[@Ch3]. This construction is especially simple in the case $q=1$. In this case the algebra $H_{1,\tau}$ is a quotient of the so called double affine braid group $\mathfrak
{B}_n$.

For better understanding of this group the reader may have in mind the picture analogous to the geometric interpretation of the usual braid group but in the case when the points live on the two dimensional torus.

In this picture the elements $T_i$, $i=1,\dots,n-1$ correspond to the paths which permute of the $i$-th and the $i+1$-th nearby points (just like in the case of the usual braid group) and $X_i$, $Y_i$ correspond to the paths in which the $i$-th point goes along the parallel, respectively the meridian of the torus. In this geometric setting it is obvious that formula (\[com\]) holds in $\mathfrak{ B}_n$. The double affine Hecke algebra is a quotient of $\mathfrak{B}_n$ by the relations (\[T\^2\]). Hence (\[com\]) holds in $H_{1,\tau}$. Below we give formal definitions to justify this reasoning.

Let $$U=\{z\in\mathbb C^n| z_k-z_l\notin \mathbb Z+ {\rm
i}\mathbb Z, k\ne l \},$$ and $\bar W=S_n\ltimes(\mathbb
Z\oplus\mathbb Z {\rm i})$ acts on $z\in{U}$ by the formula: $$\bar{w}({z})=w(z+a+{\rm i} b), \quad \bar{w}=w(a+{\rm i}b), \quad
a,b\in \mathbb Z^n,\quad w\in S_n.$$

We fix 