On Cherednik-Macdonald-Mehta identities

Authors:
Pavel Etingof and Alexander Kirillov, Jr.

Journal:
Electron. Res. Announc. Amer. Math. Soc. **4** (1998), 43-47

MSC (1991):
Primary 05E35

DOI:
https://doi.org/10.1090/S1079-6762-98-00045-6

Published electronically:
June 11, 1998

MathSciNet review:
1626789

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Abstract: In this note we give a proof of Cherednik’s generalization of Macdonald–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.

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*On an inner product in modular tensor categories*, J. Amer. Math. Soc. **9** (1996), no. 4, 1135–1169. MR **1358983**, DOI 10.1090/S0894-0347-96-00210-X
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- I. Cherednik,
*Double affine Hecke algebras and Macdonald’s conjectures*, Annals of Math. **141** (1995), 191–216.
- —,
*Difference Macdonald–Mehta conjecture*, Internat. Math. Res. Notices **1997**, 449–467.
- P. Etingof and A. Kirillov, Jr.,
*Macdonald’s polynomials and representations of quantum groups*, Math. Res. Let. **1** (1994), 279–296.
- —,
*Representation-theoretic proof of the inner product and symmetry identities for Macdonald’s polynomials*, Compos. Math. **102** (1996), 179–202.
- C. Kassel,
*Quantum groups*, Springer-Verlag, New York, 1995.
- A. Kirillov, Jr.,
*On an inner product in modular tensor categories*, J. Amer. Math. Soc. **9** (1996), 1135–1169.
- —,
*Lectures on affine Hecke algebras and Macdonald’s conjectures*, Bull. Amer. Math. Soc. **34** (1997), 251–292.
- B. Kostant,
*On Macdonald’s $\eta$-function formula, the Laplacian and generalized exponents*, Advances in Math. **20** (1976), 179–212.
- I. G. Macdonald,
*A new class of symmetric functions*, Publ. I.R.M.A. Strasbourg, 372/S-20, Actes 20 Séminaire Lotharingien (1988), 131–171.
- —,
*Orthogonal polynomials associated with root systems*, preprint (1988).
- —,
*Symmetric functions and Hall polynomials*, 2nd ed., The Clarendon Press, Oxford University Press, New York, 1995.

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Additional Information

**Pavel Etingof**

Affiliation:
Department of Mathematics, Harvard University, Cambridge, MA 02138

MR Author ID:
289118

Email:
etingof@math.harvard.edu

**Alexander Kirillov, Jr.**

Affiliation:
Department of Mathematics, MIT, Cambridge, MA 02139

Email:
kirillov@math.mit.edu

Keywords:
Macdonald polynomials

Received by editor(s):
April 14, 1998

Published electronically:
June 11, 1998

Communicated by:
David Kazhdan

Article copyright:
© Copyright 1998
American Mathematical Society