## 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

### Citations:

Zbl 0498.17006; Zbl 0669.33008
Full Text:

### References:

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