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A note on quasi-Hamiltonian geometry and representation spaces of surfaces groups

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Abstract

In this note, we gather known applications of quasi-Hamiltonian geometry to the study of representations spaces of surface groups. We consider three apects of the geometry of representation spaces of surface groups: the symplectic structure that they carry, the number of connected components of representation spaces and the construction of Lagrangian submanifolds. The present survey is based on the work of A. Alekseev, A. Malkin and E. Meinrenken [J. Differ. Geom. 48, No. 3, 445–495 (1998; Zbl 0948.53045)], N.-K. Ho and C.-C. M. Liu [Int. Math. Res. Not. 2003, No. 44, 2359–2372 (2003; Zbl 1043.53064); Int. Math. Res. Not. 2005, No. 16, 959–979 (2005; Zbl 1076.57017)], N.-K. Ho [Int. Math. Res. Not. 2004, No. 61, 3263–3285 (2004; Zbl 1075.53086)] and of the author [Can. J. Math. 59, No. 4, 845–879 (2007; Zbl 1128.53058), preprint available at arxiv:math.SG/0608682; Math. Ann. 342, No. 2, 405–447 (2008; Zbl 1146.53063); Introduction to quasi-Hamiltonian geometry (after Alekseev, Malkin and Meinrenken). submitted, preprint available at http://www.math.jussieu.fr∼florent]. This note grew out of a talk given at Keio University in September 2006.
A NOTE ON QUASI-HAMILTONIAN GEOMETRY AND
REPRESENTATION SPACES OF SURFACES GROUPS
FLORENT SCHAFFHAUSER
Abstract. In this note, we gather known applications of quasi-Hamiltonian geometry to the study
of representations spaces of surface groups. We consider three apects of the geometry of represen-
tation spaces of surface groups: the symplectic structure that they carry, the number of connected
components of representation spaces and the construction of Lagrangian submanifolds. The present
survey is based on the work of Alekseev, Malkin and Meinrenken in [3], of Ho and Liu in [12, 13, 11]
and of the author in [21, 20, 19]. This note grew out of a talk given at Keio University in September
2006.
1. Introduction
Given a compact Riemann surface Σgof genus g0, we consider the surface
Σg,l := Σg\{s1, ... , sl}
obtained from Σgby removing lpairwise disctinct points s1, ... , slΣg. The fundamental group of
the surface Σg,l thus obtained has the following finite presentation:
πg,l := π1g\{s1, ... , sl}) =< α1, β1, ... , αg, βg, γ1, ... , γl|
g
Y
i=1
[αi, βi]
l
Y
j=1
γj= 1 >
The object of study in this note will be the space (of equivalence classes) of representations of this
group πg,l into a compact connected Lie group U. More precisely, given a compact connected Lie
group Uand lconjugacy classes C1, ... , Clof U, we fix a set of generators of πg,l and consider the set
HomC(πg,l , U ) := {(a1, b1, ... , ag, bg, c1, ... , cl)(U×U)g×C1× · · ·×Cl|
g
Y
i=1
[ai, bi]
l
Y
j=1
cj= 1} ⊂ U2g+l
which is identified, via the choice of generators of πg,l , to the set of group morphisms from πg,l to U
satisfying (γj)∈ Cjfor all j∈ {1, ... , l}. This set is called the set of representations of πg,l into U.
A first remark here is that depending on the choice of conjugacy classes C1, ... , Clin U, this set might
very well be empty. In fact, it is proved in [13] that when g1and the compact connected group U
is semi-simple, the representation space is always non-empty. Conditions for the representation space
to be non-empty when Uis an arbitrary compact connected Lie group can be found in [13]. For the
g= 0 case, we refer to [1, 23], where the situation is seen to be much more complicated.
Observe then that the group Uacting diagonally by conjugation on (U×U)g× C1× · · · × Clpreserves
the relation Qg
i=1[ai, bi]Ql
j=1 cj= 1, hence the set HomC(πg,l , U )of representations of πg,l into U.
We may the consider the orbit space
Mg,l := HomC(πg,l , U )/U
of this action, usually called the representation space or representation variety, or yet, to avoid con-
fusion, the moduli space associated to πg,l.
The moduli spaces Mg,l are an important object of study and are connected to various areas of math-
ematics (see for instance [17, 5, 6]). In this note, we will consider three aspects of the geometry of
these spaces:
1. the symplectic structure that they carry
Supported by the Japanese Society for Promotion of Science (JSPS).
