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Variation on a theme of Macdonald. (English) Zbl 0763.33006

In 1982, I. G. Macdonald [Siam J. Math. Anal. 13, 988-1007 (1982; Zbl 0498.17006] published the conjecture that if \(R\) is a reduced root system, \(k\) is a positive integer, and \(d_ 1, \ldots, d_ n\) are the degrees of the fundamental invariants of the Weyl group of \(R\), then \[ [e^ 0] \prod_{\alpha \in R} (1-e^{\alpha})^ k = \prod_{i=1}^ n \left( { kd_ i \atop k} \right),\tag{1} \] where \([e^ 0]F\) denotes the coefficient of \(e^ 0\) in the expansion of \(F\). A more general conjecture in which the root system need not be reduced and in which the exponent of \((1-e^{\alpha})\) may depend on the length of \(\alpha\) was also made. For the root system \(BC_ n\), Macdonald’s conjecture is equivalent to a multidimensional \(\beta\)-integral evaluation discovered and proved by Selberg in 1944. The conjecture was subsequently verified for each root system. In 1988, E. M. Opdam [Comp. Math. 67, 191-209 (1988; Zbl 0669.33008] gave the first uniform proof.
In this paper, the authors point out that equation (1) is equivalent to \[ \int_ T |\Delta(t)|^{2k}dt = \prod_{i=1}^ n \left( { kd_ i \atop k} \right),\tag{2} \] where \(T\) is a compact torus whose unit lattice is an appropriate multiple of the lattice generated by the root system dual to \(R\), \(\Delta := \prod_{\alpha \in R}(e^{\alpha/2}- e^{-\alpha/2})\). They extend this result to prove that \[ \int_ A |\Delta(a)|^{2k}da = \prod_{j=1}^ n {1 \over{2 \sin[\pi k(1-d_ j)]}}\prod_{i=1}^ n \left( {kd_ i \atop k} \right),\tag{3} \] where \(A\) is a connected simply connected Lie group. They also prove a more general result in which the root system need not be reduced and in which the exponent may depend on the length of the root.

MSC:

33C80 Connections of hypergeometric functions with groups and algebras, and related topics
33D05 \(q\)-gamma functions, \(q\)-beta functions and integrals
22E30 Analysis on real and complex Lie groups
11P83 Partitions; congruences and congruential restrictions
05A17 Combinatorial aspects of partitions of integers
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References:

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