$T_i$, $i=1,\dots,n-1$ generate an algebra of dimension $n!$ which is called the finite Hecke algebra. We will denote it by the symbol $A^n_\tau$.

If $\rm{e}$ is the unit in $S_n$ then by the action of elements $T_i$ we can get from the vector $\rm{e}\otimes 1$ the whole space $V_{\mu,\nu}$. Hence the map $j$: $A^n_\tau\to V_{\mu,\nu}$, $j(T_{i_1}\dots T_{i_k})=T_{i_1}\dots T_{i_k} {\rm e}\otimes 1$ is an isomorphism of (left) $A^n_\tau$ modules.

We denote the subset of all finite dimensional irreducible $H_{1,\tau}$-modules which are regular $A^n_\tau$-modules by the symbol $Irrep^{n!}$.

Let us denote the subset of $Irrep^{n!}$ consisting of $V_{\mu,\nu}$ $\mu,\nu\in ({\mathbb C}^*)^n$, $\delta(\nu)\ne 0$ by $\mathcal U$. Later we will show that all finite dimensional irreducible modules are from $Irrep^{n!}$.

The $GL(2,\mathbb Z)$ action on double affine Hecke algebras {#Four}
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One of the most important properties of the double affine Hecke algebra $H_{q,\tau}$ is the existence of the action of $GL(2,\mathbb Z)$ [@Ch4]. To explain how this group acts on the double affine Hecke algebra we need to introduce pairwise commutative elements $Y_i\in H_{q,\tau}$: $$\label{Y}
Y_i=T_1\dots T_{n-i}\pi^{-1} T^{-1}_{n-i+1}\dots T^{-1}_{n-1},
\quad i=1,\dots,n-1.$$ These elements satisfy the relations $$\begin{gathered}
\label{YT1}
T_i Y_{i+1} T_i= Y_i, \quad (1\le i<n)\\ T_i Y_j=Y_jT_i,\mbox{ if
} j-i\ne 0,1.\label{YT2}\end{gathered}$$

The 