Irreducible decomposition for local representations of quantum Teichmüller space

arXiv:1404.4938v1 [math.GT] 19 Apr 2014

Irreducible decomposition for local representations of quantum Teichmüller space

Jérémy Toulisse April 22, 2014

Abstract

We give an irreducible decomposition of the so-called local represen- tations [HBL07] of the quantum Teichmüller space Tq(Σ) where Σ is a punctured surface of genus g > 0 and q is a N-th root of unity with N odd.

Let Σ be an oriented surface of genus g > 0 with s punctures v₁, ..., v_s such that 2g−2 +s > 0 (this condition is equivalent to the existence of an ideal triangulation ofΣ, ie. a triangulation whose vertices are exactly the v_i).

LetT(Σ) be the Teichmüller space of Σ, that is the moduli space of complete hyperbolic metrics on Σ. Given λ an ideal triangulation of Σ, W.P. Thurston [Thu98] constructed a parameterization ofT(Σ)by associating a strictly positive real number to each edge λi of the ideal triangulation, i ∈ {1, ..., n} (where n = 6g−6 + 3s is the number of edges of λ). These coordinates are called shear coordinatesassociated toλ. In this coordinates system, the coefficients of the Weil-Petersson form onT(Σ)depend only on the combinatoric of λand are easy to compute.

For a parameter q ∈ C^∗, L.O. Chekhov, V.V. Fock [FC99] and indepen- dently R. Kashaev [Kas98] defined the so-calledquantum Teichmüller space T_q(Σ)of Σ(the construction of R. Kashaev differs a little from the one of L.O.

Chekhov and V.V. Fock), which is a deformation of the Poisson algebra of ra- tional functions overT(Σ). This algebraic object is obtained by gluing together a collection of non-commutative algebra T_q(λ) (called Chekhov-Fock algebra) canonically associated to each ideal triangulation of Σ. A representation of T_q(Σ) is then a family of representation {ρ_λ :T_q(λ) → End(V)}_λ∈Λ(Σ), where Λ(Σ) is the space of all ideal triangulations of Σ, and ρ_λ and ρ_λ′ satisfy com- patibility conditions whenever λ6=λ^′. For λ∈ Λ(Σ), the representation ρ_λ is an avatar of the representation ofT_q(Σ)and carries almost all the information.

Whenq is aN-th root of unity, T_q(λ) admits finite-dimensional representa- tions. In this paper, we will considerN odd. The irreducible representations of T_q(λ) have been studied in [BL07]. In particular, they show that an irreducible representation of T_q(λ) is classified (up to isomorphism) by a weight x_i ∈ C^∗ assigned to each edge λ_i, a choice of N-th rootp_j = (x₁^kj1...x^k_n^jn)^1/N associated to each puncturev_j (wherek_ji is the number of times a small simple loop around

v_j intersects λ_i) and a square root c = (p₀...p_s)^1/2. Such a representation has dimension N^3g−3+s.

In [HBL07], the authors introduced another type of representations ofT_q(λ), calledlocal representations, which are well behaved under cut and paste. A local representation of T_q(λ) is defined by a an embedding into the tensorial product of triangle algebras(see definitions below). The local representations of Tq(λ) are classified (up to isomorphism) by a weight xi ∈ C^∗ associated to each edge λ_i and a choice ofN-th rootc= (x₁...x_n)^1/N. Such a representation has dimension N^4g−4+2s.

It follows that a local representation of Tq(λ) is not irreducible. In this paper, we adress the question of the decomposition of a local representation into its irreducible components. In particular, we prove the following result:

Theorem 1. Let λbe an ideal triangulation ofΣandρbe a local representation of T_q(λ) classified by weight x_j ∈C^∗ associated to each edge λ_j and a choice of N-th root c= (x1...xn)^1/N. We have the following decomposition:

ρ=M

i∈I

ρ⁽ⁱ⁾.

Here, ρ⁽ⁱ⁾ is an irreducible representation classified by the samex_j, a N-th root p⁽ⁱ⁾_j = (x^k₁^j1...x^k_n^jn)^1/N associated to each puncture, and the same c. Moreover, for each choice of N-th root pj = (x₁^kj1...x^kn^jn)^1/N for each puncture and c = (p0...ps)^1/2, there exists exactly N^g elements i ∈ I with p⁽ⁱ⁾_j = pj for all j ∈ {0, ..., s}.

