A Selberg integral for the Lie algebra A_n

TL;DR

This paper proves an A_n analogue of the Selberg integral using Macdonald polynomials, confirming a conjecture about the existence of such integrals for all simple Lie algebras.

Contribution

It introduces a new q-binomial theorem for Macdonald polynomials to establish the Selberg integral for the Lie algebra A_n, advancing the understanding of integrals associated with Lie algebras.

Findings

Established an A_n analogue of the Selberg integral

Confirmed the g=A_n case of Mukhin and Varchenko's conjecture

Utilized a new q-binomial theorem for Macdonald polynomials

Abstract

A new q-binomial theorem for Macdonald polynomials is employed to prove an A_n analogue of the celebrated Selberg integral. This confirms the g=A_n case of a conjecture by Mukhin and Varchenko concerning the existence of a Selberg integral for every simple Lie algebra g.