1
2 FLORENT SCHAFFHAUSER
2. the number of conected components of these moduli spaces
3. the construction of Lagrangian submanifolds of these moduli spaces
The symplectic structure of the moduli spaces Mg,l was first studied in [5] and [6], and extensively af-
ter that. In this note, we will present in section 2 the quasi-Hamiltonian description of this symplectic
structure. This description was obtained by Alekseev, Malkin and Meinrenken in [3]: the representa-
tion space HomC(πg,l , U )/U carries a symplectic structure because it is the quasi-Hamiltonian quotient
associated to the space (U×U)g× C1× · · · × Cl:
Mg,l =µ1({1})/U
where µis the momentum map
µ: (U×U)g× C1× · ·· × ClU
(a1, b1, ... , ag, bg, c1, ... , cl)7−[a1, b1]...[ag, bg]c1...cl
As a further remark on the sympectic structure of representation spaces, we would like to recall here
that it is necessary to prescribe the conjugacy classes of generators cjof πg,l correponding to loops
around removed points of Σg, otherwise one only obtains Poisson structures (see [2]). Let us now
come back to the three themes above. Although the second one is a question of a purely topological
nature, the method used to compute the number of connected components of Mg,l , which is due to
Ho and Liu that we will present in section 3, uses the description of Mg,l as a quasi-Hamiltonian
quotient. So will the construction of a Lagrangian submanifold of Mg ,l presented in section 4.
2. Quasi-Hamiltonian geometry
In this section, we recall the notions of quasi-Hamiltonian geometry that we will need to study the
moduli spaces Mg,l = HomC(πg ,l, U )/U.
2.1. Quasi-Hamiltonian quotients. We have seen that we have the following set-theoretic descrip-
tion:
Mg,l = HomC(πg,l , U )/U =µ1({1})/U
with
µ: (U×U)g× C1× · ·· × ClU
(a1, b1, ... , ag, bg, c1, ... , cl)7−[a1, b1]...[ag, bg]c1...cl
The work of Alekseev, Malkin and Meinrenken in [3] shows that the map µabove is a momentum
map for the diagonal conjugacy action of Uon (U×U)g× C1× · ·· × Clin the following sense: there
exists a 2-form ωon (U×U)g× C1× · ·· × Clsuch that ((U×U)g× C1× · · · × Cl, ω, µ)satisfies the
axioms of definition 2.1 below.
Definition 2.1 (Quasi-Hamiltonian space, [3]).Let (M, ω)be a manifold endowed with a 2-form ω
and an action of the Lie group (U, (.|.)) leaving the 2-form ωinvariant. We denote by (.|.)an Ad-
invariant non-degenerate symmetric bilinear form on u=Lie(U), by θL=g1.dg and θR=dg.g1
the Maurer-Cartan 1-forms of U, and by χ=1
2([θL, θL]|θL)the Cartan 3-form of U. Final ly, we
denote by X]the fundamental vector field on Massociated to Xu. Its value at xMis :
X]
x:= d
dt |t=0(exp(tX ).x). Let µ:MUbe a U-equivariant map (for the conjugacy action of Uon
itself).
Then (M, ω, µ :MU)is said to be a quasi-Hamiltonian space (with respect to the action of U) if
the map µ:MUsatisfies the following three conditions:
(i) =µχ
(ii) for all xM,ker ωx={X]
x:Xu|(Ad µ(x) + Id).X = 0}
(iii) for all Xu,ιX]ω=1
2µ(θL+θR|X)
where (θL+θR|X)is the real-valued 1-form defined on Ufor any Xuby (θL+θR|X)u(ξ) :=
(θL
u(ξ) + θR
u(ξ)|X)(where uUand ξTuU).
The map µis called the momentum map.
QUASI-HAMILTONIAN GEOMETRY AND REPRESENTATION SPACES OF SURFACES GROUPS 3
We then have the following theorem, due to Alekseev, Malkin and Meinrenken.
Theorem 2.2 (Reduction of quasi-Hamiltonian spaces,[3]).Let (M, ω, µ :MU)be a quasi-
Hamiltonian U-space. If the compact connected Lie group Uacts freely on the fiber µ1({1}), then:
(i) 1Uis a regular value of the momentum map µ(consequently, the set µ1({1})is a sub-
manifold of M).
(ii) the set µ1({1})/U is a manifold.
(iii) if we denote by i:µ1({1})Mthe inclusion of the level submanifold µ1({1})in Mand
by pthe principal fibration p:µ1({1})µ1({1})/U, the 2-form iωis basis with respect
to p: there exists a (unique) 2-form ωred on µ1({1})/U such that iω=pωred.
(iv) the 2-form ωred is symplectic.
The symplectic manifold µ1({1})/U is often denoted M//U and called the quasi-Hamiltonian quo-
tient associated to the quasi-Hamiltonian space (M, ω, µ :MU).
