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