The adjoint representation of a reductive group and hyperplane arrangements
Authors: Douglass J. 
Published: July 2002

Let

G
G

be a connected reductive algebraic group with Lie algebra

g

\mathfrak g

defined over an algebraically closed field,

k
k

, with

char
⁡
k
=
0

\operatorname {char} k=0

. Fix a parabolic subgroup of

G
G

with Levi decomposition

P
=
L
U

P=LU

where

U
U

is the unipotent radical of

P
P

. Let

u

=
Lie
⁡
(
U
)

\mathfrak u=\operatorname {Lie}(U)

and let

z

\mathfrak z

denote the center of

Lie
⁡
(
L
)

\operatorname {Lie}(L)

. Let

T
T

be a maximal torus in

L
L

with Lie algebra

t

\mathfrak t

. Then the root system of

(

g

,

t

)

(\mathfrak g, \mathfrak t)

is a subset of

t

∗

\mathfrak t^*

and by restriction to

z

\mathfrak z

, the roots of

t

\mathfrak t

in

u

\mathfrak u

determine an arrangement of hyperplanes in

z

\mathfrak z

we denote by

A

z

\mathcal A^{\mathfrak z}

. In this paper we construct an isomorphism of graded

k
[

z

]

k[\mathfrak z]

-modules
<mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" alttext="upper H o m Subscript upper G Baseline left-parenthesis German g Superscript asterisk Baseline comma k left-bracket upper G times Superscript upper P Baseline left-parenthesis German z plus German u right-parenthesis right-bracket right-parenthesis approximately-equals upper D left-parenthesis script upper A Superscript German z Baseline right-parenthesis">

Hom
G

⁡
(

g

∗

,
k
[

G

×
P

(

z

+

u

)

]
)
≅
D
(

A

z

)

\operatorname {Hom}_G(\mathfrak g^*, k[{G\times ^P(\mathfrak z+\mathfrak u)}]) \cong D(\mathcal A^{\mathfrak z})

, where

D
(

A

z

)

D(\mathcal A^{\mathfrak z})

is the

k
[

z

]

k[\mathfrak z]

-module of derivations of

A

z

\mathcal A^{\mathfrak z}

. We also show that
<mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" alttext="upper H o m Subscript upper G Baseline left-parenthesis German g Superscript asterisk Baseline comma k left-bracket upper G times Superscript upper P Baseline left-parenthesis German z plus German u right-parenthesis right-bracket right-parenthesis">

Hom
G

⁡
(

g

∗

,
k
[

G

×
P

(

z

+

u

)

]
)

\operatorname {Hom}_G(\mathfrak g^*, k[{G\times ^P(\mathfrak z+\mathfrak u)}])

and
<mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" alttext="k left-bracket German z right-bracket circled-times upper H o m Subscript upper G Baseline left-parenthesis German g Superscript asterisk Baseline comma k left-bracket upper G times Superscript upper P Baseline German u right-bracket right-parenthesis">

k
[

z

]
⊗

Hom
G

⁡
(

g

∗

,
k
[
G

×
P

u

]
)

k[\mathfrak z] \otimes \operatorname {Hom}_G(\mathfrak g^*, k[G \times ^P \mathfrak u])

are isomorphic graded

k
[

z

]

k[\mathfrak z]

-modules, so

D
(

A

z

)

D(\mathcal A^{\mathfrak z})

and
<mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" alttext="k left-bracket German z right-bracket circled-times upper H o m Subscript upper G Baseline left-parenthesis German g Superscript asterisk Baseline comma k left-bracket upper G times Superscript upper P Baseline German u right-bracket right-parenthesis">

k
[

z

]
⊗

Hom
G

⁡
(

g

∗

,
k
[
G

×
P

u

]
)

k[\mathfrak z] \otimes \operatorname {Hom}_G(\mathfrak g^*, k[G \times ^P \mathfrak u])

are isomorphic, graded

k
[

z

]

k[\mathfrak z]

-modules. It follows immediately that

A

z

\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

g

\mathfrak g

affords a self-dual representation of

G
G

, we recover a result of Sommers, Trapa, and Broer which states that the degrees in which the adjoint representation of

G
G

occurs as a constituent of the graded, rational

G
G

-module

k
[
G

×
P

u

]

k[G\times ^P \mathfrak u]

are the exponents of

A

z

\mathcal A^{\mathfrak z}

. This result has also been proved, again using case-by-case arguments, by Sommers and Trapa and independently by Broer.