This result may be used to define representations of the so-calledKauffman skein algebraS^A(Σ)[Tur91] (whereΣis the surfaceΣwithout marked points) which corresponds to a quantization by deformation of the character variety

R:=Hom(π₁(Σ), P SL(2,C))// P SL(2,C))

where P SL(2,C) acts by conjugation on the morphisms and the double slash means that we take the quotient in the sense of Geometric Invariant Theory. In [BW11, Theorem 1], the authors constructed a morphism

T r_ω(λ) :S_A(Σ)−→Zˆ_ω(λ),

where q = ω⁴,A = ω⁻² and Zˆω(λ) is an algebra of non-commutative rational fractions such that T_q(λ) consists of rational fractions in Zˆ_ω(λ) involving only even powers of the variables. This morphism, composed with a representationρ ofTq(λ)is studied in [BW12a] and [BW12b] to define a new kind of representa- tions of S^A(Σ). However, if one wants to define representation of S^A(Σ)in the same way, one has to consider the direct sum ofN^g irreducible components of a local representation ρ:T^q(λ)→End(V) arising in the decomposition of Theo- rem 1 and find a subspaceE ⊂V stable byρ◦T r_ω(λ) such that(ρ◦T r_ω(λ))_|E defines a representation ofS^A(Σ)(see [BW14] for the construction). Hopefully, this representation ofS^A(Σ)should be used to define a more intrinsic version of

the Kashaev-Baseilhac-Benedetti TQFT (see [Kas95], [Kas99], [BB04], [BB05]

and [BB07]).

In the first section, we recall the definition of the Chekhov-Fock algebra, the quantum Teichmüller space, the triangle algebra and the local representations.

In the second one, we prove the Theorem 1. The proof is done in two steps: we first prove the result for a special triangulationλ₀ and special weights; we then extend to the general case.

Acknowledgment : I would like to thank Francis Bonahon for the time he spent explaining me the subtilities of the quantum Teichmüller theory. This work has been supported by a "Summer Regs" grant from GEAR in summer 2013.

1 The Chekhov-Fock algebra and the irreducible rep- resentations of T

(Σ)

The results of this section come from [BL07] and [HBL07]. From now, forn∈N, define N_n:=Z/nZ and denote byU(N) the group of N-th root of unity.

1.1 The Chekhov-Fock algebra

Letλ be an ideal triangulation of Σ. The Chekhov-Fock algebra T_q(λ) as- sociated toλ is the algebra generated by the elements X_i^±1 associated to each edgeλ_i of the triangulation λ. These elements are subjects to the relations:

X_iX_j =q^σijX_jX_i,

where the coefficients σ_ij are the coefficients of the Weil-Petersson form in the shear coordinates associated to λ and depend only on the combinatoric of λ.

Namely, we haveσ_ij =a_ij−a_ji wherea_ij is the number of angular sector delim- ited by λi and λj in the faces of λwith λi coming before λj counterclockwise.

In practice, elements ofT_q(λ) are just Laurent polynomials in the variables X_i satisfying non-commutativity conditions. We will sometimes denote T_q(λ) by C[X₁^±1, ..., X_n^±1]q to reflect this fact.

Letk= (k₁, ..., k_n)∈Zⁿ be a multi-index; to a monomial X composed of a product ofX_i^ki, we associate its quantum ordering:

[X] :=q^−Pi<jσijX₁^k1...X_n^kn.

It allows us to associate a monomial X_k ∈ T_q(λ) to each each multi-index k∈Zⁿ.

To study finite-dimensional representations ofT_q(λ), one needs to determine its center.

Proposition 1 ([BL07], Proposition 15). The center of Tq(λ) is generated by:

• X_i^N for each i∈ {1, ..., n}.

• For each puncturev_j, thepuncture invariantP_j associated to the multi- index k_j = (k_j1, ..., k_jn)(wherek_ji is the number of intersections ofλ_i with a small simple loop around v_j).

• The element H associated to the multi index k= (1, ...,1).

Note that[P₁...P_s] =H². 1.2 Triangle algebra

Let T be a disk with three puncturesv₁, v₂, v₃ ∈∂T endowed with the natural triangulation λcomposed of three counterclokwise directed edges λ₁, λ₂ andλ₃ (as in Figure 1).