Dropping the assumption on the freeness of the action of Uon µ1({1})/U, we also have the
following generalization of theorem 2.2, saying that the set M//U =µ1({1})/U is a disjoint union
of symplectic manifolds, each of which is in fact obtained (see [20]) by the reduction procedure above
from a quasi-Hamiltonian space endowed with a free action of a compact Lie group.
Theorem 2.3 (Structure of a quasi-Hamiltonian quotient, [20]).Let (M, ω, µ :MU)be a quasi-
Hamiltonian U-space. For any closed subgroup KU, denote by MKthe isotropy manifold of type
Kin M:
MK={xM|Ux=K}.
Denote by N(K)the normalizer of Kin Uand by LKthe quotient group LK:= N(K)/K. Then the
orbit space
(µ1({1U})MK)/LK
is a smooth symplectic manifold.
Denote by (Kj)jJa system of representatives of closed subgroups of U. Then the orbit space Mred :=
µ1({1U})/U is the disjoint union of the following symplectic manifolds:
µ1({1U})/U =G
jJ
(µ1({1U})MKj)/LKj.
Theorems 2.2 and 2.3 show that the representation spaces HomC(πg,l, U )/U =µ1({1})/U indeed
carry a symplectic structure. We refer to [3] for a proof that this symplectic structure is the same as
the one previously obtained in [5], [6], [8] and [16].
2.2. Convexity theorems. When the compact connected Lie group Uis in addition simply con-
nected, there exists a subset W u=Lie(U)called a Weyl alcove such that:
(i) Wis convex.
(ii) the exponential map restricts to an homeomorphism exp |W:W → exp(W)U.
(iii) the set exp(W)is a fundamental domain for the conjugacy action of Uon itself: it contains
exactly one point of each conjugacy class.
As a consequence of the existence of such a set W ⊂ u, we see that it makes sense to ask whether a
given subset Aexp(W)' W ⊂ uis convex. We then have the following theorem, due to Meinrenken
and Woodward in [15].
Theorem 2.4 (Convexity theorem for group-valued momentum maps, [15, 3]).Let Ube a compact
connected simply connected Lie group and let (M, ω, µ :MU)be a connected quasi-Hamiltonian
space with proper momentum map µ. Then, for any choice of a maximal torus TUand any choice
of a closed Weyl alcove W t=Lie(T), the set µ(M)exp(W)exp(W)is a convex subpolytope of
exp(W)' W, called the momentum polytope. Moreover, the fibres of µare connected. In particular,
the set µ1({1})is a connected subset of M.
In the presence of an involution βon the quasi-Hamiltonian space M, we have the following result:
4 FLORENT SCHAFFHAUSER
Theorem 2.5 (A real convexity theorem for group-valued momentum maps, [22]).Let (U, τ )be
a compact connected simply connected Lie group endowed with an involutive automorphism τsuch
that the involution τ:uU7→ τ(u1)leaves a maximal torus Tof Upointwise fixed, and let
W t:= Lie(T)be a Weyl alcove. Let (M, ω, µ :MU)be a connected quasi-Hamiltonian U-space
with proper momentum map µ:MUand let β:MMbe an involution on Msatisfying:
(i) βω=ω
(ii) β(u.x) = τ(u)(x)for al l xMand all uU
(iii) µβ=τµ
(iv) Mβ:= F ix(β)6=
(v) µ(Mβ)has a non-empty intersection with the fixed-point set Q0of 1in F ix(τ)U
Then:
µ(Mβ)exp(W) = µ(M)exp(W)
In particular, µ(Mβ)exp(W)is a convex subpolytope of exp(W)' W t, equal to the full momentum
polytope µ(M)exp(W).
Observe that an involutive automorphism τof Usuch that τleaves a maximal torus of Upointwise
fixed always exists (it is a consequence of the existence of a split real form of the complexified Lie group
UC). Theorems 2.4 and 2.5 are quasi-Hamiltonian analogues of convexity theorems for momentum
maps in the usual Hamiltonian setting. We will see in sections 3 and 4 how they imply results
on the number of connected components of representation spaces and construction of Lagrangian
submanifolds of these spaces.
