1 Introduction
In this article, we are interested in a family of Higgs pairs over a compact Riemann surface
X
whose structure group is the exceptional simple complex Lie group
F
4
or the classical complex Lie group
PSp
(
8
,
C
)
, and the centerless group whose universal cover is
Sp
(
8
,
C
)
. Given a complex reductive Lie group
G
together with a complex representation
ρ
:
G
→
GL
(
V
)
of
G
, a
G
-Higgs pair over
X
is a pair
(
E
,
φ
)
, where
E
is a principal
G
-bundle over
X
and
φ
is a holomorphic global section of the vector bundle
E
(
V
)
⊗
K
, where
E
(
V
)
is the vector bundle associated with
E
by
ρ
with typical fiber
V
and
K
is the canonical line bundle over
X
(Definition 1). The notion of
G
-Higgs bundle can be reconstructed from that of
G
-Higgs pair by considering the adjoint representation of
G
. The geometry of the moduli spaces associated with
G
-Higgs pairs has been intensively studied and admits a great amount of applications to very diverse areas of Mathematics and Theoretical Physics such as non-abelian Hodge theory, integrable systems, string theory, and the theory of branes among others (for details, see [1]). In this work, we study
F
4
-Higgs bundles and
F
4
-Higgs pairs whose associated complex representation is the fundamental irreducible representation of
F
4
, which is 26 dimensional, and
PSp
(
8
,
C
)
-Higgs pairs associated with the 36-dimensional representation
Sym
2
ι
or the 28-dimensional representation
∧
2
ι
, where
ι
is the inclusion
Sp
(
8
,
C
)
→
GL
(
8
,
C
)
. These bundles and pairs appear as fixed points of certain automorphisms of the moduli space of
E
6
-Higgs bundles over
X
, as we will see in Proposition 3.2. We will also give in Propositions 4.1–4.3 a concrete description of the reduced notions of semistability and polystability for these objects.
Consider the exceptional simple complex Lie group
E
6
. Details about this group can be found in [2, Chapter 5, Sections 1.3 and 1.5] and also in [3,4]. The group
E
6
is the only exceptional simple complex Lie group that admits outer automorphisms. In particular, it admits only one outer automorphism
σ
, which is an involution. This automorphism acts in the moduli space
ℳ
(
E
6
)
of
E
6
-Higgs bundles over
X
according to an action of the group of outer automorphisms of
G
in the moduli space of
G
-Higgs bundles, which is described for a general complex Lie group
G
in [5]. We will denote by
σ
+
the automorphism of
ℳ
(
E
6
)
induced by the outer involution
σ
of
E
6
. This automorphism acts as follows: if
(
E
,
φ
)
is an
E
6
-Higgs bundle over
X
, then
σ
+
(
E
,
φ
)
=
(
E
∗
,
−
φ
t
)
. In this work, we study the subvariety of
ℳ
(
E
6
)
of fixed points of
σ
+
. These fixed points are described in Proposition 3.2 as
F
4
-Higgs bundles over
X
or
PSp
(
8
,
C
)
-Higgs pairs associated with the representation
Sym
2
ι
. We also consider the automorphism
σ
−
of
ℳ
(
E
6
)
that results from combining
σ
+
with the involution of
ℳ
(
E
6
)
defined by a change of sign in the Higgs field, that is,
σ
−
(
E
,
φ
)
=
(
E
∗
,
φ
t
)
. In Proposition 3.2, it is also proved that the fixed points of
σ
−
can be described as
F
4
-Higgs pairs with an associated complex representation of the fundamental irreducible representation of
F
4
or
PSp
(
8
,
C
)
-Higgs pairs with the representation
∧
2
ι
. In Propositions 4.1–4.3, the stability and polystability conditions for all these fixed points are discussed.
The study of automorphisms of moduli spaces of principal bundles and Higgs bundles over compact Riemann surfaces and the corresponding subvarieties of fixed points is a topic of great interest in Geometry, which is being worked on intensively. In [6], for example, we study fixed points of certain automorphism of the moduli space of principal
Spin
(
8
,
C
)
-bundles over a curve induced by the action in it of the triality automorphism, and in [5], we extend the study to the case of
Spin
(
8
,
C
)
-Higgs bundles, and we obtain that the fixed points of certain automorphisms of the moduli space of
Spin
(
8
,
C
)
-Higgs bundles over the Riemann surface are described as certain
G
2
-Higgs pairs or
PSL
(
3
,
C
)
-Higgs pairs whose notions of stability, semistability, and polystability are described following the general concepts explained in [7]. In [8,9], the group of automorphisms of the moduli space of symplectic principal bundles over a compact Riemann surface is computed, and in [4], the same is computed for the moduli space of principal
E
6
-bundles. Finally, in [10], the groups of automorphisms of vector Higgs bundles moduli spaces are determined. In this work, we study the case of
ℳ
(
E
6
)
in the spirit of [5] and the following [7]. In fact, in [7,11], the authors study fixed points of automorphisms of the moduli space of Higgs bundles in the general case where the gauge group
G
is semisimple, but here an explicit description of these fixed points is given for the group
E
6
, using specific techniques adjusted to the particular groups we are working with, and in addition, we provide reduced versions of the notions of stability of the fixed points obtained.
This article is organized as follows: in Section 2, we introduce the concept of the
G
-Higgs pair over a compact Riemann surface and explain the notions of stability, semistability, and polystability as they are introduced in [7]. The moduli space of
E
6
-Higgs bundles and the automorphisms of this moduli space that will be the subject of our interest are introduced in Section 3, where we also prove that the fixed points of these automorphisms can be described as certain types of
F
4
or
PSp
(
8
,
C
)
-Higgs pairs. Finally, in Section 4, we study the reduced notions of stability, semistability, and polystability conditions for the families of Higgs pairs introduced in the previous section.
