PARAMETRIZING SPACES OF POSITIVE REPRESENTATIONS OLIVIER GUICHARD EUGEN ROGOZINNIKOV AND ANNA WIENHARD Abstract. Using Lusztigs total positivity in split real Lie groups V. Fock and A. Goncharov

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PARAMETRIZING SPACES OF POSITIVE REPRESENTATIONS

OLIVIER GUICHARD, EUGEN ROGOZINNIKOV, AND ANNA WIENHARD

Abstract.

Using Lusztig’s total positivity in split real Lie groups V. Fock and A. Goncharov

have introduced spaces of positive (framed) representations. For general semisimple Lie groups

a generalization of Lusztig’s total positivity was recently introduced by O. Guichard and

A. Wienhard. They also introduced the associated space of positive representations. Here

we consider the corresponding spaces of positive framed representations of the fundamental

group of a punctured surface. We give several parametrizations of the spaces of framed

positive representations. Using these parametrizations, we describe their topology and their

homotopy type. We show that the number of connected components of the space of framed

positive representations agrees with the number of connected components of the space of

positive representations, and determine this number for simple Lie groups. Along the way, we

also parametrize, for an arbitrary semisimple Lie group, the space of representations of the

fundamental group of a punctured surface which are transverse with respect to a ﬁxed ideal

triangulation of the surface.

Contents

1. Introduction 2

2. Preliminary observations 4

2.1. Semisimple Lie groups and ﬂag varieties 4

2.2. Triples and quadruples of ﬂags 5

2.3. Groups with positive structure 6

3. Topological data 8

3.1. Punctured surfaces 8

3.2. Ideal triangulations 8

3.3. Graph Γ9

4. Framed representations and framed local systems 10

4.1. Transverse framed representations 10

4.2. Framed local systems on trees 10

4.3. From local systems to representations 11

4.4. From representations to local systems 12

5. Parametrization of the space of framed representations 14

6. Space of positive framed representations and its parametrization 16

7. Homotopy type of the space of positive framed representations 17

8. Connected components of the space of positive representations 19

9. Examples 20

9.1. Trivial example 20

E.R. thanks the Labex IRMIA of the Université de Strasbourg for support during the preparation of this article.

O.G. thanks the Institut univesitaire de France and acknowledges support of the Agence Nationale de la Recherche

under the grant DynGeo (ANR-16-CE40-0025). A.W. is partially funded by the European Research Council under

ERC-Advanced Grant 101018839, by the Klaus Tschira Foundation, and by the Deutsche Forschungsgemeinschaft

under Germany’s Excellence Strategy EXC-2181/1 - 390900948 (the Heidelberg STRUCTURES Cluster of

Excellence).

arXiv:2210.11605v1 [math.DG] 20 Oct 2022

2 O. GUICHARD, E. ROGOZINNIKOV, AND A. WIENHARD

9.2. Gis isogenic to SL2(R)20

9.3. G= PSLn(R)21

9.4. G= Sp2n(R)with Hermitian positive structure 21

9.5. Gis GL2(A)and some of its subgroups 21

9.6. Gis the connected adjoint form of a Hermitian Lie group of tube type 22

9.7. Gis an indeﬁnite orthogonal group SO0(1 + p, n +p)where p≥1,n≥223

9.8. Gis an exceptional from the family F4(4),E6(2),E7(−5),E8(−24) 24

References 25

1. Introduction

In the papers [GW18,GW22], O. Guichard and A. Wienhard introduced a notion of Θ-positivity

for Lie groups that generalizes the notion of the Lusztig’s total positivity for split real Lie

groups [Lus94] to a larger class of semisimple Lie groups. The groups admitting such positive

structures are of particular interest for higher rank Teichmüller theory because spaces of positive

representations of the fundamental group of a closed surface

into a semisimple Lie group

with

a positive structure provide examples of higher rank Teichmüller spaces: they provide subspaces

of the character variety

Rep

(

π1

(

)

, G

)

Hom

(

π1

(

)

, G

)

that consist entirely of discrete and

faithful representations.