2.3. Coverings of quasi-Hamiltonian spaces. Before ending this section, we observe that theo-
rems 2.4 and 2.5 only apply to quasi-Hamiltonian U-spaces with Ucompact connected and simply
connected. To be able, in sections 3 and 4, to study connected components and construct Lagrangian
submanifolds of Mg,l = HomC(πg,l , U )/U when the compact connected Lie group is not simply con-
nected, we will need the following result, due to Alekseev, Meinrenken and Woodward:
Proposition 2.6 ([4]).Let Ube a compact connected Lie group and let π:e
UUbe a covering
map. Set
f
M:= M×Ue
U={(x, eu)|µ(x) = π(eu)}
and
p:f
MMeµ:f
Me
U
(x, eu)7−x(x, eu)7−eu
so that we have the following commutative diagram:
f
M=M×Ue
Ueµ
e
U
p
y
yπ
Mµ
U
Finally, let us set eω:= pωand observe that e
Uacts on f
Mvia
eu0.(x, eu) := (π(eu0).x, eu0eueu1
0)
and that eµis equivariant for this action. Then we have: (f
M, eω, eµ:f
Me
U)is a quasi-Hamiltonian
e
U-space.
The proof shows that this works because π:e
Ue
Uis a covering homomorphism. In particular,
e
Uand Uhave isomorphic Lie algebras. Further, we have:
Proposition 2.7 ([4]).The quasi-Hamiltonian quotients associated to Mand f
Mare isomorphic: the
map p:f
MMsends eµ1({1e
U})to µ1({1U})and induces an isomorphism
eµ1({1e
U})/e
U'µ1({1U})/U
QUASI-HAMILTONIAN GEOMETRY AND REPRESENTATION SPACES OF SURFACES GROUPS 5
In particular, if µ1({1U})6=then eµ1({1e
U})6=.
3. Connected components of representation spaces
In this section, we will outline the proof, due to Ho and Liu in [12] of the following theorem, first
proved by Goldman in [7] for U=SU (2) and U=SO(3), and by Li in [14] for an arbitrary compact
connected semisimple Lie group.
Theorem 3.1 (Connected components of representation spaces, [7, 14]).Let Σgbe a compact Rie-
mann surface of genus g1and let Ube a compact connected semi-simple Lie group. Denote by
π0(Hom(π1g), U )/U)the set of connected components of the representation space
Mg,0= Hom(π1g), U )/U
and by π1(U)the fundamental group of U, which, since the compact connected Lie group Uis semi-
simple, is a finite abelian group.
Then, we have a bijection:
π0(Hom(π1g), U )/U)'
π1(U)
When g= 0, the group π1g)is π1(S2) = {1}, so that Hom(π1g), U )is a single point, hence
the moduli space Mg,0is always connected and the above theorem is no longer true.
Remark 3.2. Recall that, in this note, our purpose is to show how one can use quasi-Hamiltonian
geometry to study the geometry and the topology of representation spaces. To be able to illustrate this
with simple examples, we limit ourselves, in this section, to compact surfaces and semi-simple Lie
groups. We refer to [13] for the computation of the number connected components for surfaces with
removed points and arbitrary compact connected Lie groups.
It is remarkable in the above theorem that the number of connected components of the moduli
space Mg,0= Hom(πg,0, U )/U depends only on the Lie group Uand not on the genus g1. Such
a phenomenon also occurs (as a matter of fact, the exact statement of theorem 3.1 still holds) for
complex semi-simple Lie groups, as shown in [7] for U=SL(2,C)and in [14] for arbitrary complex
semi-simple Lie groups. This is no longer true for non-compact real semi-simple Lie groups. For
instance, Goldman showed in [7] that if U=P SL(2,R)then Mg,0has 4g3connected components.
Likewise, if U=SL(2,R), the number of connected components of Mg,0is shown in [7] to be equal to
22g+1 +2g3. Similar results for non-compact real Lie groups such as P U (n, 1) can be found in [25, 26]
(see also [9, 10, 24]). It would be interesting to know if one can write a quasi-Hamiltonian proof of these
results. As we shall soon see, this would require an analogue of theorem 2.4. Finally, Goldman also
showed that if Uis an algebraic semi-simple group then Mg,0has finitely many connected components,
but that this is no longer true for non-simply connected nilpotent Lie groups (such as the Heisenberg
group for instance).
We can now come back to giving a proof of theorem 3.1:
π0(Hom(π1g), U )/U)'
π1(U)
This proof is due to Ho and Liu in [12]. Observe that theorem 3.1 says that if Uis simply connected
then the moduli space Mg,0is connected. This is a direct consequence of the convexity theorem
2.4: the moduli space Mg,0is the quasi-Hamiltonian quotient Mg,0=µ1({1})/U and since Uis
simply connected the fiber µ1({1})of the momentum map µis connected. To be able to reduce the
general case to the case where Uis simply connected, we will use proposition 2.6 when ρ:e
UUis
the universal cover of U. Since Uis semi-simple, the simply connected Lie group e
Uis still compact.
