group $GL(2,\mathbb Z)$ is generated by the elements: $$\varepsilon=\begin{pmatrix}0&-1\\-1&0\end{pmatrix}, \quad
\sigma=\begin{pmatrix}1&1\\0&1\end{pmatrix}.$$

These generators act by the following formulas: $$\begin{gathered}
\varepsilon: X_i\mapsto Y_i, Y_i\mapsto X_i, T_i\mapsto
T_i^{-1},\\ \sigma: X_i\mapsto X_i, Y_i\mapsto X_iY_iq^{-1},
T_i\mapsto T_i,\end{gathered}$$ where $\varepsilon: H_{q,\tau}\to H_{q^{-1},\tau^{-1}}$, $\sigma:
 H_{q,\tau}\to H_{q,\tau}$. The transformation $\varepsilon$ is called the Fourier-Cherednik transform.

Using these transformations we can construct new finite dimensional representations. Indeed if $\gamma\in GL(2,\mathbb
 Z)$ is such that $\gamma(H_{1,\tau})=H_{1,\tau'}$ and $\phi':
 H_{1,\tau'}\to GL(V'_{\mu,\nu})$ is the corresponding representation of $H_{1,\tau'}$ (here $\tau'$ is either $\tau$ or $\tau^{-1}$ ) then the map $\phi'\circ\gamma$ is a representation of $H_{1,\tau}$. We denote the set of such representations by $\gamma(\mathcal U)$.

The map from $Irrep^{n!}$ to $CM_\tau$ {#constr}
======================================

In this section we construct a map $\Phi$: $Irrep^{n!}\to
CM_\tau$. Later we will show that it is an isomorphism. Constructions of this section generalize constructions of section 11 of [@EG].

Construction of the map
-----------------------

Let us denote by $A^{n-1}_\tau\subset A^n_\tau$ the subalgebra generated by the elements $T_{2},\dots, T_{n-1}$. It is the finite Hecke algebra of rank $n-2$. The element $v$ of an $A^n_\tau$-module is said to be $A^{n-1}_\tau$-invariant if $xv=\tau v$ for all $x\in A^{n-1}_{\tau}.$

The $H_{1,\tau}$-module $V\in Irrep^{n!}$ by definition is a regular $A^n_\tau$-module. Hence the 