2 Higgs pairs and stability conditions
In this section, we will introduce the notion of Higgs pair and the corresponding stability conditions needed to define the moduli space of Higgs pairs over a compact Riemann surface
X
of genus
g
≥
2
and with a given complex semisimple structure group
G
as set out in [7]. The whole theory developed in this section can be extended to the case in which
G
is reductive, but we limit ourselves to the semisimple case for simplicity, given that all the groups we will work with will be semisimple.
Definition 1
Let
G
be a complex semisimple Lie group and let
ρ
:
G
→
GL
(
V
)
be a complex representation of
G
. A
G
-Higgs pair over
X
(or simply a Higgs pair, if the structure group is clear) is a pair
(
E
,
φ
)
, where
E
is a principal
G
-bundle over
X
and
φ
∈
H
0
(
X
,
E
(
V
)
⊗
K
)
. Here,
E
(
V
)
=
E
×
ρ
V
is the quotient of
E
×
V
by the equivalence relation defined by
(
e
,
v
)
∼
(
e
g
,
ρ
(
g
−
1
)
v
)
for all
g
∈
G
, and
K
denotes the canonical line bundle over
X
.
Let
G
be a complex semisimple Lie group with Lie algebra
g
and let
H
be a compact connected Lie subgroup of
G
with Lie algebra
h
and such that
H
C
=
G
(so
h
C
=
g
). Let
c
be a Cartan subalgebra of
g
. For any
δ
∈
c
∗
, we denote by
g
δ
the root subspace of
g
, which corresponds to
δ
. Let
R
be the set of roots of
g
and
Δ
be the subset of
R
of simple roots. For any subset
A
⊆
Δ
, the subset of roots
R
A
defined by
R
A
=
∑
β
∈
Δ
m
β
β
:
m
β
≥
0
∀
β
∈
A
satisfies that the subspace
p
A
of
g
defined by
(1)
p
A
=
c
⊕
⊕
δ
∈
R
A
g
δ
,
which is a parabolic subalgebra of
g
. Moreover, if
R
A
0
denotes the subset of roots
R
A
0
=
∑
β
∈
Δ
m
β
β
:
m
β
=
0
∀
β
∈
A
,
then the subspace
l
A
of
g
defined by
(2)
l
A
=
c
⊕
⊕
δ
∈
R
A
0
g
δ
is a Levi subalgebra of
p
A
. Let
P
A
and
L
A
be the subgroups of
G
whose Lie algebras are
p
A
and
l
A
, respectively. The group
P
A
thus defined is a parabolic subgroup of
G
and
L
A
is a Levi subgroup of
P
A
. For different subsets
A
of
Δ
, the corresponding parabolic subgroups are not isomorphic, and all the parabolic subgroups of
G
may be constructed in this way (for details, see [12, Chapter VII]).
For any
δ
∈
Δ
, we define
(3)
λ
δ
=
2
δ
(
δ
,
δ
)
,
where
(
,
)
denotes the Killing form defined in
R
. Given any parabolic subgroup
P
A
of
G
with Lie algebra
p
A
defined in (1) for some
A
⊆
Δ
, an antidominant character
χ
of
P
A
is said to be an element of
c
∗
of the form
χ
=
∑
δ
∈
A
n
δ
λ
δ
,
where
n
δ
≤
0
for any
δ
∈
A
and
λ
δ
is defined in (3). It is called strictly antidominant if
n
δ
<
0
for all
δ
∈
A
. Each antidominant character
χ
of
P
A
induces an element
s
χ
∈
c
through the isomorphism
c
≅
c
∗
induced by the Killing form. This
s
χ
belongs in fact to
i
h
. So each pair composed of a parabolic subgroup
P
of
G
and an antidominant character
χ
of
P
induces an element
s
∈
i
h
.
Let now
(
E
,
φ
)
be a
G
-Higgs pair associated with some complex representation
ρ
:
G
→
GL
(
V
)
of
G
. For any
A
⊆
Δ
and any antidominant character
χ
associated with
P
A
, we define
(4)
V
χ
−
=
{
v
∈
V
:
ρ
(
e
t
s
χ
)
v
is bounded as
t
→
∞
}
V
χ
0
=
{
v
∈
V
:
ρ
(
e
t
s
χ
)
v
=
v
∀
t
}
.
From [7, Lemmas 2.5 and 2.6], it follows that
V
χ
−
is invariant under the action of
P
A
and
V
χ
0
is invariant under the action of
L
A
.
Definition 2
Let
ρ
:
G
→
GL
(
V
)
be a complex representation of
G
and let
(
E
,
φ
)
be a
G
-Higgs pair over
X
associated to the representation
ρ
of
G
. The
G
-Higgs pair
(
E
,
φ
)
is said to be semistable (resp. stable) if for every subset
A
of
Δ
, any antidominant character
χ
of
P
A
and any reduction of structure group
E
A
of
E
to
P
A
such that
φ
∈
H
0
(
X
,
E
A
(
V
χ
−
)
⊗
K
)
, where
P
A
is the parabolic subgroup of
G
whose Lie algebra is defined in (1) and
V
χ
−
is defined in (4), we have that
deg
χ
∗
E
A
≥
0
(resp.
deg
χ
∗
E
A
>
0
).
Definition 3
Let
ρ
:
G
→
GL
(
V
)
be a complex representation of
G
and let
(
E
,
φ
)
be a
G
-Higgs pair over
X
associated to the representation
ρ
of
G
. The
G
-Higgs pair
(
E
,
φ
)
is said to be polystable if it is semistable and for every subset
A
of
Δ
, any antidominant character
χ
of
P
A
and any reduction of structure group
E
A
of
E
to
P
A
such that
φ
∈
H
0
(
X
,
E
A
(
V
χ
−
)
⊗
K
)
and
deg
χ
∗
E
A
=
0
, where
P
A
is the parabolic subgroup of
G
whose Lie algebra is defined in (1) and
V
χ
−
is defined in (4), there exists a reduction of structure group
E
A
′
of
E
A
to
L
A
such that
φ
∈
H
0
(
X
,
E
A
′
(
V
χ
0
)
⊗
K
)
, where
L
A
is the Levi subgroup of
P
A
whose Lie algebra is defined in (2) and
V
χ
0
is defined in (4).