Further, we have an identification π1(U)'ker ρ⊂ Z(e
U) :=center of e
U. To prove that we have a
bijection between π0(Mg,0)and π1(U)'ker ρ, the strategy of Ho and Liu consists, following Goldman
in [7], in constructing a continuous map
σ: Hom(π1g), U )ker ρ
(this map σis called the obstruction map in [7]) and showing, by methods of quasi-Hamiltonian
geometry, that this continuous map σis surjective with connected fibres, which will eventually imply
6 FLORENT SCHAFFHAUSER
theorem 3.1.
Recall that the moduli space Mg,0is the quasi-Hamiltonian quotient
Mg,0= Hom(πg,0, U )/U =µ1
U({1})/U
where µUis the momentum map
µU:M= (U×U)× · · · × (U×U)U
(a1, b1, ... , ag, bg)7−
g
Y
i=1
[ai, bi]
Applying proposition 2.6 to the universal cover ρ:e
UUof U, the situation is as follows:
(e
U×e
U)× · · · × (e
U×e
U)
µeU
e
U
ρ2g
y
yρ
(U×U)× · · · × (U×U)µU
U
Following Goldman, Ho and Liu define the obstruction map
σ: (U×U)× · · · × (U×U)e
U
in the following way:
Definition 3.3. Let σ: (U×U)× · · · × (U×U)e
Ube the map defined by
σ(a1, b1, ... , ag, bg) :=
g
Y
i=1
[eai,e
bi]
where ρ(eai) = aiand ρ(e
bi) = bifor all i∈ {1, ... , g}.
Lemma 3.4. The map σis wel l-defined and satisfies σρ2g=µe
U. In particular, since the covering
map ρ2gis an open surjective map, the obstruction map σis continuous.
Proof. If ρ(eai) = ρ(ea0
i)and ρ(e
bi) = ρ(e
b0
i), then ea0
i=xieaiand e
b0
i=yie
biwith xi, yiker ρ⊂ Z(e
U). It
follows that [ea0
i,e
b0
i]=[ai, bi]for all i, hence that σis well-defined and satisfies σρ2g=µe
U.
To sum up, we have:
(e
U×e
U)g
µeU//
ρ2g
e
U
ρ
(U×U)gµU//
σ
;;
w
w
w
w
w
w
w
w
w
w
U
Further:
Lemma 3.5. We have σ(µ1
U({1})) ker ρ.
Proof. If Qg
i=1[ai, bi]=1, then:
ρ(a1, b1, ... , ag, bg) = ρµe
U((ea1,e
b1, ... , eag,e
bg))
=µUρ2g((ea1,e
b1, ... , eag,e
bg))
=µ(a1, b1, ... , ag, bg)
=
g
Y
i=1
[ai, bi]
= 1
We now begin the study of the fibres of the obstruction map σ.
QUASI-HAMILTONIAN GEOMETRY AND REPRESENTATION SPACES OF SURFACES GROUPS 7
Lemma 3.6. For any zker ρe
U, the fiber µe
U({z})is non-empty and connected. The map
ρ2g: ( e
U×e
U)g(U×U)grestricts to a continuous surjective map
αz:µ1
e
U({z})σ1({z})(U×U)g
Proof. The fibres of µe
Uare non-empty because e
Uis a compact connected semi-simple Lie group, hence
[e
U, e
U] = e
Uand z= 1 × · · · × 1×[eag,e
bg]for some eag,e
bge
U. Since e
Uis in addition simply connected,
theorem 2.4 shows that the fiber µ1
e
U({z})is connected.
Consider now (ea1,e
b1, ... , eag,e
bg)µ1
e
U({z})(that is: Qg
i=1[eai,e
bi] = z) and set ai:= ρ(eai)and bi:=
ρ(e
bi)for all i. Then σ(a1, b1, ... , ag, bg) = Qg
i=1[eai,e
bi] = zso that ρ2gindeed restricts to a continuous
map αz:µ1
e
U({z})σ1({z}). Surjectivity of αzfollows from the construction of σ.
From this we deduce immediately:
Proposition 3.7. The fibres of the continuous map σ|µ1
U({1U}):µ1
U({1U})ker ρare non-empty
and connected. Since ker ρis a finite set, the connected components of µ1
U({1U})are precisely the
fibres of σabove ker ρ. Consequently the number of connected components of µ1({1U})and therefore
of Mg,0= Hom(πg,0, U )/U is equal to the cardinal of ker ρ'π1(U). More precisely, the map
σ|µ1
U({1U}):µ1
U({1U})ker ρ
induces a map
σ:Mg,0=µ1
U({1})/U ker ρ'π1(U)
whose fibres are the connected components of Mg,0, thereby proving theorem 3.1.
