A uniform proof of the Macdonald-Mehta-Opdam identity for finite Coxeter groups

A Uniform Proof of the Macdonald-Mehta-Opdam Identity for Finite Coxeter Groups

Citation Etingof, Pavel. "A uniform proof of the Macdonald-Mehta-Opdam identity for finite Coxeter groups. The MIT Faculty has made this article openly available. Please share how this access benefits you. Your story matters.

Macdonald-Mehta-Opdam identity for finite Coxeter groups

In this note we give a new proof of the Macdonald-Mehta-Opdam integral identity for finite Coxeter groups. This identity was conjectured by Macdonald and proved by Opdam in [O1, O2] using the theory of multivariable Bessel functions, but in non-crystallographic cases the proof relied on a computer calculation by F. Garvan. Our proof is somewhat more elementary (in particular, it does not use mu...

Difference Macdonald - Mehta conjecture

with respect to the Gaussian measure. Macdonald extended it from An−1 to other root systems and verified his conjecture for classical ones by means of the Selberg integrals [M1]. It was established by Opdam in [O1] in full generality using the shift operators. The integral is an important normalization constant for a k-deformation of the Hankel transform introduced by Dunkl [D]. The generalized...

Diierence Macdonald-mehta Conjecture

On Cherednik–macdonald–mehta Identities

In this note we give a proof of Cherednik’s generalization of Macdonald–Mehta identities for the root system An−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 represen...