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On Cherednik--Macdonald--Mehta identities
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## On Cherednik--Macdonald--Mehta identities

### Pavel Etingof and Alexander Kirillov, Jr.

**Abstract.**
In this note we give a proof of Cherednik's generalization of
Mac\-donald--Mehta identities for the root system $A_{n-1}$, using
representation theory of quantum 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 gives a nice interpretation for these identities in terms of
representations of quantum groups and seems to be simpler than
that of Cherednik.

*Copyright 1998 American Mathematical Society*

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#### Article Info

- ERA Amer. Math. Soc.
**04** (1998), pp. 43-47
- Publisher Identifier: S 1079-6762(98)00045-6
- 1991
*Mathematics Subject Classification*. Primary 5E35
*Key words and phrases*. Macdonald polynomials
- Received by the editors April 14, 1998
- Posted on June 11, 1998
- Communicated by David Kazhdan
- Comments (When Available)

**Pavel Etingof**

Department of Mathematics, Harvard University, Cambridge, MA 02138

*E-mail address:* `etingof@math.harvard.edu`

**Alexander Kirillov, Jr.**

Department of Mathematics, MIT, Cambridge, MA 02139

*E-mail address:* `kirillov@math.mit.edu`

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