Proof. The fact that the map σ|µ1({1U})has non-empty connected fibres follows from lemma 3.6: the
continuous image αz(µ1
e
U({z})of a connected set is connected. The fact that σ:µ1
U({1U})ker ρ
induces a map σ:µ1
U({1U})ker ρfollows from the fact that ker ρ⊂ Z(e
U).
As an application, we state the following result, first proved by Goldman in [7]: if U=SO(3) (so
that π1(U) = Z/2Z) the moduli space Hom(πg,0, U )/U has 2connected components.
4. Lagrangian submanifolds of representation spaces
In this section, we outline a general strategy for constructing Lagrangian submanifolds of quasi-
Hamiltonian submanifolds of a quasi-Hamiltonian quotient M//U =µ1({1})/U starting from a
quasi-Hamiltonian space (M, ω, µ :MU)and provide an example by applying this strategy to
moduli spaces associated to surface groups.
Henceforth we shall assume that Uacts freely on µ1({1}), so that theorem 2.2 applies and µ1({1})/U
is a symplectic manifold. Our strategy consists in obtaining a Lagrangian submanifold of the quasi-
Hamiltonian quotient M//U =µ1({1})/U by constructing an anti-symplectic involution νon the
symplectic space M//U . Then, if the fixed-point set of νis non-empty, it is a Lagrangian submanifold
of M//U . More precisely, we give sufficient conditions on an involution βon the quasi-Hamiltonian
space (M, ω, µ :MU)for it to induce an anti-symplectic involution ν:= ˆ
βon the associated
quasi-Hamiltonian quotient M//U =µ1({1})/U . To state such a result, we draw on the usual
Hamiltonian case considered in [18] and assume that the compact connected Lie group Uis endowed
with an involutive automorphism τ. We then have:
Proposition 4.1 ([21]).Let (M, ω, µ :MU)be a quasi-Hamiltonian space and let τbe an
involutive automorphism of U. Denote by τthe involution on Udefined by τ(u) = τ(u1)and let
βbe an involution on Msuch that:
(i) uU,xM,β(u.x) = τ(u)(x)
(ii) xM,µβ(x) = τµ(x)
(iii) βω=ω
8 FLORENT SCHAFFHAUSER
then βinduces an anti-symplectic involution ˆ
βon the quasi-Hamiltonian quotient
M//U := µ1({1})/U
defined by ˆ
β([x]) = [β(x)]. If ˆ
βhas fixed points, then F ix(ˆ
β)is a Lagrangian submanifold of M//U .
From now on, we assume additionally that the involution τleaves a maximal torus TU
pointwise fixed, so that the assumptions (U, τ)appearing in theorem 2.5 are satisfied. Recall that
such an involutive automorphism τof Ualways exists, as was recalled earlier. The rest of this section
will be devoted to proving that the assumption F ix(ˆ
β)6=is in fact always satisfied if Uis a compact
connected semi-simple Lie group, provided that the involution βon Mhas fixed points whose image
lies in the connected component of F ix(τ)Ucontaining 1(so that we can apply the real convexity
theorem for group-valued momentum maps -theorem 2.5- stated in subsection 2.2). In fact, we will
prove the following stronger result:
F ix(β)µ1({1})6=(1)
which immediately implies:
F ix(ˆ
β)6=
by definition of ˆ
β.
In order to prove that F ix(β)µ1({1})6=, we will distinguish two cases. We begin with the case
where the compact connected Lie group Uis in addition simply connected and then deal with the
case of a compact connected semi-simple Lie group. In this last case, we will reduce the situation to
the case of simply connected groups by using proposition 2.6, much like what was done in section 3
in order to compute the number of connected components of the representation spaces.
4.1. The case where Uis simply connected. When Uis a compact connected simply connected
Lie group, theorem 2.5 holds. We then have the following corollary, which is exacly the result we set
out to prove (see (1)).
Proposition 4.2 (F ix(β)µ1({1})6=).If βsatisfies the assumptions of theorem 2.5 and ˆ
βdesig-
nates the induced involution ˆ
β([x]) := [β(x)] on the quasi-Hamiltonian quotient M//U =µ1({1})/U ,
we have: F ix(β)µ1({1})6=and therefore F ix(ˆ
β)6=.
Proof. Since µ1({1})6=and since we always have 1exp(W), we obtain, using theorem 2.5:
1µ(M)exp(W) = µ(Mβ)exp(W)
that is:
F ix(β)µ1({1})6=
If xF ix(β)µ1({1})6=, then by definition ˆ
β([x]) = [β(x)] = [x].