Spaces of positive representations are not only interesting for closed surfaces but also for

surfaces with punctures or boundary components. In fact, in these cases, the space of framed (and

of decorated) representations admits particularly nice cluster structures. The space of framed

representations of the fundamental group of a punctured surface of negative Euler characteristic

into complex Lie groups and split real Lie groups was introduced by V. Fock and A. Goncharov

in [FG06]. They study these spaces using ideal triangulations of surfaces, i.e. triangulations whose

set of vertices agrees with the set of punctures of the surface. They introduce so-called

-type and

-type cluster coordinates associated with ideal triangulations and parametrize spaces of framed

representations that are transverse to a given ideal triangulation. Moreover, they show that the

positive locus of these coordinates is independent on the choice of triangulation and parametrizes

higher rank Teichmüller spaces. In [BD14, BD17] F. Bonahon and G. Dreyer generalize this

approach and give parametrizations of higher rank Teichüller spaces if

PSLn

(

)for closed

surfaces using coordinates associated to geodesic laminations.

In the case of a split real Lie group G, the positive structure introduced by O. Guichard and

A. Wienhard, agrees with Lusztig’s total positivity if Θis the set of all simple roots of

. In

this case, when the surface is closed the space of positive representations agrees with the Hitchin

components. The structure of Hitchin components is well-studied for closed surfaces [Hit92]

as well as for surfaces with punctures and boundary components [FG06]. In particular, their

topology is well understood. More precisely, every Hitchin component is homeomorphic to an

open ball in a Euclidean vector space. In particular, they are contractible manifolds.

Another example of Lie groups with a positive structure are Hermitian Lie groups of tube

type. In this case positive representations are maximal representations which were introduced

and studied in [BIW10, BILW05, Str15]. The topology of spaces of maximal representation

for closed surfaces was studied in [Got01, GW10, BGPG06, AC19], partly using the theory of

Higgs bundles. In [AGRW22], the spaces of framed and decorated maximal representations into

the real symplectic group

n, R

)are parametrized using a noncommutative analog of the

Fock–Goncharov parametrization [FG06] and the topology of them is studied. In contrast to the

Hitchin components, these spaces are not contractible, and their topology is more complicated.

PARAMETRIZING SPACES OF POSITIVE REPRESENTATIONS 3

In this article, we generalize the approach of [FG06,AGRW22] to a larger class of Lie groups.

For a semisimple Lie group

and a self-opposite parabolic subgroup

P+

, we introduce

the space of representations of the fundamental group of a punctured surface into

framed by

elements of the ﬂag variety

G/P+

. We parametrize these spaces using parameters associated

to ideal triangulations of the surface. If the group

carries a positive structure in the sense

of [GW22], we parametrize spaces of positive representations as well. Using this parametrization,

we study the topology of spaces of positive representations, its homotopy type and count the

number of connected components.

We now describe our results in more detail.

Let

be a surface without boundary of negative Euler characteristic

(

)and with

k >

punctures (we refer to Section 3 for the wider generality that can be allowed for

, for example

disks with marked points on the boundary). Let

be a semisimple Lie group and

P+

be a

self-opposite parabolic subgroup of G.

Aframed representation is a representation

π1

(

)

→G

together with a

-tuples of ﬂags

(

F1, . . . , Fk

)of

G/P+

that are ﬁxed by the images of the peripheral elements

c1, . . . , ck

π1(S).

Fixing an ideal triangulation

, we consider a subspace of the space of framed representa-

tions, that are transverse with respect to

(we refer to Deﬁnition 4.3 for more details). Further,

we introduce two collections of parameters: The ﬁrst collection is associated to the triangles of

and is given by the elements of the unipotent subgroup

U+

P+

that parametrize triples of ﬂags

associated to the vertices of the triangles. There are

−

(

)parameters in this ﬁrst collection.

To describe the second collection of parameters, we need to lift the ideal triangulation

to the

universal covering

and to ﬁx a connected fundamental domain

that consists of

triangles of the lifted triangulation. We associate to every pair of edges in the boundary of

that

are identiﬁed by an element of the fundamental group of

an element of the Levi subgroup of

P+

. There are 1

−χ

(

)parameters in this second collection. The following theorem shows

that given these two collections of parameters up to a common conjugation by an element of

, a

framed representation can be uniquely reconstructed:

Theorem 1.1.

