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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 g≥0, we consider the surface

Σg,l := Σg\{s1, ... , sl}

obtained from Σgby removing lpairwise disctinct points s1, ... , sl∈Σg. The fundamental group of

the surface Σg,l thus obtained has the following ﬁnite presentation:

πg,l := π1(Σg\{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 g≥1and 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 =µ−1({1})/U

where µis the momentum map

µ: (U×U)g× C1× · ·· × Cl−→ U

(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 =µ−1({1})/U

with

µ: (U×U)g× C1× · ·· × Cl−→ U

(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=g−1.dg and θR=dg.g−1

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 X∈u. Its value at x∈Mis :

X]

x:= d

dt |t=0(exp(tX ).x). Let µ:M→Ube a U-equivariant map (for the conjugacy action of Uon

itself).

Then (M, ω, µ :M→U)is said to be a quasi-Hamiltonian space (with respect to the action of U) if

the map µ:M→Usatisﬁes the following three conditions:

(i) dω =−µ∗χ

(ii) for all x∈M,ker ωx={X]

x:X∈u|(Ad µ(x) + Id).X = 0}

(iii) for all X∈u,ιX]ω=1

2µ∗(θL+θR|X)

where (θL+θR|X)is the real-valued 1-form deﬁned on Ufor any X∈uby (θL+θR|X)u(ξ) :=

(θL

u(ξ) + θR

u(ξ)|X)(where u∈Uand ξ∈TuU).

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 :M→U)be a quasi-

Hamiltonian U-space. If the compact connected Lie group Uacts freely on the ﬁber µ−1({1}), then:

(i) 1∈Uis a regular value of the momentum map µ(consequently, the set µ−1({1})is a sub-

manifold of M).

(ii) the set µ−1({1})/U is a manifold.

(iii) if we denote by i:µ−1({1})→Mthe inclusion of the level submanifold µ−1({1})in Mand

by pthe principal ﬁbration p:µ−1({1})→µ−1({1})/U, the 2-form i∗ωis basis with respect

to p: there exists a (unique) 2-form ωred on µ−1({1})/U such that i∗ω=p∗ωred.

(iv) the 2-form ωred is symplectic.

The symplectic manifold µ−1({1})/U is often denoted M//U and called the quasi-Hamiltonian quo-

tient associated to the quasi-Hamiltonian space (M, ω, µ :M→U).

Dropping the assumption on the freeness of the action of Uon µ−1({1})/U, we also have the

following generalization of theorem 2.2, saying that the set M//U =µ−1({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 :M→U)be a quasi-

Hamiltonian U-space. For any closed subgroup K⊂U, denote by MKthe isotropy manifold of type

Kin M:

MK={x∈M|Ux=K}.

Denote by N(K)the normalizer of Kin Uand by LKthe quotient group LK:= N(K)/K. Then the

orbit space

(µ−1({1U})∩MK)/LK

is a smooth symplectic manifold.

Denote by (Kj)j∈Ja system of representatives of closed subgroups of U. Then the orbit space Mred :=

µ−1({1U})/U is the disjoint union of the following symplectic manifolds:

µ−1({1U})/U =G

j∈J

(µ−1({1U})∩MKj)/LKj.

Theorems 2.2 and 2.3 show that the representation spaces HomC(πg,l, U )/U =µ−1({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 A⊂exp(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 :M→U)be a connected quasi-Hamiltonian

space with proper momentum map µ. Then, for any choice of a maximal torus T⊂Uand 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 µ−1({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−:u∈U7→ τ(u−1)leaves a maximal torus Tof Upointwise ﬁxed, and let

W ⊂ t:= Lie(T)be a Weyl alcove. Let (M, ω, µ :M→U)be a connected quasi-Hamiltonian U-space

with proper momentum map µ:M→Uand let β:M→Mbe an involution on Msatisfying:

(i) β∗ω=−ω

(ii) β(u.x) = τ(u).β(x)for al l x∈Mand all u∈U

(iii) µ◦β=τ−◦µ

(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 τ−leaves 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

U→Ube a covering

map. Set

f

M:= M×Ue

U={(x, eu)|µ(x) = π(eu)}

and

p:f

M−→ Meµ:f

M−→ e

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:= p∗ωand observe that e

Uacts on f

Mvia

eu0.(x, eu) := (π(eu0).x, eu0eueu−1

0)

and that eµis equivariant for this action. Then we have: (f

M, eω, eµ:f

M→e

U)is a quasi-Hamiltonian

e

U-space.

