useful consultations about Hecke algebras and to Yuri Berest for his consultations on the Calogero-Moser space.

Definitions
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Definition of the double affine Hecke algebra corresponding to $GL(n,{\mathbb C})$
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We denote this algebra by the symbol $H_{q,t}$. It is generated by the elements $T_i$, $1\le i\le n-1$, $\pi$, $X_i^{\pm 1}$, $1\le i\le n$ with relations $$\begin{gathered}
X_iX_j=X_jX_i,\quad (1\le i,j\le n),\label{dbre1}\\ T_i X_i
T_i=X_{i+1},\quad (1\le i<n),\\ T_iX_j=X_jT_i,\mbox{ if } j-i\ne
0,1\\ [T_i,T_j]=0,\mbox{ if } |i-j|>1\\ T_i T_{i+1} T_i=T_{i+1}T_i
T_{i+1}, (1\le i<n),\\ \pi X_i=X_{i+1}\pi \quad (1\le i\le
n-1),\quad \pi X_n=q^{-1}X_1\pi,\\ \pi T_i=T_{i+1}\pi,\quad \pi^n
T_j=T_j\pi^n,\quad (1\le i<n-1, 1\le j<n)
,\label{dbre2}\\(T_i-\tau)(T_i+\tau^{-1})=0,\quad (1\le i\le
n).\label{T^2}\end{gathered}$$

To identify this definition with the standard definition from the papers of Cherednik one should replace $\tau$ by $t^{\frac12}$ and $q$ by $q^{\frac12}$. Also, some definitions use the element $T_0=\pi T_{n-1}\pi^{-1}$.

The double affine Hecke algebra corresponding to $SL(n,{\mathbb C})$ is a quotient of the subalgebra of $H_{q,\tau}$ generated by $X_i/X_{i+1}$, $T_i$, $\pi$, $1\le i\le n-1$, by one extra relation: $$\pi^n=1.$$

Definition of the Calogero-Moser space {#CMsec}
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Let $E$ be an $n$-dimensional vector space (over ${\mathbb C}$). We denote by the symbol $CM'_\tau$ the subset of $GL(E)\times GL(E) \times
E\times  E^*$ consisting of the elements $(X,Y,U,V)$ satisfying the 