the space of all finite dimensional representations. The main tool of this section is the faithful representation of $H_{1,\tau}$ which is the quasiclassical limit of the standard realization of $H_{q,\tau}$ as a subring of the ring of reflection difference operators [@Ch4].

Limit of the Lusztig-Demazure operators
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Let us introduce the ring $\tilde{R}={\mathbb C}[P_1^{\pm 1}, \dots,
P_n^{\pm 1},X_1^{\pm 1},\dots,X_n^{\pm 1}]_{\delta(X)}\# S_n$, where the subscript $\delta(X)$ means localization by the ideal generated by $\delta(X)=\prod_{1\le i<j\le n}(X_i-X_j)$ and $\#$ is a notation for the smash product. Let us explain what the smash product is. For brevity we will use notation ${\mathbb C}[P^{\pm 1},
X^{\pm 1}]$ for the ring $\mathbb C[P_1^{\pm 1}, \dots, P_n^{\pm
1},X_1^{\pm 1},\dots,X_n^{\pm 1}]$

An element of the ring $\tilde{R}$ has the form $\sum_{w\in S_n}
F_w(P,X) w$. The group $S_n$ acts on the ring $R=\mathbb C[P^{\pm
1},X^{\pm 1}]_{\delta(X)}$ by the formulas $$P_i^w=P_{w^{-1}(i)},\quad X_i^w=X_{w^{-1}(i)},$$ and $$F(P,X)wF'(P,X)w'=F(P,X)(F')^{w}(P,X)ww'.$$

[@Ch4] The following formulas give an injective homomorphism of $H_{1,\tau}\to \tilde{R}$: $$\begin{gathered}
X^\mu\mapsto X^\mu,\\ T_i\mapsto \tau s_i+ \frac{\tau-\tau^{-1}}
{X_{i}/X_{i+1}-1}(s_i-1), \quad i=1,\dots,n-1,\\ \pi\mapsto
P_1^{-1}c,\end{gathered}$$ where $s_i=(i,i+1)\in S_n$ is a transposition and $c\in S_n$ is a cyclic transformation: $c(i)=i+1, i=1,\dots,n-1$, $c(n)=1$.

The homomorphism from the proposition is a quasiclassical limit of the Lusztig-Demazure representation [@Ch4]. For brevity 