Figure 1: The triangleT

Define the triangle algebra as the the Chekhov-Fock algebra T :=T_q(λ).

It is generated by X₁^±1, X₂^±1, X₃^±1 with relations XiXi+1 = q²Xi+1Xi for all i ∈ N₃. The center of T is given by X₁^N, X₂^N, X₃^N and H = q⁻¹X₁X₂X₃. Irreducible finite dimensional representations of T have dimension N and are classified (up to isomorphism) by a choice of weight x_i∈C^∗ associated to each edge λ_i and a central charge, that is a choice of a N-th root c = (x₁x₂x₃)^1/N (see [HBL07, Lemma 2]).

To be more precise, forV theN-dimensional complex vector space generated by {e₁, ..., e_N} andρ an irreducible repesentation of T classified byx₁, x₂, x₃ ∈ C^∗ and c= (x₁x₂x₃)^1/N. Up to isomorphism, the action of T onV defined by ρ is given by:





X₁e_i = ex₁q²ⁱe_i X₂e_i = ex₂e_i+1 X₃e_i = ex₃q¹⁻²ⁱe_i−1

where xe_i is anN-th root of x_i such thatxe₁xe₂xe₃ =c. Note that, up to isomor- phism, ρ is independent of the choice of the N-th rootex_i withxe₁xe₂xe₃=c.

In particular, for the representation ρ classified by x₁ = x₂ = x₃ = 1 and c ∈ U(N), as ρ(X_i)^N = Id_V, the spectrum of ρ(X_i) is a subset of U(N). For h∈ U(N), denote byV_h(X_i)the eigenspace ofρ(X_i)associated to the eigenvalue h. We have the following lemma which will be usefull in the next section:

Lemma 1. For each i∈ {1,2,3} and h∈ U(N), dim(V_h(X_i)) = 1.

Proof. We use the explicit form of the representationρinV =span{e₁, ..., e_N}.

Takexe₁ =xe₂= 1 and ex₃ =c.

Fori= 1, one sees that V_h(X₁) =span{e_k} where h=q^2k. For i = 2, the vector α_k := P

i∈N_Nq^−2kiei satisfies X2α_k = q^2kα_k and {α₁, ..., α_k}form a basis of V. ThenV_h(X₂) is generated byα_k.

Fori= 3, we use the fact that X₁X₂X₃e_i=ce_i, where c, the central charge ofρ, lies in U(N).

1.3 Local representation of Tq(λ)

Letλbe an ideal triangulation ofΣ. Such a triangulation is composed ofmfaces T₁, ..., T_m and each face T_j determines a triangle algebra T_j whose generators are associated to the three edges ofT_j. It provides a canonical embedding iof T_q(λ) into T₁⊗...⊗ T_m defined on the generators as follow:

• i(X_i) = X_ji ⊗X_ki if λ_i belongs to two distinct triangles T_j and T_k and X_ji ∈ T_j,X_ki ∈ T_k are the generators associated to the edge λ_i ∈T_j and λi ∈T_k respectively.

• i(X_i) = [X_ji1X_ji2] if λ_i corresponds to two sides of the same face T_j and X_ji1, X_ij2 ∈ T_j are the associated generators.

Now, a local representation of T_q(λ) is a representation which factorizes as (ρ₁ ⊗...⊗ρ_m) ◦i where ρ_i : T_i → V_i is an irreducible representation of the triangle algebra Ti. In particular, such a representation has dimension N^m where m= 4g−4 + 2sis the number of faces of the triangulation.

1.4 Classification of these representations

Here we recall [BL07, Theorem 21] and [HBL07, Proposition 6] respectively:

Theorem 2. (F. Bonahon, X. Liu) An irreducible representation of T_q(λ) is determined by its restriction to the center ofTq(λ)and is classified by a non-zero complex numberx_i associated to each edges λ_i, for each puncturev_j, a choice of a N-th root pj = (x^k₁^j1...xn^kjn)^1/N and a choice of a square root c= (p0...ps)^1/2.

Such a representation satisfies:

• ρ(X_i^N) =x_iId,

• ρ(P_j) =p_jId,

• ρ(H) =cId.

Theorem 3. (H. Bai, F. Bonahon, X. Liu) Up to isomorphism, a local repre- sentation of Tq(λ) is classified by a non-zero complex number xi associated to the edge λ_i and a choice of a N-th root c= (x₁...x_n)^1/N. Such a representation satisfies:

• ρ(X_i^N) =x_iId,

• ρ(H) =cId.

