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Representation Theory

Published by the American Mathematical Society, the Representation Theory (ERT) is devoted to research articles of the highest quality in all areas of pure and applied mathematics.

ISSN 1088-4165

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The adjoint representation of a reductive group and hyperplane arrangements
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by J. Matthew Douglass PDF
Represent. Theory 3 (1999), 444-456 Request permission


Let $G$ be a connected reductive algebraic group with Lie algebra $\mathfrak g$ defined over an algebraically closed field, $k$, with $\operatorname {char} k=0$. Fix a parabolic subgroup of $G$ with Levi decomposition $P=LU$ where $U$ is the unipotent radical of $P$. Let $\mathfrak u=\operatorname {Lie}(U)$ and let $\mathfrak z$ denote the center of $\operatorname {Lie}(L)$. Let $T$ be a maximal torus in $L$ with Lie algebra $\mathfrak t$. Then the root system of $(\mathfrak g, \mathfrak t)$ is a subset of $\mathfrak t^*$ and by restriction to $\mathfrak z$, the roots of $\mathfrak t$ in $\mathfrak u$ determine an arrangement of hyperplanes in $\mathfrak z$ we denote by $\mathcal A^{\mathfrak z}$. In this paper we construct an isomorphism of graded $k[\mathfrak z]$-modules $\operatorname {Hom}_G(\mathfrak g^*, k[{G\times ^P(\mathfrak z+\mathfrak u)}]) \cong D(\mathcal A^{\mathfrak z})$, where $D(\mathcal A^{\mathfrak z})$ is the $k[\mathfrak z]$-module of derivations of $\mathcal A^{\mathfrak z}$. We also show that $\operatorname {Hom}_G(\mathfrak g^*, k[{G\times ^P(\mathfrak z+\mathfrak u)}])$ and $k[\mathfrak z] \otimes \operatorname {Hom}_G(\mathfrak g^*, k[G \times ^P \mathfrak u])$ are isomorphic graded $k[\mathfrak z]$-modules, so $D(\mathcal A^{\mathfrak z})$ and $k[\mathfrak z] \otimes \operatorname {Hom}_G(\mathfrak g^*, k[G \times ^P \mathfrak u])$ are isomorphic, graded $k[\mathfrak z]$-modules. It follows immediately that $\mathcal A^{\mathfrak z}$ 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 $\mathfrak g$ affords a self-dual representation of $G$, we recover a result of Sommers, Trapa, and Broer which states that the degrees in which the adjoint representation of $G$ occurs as a constituent of the graded, rational $G$-module $k[G\times ^P \mathfrak u]$ are the exponents of $\mathcal A^{\mathfrak z}$. 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:
  • Received by editor(s): March 8, 1999
  • Received by editor(s) in revised form: September 28, 1999
  • Published electronically: November 9, 1999
  • © Copyright 1999 American Mathematical Society
  • Journal: Represent. Theory 3 (1999), 444-456
  • MSC (1991): Primary 22E46
  • DOI:
  • MathSciNet review: 1722107