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
Full-text PDF Free Access
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 $A_{n-1}$, 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.
- Ivan Cherednik, Double affine Hecke algebras and Macdonald’s conjectures, Ann. of Math. (2) 141 (1995), no. 1, 191–216. MR 1314036, DOI https://doi.org/10.2307/2118632
- ---, Difference Macdonald–Mehta conjecture, Internat. Math. Res. Notices 1997, 449–467.
- Pavel I. Etingof and Alexander A. Kirillov Jr., Macdonald’s polynomials and representations of quantum groups, Math. Res. Lett. 1 (1994), no. 3, 279–296. MR 1302644, DOI https://doi.org/10.4310/MRL.1994.v1.n3.a1
- Pavel I. Etingof and Alexander A. Kirillov Jr., Representation-theoretic proof of the inner product and symmetry identities for Macdonald’s polynomials, Compositio Math. 102 (1996), no. 2, 179–202. MR 1394525
- Christian Kassel, Quantum groups, Graduate Texts in Mathematics, vol. 155, Springer-Verlag, New York, 1995. MR 1321145
- Alexander A. Kirillov Jr., On an inner product in modular tensor categories, J. Amer. Math. Soc. 9 (1996), no. 4, 1135–1169. MR 1358983, DOI https://doi.org/10.1090/S0894-0347-96-00210-X
- ---, Lectures on affine Hecke algebras and Macdonald’s conjectures, Bull. Amer. Math. Soc. 34 (1997), 251–292.
- Bertram Kostant, On Macdonald’s $\eta $-function formula, the Laplacian and generalized exponents, Advances in Math. 20 (1976), no. 2, 179–212. MR 485661, DOI https://doi.org/10.1016/0001-8708%2876%2990186-9
- 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).
- I. G. Macdonald, Symmetric functions and Hall polynomials, 2nd ed., Oxford Mathematical Monographs, The Clarendon Press, Oxford University Press, New York, 1995. With contributions by A. Zelevinsky; Oxford Science Publications. MR 1354144
- 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.
Similar Articles
Retrieve articles in Electronic Research Announcements of the American Mathematical Society
with MSC (1991):
05E35
Retrieve articles in all journals
with MSC (1991):
05E35
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