The adjoint representation of a reductive group and hyperplane arrangements

Author:
J. Matthew Douglass

Journal:
Represent. Theory **3** (1999), 444-456

MSC (1991):
Primary 22E46

DOI:
https://doi.org/10.1090/S1088-4165-99-00066-7

Published electronically:
November 9, 1999

MathSciNet review:
1722107

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Abstract: Let be a connected reductive algebraic group with Lie algebra defined over an algebraically closed field, , with . Fix a parabolic subgroup of with Levi decomposition where is the unipotent radical of . Let and let denote the center of . Let be a maximal torus in with Lie algebra . Then the root system of is a subset of and by restriction to , the roots of in determine an arrangement of hyperplanes in we denote by . In this paper we construct an isomorphism of graded -modules , where is the -module of derivations of . We also show that and are isomorphic graded -modules, so and are isomorphic, graded -modules. It follows immediately that is a free hyperplane arrangement. This result has been proved using case-by-case arguments by Orlik and Terao. By keeping track of the gradings involved, and recalling that affords a self-dual representation of , we recover a result of Sommers, Trapa, and Broer which states that the degrees in which the adjoint representation of occurs as a constituent of the graded, rational -module are the exponents of . This result has also been proved, again using case-by-case arguments, by Sommers and Trapa and independently by Broer.

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Additional Information

**J. Matthew Douglass**

Affiliation:
Department of Mathematics, University of North Texas, Denton, Texas 76203

Email:
douglass@unt.edu

DOI:
https://doi.org/10.1090/S1088-4165-99-00066-7

Received by editor(s):
March 8, 1999

Received by editor(s) in revised form:
September 28, 1999

Published electronically:
November 9, 1999

Article copyright:
© Copyright 1999
American Mathematical Society