1.5 The quantum Teichmüller spaces and its representations If one wants to quantize the Teichmüller space, he has to do it in a canonical way.

The definition of the Chekhov-Fock algebraT_q(λ) involves the choice of an ideal triangulation. So we have to understand the behavior when one changes from an ideal triangulation λ to another one λ^′. Set Tq(λ) = C[X₁^±1, ..., X_n^±1]q and T_q(λ^′) =C[X₁^′±1, ..., X_n^′±1]_q. These algebras admit a division algebras, denoted byTˆ_q(λ)andTˆ_q(λ^′)respectively, consisting of rational fractions in the variables X_i (respectively X_i^′) satisfying some non-commutativity relations.

For each pair of ideal triangulationλ and λ^′, L.O. Chekhov and V.V. Fock constructed coordinates change isomorphisms

Ψ^q_λλ′ : ˆT_q(λ^′)−→Tˆ_q(λ),

which are the unique isomorphism satisfying naturals conditions (as for example Ψ^q_λλ′′= Ψ^q_λλ′◦Ψ_λ^q′λ^′′ for eachλ, λ^′ andλ^′′ideal triangulations of Σ). See [Liu09]

for more details and explicit formulae of Ψ^q_λλ′.

Now, thequantum Teichmüller space T_q(Σ)is defined by:

Tq(Σ) := G

λ∈Λ(Σ)

Tˆq(λ)/∼,

where Λ(Σ) is the set of ideal triangulation of Σ, and the equivalence relation

∼identifies each pair of Tˆ_q(λ) and Tˆ_q(λ^′) by the isomorphism Ψ^q_λλ′. Note that, as each coordinates change Ψ^q_λλ′ is an algebra isomorphism, T_q(Σ) inherits an algebra structure, and theTˆ_q(λ)can be thought as ”global coordinates” onT_q(Σ).

A natural definition for a finite dimensional representation ofT_q(Σ) would be a family of finite dimensional representation {ρ_λ : ˆT_q(λ)−→End(V_λ)}_λ∈Λ(Σ) such that for each pair of ideal triangulation λ and λ^′, ρ_λ^′ is isomorphic to ρ_λ◦Ψ^q_λ,λ′. Note that, as pointed out in [HBL07, Section 4.2], there exists no algebra homomorphism ρ_λ : ˆT_q(λ) −→ End(V_λ) for V_λ finite dimensional. In fact, as Tˆ_q(λ) is infinite dimensional as a vector space and End(V_λ) is finite dimensional, such a homomorphism ρ_λ would have non-zero kernel. Hence, there would exists elements x ∈ Tˆq(λ) such that ρ_λ(x) = 0 and so, ρ_λ(x⁻¹) would make no sense.

So one defines alocal representation (respectively irreducible repre- sentation) ofT_q(Σ)as a family of representation{ρ_λ :T_q(λ)−→End(V_λ)}_λ∈Λ(Σ) such that for each λ, λ^′ ∈Λ(Σ), ρ_λ is a local representation (respectively irre- ducible representation) of T_q(λ), and ρ_λ′ is isomorphic (as representation) to ρ_λ◦Ψ^q_λλ′ whenever ρ_λ◦Ψ^q_λλ′ makes sense. We say that ρ_λ◦Ψ^q_λλ′ makes sense, if for each Laurent polynomialX^′ ∈ T_q(λ^′), there existsP, P^′, Q andQ^′∈ T_q(λ) such that:

Ψ_λλ′(X^′) =P Q⁻¹=Q^′−1P^′ ∈Tˆ_q(λ);

now, asρ_λ(T_q(λ))⊂GL(V_λ),ρ_λ(Q)andρ_λ(Q^′)are invertibles, so we can define:

ρ_λ◦Ψ_λλ′(X^′) :=ρ_λ(P)ρ_λ(Q)⁻¹=ρ_λ(Q^′)⁻¹ρ_λ(P^′).