The space of framed representations that are transverse with respect to

homeomorphic to:

(U+

∗)−2χ(S)× L1−χ(S)/L

where

U+

∗

is the open and dense subspace of

U+

that parametrizes pairwise transverse triples of

ﬂags in Fand Lacts by conjugation in every factor.

If the group

admits a positive structure, then we can similarly parameterize spaces of framed

positive representations in the sense of [GW18, GLW21]:

Theorem 1.2. The space of positive framed representations is homeomorphic to:

(U+

>0)−2χ(S)× L1−χ(S)

0/L0,

where

U+

is the positive semigroup of

U+

parametrizing positive triples of ﬂags in

and

the subgroup of the Levi subgroup

stabilizing

U+

under the action by conjugation that acts by

conjugation in every factor.

As corollary of this result, we obtain a description of the homotopy type of the space of positive

framed representations:

Corollary 1.3.

The space of positive framed representations is homotopy equivalent to

K1−χ(S)

0/K0, where K0is a maximal compact subgroup of L0.

4 O. GUICHARD, E. ROGOZINNIKOV, AND A. WIENHARD

This immediately gives a count of the number of connected components of the space of positive

framed representations. If

is a connected Lie group, by the following Corollary, we also obtain a

count of the connected components of the space of positive representations (without any framing),

see Table 1.

Corollary 1.4.

The space of positive framed representations and the space of positive non-framed

representations have the same number of connected components.

Group Number of connected components

Adjoint form of split real groups 1

Hermitian Lie groups

Sp2n(R) 21−χ(S)

PSp2n(R),neven 21−χ(S)

PSp2n(R),nodd 1

U(n, n)1

SU(n, n)1

SO∗(4n)1

E7(−25) 1

Orthogonal indeﬁnite groups

SO0(1 + p, n +p),podd 21−χ(S)

SO0(1 + p, n +p),peven 1

Exceptional family

F4(4) 1

E6(2) 1

E7(−5) 1

E8(−24) 1

Table 1. Number of connected components of spaces of positive representations

For exceptional Lie groups, we always consider the adjoint form. For further details about the

topology of spaces of positive representations, we refer the reader to Section 9.

Structure of the paper:

In Section 2 we recall some facts on the Lie theory and the notion of

positivity for Lie group following [GW18,GW22, GLW21]. In Section 3 we introduce punctured

surfaces, ideal triangulations and some special kind of graphs on surfaces associated with triangu-

lations that we are using later. In Section 4, we introduce the spaces of framed representations

and local systems on trees and describe the connection between them. In Section 5 we prove

Theorem 1.1 on the parametrization of framed representations. In Section 6 we introduce positive

framed representations and describe several parametrizations of them, in particular, we prove

Theorem 1.2. In Section 7 we describe the homotopy type of the space of positive framed

representations. In Section 8 we prove that the number of connected components of the space

of positive framed representations coincides with the number of connected components of the

space of positive non-framed representations. In Section 9 we apply the parametrizations from

Sections 5 and 6 to understand explicitly the topology of spaces of transverse framed and positive

framed representations for some Lie groups.

2. Preliminary observations

2.1.

Semisimple Lie groups and ﬂag varieties.

Let

be a connected semisimple Lie group

with identity element

. Let Θbe a subset of the set of simple roots ∆of

Lie

(

). Let

P+

PARAMETRIZING SPACES OF POSITIVE REPRESENTATIONS 5

(resp.

P−

) be the parabolic subgroup associated with Θ. We assume that

P+

is self-opposite. By

the Levi decomposition,

P+

(resp.

P−

) is a semidirect product of its unipotent radical

U+

(resp.

U−) and the Levi factor L=P+∩ P−.

Let

be the Weyl group of

and

be the longest element of

. Let

ω∈G

be a lift of

We ﬁx

once and for all. Since

P+

is self-opposite,

ωP+ω−1

P−

and

ω2∈ L

. We consider

the ﬂag variety: F=G/P+. The group Gacts on Ftransitively. We call elements of Fﬂags.

