n\] implies the irreducibility.

Let us denote by $J_k(\lambda)$ the Jordan block of size $k$ with the eigenvalue $\lambda$ and by the symbol $J_{\vec{k}}(\lambda)$ the matrix $diag(J_{k^1}(\lambda),\dots,J_{k^t}(\lambda))$, $\vec{k}\in\mathbb N^t$ and $k^i\ge k^{i+1}$, $i=1,\dots,t-1$. Let us formulate without a proof an elementary statement from linear algebra.

The dimension of $$Stab(J_{\vec{k}}(\lambda))=
\{X\in GL(n,\mathbb C)|[X,J_{\vec{k}}(\lambda)]=0\}$$ is equal to $\sum_{1\le i,j\le t} min\{k^i,k^j\}$.

Let us denote by $J_{\bold{k}}(\lambda)$ the matrix $$diag(J_{\vec{k_1}}(\lambda),J_{\vec{k_2}}(\lambda\tau^2),\dots,
J_{\vec{k_r}}(\lambda\tau^{2r})),$$ $\vec{k_i}\in\mathbb N^{t_i}$. We use notations $|\vec{k}_i|=\sum_{j=1}^{t_i} k_i^j$, $|\bold{k}|=\sum_{j=1}^r|\vec{k}_j|$.

Let $\lambda_1,\dots,\lambda_s\in\mathbb C$ be such that $\lambda_i/\lambda_j\ne \tau^{2c}$, $c\in\mathbb Z$, $|c|\le n$ and $$\label{J}
J=
diag(J_{\bold{k}_1}(\lambda_1),\dots,J_{\bold{k}_s}(\lambda_s)).$$

We denote by $\pi'_{34}$: $CM'_\tau\to E\times E^*$ the slightly modified projection on the last two components: $\pi'_{34}(X,Y,U,V)=(YXU,V)$. The fiber of the map $\pi'_{34}$ over the point $(U,V)$ of the subset $\hat{J}=\pi'_{34}((\pi'_1)^{-1}(J))$ consists of the points $(J,Y+F,J^{-1}(Y+F)^{-1}U,V)$ where $F$ is an element of the kernel of the linear map: $$S_J(F)=\tau J F-\tau^{-1} F J,\quad F\in\mathfrak{gl}(E),$$ $Y+F$ is invertible, and $(J,Y,J^{-1}Y^{-1}U,V)\in CM'_\tau$. Obviously $(\pi'_{34})^{-1}(U,V)$ is a Zariski open nonempty subset inside $ker(S_J)$ hence they have the same dimension.

First let us study the map $S_J$ in the simple case when in the 