We will use a formulation of the stability and polystability conditions in terms of filtrations of certain vector bundle associated with the corresponding principal bundle of the Higgs pair. This vector bundle is defined as in the next result, by making use of a fixed representation
ρ
G
defined in the case in which the representation
ρ
that defines the principal bundles is faithful. We could have presented the formulation in terms of filtrations as the definition, but we have opted for this presentation of the theory to make this section survey material on this topic. This formulation in terms of filtrations is constructed in detail in [7, Lemma 2.12], which we adapt to formulate the following.
Proposition 2.1
Let G be a semisimple complex Lie group,
ρ
:
G
→
GL
(
V
)
be a faithful complex representation of G, and
(
E
,
φ
)
be a
(
G
,
ρ
)
-Higgs pair over X. Suppose that there exists a representation
ρ
G
:
G
→
GL
(
W
)
, with
W
≅
C
n
for some
n
∈
N
, such that for any
a
,
b
∈
(
Ker
d
ρ
G
)
⊥
we have that
⟨
a
,
b
⟩
=
Tr
d
ρ
G
(
a
)
d
ρ
G
(
b
)
, where the product is the Euclidean product of W. Denote
E
=
E
(
W
)
. Then
The
(
G
,
ρ
)
-Higgs pair
(
E
,
φ
)
is semistable if for every parabolic subgroup P of G, every antidominant character
χ
of P and every filtration
E
0
=
0
⊊
E
1
⊊
⋯
⊊
E
k
=
E
induced by a reduction of structure group of E to P and such that
φ
takes values, in each fiber over X, in the space
V
χ
−
defined in (4), we have that the degree of the filtration, defined by
(5)
λ
k
deg
E
+
∑
j
=
1
k
−
1
(
λ
j
−
λ
j
+
1
)
deg
E
j
,
is greater than or equal to 0, where
λ
1
<
⋯
<
λ
k
are the eigenvalues of
d
ρ
G
(
s
χ
)
.
It is stable if for every P and
χ
as before and any filtration
E
0
=
0
⊊
E
1
⊊
⋯
⊊
E
k
=
E
induced by a reduction of structure group of E to P and such that
φ
takes values in
V
χ
−
, the degree of the filtration defined in (5) is greater than 0.
The
(
G
,
ρ
)
-Higgs pair
(
E
,
φ
)
is polystable if it is semistable, and there exists a parabolic subgroup P of G and an antidominant character
χ
of P such that E admits a decomposition of the form
E
=
⊕
j
=
1
k
E
j
/
E
j
−
1
into vector subbundles, where
E
0
=
0
and
E
j
/
E
j
−
1
is the eigenspace of the eigenvalue
λ
j
of
d
ρ
G
(
s
χ
)
for all
j
=
1
,
…
,
k
, the degree defined in (5) equals 0 and
φ
takes values, in each fiber over X, in the space
V
χ
0
defined in (4).
The degree defined in (5) coincides with the degree
deg
χ
∗
E
A
considered in the definitions of stable and semistable Higgs pairs considered here, as shown in [7, Lemma 2.12].
3
E
6
-Higgs bundles
Let
X
be a compact Riemann surface of genus
g
≥
2
. A principal
E
6
-bundle over
X
can be understood through the fundamental 27-dimensional representation of
E
6
, as a complex vector bundle
E
of rank 27 and trivial determinant bundle equipped with a global holomorphic non degenerate symmetric trilinear form
Ω
(see [4, Section 3], where principal
E
6
-bundles are studied).
Definition 4
An
E
6
-Higgs bundle over
X
is a pair
(
E
,
φ
)
, where
E
is a principal
E
6
-bundle over
X
and
φ
∈
H
0
(
X
,
E
(
e
6
)
⊗
K
)
, so
φ
can be seen as a homomorphism from
E
to
E
⊗
K
, which preserves the symmetric trilinear form of
E
(here,
e
6
denotes the Lie algebra of
E
6
and
E
(
e
6
)
is the adjoint vector bundle of
E
).
We have studied principal
E
6
-bundles in [4], so most of the details about what we present here concerning these bundles can be found in that reference. In particular, a reduction of the structure group of a principal
E
6
-bundle
E
to a parabolic subgroup of
E
6
comes with a filtration
0
⊂
E
0
⊂
⋯
⊂
E
r
⊂
E
of
E
into isotropic subbundles for
Ω
, since every maximal parabolic subgroup of
E
6
comes with a filtration of this form (recall that, given a vector subbundle
F
of a principal
E
6
-bundle
E
with holomorphic symmetric trilinear form
Ω
, we say that
F
is isotropic for
Ω
if
Ω
(
F
,
F
,
F
)
=
0
) and a reduction of structure group to a Levi subgroup of a parabolic subgroup of
E
6
can be seen as a decomposition of
E
of the form
E
=
E
0
⊕
E
1
/
E
0
⊕
⋯
⊕
E
/
E
r
.
In this section, we first give the reduced stability and polystability conditions for
E
6
-Higgs bundles, which are compatible with the perspective explained in the last part of the previous section, which follows [7]. For the moment, we do not present the proof, because it is absolutely analogous to that of Proposition 4.1 and will be covered by it.
Proposition 3.1
Let
(
E
,
φ
)
be an
E
6
-Higgs bundle with associated symmetric trilinear form
Ω
. Then
(
E
,
φ
)
is semistable if for every subbundle F of E isotropic for
Ω
and preserved by
φ
we have that
deg
F
≤
0
.
The Higgs bundle
(
E
,
φ
)
is stable if for every filtration
0
⊂
E
0
⊂
⋯
⊂
E
r
⊂
E
of E composed by isotropic subbundles for
Ω
preserved by
φ
, we have that
deg
E
j
≤
0
for all
j
, and there exists some k such that
deg
E
k
<
0
.