A fundamental result in [BL07] and [HBL07, Proposition 10] is that for each pair of ideal triangulations λand λ^′, there exists a rational map

ϕ_λλ^′ :Cⁿ−→Cⁿ

such that a local representationρ_λ′ ofT_q(λ^′) classified byx^′_i ∈C^∗ associated to λ^′_i and c^′ = (x^′₁...x_n^′)^1/N is isomorphic to ρ_λ◦Ψ_λλ^′ (whenever it makes sense) for a representation ρ_λ of T_q(λ) classified by x_i ∈ C^∗ associated to λ_i and c= (x₁...x_n)^1/N if and only ifc=c^′ and

(x^′₁, ..., x^′_n) =ϕ_λλ′(x₁, ..., x_n).

2 Proof of Theorem 1

2.1 Special case

Here we prove Theorem 1 for a local representation ρ : T_q(λ₀) −→ End(V) where λ₀ is special triangulation ofΣ and ρ is classified by weightsx_i = 1and c ∈ U(N). Here, Σ is a genus g > 0 surface with s+ 1 punctures v₀, ..., v_s. Recall that m = 4g−4 + 2(s+ 1) and n = 6g−6 + 3(s+ 1) are respectively the number of faces and edges ofλ₀. Moreover, we denote by Xuthe action of X∈ T_q(λ₀) on u∈V defined byρ.

To decomposeρ into irreducible factors, one has to look at the eigenspaces of ρ(P_j) for each puncture invariant P_j associated to the puncture v_j. Note that, asρ(P_j)^N =Id, the spectrum of P_j is contained in U(N).

The idea of the proof is to look at the action of the Pj on each factor of a nice decomposition of V into a tensorial product of vector spaces. It is based on the following remark:

Remark 1. For a decomposition V = E₁ ⊗E₂, if x_j ∈ E_j satisfies P x_j = h_jx_j for j = 1,2 where P ∈ {P₀, ..., P_s} and h_j ∈ U(N), then P(x₁ ⊗x₂) = h₁h₂x₁ ⊗x₂. That is, the eigenspace of P in V associated to the eigenvalue h∈ U(N) contains the tensorial product of eigenspaces ofP inE_j associated to the eigenvalues h_j, forj = 1,2, whenever h=h₁h₂.

Forh= (h₁, ..., h_s)∈ U(N)^s, set

V_h :={u∈V, P_iu=h_iu, i= 1, ..., s}.

Proposition 2. For eachh∈ U(N)^s, dimVh=N^m−s.

Proof. Take an ideal triangulationλeofΣ\ {v1, ..., vs}(which is a one punctured surface), and for a triangleT ofλ, consider the triangulation ofe T ∪ {v₁, ..., v_s} as in Picture 2.

The union of these two triangulations gives an ideal triangulation λ0 of Σ.

Denote by Ve the tensorial product of all the vector spaces associated to the triangles ofeλ\T. As the triangulation λe contains 3g−1 triangles, dim(eV) = N^3g−2 (because we do not consider the vector space associated to T). Denote by V^j and V^′k the j^th (resp. k^th) vector space associated to the triangle T_j (resp. T_k^′) as in Figure 2 (here, j∈ {0, ..., s} and k∈ {1, ..., s}).

Forh= (h₁, ..., h_s)∈ U(N)^s and j∈ {1, ..., s}, define:

Figure 2: Triangulation of T∪ {v₁, ..., v_s} . V_h^j ={x∈V^j ⊗V^′j, P_kx=h_kx, k= 1, ..., s}.

. V_h⁰ ={x∈V⁰, P_kx =h_kx, k= 1, ..., s}.

We have the following lemma:

Lemma 2.

i. dimVh⁰ =

1 if h_k = 1 ∀k6= 1 0 otherwise.

ii. ∀j∈ {1, ..., s−1} dimVh^j =

1 if h_k= 1 ∀k /∈ {j, j+ 1}

0 otherwise.

iii. dimVh^s =

N if h_k= 1 ∀k6=s 0 otherwise.

Proof. i. Ifk6= 1,v_k is not a vertex of T₀. It follows thatP_k acts onV⁰ by the identity; so if h_k6= 1,V_h⁰ ={0}.

Now, if h_k = 1for all k6= 1, thenV_h⁰ =V_h⁰₁(P₁) (as defined in Lemma 1) which is one dimensional.

ii. Fix j ∈ {1, ..., s−1}. For k /∈ {j, j+ 1}, v_k is neither a vertex of T_j nor of T_j^′. So P_j acts on V^j ⊗V^′j as the identity. Hence, if h_k 6= 1, then V_h^j ={0}.

