Macdonald's Constant-Term Conjecture

Macdonald's constant term conjectures are related to root systems of Lie algebras (Macdonald 1982, Andrews 1986). They can be regarded as generalizations of Dyson's conjecture (Dyson 1962), its -analog due to Andrews, and Mehta's conjecture (Mehta 2004). The simplest of these states that if is a root system, then the constant term in , where is a nonnegative integer, is , where the are fixed integer parameters of the root system corresponding to the fundamental invariants of the Weyl group of (Andrews 1986, p. 41).

Opdam (1989) proved the case for all root systems. The general conjecture had remained "almost proved" for some time, since the infinite families were accomplished by Zeilberger-Bressoud (), Kadell (, ), and Gustafson (, ), while the exceptional cases were done by Zeilberger and (independently) Habsieger (), Zeilberger ( dual), and Garvan and Gonnet ( and dual), using Zeilberger's method. This left only the three root systems (, , ) which were infeasible to address using existing computers. In the meanwhile, however, Cherednik (1993) proved the constant term conjectures for all root systems using a methodology not dependent on classification.

A special case of the constant-term conjecture is given by the assertion that the constant term in

(1)

is . Another special case asserts that the constant term in

(2)

is

(3)

(Andrews 1986, p. 41).