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.


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
Full Text: DOI EuDML


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