Takeh_k= 1 for allk /∈ {j, j+ 1}and denote byT_j =C[X^±1, Y^±1, Z^±1]_q, T_j^′ =C[X^′±1, Y^′±1, Z^′±1]_q the triangle algebras associated to the triangles T_j andT_j^′ respectively (as in Figure 3).

Figure 3: The generators ofT_j and T_j^′

For c_j, c^′_j ∈ U(N) the central charges of the restriction of the representa- tion to T_j and T_j^′ respectively, P_j acts on V^j := span{e₀, ..., e_N−1} like cjZ⁻¹, on V^′j = span{e^′₀, ..., e^′_N−1} like c^′_jZ^′−1 and Pj+1 acts on Vj like c_jY⁻¹, on V_j^′ like c^′_jY^′−1. Setc_j =q^p and c^′_j =q^p′, we get the following:

P_je_k =q^2k−1+pe_k+1 P_je^′_l =q^1−2l+p′e_l+1

It follows that the action of Pj onV^j⊗V^′j is given by:

P_jǫ_k,l=q^{2(k−l)+p+p′}ǫ_k+1,l+1 where ǫ_k,l :=e_k⊗e^′_l.

In the same way, one sees that the action of P_j+1 onV^j⊗V^′j is given by:

P_j+1ǫ_k,l =q^p+p′ǫ_k−1,l−1.

Now, form, n∈N, setα_m,n :=

N−1X

k=0

q^2kmǫ_k,k+n, an easy calculation shows

that:

P_jα_m,n = q−2(m+n)+p+p^′α_m,n P_j+1α_m,n = q^2m+p+p′α_m,n.

It follows that{αn,m, n, m∈N}is a base ofV^j⊗V^′j and, for allhj, hj+1∈ U(N), there exists a unique couple(m, n)∈N²

N withh_j =q−2(m+n)+p+p^′

and h_j+1 = q^2m+p+p′. So dimV_h^j = 1 if and only if h_k = 1 for all k /∈ {j, j+ 1}.

iii. Ifk6=s,v_k is neither a vertex of T_s nor T_s^′, so ifh_k6= 1,V_h^s ={0}.

Suppose thath_k= 1 for allk∈ {1, ..., s−1}, then V_h^s ⊃ M

hahb=hs

V_h^s_a(P_s)⊗V_h^′s

b(P_s), (where V_h^s

a(P_s) and V_h^s

b are defined as in Lemma 1). The direct sum contains N terms of dimension one, hence dimV_h^s ≥N. But, we have

dim(V^s⊗V^′s) =N² = X

h∈U(N)^s

dim(V_h^s)≥N ×N.

So V_h^s isN-dimensional.

Now, the proof of Proposition 2 is straightforward: from Remark 1, we have M

h⁰h¹...h^s=h

V_h⁰0 ⊗...⊗ V_h^ss ⊗Ve ⊂V_h.

Writing h^j = (h^j₁, ..., h_s^j) and h = (h₁, ..., h_s), one notes that the only non-zero terms in the direct sum are those who satisfy:











h⁰₁h¹₁ =h₁ h¹₂h²₂ =h₂ ...

h^s−1_s h^s_s=h_s

There exists exactlyN^sdifferent choices forh⁰, ...,h^s∈ U(N)^ssatisfying the above relations, and each non-zero vector space of the direct sum has dimension N^m−2s. So dimV_h ≥N^m−s. Now, we have

dimV =N^m= X

h∈U(N)^s

dimV_h≥N^s×N^m−s,

and so dimV_h =N^m−s for each h∈ U(N).

In particular, it proves the decomposition of Theorem 1 for ρ. In fact, let ρ⁽ⁱ⁾:T_q(λ₀)−→End(V⁽ⁱ⁾)be an irreducible representation in the decomposition of ρ. It must satisfies ρ⁽ⁱ⁾(X_i)^N =Id_V(i) and ρ⁽ⁱ⁾(H) =cId_V(i), in other word, ρ⁽ⁱ⁾ must be associated to the same weights x_i = 1 and global chargec∈ U(N) than ρ.

