Dynamics on Wild Character Varieties

In the present paper, we will first present briefly a general research program about the study of the"natural dynamics"on character varieties and wild character varieties. Afterwards, we will illustrate this program in the context of the Painlev\'e differential equations $P_{\rm VI}$ and $P_{\rm V}$.

Example 1.1. Let Y be a (topological) manifold. The fundamental groupoid π 1 (Y ) of Y is the groupoid whose objects are the elements y of Y , and whose morphisms are the paths between elements of Y up to homotopy.
We have the following generalization. Definition 1.2. Let Y be a (topological) manifold and S ⊂ Y . The fundamental groupoid π 1 (Y, S) of the pair (Y, S) is the groupoid whose objects are the elements s of S in Y , and whose morphisms are the paths between elements of S up to homotopy.
2 Character varieties in the Painlevé context: the regular singular case revisited We focus now on the "Painlevé context": let E be the set of linear rank 2 connections on the trivial bundle over P 1 (C), with coefficients in sl 2 (C) and such that its singular locus contains at most 4 singular points. In this section, we first consider the "classical" case with 4 regular singular points, in order to be more familiar with the groupoid point of view which is essential to deal with the irregular cases. Furthermore it turns out that this point of view is yet useful to obtain the dynamics in the regular singular case. In this section, ∆ is a linear differential system which represents the connection ∇.

The fundamental groupoid
We consider the "extended" singular locus S of ∆: S is the set of pairs s = (p, d) where p is a singular point of ∆ and d is a ray based in p. Therefore s is also a point on the divisor D p of the real blowing up E p at p. Let X be the manifold obtained by the real blowing up of each singular point. We denote by γ s,s a loop from s to s in X with positive orientation, homotopic to the exceptional divisor: Definition 2.1. The fundamental groupoid π 1 (X, S) is the groupoid whose objects are the elements s of S in X, and whose morphisms are the paths between elements of S up to homotopy. The subgroupoid π loc 1 (X, S) is the groupoid with same objects, whose morphisms are generated only by the loops γ i,i homotopic to each exceptional divisor at p i .
We denote: -Aut(π 1 (X, S)) the group of the automorphisms of the groupoid, and Aut 0 (π 1 (X, S)) the subgroup of the "pure" automorphisms, which fix each object.
The local fundamental subgroupoid is generated by the loops γ i,i , and is a disjoint union of four monogeneous groups.
Representations of the groupoid π 1 (X, S). A representation of π 1 (X, S) in a group G is a morphism of groupoids ρ from π 1 (X, S) into G. The group G is here a groupoid with only one object, whose morphisms are the elements of G. Therefore ρ is characterized by its action on the morphisms of π 1 (X, S).
Analytic representations of the groupoid π 1 (X, S) in G induced by a connection ∇ in E.
-For each object s = (p, d), we consider a fundamental system of holomorphic solutions X s in a neighborhood of s in X, i.e., in a small sector at p around the direction d, admitting an asymptotic expansion at p.
-At each morphism γ i,j joining s i to s j , corresponds a connection matrix M i,j between the fundamental systems of solutions X i and X j chosen at s i and s j defined by where X i γ i,j is the analytic continuation of X i along γ i,j . With this notation ρ(γ i,j γ j,k ) = ρ(γ i,j )ρ(γ j,k ).