Deﬁnition 2.1.

Two ﬂags

F1, F2∈ F

are transverse if there exists

g∈G

such that

g(F1, F2)=(P+, ωP+).

Remark 2.2.•

The relation on

to be transverse is symmetric because

ω(P+, ωP+) = (ωP+, ω2P+)=(ωP+,P+).

•The stabilizer in Gof the pair (P+, ωP+)is equal to the Levi subgroup L.

Proposition 2.3.

For every ﬂag

F∈ F

which is transverse to

P+

there exists a unique

u∈ U+

such that F=uωP+.

Proof.

Since the pair (

P+, F

)is transverse, there exists

g∈G

such that

(

P+, F

) = (

P+, ωP+

In particular,

g∈ P+

StabG

(

P+

). Further,

g−1ωP+

. By the Levi decomposition,

g−1=u` where u∈ U+,`∈ L. Since `ωP+=ωP+, we obtain F=uωP+.

Let

u0∈ U+

be another element such that

u0ωP+

. Therefore,

ωP+

u−1u0ωP+

. This

means that u−1u0∈ P−∩ U+={e}, i.e. u=u0.

2.2. Triples and quadruples of ﬂags.

Lemma 2.4.

Let (

F1, F2, F3

)be a triple of pairwise transverse ﬂags. There exist an element

g∈Gsuch that g(F1) = P+,g(F2) = ωP+,g(F3) = uωP+where u∈ U+∩ P−ωP−.

Proof.

Indeed, up to

-action, we can assume

P+

ωP+

uωP+

for some

u∈ U+

Since

and

are transverse, there exists

g∈G

such that

gF2

P+

gF3

ωP+

. From the

ﬁrst equality, we obtain

pω−1

for

p∈ P+

. From the second one:

pω−1uωP+

ωP+

, i.e.

pω−1u∈ P−

. This means there exists

p0∈ P−

such that

pω−1u

, i.e.

ωpp0

p00ωp0

where p00 =ωpω−1∈ P−. So u∈ P−ωP−.

We denote

U+

∗:

U+∩ P−ωP−

. This is an open dense subset of

U+

. The Levi factor

acts

on U+

∗.

Deﬁnition 2.5.

We denote by

Conf∗

(

)the space of transverse triples of ﬂags in

up to the

action of G.

The following Propositions are immediate:

Proposition 2.6.

The space

Conf∗

(

)is homeomorphic to

U+

∗/L

where

acts by conjugation

on U+

∗.

Proposition 2.7.

Let (

F1, F2, F3, F4

)be a quadruple of ﬂags such that the triples (

F1, F3, F4

)

and (F1, F2, F3)are transverse. Then there exists g∈Gsuch that

g(F1, F2, F3, F4) = (P+, ωu0ωP+, ωP+, uωP+)=(P+,(u00)−1ωP+, ωP+, uωP+)

where

u, u0, u00 ∈ U+

∗

. The element

is unique up to the left multiplication by an element of

i.e. as an element of L \ G={Lg|g∈G}.

The following proposition is quite technical, we will need it later to understand groups with

positive structure. But since this proposition holds in general, we state it here.

Proposition 2.8.

Let

u∈ U+

such that (

uω

)

k∈ P+

for some

k∈N

. Then

(uω)k= (ωu)k∈NormL(u).

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PARAMETRIZINGSPACESOFPOSITIVEREPRESENTATIONSOLIVIERGUICHARD,EUGENROGOZINNIKOV,ANDANNAWIENHARDAbstract.UsingLusztig'stotalpositivityinsplitrealLiegroupsV.FockandA.Goncharovhaveintroducedspacesofpositive(framed)representations.ForgeneralsemisimpleLiegroupsageneralizationofLusztig'stotalpositivitywasre...

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PARAMETRIZING SPACES OF POSITIVE REPRESENTATIONS OLIVIER GUICHARD EUGEN ROGOZINNIKOV AND ANNA WIENHARD Abstract. Using Lusztigs total positivity in split real Lie groups V. Fock and A. Goncharov

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