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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- ---, 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
