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**Double affine Hecke algebras and Macdonald’s conjectures.**
*(English)*
Zbl 0822.33008

This paper proves Ian Macdonald’s constant term conjectures which express the constant term in \(\prod_{\alpha \in R} (1 - e^ \alpha)^{k_ \alpha}\) as an explicit product. \(R\) is a reduced root system which may be either finite or affine, and in general the constants \(k_ \alpha\) must be the same for roots of the same length. As an example, if \(R\) is finite, then
\[
CT \prod_{\alpha \in R} (1 - e^ \alpha)^ k = \prod^ n_{i = 1} {kd_ i \choose k},
\]
where \(d_ i\) are the degrees of the fundamental invariants of the Weyl group of \(R\). The corresponding affine case can be written as
\[
CT \prod_{\alpha \in R^ +} \prod^ k_{j = 1} (1 - q^{j - 1} e^{- \alpha}) (1 - q^ j e^ \alpha) = \prod^ n_{i = 1} \prod^ k_{j = 1} {1 - q^{kd_ i - k + j} \over 1 - q^ j}.
\]
Much of the initial impetus for studying these constant term identities is that they imply evaluations of multi- dimensional beta integrals. Several special cases were known before Macdonald stated his conjectures. Throughout the 1980s, the affine forms of Macdonald’s conjectures were tackled one root system at a time: \(A_ n\) (the only root systems for which the \(k_ \alpha\) may be independent) by Doron Zeilberger and this reviewer, \(BC_ n\) and \(D_ n\) by Kevin Kadell, \(G_ 2\) by Laurent Habsieger and Doron Zeilberger, \(F_ 4\) by Frank Garvan and Gaston Gonnet. In 1988, Macdonald suggested a more general setting for these conjectures in terms of the value of a certain scalar product. It is this more general conjecture that Cherednik proves, building on the work on Heckman, Opdam, and Dunkl. The actual technique is based on the realization of the algebras under investigation in terms of Demazure-Lusztig operators.

Reviewer: D.M.Bressoud (St.Paul)