Observe that, as in section 3, to prove the proposed statement (1) for simply connected compact
connected Lie groups, one applies directly a theorem from quasi-Hamiltonian geometry.
4.2. The case where Uis semi-simple. To prove that the statement F ix(β)µ1({1})still holds
when Uis assumed to be semi-simple but not necessarily simply connected, we use proposition 2.6 to
construct a quasi-Hamiltonian e
U-space (f
M=M×Ue
U, eω, µ :f
Me
U), where e
Uis the universal cover
of U. Since Uis semi-simple, the simply connected group e
Uis still compact and we can therefore
apply proposition 4.2 to the quasi-Hamiltonian space (f
M, eω, µ :f
Me
U). This will turn out to be
sufficient.
First, we need to observe that if βis a form-reversing involution on M, it induces a form-reversing
involution e
βon f
M. As a first step, observe that since the compact connected groups Uand e
Uhave
isomorphic Lie algebras, the involutive automorphism τof Uinduces an involutive automorphism of
e
U, that we denote by eτ. In particular, we have πeτ=τπ, where πis the covering map π:e
UU.
We will denote by eτthe involution eτ(eu) := eτ(eu1). If τis of maximal rank, so is eτ. If we denote
by Q0the connected component of 1Uin F ix(τ)Uand by e
Q0the connected component of 1e
Uin
F ix(eτ)e
U, the covering map π:e
UUrestricts to a covering map π|e
Q0:e
Q0Q0. Then:
QUASI-HAMILTONIAN GEOMETRY AND REPRESENTATION SPACES OF SURFACES GROUPS 9
Proposition 4.3. Let βbe a form-reversing involution on the quasi-Hamiltonian space (M, ω , µ :
MU), compatible with the action of (U, τ )and the momentum map µ. Then the map
e
β:f
Mf
M
(x, eu)7−(β(x),eτ(eu))
is a form-reversing involution on the quasi-Hamiltonian space (f
M, eω, eµ:f
Me
U), satisfying e
β(eu.x) =
eτ(u).e
β(x)and eµe
β=eτeµ.
We then have:
Theorem 4.4. Let (U, τ )be a compact connected semi-simple Lie group endowed with an involutive
automorphism τof maximal rank, and let (M, ω, µ :MU)be a connected quasi-Hamiltonian
U-space such that µ1({1U})6=. Let βbe a form-reversing compatible involution βon M, whose
fixed-point set F ix(β)is not empty and has an image under µthat intersects the connected component
of 1Uin F ix(τ)U. Then:
F ix(β)µ1({1U})6=
Proof. We will show that there exists a connected component of f
M=M×Ue
Uwhich contains points
of eµ1({1e
U})and fixed points of e
β, and apply the corollary of the convexity theorem (corollary 4.2)
to this connected component, which is a quasi-Hamiltonian space. From this we will deduce the
statement of the theorem.
Since µ1({1})6=and µ(F ix(β))Q06=, there exist x0Msuch that µ(x0)=1Uand x1Msuch
that β(x1) = x1and µ(x1)Q0. Since Mis connected, there is a path (xt)t[0,1] from x0to x1. Set
ut:= µ(xt)Ufor all t[0,1]. Since π:e
UUis a covering map, we can lift the path (ut)t[0,1] to
a path (eut)t[0,1] on e
Usuch that π(eut) = ut=µ(xt)and eu0= 1e
U. Then (xt,eut)f
M=M×Ue
Uand it
is a path going from (x0,eu0)=(x0,1e
U)to (x1,eu1), which are therefore contained in a same connected
component f
M0of f
M. Then, we have eµ(x0,1e
U)=1e
Uand, since π(eu1) = u1=µ(x1)Q0F ix(τ),
we have eu1e
Q0F ix(eτ), hence
e
β(x1,eu1)=(β(x1),eτ(eu1)) = (x1,eu1)
and eµ(x1,eu1) = eu1e
Q0. Therefore, the connected component f
M0of f
M, which is a quasi-Hamiltonian
e
U-space, contains points of eµ1({1e
U})and points of F ix(e
β)whose image is contained in e
Q0. Since
e
Uis simply connected, we can apply corollary 4.2 and conclude that F ix(e
β)eµ1({1e
U})6=. Take
now (x, eu)F ix(e
β)eµ1({1e
U}). In particular, eu= 1e
U. Since e
β(x, eu) = (x, eu), we have β(x) = x
and µ(x) = µp(x, eu) = πeµ(x, eu) = π(eu) = π(1e
U)=1U. That is: xF ix(β)µ1({1U}), which
is therefore non-empty.
