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Invariants of reflection groups, arrangements, and normality of decomposition classes in Lie algebras

Part of: Projective and enumerative geometry General commutative ring theory Special aspects of infinite or finite groups

Published online by Cambridge University Press:  19 March 2012

J. Matthew Douglass  and
Gerhard Röhrle
J. Matthew Douglass
Affiliation:
Department of Mathematics, University of North Texas, Denton, TX 76203, USA (email: douglass@unt.edu)
Gerhard Röhrle
Affiliation:
Fakultät für Mathematik, Ruhr-Universität Bochum, D-44780 Bochum, Germany (email: gerhard.roehrle@rub.de)
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Abstract

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Suppose that W is a finite, unitary, reflection group acting on the complex vector space V and X is a subspace of V. Define N to be the setwise stabilizer of X in W, Z to be the pointwise stabilizer, and C=N/Z. Then restriction defines a homomorphism from the algebra of W-invariant polynomial functions on V to the algebra of C-invariant functions on X. In this note we consider the special case when W is a Coxeter group, V is the complexified reflection representation of W, and X is in the lattice of the arrangement of W, and give a simple, combinatorial characterization of when the restriction mapping is surjective in terms of the exponents of W and C. As an application of our result, in the case when W is the Weyl group of a semisimple, complex Lie algebra, we complete a calculation begun by Richardson in 1987 and obtain a simple combinatorial characterization of regular decomposition classes whose closure is a normal variety.

Keywords

arrangementsCoxeter groupsdecomposition classesinvariants

MSC classification

Secondary: 20F55: Reflection and Coxeter groups 14N20: Configurations and arrangements of linear subspaces 13A50: Actions of groups on commutative rings; invariant theory
Type
Research Article
Information
Compositio Mathematica , Volume 148 , Issue 3 , May 2012 , pp. 921 - 930
Copyright
Copyright © Foundation Compositio Mathematica 2012

References

[Bor81]Borho, W., Über Schichten halbeinfacher Lie-Algebren, Invent. Math. 65 (1981), 283–317.CrossRefGoogle Scholar
[Bou68]Bourbaki, N., Éléments de mathématique. Groupes et algèbres de Lie. Chapitres IV–VI, in Actualités scientifiques et industrielles, no. 1337 (Hermann, Paris, 1968).Google Scholar
[Bro98]Broer, A., Decomposition varieties in semisimple Lie algebras, Canad. J. Math. 50 (1998), 929–971.CrossRefGoogle Scholar
[DL95]Denef, J. and Loeser, F., Regular elements and monodromy of discriminants of finite reflection groups, Indag. Math. (N.S.) 6 (1995), 129–143.CrossRefGoogle Scholar
[Dou99]Douglass, J. M., The adjoint representation of a reductive group and hyperplane arrangements, Represent. Theory 3 (1999), 444–456.CrossRefGoogle Scholar
[GHLMP96]Geck, M., Hiß, G., Lübeck, F., Malle, G. and Pfeiffer, G., CHEVIE – A system for computing and processing generic character tables, Appl. Algebra Engrg. Comm. Comput. 7 (1996), 175–210.CrossRefGoogle Scholar
[How80]Howlett, R. B., Normalizers of parabolic subgroups of reflection groups, J. Lond. Math. Soc. (2) 21 (1980), 62–80.CrossRefGoogle Scholar
[LS99]Lehrer, G. I. and Springer, T. A., Intersection multiplicities and reflection subquotients of unitary reflection groups. I, Geometric Group Theory Down Under (Canberra, 1996) (de Gruyter, Berlin, 1999), 181–193.Google Scholar
[OS83]Orlik, P. and Solomon, L., Coxeter arrangements, in Singularities, Proceedings in Symposium in Pure Mathematics, vol. 40 (American Mathematical Society, Providence, RI, 1983), 269–292.CrossRefGoogle Scholar
[OT92]Orlik, P. and Terao, H., Arrangements of hyperplanes (Springer, Berlin, 1992).CrossRefGoogle Scholar
[OT93]Orlik, P. and Terao, H., Coxeter arrangements are hereditarily free, Tôhoku Math. J. (2) 45 (1993), 369–383.CrossRefGoogle Scholar
[Ric87]Richardson, R. W., Normality of G-stable subvarieties of a semisimple Lie algebra, in Algebraic groups: Utrecht 1986, Lecture Notes in Mathematics, vol. 1271 (Springer, Berlin, 1987), 243–264.CrossRefGoogle Scholar
[Sch97]Schönert, M.et al., GAP – Groups, Algorithms, and Programming – version 3 release 4, RWTH Aachen (1997).Google Scholar
[Spr74]Springer, T. A., Regular elements of finite reflection groups, Invent. Math. 25 (1974), 159–198.CrossRefGoogle Scholar
[Ter80]Terao, H., Arrangements of hyperplanes and their freeness I, J. Fac. Sci. Univ. Tokyo Sect. IA Math. 27 (1980), 293–320.Google Scholar