[Paper Review] A uniform proof of the Macdonald-Mehta-Opdam identity for finite Coxeter groups

Pavel Etingof|ArXiv.org|Mar 29, 2009

Advanced Combinatorial Mathematics2 references3 citations

TL;DR

This paper presents a uniform, classification-free proof of the Macdonald-Mehta-Opdam identity for finite Coxeter groups using rational Cherednik algebras and bilinear forms on polynomial representations. The authors establish a new integral identity involving the discriminant polynomial and gamma functions, avoiding multivariable Bessel functions and computer-assisted calculations, and prove the identity holds uniformly across all finite Coxeter groups by analyzing the structure of the contravariant form and its roots via contour integration and asymptotic analysis.

ABSTRACT

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 1993 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 multivariable Bessel functions), and uniform (does not refer to the classification of finite Coxeter groups and does not use computers).

Motivation & Objective

• To provide a uniform and elementary proof of the Macdonald-Mehta-Opdam integral identity for all finite Coxeter groups, independent of the classification of these groups.
• To eliminate reliance on multivariable Bessel functions and computer-assisted computations, particularly in non-crystallographic cases.
• To establish the identity via the structure of the contravariant bilinear form on the polynomial representation of rational Cherednik algebras.
• To show that the roots of the form $ b(k) = \beta_k(\Delta, \Delta) $ are exactly $ -m/d_i - p_{i,m} $, and prove $ p_{i,m} = 0 $, thereby confirming the full identity.

Proposed method

• Use of the rational Cherednik algebra $ H_k $ and its polynomial representation $ M_k $, with Dunkl operators acting as $ y_a $-derivatives.
• Construction of a unique $ W $-invariant symmetric bilinear form $ \beta_k $ on $ M_k $ with $ \beta_k(1,1) = 1 $, satisfying a contravariance condition.
• Analysis of the form $ b(k) = \beta_k(\Delta, \Delta) $, showing it is a polynomial of degree $ |S| $ with negative rational roots.
• Application of contour deformation and analytic continuation to relate the integral $ \int_{\mathfrak{h}_\mathbb{R}} e^{-(x,x)/2} |\Delta(x)|^{2k} dx $ to the gamma function product.
• Reduction of the identity to rank 2 cases via parabolic subgroups of rank 2, using a combinatorial identity involving $ \psi(W) = 3|S|^2 - \sum_i (d_i^2 - 1) $.
• Use of asymptotic analysis and Taylor expansion up to order $ k^3 $ to show that the second derivative of the integral matches the gamma function side, proving $ p_{i,m} = 0 $.

Experimental results

Research questions

• RQ1 Can the Macdonald-Mehta-Opdam identity be proven uniformly across all finite Coxeter groups without relying on classification or case-by-case analysis?
• RQ2 Is it possible to avoid the use of multivariable Bessel functions and computer calculations in proving the identity, especially in non-crystallographic cases?
• RQ3 What is the precise structure of the roots of the polynomial $ b(k) = \beta_k(\Delta, \Delta) $, and how do they relate to the degrees $ d_i $ of the invariants?
• RQ4 Does the identity hold in $ \mathbb{C}[k]/k^3 $, and can this imply the full identity via the strict monotonicity of $ (\log \Gamma)'' $?
• RQ5 Can the identity be proven in a way that is independent of the classification of finite Coxeter groups, using only algebraic and analytic tools on Cherednik algebras?

Key findings

• The Macdonald-Mehta-Opdam identity holds uniformly for all finite Coxeter groups: $ (2\pi)^{-r/2} \int_{{\mathfrak{h}}_{\mathbb{R}}} e^{-(x,x)/2} |\Delta(x)|^{2k} dx = \prod_{i=1}^r \frac{\Gamma(1 + k d_i)}{\Gamma(1 + k)} $.
• The form $ b(k) = \beta_k(\Delta, \Delta) $ is a polynomial of degree $ |S| $, with roots at $ -m/d_i - p_{i,m} $, where $ 1 \leq m \leq d_i - 1 $, and $ p_{i,m} \in \mathbb{Z}_{\geq 0} $.
• It is shown that $ p_{i,m} = 0 $ for all $ i,m $, which implies the exact form $ b(k) = |W| \prod_{i=1}^r \prod_{m=1}^{d_i - 1} (k d_i + m) $, confirming the identity.
• The second derivative of the integral at $ k = 0 $ matches the second derivative of the gamma product, proving the identity up to $ k^3 $, which suffices to fix the normalization.
• The proof is classification-free and avoids multivariable Bessel functions, providing a uniform and self-contained argument valid for all finite Coxeter groups.
• The method reduces the identity to rank 2 cases via parabolic subgroups, and uses a combinatorial identity involving $ \psi(W) $, which sums over rank 2 parabolic subgroups.