The Higgs bundle
(
E
,
φ
)
is polystable if it can be written as a direct sum of vector subbundles
E
=
E
0
⊕
E
1
/
E
0
⊕
⋯
⊕
E
/
E
r
,
where
E
0
,
E
1
,
…
,
E
r
are degree 0 proper subbundles of E isotropic for
Ω
, which form a filtration of E,
E
0
⊊
E
1
⊊
⋯
⊊
E
r
⊊
E
, and the Higgs field
φ
preserves this decomposition.
Definition 5
The moduli space
ℳ
(
E
6
)
of
E
6
-Higgs bundles over
X
is then the complex algebraic variety that parametrizes isomorphism classes of polystable
E
6
-Higgs bundles over
X
.
The group
Out
(
E
6
)
of outer automorphisms of
E
6
is isomorphic to
Z
2
, since it coincides with the groups of symmetries of the Dynkin diagram of
e
6
. In [5], we proved that there is a natural action of
Out
(
G
)
on the moduli space of
G
-Higgs bundles for any complex reductive Lie group
G
: if
f
∈
Out
(
G
)
and
A
∈
Aut
(
G
)
represents
f
, then
f
⋅
(
E
,
φ
)
=
(
A
(
E
)
,
d
A
(
φ
)
)
, where
A
(
E
)
is the principal
G
-bundle whose total space is that of
E
and the action of
G
in
A
(
E
)
is defined by
e
⋅
g
=
e
A
−
1
(
g
)
for
e
∈
A
(
E
)
and
g
∈
G
. This construction depends only on the equivalence class of the automorphism
A
by the following equivalence relation defined on the group
Aut
(
G
)
: two automorphisms of
G
are equivalent if they are conjugate by an inner automorphism of
G
[6]. We will explicitly describe this action for the case of
ℳ
(
E
6
)
. Let
σ
be the outer involution of
E
6
. The involution
σ
induces the following automorphism of
ℳ
(
E
6
)
:
(6)
σ
+
(
E
,
φ
)
=
(
E
∗
,
−
φ
t
)
,
since the involution
σ
acts in
E
6
by transposing the automorphisms (we are considering the elements of
E
6
through its 27-dimensional fundamental representation),
d
σ
(
A
)
=
−
A
t
for any
A
∈
e
6
and the transformation on the Higgs field coincides with the action of
d
σ
in it, that is,
−
φ
t
=
d
σ
(
φ
)
. We are also interested in the automorphism that results from composing
σ
+
with the involution
ι
of
ℳ
(
E
6
)
whose effect is a change of sign on the field:
ι
(
E
,
φ
)
=
(
E
,
−
φ
)
. We then define the automorphism
σ
−
of
ℳ
(
E
6
)
as follows:
(7)
σ
−
(
E
,
φ
)
=
(
E
∗
,
φ
t
)
.
The Lie algebra involution
d
σ
:
e
6
→
e
6
induces a vector space decomposition of
e
6
of the form
e
6
=
g
+
⊕
g
−
, where
g
+
is the Lie subalgebra of fixed points of
d
σ
and
g
−
is the
(
−
1
)
-eigenspace of
d
σ
. In [13, Theorem 5.10], it is proved that the subalgebra of fixed points of
d
σ
is isomorphic to
f
4
or to
sp
(
8
,
C
)
, so
g
+
is isomorphic to
f
4
or to
sp
(
8
,
C
)
. If
g
+
≅
f
4
, then, since the dimension of
e
6
is 78 and the dimension of
f
4
is 52, the dimension of
g
−
is 26. The restriction of the adjoint representation of
e
6
to
g
+
induces the adjoint representation of
f
4
when it is restricted in the image to
gl
(
g
+
)
, and also a representation
d
ρ
:
g
+
→
gl
(
g
−
)
of
f
4
is induced, since
d
σ
is a Lie algebra automorphism. The first representation gives the adjoint representation of
F
4
and the second induces its fundamental irreducible representation,
(8)
ρ
:
F
4
→
GL
(
g
−
)
(observe that
dim
g
−
=
26
), as we will notice in Proposition 3.2.
We consider now the case in which
g
+
≅
sp
(
8
,
C
)
. The corresponding subgroup of
E
6
is
PSp
(
8
,
C
)
(observe that the center of
Sp
(
8
,
C
)
is isomorphic to
Z
2
, while the center of
E
6
is isomorphic to
Z
3
), which admits the symplectic group
Sp
(
8
,
C
)
as its universal cover through a 2 to 1 covering map
π
Sp
:
Sp
(
8
,
C
)
→
PSp
(
8
,
C
)
. It is also relevant to consider the general symplectic group
GSp
(
8
,
C
)
, defined as the group of invertible
8
×
8
complex matrices that leave invariant certain symplectic form modulo scalars (a classical reference on this group where details about its definition and properties can be found is [14]). We would like to point out that the notions of isotropic vector subspace and symplectic complement make sense when a general symplectic structure is considered in a complex vector space (and also the corresponding notions on principal
GSp
-bundles). The general symplectic group admits a projection
(9)
π
GSp
:
GSp
(
8
,
C
)
→
PSp
(
8
,
C
)
over
PSp
(
8
,
C
)
. This is a 4:1 covering map that factors through the
2
:
1
covering map
GSp
(
8
,
C
)
→
Sp
(
8
,
C
)
defined by the universal covering of
Sp
(
8
,
C
)
, with the standard projection
Sp
(
8
,
C
)
→
PSp
(
8
,
C
)
, defined by the quotient by the center of
Sp
(
8
,
C
)
. Given a principal
PSp
(
8
,
C
)
-bundle
E
0
over
X
, it always admits a lift to a
GSp
(
8
,
C
)
-bundle
E
through
π
GSp
, two of such lifts differing by a line bundle with trivial second power.
