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Modules of Differential Operators of Order 2 on Coxeter Arrangements

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Abstract

The collection of reflection hyperplanes of a finite reflection group is called a Coxeter arrangement. A Coxeter arrangement is known to be free. K. Saito has constructed a basis consisting of invariant elements for the module of derivations on a Coxeter arrangement. We study the module of \(\mathcal{A}\)-differential operators as a generalization of the study of the module of \(\mathcal{A}\)-derivations. In this article, we prove that the modules of differential operators of order 2 on Coxeter arrangements of types A, B and D are free, by exhibiting their bases. We also prove that the modules cannot have bases consisting of only invariant elements. Two keys for the proof of freeness are the “Cauchy-Sylvester theorem on compound determinants” and the “Saito-Holm criterion for freeness.”

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  1. Hokkaido University, Sapporo, Hokkaido, Japan

    Norihiro Nakashima

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  1. Norihiro Nakashima
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Correspondence to Norihiro Nakashima.

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Nakashima, N. Modules of Differential Operators of Order 2 on Coxeter Arrangements. Algebr Represent Theor 17, 1163–1180 (2014). https://doi.org/10.1007/s10468-013-9440-0

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  • DOI: https://doi.org/10.1007/s10468-013-9440-0

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