previous reasoning imply: $$\dim\pi_1^{-1}(J)\le \dim\pi'_{34}({\pi'}_1^{-1}(J))+\dim \ker
S_J- \dim Stab(J).$$ Together with the inequality from Lemma \[ineq\] it proves the statement.

Another case (i.e. $\bold{k}=\vec{k}_1$) is even easier because in this case we have $$\dim\pi_1^{-1}(J)\le 2n -\dim Stab(J)<2n-1.$$

The case when in formula (\[J\]) $s>1$ can be easily reduced to the previous case. For that let us introduce the embedding $i_l$: $\mathfrak{gl}(|\bold{k}_l|,\mathbb C)\to
\mathfrak{gl}(n,\mathbb C)$ and the projection $pr_l$: $\mathfrak{gl}(n,\mathbb C)\to \mathfrak{gl}(|\bold{k}_l|,\mathbb
C)$: $i_l(Y)_{p',q'}=Y_{pq}$, $pr_l(Y)_{pq}=Y_{p',q'}$, $p'=p+\sum_{m=1}^{l-1}|\vec{k}_m|$, $q'=q+\sum_{m=1}^{l-1}|\vec{k}_m|$, $0\le p,q\le |\vec{k}_l|$, and $i_l(Y)_{ij}=0$ for the rest of the entries of $i_l(Y)$.

Using arguments analogous to the ones from Lemma \[kerSJ\] one gets

Let $J$ be given by formula (\[J\]). Then

1.  $ker S_J=\oplus_{l=1}^s i_l(ker
    S_{J_{\bold{k}_l}}(\lambda_i))$

2.  for $l=1,\dots,s$, $pr_l(Im S_J)\subset Im S_{J_{\bold{k}_l}}$.

This lemma immediately implies

\[dimfib\] Let $J$ be given by formula (\[J\]) and exists $l$, $1\le l\le s$ such that $|\bold{k}_l|>1$ then $\dim\pi_1^{-1}(J)<2n-s$.

And we eventually achieved the goal of the subsection:

Indeed Proposition \[dimfib\] implies $\dim\pi_1^{-1}(JNF\setminus\tilde{U})<2n$. Hence by Corollary \[sm2n\] $\pi_1^{-1}(JNF\setminus\tilde{U})$ lies inside the Zariski closure of $\pi_1^{-1}(\tilde{U})$. But $\pi_1^{-1}(\tilde{U})$ is irreducible.

The Poisson structure on the CM space {#Pois}
-------------------------------------

In the paper [@FR] the Poisson structure on the space $CM_\tau$ was constructed. This Poisson structure on $CM_\tau$ 