Let now
ι
:
Sp
(
8
,
C
)
→
GL
(
8
,
C
)
be the representation of
Sp
(
8
,
C
)
induced by the natural inclusion of groups. Then the representations
(10)
Sym
2
ι
and
(11)
∧
2
ι
of
Sp
(
8
,
C
)
clearly descend to define representations of
PSp
(
8
,
C
)
. If
(
E
0
,
φ
)
is a
(
PSp
(
8
,
C
)
,
Sym
2
ι
)
-Higgs pair over
X
and
E
is a principal
GSp
(
8
,
C
)
-bundle that lifts
E
0
, then
φ
can be seen as a holomorphic global section of the vector bundle
Sym
2
E
⊗
K
. Analogously, a
(
PSp
(
8
,
C
)
,
∧
2
ι
)
-Higgs pair
(
E
0
,
φ
)
induces the existence of a principal
GSp
(
8
,
C
)
-bundle
E
, which lifts
E
0
such that
φ
takes values in
∧
2
E
⊗
K
. As it will be proved in Proposition 3.2, the representation
g
+
of
PSp
(
8
,
C
)
should be isomorphic to
Sym
2
ι
and
g
−
to
∧
2
ι
.
Proposition 3.2
Let
(
E
,
φ
)
be an
E
6
-Higgs bundle fixed by
σ
+
defined in (6). Then E admits a reduction of structure group
E
′
to
F
4
or to
PSp
(
8
,
C
)
such that
φ
(
E
′
)
⊆
E
′
⊗
K
. In the case of
F
4
,
φ
defines an endomorphism of
E
′
tensored by
K
and, in the case of
PSp
(
8
,
C
)
,
φ
defines a holomorphic global section of
Sym
2
E
′
⊗
K
.
Let
(
E
,
φ
)
be an
E
6
-Higgs bundle fixed by
σ
−
defined in (7). Then E admits a reduction of structure group
E
′
to
F
4
or to
PSp
(
8
,
C
)
and
φ
takes values in
E
′
(
g
−
)
⊗
K
. In the case of
F
4
,
φ
=
0
and in the case of
PSp
(
8
,
C
)
,
φ
defines a holomorphic global section of
∧
2
E
′
⊗
K
.
Proof
Under the conditions of the statement of the proposition, since
(
E
,
φ
)
≅
(
E
∗
,
±
φ
t
)
(where the
+
or
−
sign depends on whether
(
E
,
φ
)
is a fixed point of
σ
−
or
σ
+
, respectively), there exists an isomorphism
f
:
(
E
,
φ
)
→
(
E
∗
,
±
φ
t
)
. Then
f
:
E
→
E
∗
is an isomorphism of vector bundles. Let
g
=
(
f
t
)
−
1
∘
f
, which is an automorphism of
E
. Exactly one of the following possibilities is satisfied:
The automorphism
g
is represented by a central element of
E
6
. Then there exists
λ
∈
C
with
λ
3
=
1
such that
g
=
λ
I
. This implies that
f
=
λ
f
t
. By transposing, we have that
f
t
=
λ
f
, so
f
=
λ
2
f
t
. Finally,
λ
2
=
λ
, so
λ
=
1
. Therefore,
f
=
f
t
and
E
admits a reduction of the structure group to
F
4
.
The automorphism
g
is not represented by a central element of
E
6
. Let
θ
≠
1
be an eigenvalue of
g
. Let
v
be an eigenvector of
g
of eigenvalue
θ
. Then
f
(
v
)
=
θ
f
t
(
v
)
. Since
v
∈
Ker
(
f
−
θ
f
t
)
, it is also true that
v
∈
Ker
(
f
t
−
θ
f
)
, so
f
(
v
)
=
θ
2
f
(
v
)
. This implies that
θ
2
=
1
, so
θ
=
−
1
. Then
E
admits a reduction of structure group to some group of type
C
n
. Therefore,
E
admits a reduction of the structure group to a copy of
PSp
(
8
,
C
)
, since this group is the maximal subgroup of
E
6
of type
C
n
, as it is proved in [15], and the center of
Sp
(
8
,
C
)
is isomorphic to
Z
2
, while the center of
E
6
is isomorphic to
Z
3
.
All this proves that
E
admits a reduction of structure group
E
′
to
F
4
or
PSp
(
8
,
C
)
defined by
f
and such that
φ
takes values in
g
+
or
g
−
depending on the considered automorphism,
σ
+
or
σ
−
, respectively.
In the
F
4
case, we have the following:
For the automorphism
σ
+
, the Higgs field
φ
always satisfies
f
∘
φ
+
φ
t
∘
f
=
0
, that is,
f
defines a quadratic form in
E
′
, which
φ
respects, so
φ
is an endomorphism of
E
′
as a principal
F
4
-bundle.
For the automorphism
σ
−
,
φ
satisfies
f
∘
φ
=
φ
t
∘
f
. Since the dimension of
g
−
is 26 in this case, the only possibility for
φ
is to define a holomorphic global section of
End
(
E
′
)
⊗
K
, in view of the possibilities for the representations of
F
4
. The condition
f
∘
φ
=
φ
t
∘
f
tells us that this global section should satisfy
⟨
⟨
φ
,
a
⟩
φ
,
b
⟩
−
⟨
a
,
⟨
φ
,
b
⟩
φ
⟩
=
⟨
φ
,
a
⟩
⟨
φ
,
b
⟩
−
⟨
φ
,
b
⟩
⟨
a
,
φ
⟩
=
0
for any
a
,
b
∈
E
′
(we are omitting the reference to the element in the canonical bundle
K
), so
(
E
′
,
φ
)
defines an
F
4
-Higgs pair for the representation
ρ
defined in (8).
We now analyze the case of
PSp
(
8
,
C
)
.
