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 | References | Similar Articles | Additional Information

Abstract: In this note we give a proof of Cherednik's generalization of Macdonald-Mehta identities for the root system , 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.

**[Ch1]**I. Cherednik,*Double affine Hecke algebras and Macdonald's conjectures*, Annals of Math.**141**(1995), 191-216. MR**96m:33010****[Ch2]**-,*Difference Macdonald-Mehta conjecture*, Internat. Math. Res. Notices**1997**, 449-467. CMP**97:12****[EK1]**P. Etingof and A. Kirillov, Jr.,*Macdonald's polynomials and representations of quantum groups*, Math. Res. Let.**1**(1994), 279-296. MR**96m:17025****[EK2]**-,*Representation-theoretic proof of the inner product and symmetry identities for Macdonald's polynomials*, Compos. Math.**102**(1996), 179-202. MR**97j:17013****[Kas]**C. Kassel,*Quantum groups*, Springer-Verlag, New York, 1995. MR**96e:17041****[Kir1]**A. Kirillov, Jr.,*On an inner product in modular tensor categories*, J. Amer. Math. Soc.**9**(1996), 1135-1169. MR**97f:18007****[Kir2]**-,*Lectures on affine Hecke algebras and Macdonald's conjectures*, Bull. Amer. Math. Soc.**34**(1997), 251-292. CMP**97:13****[Kos]**B. Kostant,*On Macdonald's -function formula, the Laplacian and generalized exponents*, Advances in Math.**20**(1976), 179-212. MR**58:5484****[M1]**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.**[M2]**-,*Orthogonal polynomials associated with root systems*, preprint (1988).**[M3]**-,*Symmetric functions and Hall polynomials*, 2nd ed., The Clarendon Press, Oxford University Press, New York, 1995. MR**96h:05207**

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Additional Information

**Pavel Etingof**

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

Email:
etingof@math.harvard.edu

**Alexander Kirillov Jr.**

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

Email:
kirillov@math.mit.edu

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

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