a point $z^0$ such that its real and imaginary part is sufficiently small.

Paths $\gamma\subset U$ joining $z^0$ with points from $\{ \bar{w}(z^0),\bar{w}\in \bar{W}\}$ modulo homotopy and the action of $\bar{W}$ form the double affine braid group $\mathfrak{
B}_n$ with the multiplication induced by the usual composing operation for the paths.

This group is generated by the elements: $$\begin{gathered}
T_j=t_j(\psi)=z^0+(exp(\pi {\rm
i}\psi)-1)(z_j-z_{j+1})(e_j-e_{j+1})\\ X_j=x_j(\psi)=z^0+\psi
e_j,\quad Y_j=y_j(\psi)=z^0+\psi e_j {\rm i},\\ \pi=T_1\dots
T_{n-1} Y_1,\end{gathered}$$ where $0\le\psi\le 1$.

[@Ch3] The group $\mathfrak{ B}_n$ is generated by elements $X_i$, $Y_i$, $i=1,\dots,n$, $T_j$, $j=1,\dots,n-1$ (and $\pi$) with defining relations (\[dbre1\]-\[dbre2\]), (\[YT1\]-\[YT2\]) with $q=1$.

Thus $H_{1,\tau}$ is a quotient of $\mathfrak{B}_n$ by relations (\[T\^2\]) and Lemma \[lcom\] follows.

The spectrum of $Z=T_1\dots T_{n-2}T_{n-1}^2 T_{n-2}\dots T_1$
--------------------------------------------------------------

For a representation $V$ is from $Irrep^{n!}$ there is an isomorphism $V\simeq A^n_\tau$ of left $A^n_\tau$-modules. Hence the right multiplication on $A^n_\tau$ induces a structure of a right $A^n_\tau$-module on $V$ and as a consequence on $V^{A^{n-1}_\tau}$.

The right $A^{n}_\tau$-module $V^{A^{n-1}_\tau}$ is a sum of the $n-1$ dimensional vector representation and one-dimensional representation because it is true for $\tau=1$. Obviously, the operator $Z$ (acting by the left multiplication) commutes with the right action of $A^n_\tau$. Hence by the Schur lemma $Z$ 