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

Abstract

In this note we give a new proof of the Macdonald-Mehta-Opdam integral identity for finite Coxeter groups (in the equal parameter case). This identity was conjectured by Macdonald and proved by Opdam in \cite{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 multivariable Bessel functions), and uniform (does not refer to the classification of finite Coxeter groups). \footnote{We expect that this proof can be generalized to the case of non-equal parameters. Indeed, many of the steps of our proof, including key Proposition \ref{l2}, generalize without effort to this setting.