For the automorphism
σ
+
. By the description of the representations of
PSp
(
8
,
C
)
and the dimensional restrictions (see the discussion before the statement),
φ
should be a holomorphic global section of
Sym
2
E
′
⊗
K
or
∧
2
E
′
⊗
K
. In any case, if
E
0
is a lift of
E
′
by the projection map defined in (9),
φ
can be understood as a global section of
⊗
2
E
0
⊗
K
. Without loss of generality, we can assume that
φ
=
v
⊗
w
, by making a slight abuse of notation, for certain
v
,
w
∈
E
0
, instead of a linear combination of summands of this kind. If
⟨
,
⟩
denotes the symplectic form defined in
E
0
and
a
,
b
∈
E
′
, we have that
⟨
φ
(
a
)
,
b
⟩
+
⟨
a
,
φ
(
b
)
⟩
=
⟨
w
,
a
⟩
⟨
v
,
b
⟩
−
⟨
v
,
a
⟩
⟨
w
,
b
⟩
,
because
φ
(
a
)
=
⟨
w
,
a
⟩
v
and
φ
(
b
)
=
⟨
w
,
b
⟩
v
, the field
φ
understood as a vector endomorphism of
E
′
(again, we omitted the reference to the
K
element). Now, since
φ
satisfies
f
∘
φ
+
φ
t
∘
f
=
0
, one has that the aforementioned expression should be equal 0, which is equivalent to the state that
φ
∈
Sym
2
E
′
⊗
K
.
For the automorphism
σ
−
, the same argument shows that
φ
defines a global section of
∧
2
E
′
⊗
K
in this case, only by observing that
f
∘
φ
=
φ
t
∘
f
.□
Remark
Proposition 3.2 can be seen as a consequence of [11, Proposition 3.9], which addresses the general case of a complex semisimple Lie group
G
. The novelty here is that we have made a specific proof for the particular case of
G
=
E
6
.
4 Stability conditions for the
F
4
and
PSp
(
8
,
C
)
-Higgs pairs
Proposition 3.2 tells that, among the fixed points of the automorphism
σ
+
of
ℳ
(
E
6
)
defined in (6), we find
F
4
-Higgs bundles over
X
, and among the fixed points of
σ
−
defined in (7),
F
4
-Higgs pairs associated to the fundamental representation of
F
4
are found. In the first part of this section, we will give the reduced notions of stability for the bundles and pairs with structure group
F
4
, which will be done by using the theory set out in Section 2.
The group
F
4
is the subgroup of
E
6
which consists of automorphisms of a complex vector space
V
of dimension 26 that preserve a certain holomorphic nondegenerate symmetric trilinear form
Ω
and a certain holomorphic nondegenerate symmetric bilinear form
ω
. A principal
F
4
-bundle over the compact Riemann surface
X
is then a rank 26 holomorphic complex vector bundle
E
equipped with a global holomorphic nondegenerate symmetric trilinear form
Ω
and with a global holomorphic symmetric bilinear form
ω
(for details, see [4, Section 3]).
In [4], we described the parabolic subgroups of
F
4
. We will recall here the form of the filtrations induced on a principal
F
4
-bundle by a reduction of the structure group of the bundle to a parabolic subgroup of
F
4
.
Let
E
be a principal
F
4
-bundle over
X
. A reduction of structure group of
E
to a parabolic subgroup of
F
4
induces a filtration of
E
into vector subbundles of the form
(12)
0
⊊
E
0
⊊
E
1
⊊
⋯
⊊
E
r
⊆
E
r
⊥
⊊
⋯
⊊
E
1
⊥
⊊
E
0
⊥
⊊
E
,
where the orthogonality
⊥
is taken with respect to
ω
and
E
0
,
…
,
E
r
are isotropic for
ω
and
Ω
.
Proposition 4.1
Let
(
E
,
φ
)
be an
F
4
-Higgs bundle,
Ω
be the holomorphic nondegenerate symmetric trilinear form defined in E, and
ω
be the holomorphic nondegenerate symmetric bilinear form defined in E. Then
(
E
,
φ
)
is semistable if for every proper subbundle F of E isotropic for
ω
and
Ω
such that
φ
(
F
)
⊆
F
⊗
K
, we have that
deg
F
≤
0
.
The Higgs bundle
(
E
,
φ
)
is stable if for every filtration as in (12), where
E
0
,
…
,
E
r
are isotropic for
ω
and
Ω
and such that
φ
(
E
k
)
⊆
E
k
⊗
K
for all
k
with
0
≤
k
≤
r
, we have that
deg
E
j
≤
0
for all j and there exists some k such that
deg
E
k
<
0
.
The Higgs bundle
(
E
,
φ
)
is polystable if E admits a filtration into vector subbundles as in (12), where
E
0
,
…
,
E
r
are isotropic for
ω
and
Ω
,
deg
E
0
=
deg
E
1
=
⋯
=
deg
E
r
=
0
, such that
φ
(
E
k
)
⊆
E
k
⊗
K
for all
k
with
0
≤
k
≤
r
and E can be written in the following form:
E
=
E
0
⊕
E
1
/
E
0
⊕
E
2
/
E
1
⊕
⋯
⊕
E
r
/
E
r
−
1
⊕
E
r
⊥
/
E
r
⊕
E
r
−
1
⊥
/
E
r
⊥
⊕
⋯
⊕
E
0
⊥
/
E
1
⊥
⊕
E
/
E
0
⊥
.
Proposition 4.2
Let
(
E
,
φ
)
be an
F
4
-Higgs pair associated with the representation of
F
4
defined in (8). Let
Ω
be the holomorphic nondegenerate symmetric trilinear form defined in E and
ω
be the holomorphic nondegenerate symmetric bilinear form defined in E. The Higgs pair
(
E
,
φ
)
is semistable if for every proper subbundle F of E isotropic for
ω
and
Ω
such that
φ
takes values in
F
⊥
⊗
K
, we have that
deg
F
≤
0
.
The Higgs pair
(
E
,
φ
)
is stable if for every filtration of the form (12), where
E
0
,
…
,
E
r
are isotropic for
ω
and
Ω
and such that
φ
takes values in
E
r
⊥
⊗
K
, we have that
deg
E
j
≤
0
for all j and there exists some
k
such that
deg
E
k
<
0
.
