space $V^{A^{n-1}_\tau}$ of $A^{n-1}_\tau$-invariants has dimension $n$. The relations inside $H_{1,\tau}$ and (\[YT2\]) imply that $X_1$ and $Y_1$ commute with the action of $A^{n-1}_\tau$. Thus if we fix a basis in $V$ we get $X_1|_{V^{A^{n-1}_\tau}}, Y_1|_{V^{A^{n-1}_\tau}}\in
GL(n,\mathbb C)$. The following statement is a key statement of the section.

\[rk1\] Let $V\in Irrep^{n!}$ then the operators $\bar{X}_1=X_1|_{V^{A^{n-1}_\tau}}$, $\bar{Y}_1=Y_1|_{V^{A^{n-1}_\tau}}$ satisfy the equation: $$rk(\bar{X}_1\bar{Y}_1\bar{X}^{-1}_1\bar{Y}^{-1}_1-\tau^{-2} Id)=1.$$

Obviously the space $CM_\tau$ is isomorphic to the quotient of the space of solutions of (\[eqXY-YX\]) by the action of $GL(n,{\mathbb C})$. Thus the last proposition proves that the map $\Phi$: $Irrep^{n!}\to CM_\tau$, $\Phi(V)=(\bar{X}_1,\bar{Y}_1)$ is well defined.

In the rest of the section we prove Proposition \[rk1\]. It is done in two steps. First we prove

\[lcom\] The elements $X_1,Y_1\in H_{1,\tau}$ satisfy the relation $$\label{com}
X_1Y_1X^{-1}_1Y^{-1}_1=T_1T_2\dots T_{n-2}T_{n-1}^2T_{n-2}\dots
T_1.$$

This is done in the next subsection using the geometric interpretation of the double affine Hecke algebra. The proof of this lemma was communicated to the author by Ivan Cherednik. The last step is the analysis of the LHS of (\[com\]) using the quasiclassical limit $\tau\to 1$. It is done in the last subsection.

The double affine braid group
-----------------------------

The double affine Hecke algebra admits a simple topological interpretation 