The proof shows that this works because π:e

U→e

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

M→Msends eµ−1({1e

U})to µ−1({1U})and induces an isomorphism

eµ−1({1e

U})/e

U'µ−1({1U})/U

QUASI-HAMILTONIAN GEOMETRY AND REPRESENTATION SPACES OF SURFACES GROUPS 5

In particular, if µ−1({1U})6=∅then eµ−1({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 g≥1and let Ube a compact connected semi-simple Lie group. Denote by

π0(Hom(π1(Σg), U )/U)the set of connected components of the representation space

Mg,0= Hom(π1(Σg), 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(π1(Σg), U )/U)'

−→ π1(U)

When g= 0, the group π1(Σg)is π1(S2) = {1}, so that Hom(π1(Σg), 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 g≥1. 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 4g−3connected components.

Likewise, if U=SL(2,R), the number of connected components of Mg,0is shown in [7] to be equal to

22g+1 +2g−3. 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(π1(Σg), 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=µ−1({1})/U and since Uis

simply connected the ﬁber µ−1({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

U→Uis

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(π1(Σg), 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 =µ−1

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

U→Uof 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 σ◦ρ2g=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, yi∈ker ρ⊂ Z(e

U). It

follows that [ea0

i,e

b0

i]=[ai, bi]for all i, hence that σis well-deﬁned and satisﬁes σ◦ρ2g=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(µ−1

U({1})) ⊂ker ρ.

Proof. If Qg

i=1[ai, bi]=1, then:

ρ◦(a1, b1, ... , ag, bg) = ρ◦µe

U((ea1,e

b1, ... , eag,e

bg))

=µU◦ρ2g((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 z∈ker ρ⊂e

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:µ−1

e

U({z})−→ σ−1({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

bg∈e

U. Since e

Uis in addition simply connected,

theorem 2.4 shows that the ﬁber µ−1

e

U({z})is connected.

Consider now (ea1,e

b1, ... , eag,e

bg)∈µ−1

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:µ−1

e

U({z})→σ−1({z}). Surjectivity of αzfollows from the construction of σ.

From this we deduce immediately:

Proposition 3.7. The ﬁbres of the continuous map σ|µ−1

U({1U}):µ−1

U({1U})→ker ρare non-empty

and connected. Since ker ρis a ﬁnite set, the connected components of µ−1

U({1U})are precisely the

ﬁbres of σabove ker ρ. Consequently the number of connected components of µ−1({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|µ−1

U({1U}):µ−1

U({1U})−→ ker ρ

induces a map

σ:Mg,0=µ−1

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|µ−1({1U})has non-empty connected ﬁbres follows from lemma 3.6: the

continuous image αz(µ−1

e

U({z})of a connected set is connected. The fact that σ:µ−1

U({1U})→ker ρ

induces a map σ:µ−1

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 =µ−1({1})/U starting from a

quasi-Hamiltonian space (M, ω, µ :M→U)and provide an example by applying this strategy to

moduli spaces associated to surface groups.

