On Cherednik-Macdonald-Mehta identities

In this note we give a proof of Cherednik's generalization of Macdonald-Meh ta identities for the root system A(n-1), using representation theory of qu antum groups. These identities give an explicit formula for the integral of a product of Macdonald polynomials with respect to a "difference analogue of the Gaussian measure". They were suggested by Cherednik, who also gave a proof based on representation theory of affine Hecke algberas; our proof g ives a nice interpretation for these identities in terms of representations of quantum groups and seems to be simpler than that of Cherednik.