This completes the program announced at the beginning of this section (see (1)). We refer to [19]
for a proof of the fact that when M= (U×U)g×C1× · · ·× Cland Uis an arbitrary compact connected
Lie group, we still have F ix(β)µ1({1})6=(see also subsection 4.3 below).
4.3. An example of form-reversing involution β.We end this note with an example of a form-
reversing involution β:MMon the quasi-Hamiltonian space
M= (U×U)g× C1× · ·· × Cl
This involution satisfies the assumptions of theorem 2.5, as is shown in [19], which, as explained in
the above subsections, provides an example of Lagrangian submanifold of the representation space
Mg,l = HomC(πg,l , U )/U =M//U
for any compact connected semi-simple Lie group U. As a matter of fact, it is shown in [19] that the
condition F ix(β)µ1({1})6=is also satisfied for an arbitrary compact semi-simple Lie group (in
the case where Mis (U×U)g× C1× · · · × Cl). Indeed, in this case, the situation is reduced to the
case of a simply connected Lie group by using proposition 2.6 for the covering ρ:S×GUwhere
10 FLORENT SCHAFFHAUSER
S⊂ Z(U)is a torus and Gis a compact connected simply connected Lie group. The same technique
also works for computing the number of connected components of the moduli spaces Mg ,l (see [13]).
In [19], the involution βis otained by introducing the following notion of decomposable representation
of the fundamental group πg,l =π1g\{s1, ... , sl})into U:
Definition 4.5 (Decomposable representations of π1g\{s1, ... , sl}), [19]).Let (U, τ )be a compact
connected Lie group endowed with an involutive automorphism τof maximal rank. A representation
(a1, b1, ... , ag, bg, c1, ... , cl)of πg ,l =π1g\{s1, ... , sl})into Uis called decomposable if there exist
(g+l)elements v1, ... , vg, w1, ... , wlUsatisfying:
(i) τ(vi) = v1
ifor all iand τ(wj) = w1
jfor all j.
(ii) [a1, b1] = v1v1
2,[a2, b2] = v2v1
3, ... , [ag, bg] = vgw1
1,c1=w1w1
2,c2=w2w1
3, ... ,
cl=wlv1
1.
(iii) τ(ai) = v1
i+1bivi+1 for al l i∈ {1, ... , g}(with vg+1 =w1).
We then show that these decomposable representations are characterized in terms of an involution
βon M= (U×U)g×C1×· · ·×Clsatisfying the assumptions of theorem 2.5, from which we can deduce
(see proposition 4.1 and theorem 4.4) that F ix(ˆ
β)6=and is therefore a Lagrangian submanifold of
the moduli space Mg,l . Namely, we have:
Theorem 4.6 (A Lagrangian submanifold of the representation space [19]).There exists a form-
reversing involution βon the quasi-Hamiltonian space M= (U×U)g× C1× ·· · × Clsuch that a
representation (a1, b1, ... , ag, bg, c1, ... , cl)of πg,l into Uis decomposable in the sense of definition 4.5
if and only if there exists an element uUsuch that
β(a1, b1, ... , ag, bg, c1, ... , cl) = u.(a1, b1, ... , ag, bg, c1, ... , cl) and uF ix(τ).
This involution βsatisfies the assumptions of theorem 2.5, hence, if Uis semi-simple, by theorem 4.4
we have
F ix(β)µ1({1})6=
which proves by proposition 4.1 that βinduces an anti-symplectic involution ˆ
βon the quasi-Hamiltonian
quotient
Mg,l = HomC(πg,l , U )/U
whose fixed-point set F ix(ˆ
β)is non-empty and consists of equivalence classes of decomposable repre-
sentations of πg,l into U: it is a Lagrangian submanifold of the moduli space Mg,l.
In fact, we cannot immediately apply the results of subsections 4.1 and 4.2 because in general we
do not have Uacting freely on µ1({1})in the above example where M= (U×U)g× C1× · ·· × Cl.
We refer to [19] to see how to circumvent this difficulty. We also refer to [19] for a general expression
of β. When g= 0 and l= 3, we have the following expression:
β(c1, c2, c3)=(τ(c2c3)τ(c1)τ(c2c3), τ (c3)τ(c2)τ(c3), τ (c3))
When g= 1 and l= 0, we have:
β(a, b) = τ(b), τ (a)
Finally, we refer to [11] for another example of an anti-symplectic involution σon the representation
space Mg,0= Hom(πg,0, U )/U of the fundamental group of a compact surface.
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Keio University, Dept. of Mathematics, Hiyoshi 3-14-1, Kohoku-ku, 223-8522, Yokohama, Japon
E-mail address:florent@math.jussieu.fr
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