Henceforth we shall assume that Uacts freely on µ−1({1}), so that theorem 2.2 applies and µ−1({1})/U

is a symplectic manifold. Our strategy consists in obtaining a Lagrangian submanifold of the quasi-

Hamiltonian quotient M//U =µ−1({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 :M→U)for it to induce an anti-symplectic involution ν:= ˆ

βon the associated

quasi-Hamiltonian quotient M//U =µ−1({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 :M→U)be a quasi-Hamiltonian space and let τbe an

involutive automorphism of U. Denote by τ−the involution on Udeﬁned by τ−(u) = τ(u−1)and let

βbe an involution on Msuch that:

(i) ∀u∈U,∀x∈M,β(u.x) = τ(u).β(x)

(ii) ∀x∈M,µ◦β(x) = τ−◦µ(x)

(iii) β∗ω=−ω

8 FLORENT SCHAFFHAUSER

then βinduces an anti-symplectic involution ˆ

βon the quasi-Hamiltonian quotient

M//U := µ−1({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 τ−leaves a maximal torus T⊂U

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)∩µ−1({1})6=∅(1)

which immediately implies:

F ix(ˆ

β)6=∅

by deﬁnition of ˆ

β.

In order to prove that F ix(β)∩µ−1({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)∩µ−1({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 =µ−1({1})/U ,

we have: F ix(β)∩µ−1({1})6=∅and therefore F ix(ˆ

β)6=∅.

Proof. Since µ−1({1})6=∅and since we always have 1∈exp(W), we obtain, using theorem 2.5:

1∈µ(M)∩exp(W) = µ(Mβ)∩exp(W)

that is:

F ix(β)∩µ−1({1})6=∅

If x∈F ix(β)∩µ−1({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)∩µ−1({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

M→e

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

M→e

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 π◦eτ=τ◦π, where πis the covering map π:e

U→U.

We will denote by eτ−the involution eτ−(eu) := eτ(eu−1). 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

U→Urestricts to a covering map π|e

Q0:e

Q0→Q0. 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 :

M→U), compatible with the action of (U, τ )and the momentum map µ. Then the map

e

β:f

M−→ f

M

(x, eu)7−→ (β(x),eτ−(eu))

is a form-reversing involution on the quasi-Hamiltonian space (f

M, eω, eµ:f

M→e

U), satisfying e

β(eu.x) =

eτ(u).e

β(x)and eµ◦e

β=eτ−◦eµ.

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 :M→U)be a connected quasi-Hamiltonian

U-space such that µ−1({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)∩µ−1({1U})6=∅

Proof. We will show that there exists a connected component of f

M=M×Ue

Uwhich contains points

of eµ−1({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 µ−1({1})6=∅and µ(F ix(β))∩Q06=∅, there exist x0∈Msuch that µ(x0)=1Uand x1∈Msuch

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

U→Uis 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)∈Q0⊂F ix(τ−),

we have eu1∈e

Q0⊂F ix(eτ−), hence

e

β(x1,eu1)=(β(x1),eτ−(eu1)) = (x1,eu1)

and eµ(x1,eu1) = eu1∈e

Q0. Therefore, the connected component f

M0of f

M, which is a quasi-Hamiltonian

e

U-space, contains points of eµ−1({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)∩eµ−1({1e

U})6=∅. Take

now (x, eu)∈F ix(e

β)∩eµ−1({1e

U}). In particular, eu= 1e

U. Since e

β(x, eu) = (x, eu), we have β(x) = x

and µ(x) = µ◦p(x, eu) = π◦eµ(x, eu) = π(eu) = π(1e

U)=1U. That is: x∈F ix(β)∩µ−1({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)∩µ−1({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:M→Mon 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)∩µ−1({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×G→Uwhere

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 =π1(Σg\{s1, ... , sl})into U:

Deﬁnition 4.5 (Decomposable representations of π1(Σg\{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 =π1(Σg\{s1, ... , sl})into Uis called decomposable if there exist

(g+l)elements v1, ... , vg, w1, ... , wl∈Usatisfying:

(i) τ(vi) = v−1

ifor all iand τ(wj) = w−1

jfor all j.

(ii) [a1, b1] = v1v−1

2,[a2, b2] = v2v−1

3, ... , [ag, bg] = vgw−1

1,c1=w1w−1

2,c2=w2w−1

3, ... ,

cl=wlv−1

1.

(iii) τ(ai) = v−1

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 u∈Usuch that

β(a1, b1, ... , ag, bg, c1, ... , cl) = u.(a1, b1, ... , ag, bg, c1, ... , cl) and u∈F 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)∩µ−1({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 µ−1({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