we call this homomorphism the Lusztig-Demazure representation.

Actually the paper [@Ch4] contains the proof for the case $q\ne 1$. The proof in the case $q=1$ can be obtained from this proof by mechanical replacement of shifts operators $\tau(\lambda)$, $\lambda\in {\mathbb Z}^n$ by their quasiclassical limits $P^\lambda$. The reader may do this operation with Lecture 5 from the exposition [@K].

The representation $V_{\mu,\nu}$ {#Vml}
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Let $(\mu,\nu)\in (\mathbb C^*)^{2n}$ and $\chi_{\mu,\nu}\simeq
\mathbb C$ be a one dimensional $R$-module (character): $\chi_{\mu,\nu}(R(P,X))=R(\mu,\nu)$. We can induce a finite dimensional module $V_{\mu,\nu}$ from this module: $$V_{\mu,\nu}=\tilde{R}\otimes_{R}\chi_{\mu,\nu}.$$

This module has a $\mathbb C$ basis $w\otimes 1$, $w\in S_n$, hence $\dim V_{\mu,\nu}=n!$.

If $\nu_i\ne \nu_j$, $i\ne j$ then the $H_{1,\tau}$-module $V_{\mu,\nu}$ is irreducible.

The module $V_{\mu,\nu}$ has a natural $H_{\delta(X)}$ module structure. From the Lusztig-Demazure representation we see that $H_{\delta(X)}\simeq {\mathbb C}[P^{\pm 1}, X^{\pm 1}]_{\delta(X)}\#S_n$. The group $S_n$ acts freely on the variety $Spec({\mathbb C}[P^{\pm 1},
X^{\pm 1}]_{\delta(X)})$ hence the algebra $H_{\delta(X)}$ is Morita equivalent to the algebra ${\mathbb C}[P^{\pm 1}, X^{\pm
1}]^{S_n}_{\delta(X)}$. In particular, the module $V_{\mu,\nu}$ corresponds to the one-dimensional representation: $P\mapsto
P(\mu,\nu)$. Thus $V_{\mu,\nu}$ is an irreducible $H_{\delta(X)}$-module and hence an irreducible $H$-module.

The action of the finite Hecke algebra {#fHecke}
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The elements 