$s_{i_1,j_1}\dots s_{i_r,j_r}c^{-1}(1)=1$. This is possible only in the case $r=1$, $i_1=1,j_1=n$. Thus rewriting $T_i$ in the form: $$T_i=\frac{(\tau
X_i-\tau^{-1}X_{i+1})}{X_i-X_{i+1}}s_i+\frac{X_{i+1}(\tau^{-1}-\tau)}
{X_i-X_{i+1}},$$ we see that $$Y_1e_1=
\left(\prod_{i=1}^{n-1}\frac{(\tau
X_i-\tau^{-1}X_{i+1})}{X_i-X_{i+1}}s_i\right) c^{-1}e_1+R,$$ where $R$ is a linear combination of $e_j$ with $j>1$. This formula immediately implies the last formula from the proposition for $i=1$.

Let $D_\tau$ be a subset of $\mathcal{U}$ consisting of the representations of the form $V_{\mu,\nu}$ such that $\delta_\tau(\nu)=\prod_{i,j}(\tau \nu_i-\tau^{-1}\nu_j)=0$.

It is actually not easy to compute all coefficients $\bar{Y}_1$ using explicit formulas for $Y_1$ and $T_i$ but we do not need them. Because by proposition \[propcoor\], if the pair $(X,Y)$ satisfies equation (\[eqXY-YX\]) and $X$ is diagonal with eigenvalues satisfying the conditions of Proposition \[propcoor\] then the corresponding $GL(E)$-orbit is uniquely determined by the diagonal elements of $X$ and $Y$ (because the stabilizer of $X$ consists of diagonal matrices which do not change diagonal elements of $Y$ and we can extract $q$ from these elements). This reasoning implies

\[isoopen\] The map $\Phi$ is an isomorphism on the subset $\mathcal{U}_0=\mathcal{U}\setminus D_\tau$, and local coordinates $\lambda,q$ on $CM_\tau$ are expressed through coordinates $\mu,\nu$ on $\mathcal{U}_0\subset Irrep^{n!}$ by the formulas $$\begin{gathered}
\lambda_i=\nu_i,\quad q_i=\mu_i \prod_{j\ne
i}\frac{(\tau^{-1}\nu_j-\tau\nu_i)}{(\nu_j-\nu_i)}.\end{gathered}$$

 Results on the general double affine Hecke algebra
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