Affine root systems and Dedekind's?-function

An Approach to Script N = 4 ADE Gauge Theory onK3

“…The second one is that 1/η(τ /N ) can be written as the form θ 2 A N−1 (τ )/η(τ ) N , where θ 2 A N−1 (τ ) is a kind of A N −1 theta function [7]. This identity is easily verified from the celebrated denominator identity of the affine Lie algebra [8,12]. These two facts enabled us to rewrite the partition function of SU (N ) theory on K3 into the desired form.…”

A generalization of Selberg’s beta integral

“…Selberg's integral ( 1 ) has had diverse applications in fields ranging from number theory, physics, statistics, combinatorics, algebra and analysis. Two particular applications were a use by Bombieri to prove Mehta's conjecture [18] and by Macdonald [17] to prove some of his conjectures (q = 1 case) for the affine root systems (for definition and properties see [15]) of types S(JBC 7 ), 5 , (5 / ), 5(5 ; ) v , S{C/) 9 5(C 7 ) V and 5(Z) Z ) for all /> 1 (when defined). Just as Macdonald used integral (1) to prove some of his (q = 1) conjectures, we will use integral (2) to prove for the same set of affine root systems the corresponding Macdonald-Morris conjectures with arbitrary parameter q .…”

Macdonald’s constant term conjectures for exceptional root systems

“…Since every affine root system is a direct sum of irreducible ones [Mai,Kac,Hu2], it suffices to prove the conjecture for the latter. Recently, Gustafson [Gu] completed the proof of Macdonald's conjecture for all the infinite families by proving it for the infinite families S (C n …”