Macdonald Identities: revisited
K. Iohara, Y. Saito
TL;DR
The paper revisits Macdonald's denominator identities for untwisted affine root systems and uses folding techniques, notably K. Saito's approach and Kostant's framework, to derive Macdonald identities for twisted affine root systems (excluding BC_l^{(2)}), with BC_l^{(2)} treated separately. By normalizing and specializing the standard denominator identities, the authors obtain eta-product expressions for the left-hand sides and finite-root-system sums on the right-hand side, linking modular forms to root-system data. The results illuminate the role of folding, mean-folding, and affine Weyl groups in generating twisted identities and clarify the structure behind BC_l^{(2)} through explicit eta-quotients and lattice sums. The postscript connects these identities to non-reduced affine systems and affine superalgebras, revealing broader algebraic structures and potential avenues for further exploration in Macdonald theory and related modular phenomena.
Abstract
In this note, after recalling a proof of the Macdonald identities for untwisted affine root systems, we derive the Macdonald identities for twisted affine root systems.