the parameters $(\lambda_i,q_i)$.

Equation (\[eqXY-YX\]) is equivalent to the system $$\label{orb}
\frac{(\tau\lambda_i-\tau^{-1}\lambda_j)Y_{ij}}{\tau-\tau^{-1}}=p_is_j,\quad
1\le i,j\le n,$$ if $X=diag(\lambda_1,\dots,\lambda_n).$ If there exists $i$ such that $s_i=0$ then $Y_{ij}=0$, $j=1,\dots,n$ and $\det(Y)=0$. Thus we have $s_i\ne 0$. Analogously we get $p_i\ne 0$.

Let us fix a solution of (\[orb\]) lying in the $GL(n,E)$ orbit of $(X,Y,U,V)$. Putting $q_i=p_is_i/\lambda_i$ we get the desired representative with $X$ given by formula (\[repf\]), $Y$ by formulas (\[repm\]),(\[repl\]) and $U=(\tau-\tau^{-1})X^{-1}Y^{-1}q$.

This proposition together with Corollary \[sm2n\] implies that $(\lambda,q)$ are local coordinates on the open subset $\bold{U}
\subset CM_\tau$. In the next section we show that this subset is dense.

Irreducibility of $CM_\tau$
---------------------------

In this subsection we prove

\[irrCM\] The variety $CM_\tau$ is irreducible.

Let us consider the projection on the first component $\pi'_1$: $CM'_\tau\to GL(E)$. After the taking the quotient by the action of $GL(E)$ this map becomes a map $\pi_1$: $CM_\tau\to JNF$, where $JNF$ is a stack but we can think about it as the set of Jordan normal forms of matrices (we do not need the stack structure).

Inside $JNF$ there is an open part $\tilde{U}$ corresponding to diagonal matrices with eigenvalues $\{\lambda_1,\dots,\lambda_n\}$ such that $\lambda_i\ne\lambda_j$, $\tau\lambda_i\ne\tau^{-1}\lambda_j$ for $i\ne j$. The subset $\pi_1^{-1}(\tilde{U})$ was 