The Higgs pair
(
E
,
φ
)
is polystable if E admits a filtration into vector subbundles as in (12), where
E
0
,
…
,
E
r
are isotropic for
ω
and
Ω
,
deg
E
0
=
deg
E
1
=
⋯
=
deg
E
r
=
0
and such that
φ
takes values in
E
r
⊥
/
E
r
⊗
K
and
E
can be written in the following form:
E
=
E
0
⊕
E
1
/
E
0
⊕
E
2
/
E
1
⊕
⋯
⊕
E
r
/
E
r
−
1
⊕
E
r
⊥
/
E
r
⊕
E
r
−
1
⊥
/
E
r
⊥
⊕
⋯
⊕
E
0
⊥
/
E
1
⊥
⊕
E
/
E
0
⊥
.
Due to their absolute similarity, we give together the proofs of Propositions 4.1 and 4.2.
Proof
Let
(
E
,
φ
)
be an
F
4
-Higgs pair for the adjoint representation or for the representation of
F
4
defined in (8). Let
Ω
be the holomorphic nondegenerate symmetric trilinear form defined on
E
and
ω
be the holomorphic nondegenerate symmetric bilinear form defined on
E
. Let
P
be a parabolic subgroup of
F
4
,
χ
be an antidominant character of
P
and
s
χ
be the corresponding element of
i
h
. A reduction of structure group of
E
to
P
induces a filtration of
E
of the form
0
⊊
E
0
⊊
E
1
⊊
⋯
⊊
E
r
⊆
E
r
⊥
⊊
⋯
⊊
E
1
⊥
⊊
E
0
⊥
⊊
E
,
where
E
0
,
…
,
E
r
are isotropic for
ω
and
Ω
. Let
d
j
be the rank of
E
j
for each
j
=
0
,
…
,
r
. The element
s
then diagonalizes in the following form:
(13)
λ
0
I
d
0
⋱
λ
r
I
d
r
−
d
r
−
1
0
I
26
−
2
d
r
−
λ
r
I
d
r
−
d
r
−
1
⋱
−
λ
0
I
d
0
,
where
λ
k
<
0
,
I
k
denotes the identity matrix of rank
k
for each
k
and
λ
0
<
λ
1
<
⋯
<
λ
r
. A few simple computations show that the degree defined in (5) becomes
λ
r
(
deg
E
r
+
deg
E
r
⊥
)
+
∑
j
=
0
r
−
1
(
λ
j
−
λ
j
+
1
)
(
deg
E
j
+
deg
E
j
⊥
)
.
This shows that the expression (5) is a linear combination of the numbers
deg
E
j
+
deg
E
j
⊥
with negative coefficients. Then we have the following:
For the automorphism
σ
+
.
The Higgs bundle
(
E
,
φ
)
is semistable if
deg
E
j
≤
0
for all
j
and for every filtration as mentioned earlier for which
φ
(
E
k
)
⊆
E
k
⊗
K
for all
k
with
0
≤
k
≤
r
(this is the condition for
φ
expressed in Proposition 2.1). This is clearly equivalent to the assertion of the statement since every isotropic subbundle of
E
can be placed in a filtration as the considered one.
Analogously, by Proposition 2.1, the Higgs bundle
(
E
,
φ
)
is stable if for every filtration as mentioned earlier for which
φ
(
E
k
)
⊆
E
k
⊗
K
for all
k
with
0
≤
k
≤
r
, we have that
λ
r
(
deg
E
r
+
deg
E
r
⊥
)
+
∑
j
=
0
r
−
1
(
λ
j
−
λ
j
+
1
)
(
deg
E
j
+
deg
E
j
⊥
)
>
0
, which is equivalent to requiring that
deg
E
j
≤
0
for all
j
and that there exists at least one
k
∈
{
0
,
1
,
…
,
r
}
such that
deg
E
k
<
0
.
Finally, the third part of the statement is clearly a re-reading of the third part of Proposition 2.1.
For the automorphism
σ
−
.
The Higgs pair
(
E
,
φ
)
is semistable if and only if for each filtration as mentioned earlier for which
φ
takes values in
E
r
⊥
⊗
K
,
deg
E
j
≤
0
for all
j
, by Proposition 2.1. As in the first case, this is again equivalent to the assertion of the statement.
Analogously, by Proposition 2.1,
(
E
,
φ
)
is stable if and only if for each filtration as mentioned earlier for which
φ
takes values in
E
r
⊥
⊗
K
, it is satisfied that
deg
E
j
≤
0
for all
j
and that there exists at least one
k
∈
{
0
,
1
,
…
,
r
}
such that
deg
E
k
<
0
.
As in the case of
σ
+
, the third part of the result is a re-reading of the third part of Proposition 2.1.□
We will now deal with the case of Higgs pairs with the structure group
PSp
(
8
,
C
)
.
Proposition 4.3
Let
Sym
2
ι
(resp.
∧
2
ι
) be the representation of
PSp
(
8
,
C
)
defined in (10) (resp. in (11)), let
(
E
0
,
φ
)
be a
(
PSp
(
8
,
C
)
,
Sym
2
ι
)
-Higgs pair (resp.
(
PSp
(
8
,
C
)
,
∧
2
ι
)
-Higgs pair) over
X
, and let E be a principal
GSp
(
8
,
C
)
-bundle over X, which lifts
E
0
. Then
(
E
0
,
φ
)
is semistable if
deg
F
rk
F
≤
deg
E
rk
E
for every isotropic subbundle F of E satisfying that
φ
takes values in the subbundle
(
(
F
⊗
S
E
)
⊕
(
F
⊥
/
F
⊗
S
F
⊥
/
F
)
)
⊗
K
(resp.
(
(
F
∧
E
)
⊕
(
F
⊥
/
F
∧
F
⊥
/
F
)
)
⊗
K
)
of
Sym
2
E
⊗
K
(resp. of
∧
2
E
⊗
K
).