Set h⁽ⁱ⁾_j ∈ U(N) the weight of ρ⁽ⁱ⁾ associated to the each puncture v_j, that is, ρ⁽ⁱ⁾(P_j) = h⁽ⁱ⁾_j Id_V(i). Note that, as ρ⁽ⁱ⁾([P₀...P_s]) = ρ⁽ⁱ⁾([H²]) = h⁽ⁱ⁾₀ h⁽ⁱ⁾₁ ...h⁽ⁱ⁾_s Id_V(i) =c²Id_V(i), a necessary condition for ρ⁽ⁱ⁾ to be in the decom- position of ρ is to satisfyh⁽ⁱ⁾₀ ...h⁽ⁱ⁾_s =c². Hence, if ρ⁽ⁱ⁾ is in the decomposition of ρ, knowing h⁽ⁱ⁾_j for each j = 1, ..., s uniquely determine h⁽ⁱ⁾₀ and so fully determine ρ⁽ⁱ⁾.

Now, as for each h = (h₁, ..., h_s) ∈ U(N)^s, V_h has dimension N^m−s = N^4g−3+(s+1)and as an irreducible representation ofT_q(λ₀)has dimensionN^3g−3+(s+1), then each spaceV_h contains exactly N^g times the representation ρ⁽ⁱ⁾, classified byp0=c²h⁻¹₁ ...h_s⁻¹, p1 =h1, ..., ps=hs.

2.2 Proof in the global case

Now, to complete the proof of Theorem 1, one remarks that the decomposition of ρ into irreducible factors only depends on the decomposition of ρ(P_j) into eigenspaces (for each puncturevj), that is on the possible choices of N-th root ofx^k₁^k1...x^kn^jn (wherePj is associated to the multi-indexkj = (kj1, ..., kjn)). But this choice is discrete and depends continuously on the weightsx_i associated to the edgeλ_i, hence does not depend on the choice ofx_i ∈C^∗. It proves Theorem 1 for the triangulation λ₀ and every weight x_i ∈C^∗.

Note that the map ϕ_λ0λ defined in Subsection 1.5 is rational, hence defined on a Zariski dense open set of Cⁿ. As we extended the decomposition for all weights x_i associated to each edge of the triangulation λ₀, there exists a local representation{ρ_λ :T_q(λ)−→End(V_λ)}_λ∈Λ(Σ)ofT_q(Σ)as defined in Subsection 1.5. So, for each λ∈ Λ(Σ), ρ_λ0 ◦Ψ^q_λ0λ : T_q(λ) −→ End(V_λ0) makes sense and is isomorphic to ρ_λ : T_q(λ) −→ End(V_λ). That is, there exists a vector space isomorphism L_λ0λ :V_λ −→V_λ0 such that, for each X∈ T_q(λ),

ρ_λ0(Ψ^q_λ0λ(X)) =L_λ0λ◦ρ_λ(X)◦L⁻¹_λ0λ.

However, ρ_λ0 is a local representation of T_q(λ₀), hence there exists an ir- reducible decomposition ofρ_λ0 given by the decomposition V_λ0 =M

i∈I

V_λⁱ₀ as in Theorem 1. That is, for each i ∈ I, V_λⁱ

0 is stable by ρ_λ0 and has dimension N^3g−3+s+1.

Using the isomorphismΨ_λλ0, one gets that for eachX ∈ T_q(λ),ρ_λ0(Ψ_λ0λ(X))V_λⁱ₀ = V_λⁱ₀. SetV_λⁱ :=L⁻¹_λ0λ(V_λⁱ₀), we have dimV_λⁱ = dimV_λⁱ₀ = 3g−3 +s+ 1 (because L_λ0λ is an isomorphism) so for eachX∈ T_q(λ),ρ_λ(X)V_λⁱ=V_λⁱ. In other words, we have a decomposition

ρ_λ =M

i∈I

ρ⁽ⁱ⁾_λ ,

where ρ⁽ⁱ⁾_λ :T_q(λ)−→ End(V_λⁱ). As each V_λⁱ as the dimension of an irreducible representation, we get an irreducible decomposition ofρ_λ. One easily checks that it satisfies the conditions of Theorem 1. Now we extend this decomposition by continuity for all weightx_i∈C^∗ associated to the edge λ_i ofλ.

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Jérémy Toulisse

University of Luxembourg, Campus Kirchberg Mathematics Research Unit, BLG

6, rue Richard Coudenhove-Kalergi L-1359 Luxembourg

Grand Duchy of Luxembourg [email protected]