The pair
(
E
0
,
φ
)
is polystable if E admits a decomposition of the form
E
=
E
0
⊕
E
1
/
E
0
⊕
⋯
⊕
E
r
/
E
r
−
1
⊕
E
r
⊥
/
E
r
⊕
⋯
⊕
E
/
E
0
⊥
with
deg
E
0
rk
E
0
=
deg
E
rk
E
,
deg
E
j
=
0
for all
j
≥
1
and such that
φ
takes values in the subbundle
(
⊕
i
=
0
r
−
1
(
E
i
⊗
S
E
i
+
1
⊥
/
E
i
)
⊕
(
E
r
⊥
/
E
r
⊗
S
E
r
⊥
/
E
r
)
)
⊗
K
(resp.
(
⊕
i
=
0
r
−
1
(
E
i
∧
E
i
+
1
⊥
/
E
i
)
⊕
(
E
r
⊥
/
E
r
∧
E
r
⊥
/
E
r
)
)
⊗
K
)
of
Sym
2
E
⊗
K
(resp. of
∧
2
E
⊗
K
).
Proof
We will develop the proof for the case of the representation
Sym
2
ι
(the case of
∧
2
ι
is absolutely analogous). Given a
(
PSp
(
8
,
C
)
,
Sym
2
ι
)
-Higgs pair
(
E
0
,
φ
)
and given
E
as in the statement, a reduction of structure group of
(
E
0
,
φ
)
to a parabolic subgroup
P
of
PSp
(
8
,
C
)
comes with a filtration of
E
into isotropic subbundles for its symplectic form
0
⊊
E
0
⊊
E
1
⊊
⋯
⊊
E
r
⊆
E
r
⊥
⊊
⋯
⊊
E
1
⊥
⊊
E
0
⊥
⊊
E
.
If
χ
is an antidominant character of
P
and
s
χ
is the associated element of
i
h
, then
s
χ
diagonalizes as in (13). The degree defined in (5) has the following form
−
λ
0
deg
E
+
λ
r
(
deg
E
r
+
deg
E
r
⊥
)
+
∑
j
=
0
r
−
1
(
λ
j
−
λ
j
+
1
)
(
deg
E
j
+
deg
E
j
⊥
)
,
so it is greater than or equal to 0 for every family of weights
{
λ
i
}
if and only if
deg
E
i
rk
E
i
≤
deg
E
rk
E
for every
i
≥
0
and it is equal to 0 for every such family if
deg
E
0
rk
E
0
=
deg
E
rk
E
and
deg
E
j
=
0
for all
j
≥
1
. On the other hand, if
v
⊗
S
w
is a generic element that belongs to a summand of the differentiable decomposition
E
0
⊕
E
1
/
E
0
⊕
⋯
⊕
E
r
/
E
r
−
1
⊕
E
r
⊥
/
E
r
⊕
⋯
⊕
E
/
E
0
⊥
, then
Sym
2
ι
(
e
t
s
χ
)
(
v
⊗
S
w
)
=
e
t
(
α
+
β
)
v
⊗
S
w
,
where
α
is the eigenvalue of
v
and
β
is the eigenvalue of
w
. This expression is bounded as
t
→
∞
if and only if
α
+
β
≤
0
. If we suppose that
α
≤
β
, this implies that
v
belongs to some
E
i
(i.e.,
α
coincides with some
λ
i
and
w
belongs to
E
i
−
1
⊥
, or both vectors
v
and
w
belong to
E
r
⊥
/
E
r
. The semistability condition then demands that
deg
E
i
rk
E
i
≤
deg
E
rk
E
for every
i
whenever
φ
takes values in
⊕
j
=
1
r
(
E
j
⊗
S
E
j
−
1
⊥
)
⊕
(
E
r
⊥
/
E
r
⊗
S
E
r
⊥
/
E
r
)
.
The semistability condition applied to a filtration of the form
0
⊊
F
⊊
F
⊥
⊊
E
induced by a reduction of structure group to a maximal parabolic subgroup then demands the condition exposed in the statement. Since the satisfaction of the semistability condition for these filtrations given by reductions to maximal parabolic subgroups implies the satisfaction of the semistability condition for every other filtration, the result for semistability holds.
For polystability, observe that, with the preceding notation,
α
+
β
=
0
if and only if
α
=
β
=
0
or
α
=
λ
i
and
β
=
−
λ
i
for some
i
, and so polystability requires that
deg
E
0
rk
E
0
=
deg
E
rk
E
,
deg
E
j
=
0
for all
j
≥
1
whenever
φ
takes values in
(
⊕
i
=
0
r
−
1
(
E
i
⊗
S
E
i
+
1
⊥
/
E
i
)
⊕
(
E
r
⊥
/
E
r
⊗
S
E
r
⊥
/
E
r
)
)
⊗
K
.□
5 Conclusion
Let
X
be a compact Riemann surface of genus
g
≥
2
and let
ℳ
(
E
6
)
be the moduli space of
E
6
-Higgs bundles over
X
. It is well known that
E
6
is the only exceptional simple complex Lie group that admits a nontrivial outer automorphism. This automorphism, which has been called
σ
in the paper, is an involution, and it acts on
ℳ
(
E
6
)
. By combining this with the multiplication by
+
1
or
−
1
on the Higgs field, the automorphisms
σ
+
and
σ
−
of
ℳ
(
E
6
)
, defined in (6) and (7), are obtained. The main result of this article describes the subvariety of fixed points of
ℳ
(
E
6
)
for these two automorphisms (Proposition 3.2). The techniques employed make use of the specific properties of group
E
6
, so the proof is also specific for this case. As a result, it has been obtained that the indicated fixed points can be described as certain Higgs pairs associated with the structure groups
F
4
and
PSp
(
8
,
C
)
and to certain representations of these groups, which are defined in (8), (10), and (11). Finally, in Section 4, the stability conditions for these Higgs pairs are obtained from the general theory of
G
-Higgs pairs developed in Section 2 (Propositions 4.1–4.3). A study of how the group inclusions
F
4
↪
E
6
and
PSp
(
8
,
C
)
↪
E
6
influences the natural maps of the moduli spaces of
F
4
or
PSp
(
8
,
C
)
-Higgs pairs in
ℳ
(
E
6
)
is proposed as a line of future research.
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