The character variety
Definition 2.2. Let ρ : π 1 (X, S) → G and ρ : π 1 (X, S) → G be two analytic representations of The two representations ρ and ρ are equivalent if and only if for each object s i , there exists N i in G such that Therefore, if we change the choice of the fundamental system attached to each object s i , we obtain a new equivalent representation. All the representations induced by ∆ are equivalent. Furthermore, if ∆ and ∆ are gauge equivalent, their representations are equivalent. The class [ρ] only depends on ∇. Definition 2.3. Let R(S) (resp. R(S) loc ) be the space of the analytic representations of π 1 (X, S) (resp. of π loc 1 (X, S)) induced by some ∆ in E, and ∼ the above equivalence relation on R(S). • The character variety χ(S) is the quotient R(S)/∼.
• In the same way, the local character variety is χ(S) loc = R(S) loc /∼. The morphism π from χ(S) to χ(S) loc is induced by restriction of the representations to π loc 1 (X, S). Normalized representations. We construct a "good" representative of [ρ] in χ(S) by using the following process: -We choose freely a fundamental system of solutions X 1 in s 1 .
-In s 2 , we choose X 2 to be the analytic continuation of X 1 along the path γ 1,2 , then we choose X 3 by analytic continuation of X 2 along γ 2,3 , and finally X 4 by analytic continuation along γ 3,4 . With these choices, we have and from the exterior relation, we obtain ρ(γ 1,4 ) = I. The representation ρ is now characterized by 4 matrices M i,i = ρ(γ i,i ). From the interior relation, we have A change in the initial choice of X 1 will give rise to 4 matrices related to the previous ones by a common conjugation. Finally, we have characterized [ρ] in χ(S) by the data of 3 matrices M i,i , i = 1, 2, 3 up to a common conjugation, as in the usual presentation with only one base point. Nevertheless, this groupoid point of view is more convenient, first for computing the isomonodromic dynamics, but also to get an extension to the irregular cases.
We call it the "Fricke hypersurface". It is a quartic in C 7 endowed with the coordinates (a 1 , a 2 , a 3 , a 4 , x 1,2 , x 2,3 , x 3,1 ) and, with respect to the last 3 coordinates, a family of cubics F a indexed by a = (a 1 , a 2 , a 3 , a 4 ).
The restriction of T on χ(S) * loc is an isomorphism onto C\{±2} × C 3 and we have T π = p 1 T , where p 1 is the first projection C 4 × C 3 → C 4 .
Proof . By using a conjugation, we may suppose that This writing is not still unique: we may use a conjugation by a diagonal matrix D. Since the center do not act, we may suppose that det(D) = 1, i.e., Proof . Since the condition is necessary. Suppose now that this condition holds for two triples. Since β 2 γ 3 = 0 = β 2 γ 3 we can choose α such that α 2 = β 2 . We also have Now we have to solve in SL 2 the system We choose one of the two solutions of equation (2.2a). Since α 1 − α −1 1 = 0, we obtain from equations (2.2b), (2.2c), (2.2d) and (2.2e) a unique solution for α 2 , δ 2 , α 3 , δ 3 . Equations (2.2f), (2.2g) define a linear system in the 2 variables (β 2 γ 3 , γ 2 β 3 ) of maximal rank if α 1 −α −1 1 = 0. We obtain a unique solution for α 2 , δ 2 , α 3 , δ 3 , β 2 γ 3 , γ 2 β 3 , and therefore a unique triple (M 1 , M 2 , M 3 ) up to conjugation according to the preliminary remark. Note that this triple is not necessarily in SL 2 : the compatibility condition corresponds to the Fricke relation. Now if we begin with the second solution of (2.2a), the new matrix M 1 satisfies M 1 = P M 1 P −1 , where P is the matrix of the transposition. The system (2.2b)-(2.2g) has a unique solution M i under the same assumption α 1 − α −1 = 0. Since we know that P M i P −1 is another pre-image for T , we have: M i = P M i P −1 . Therefore this second solution is conjugated to the first one by P , and we obtain a unique pre-image of a point in F * in χ(S) * . For the second part of the statement, if each matrix M i is a semi-simple one, the trace of M i characterizes the conjugation class of M i in SL 2 .

The dynamics on χ(S)
We set where S belongs to the space C of the configurations of 4 distinct points in the plane. This fibration is endowed with a flat connection (the isomonodromic connection) whose local trivialisations are defined by identifying the generators γ i,j (S) and γ i,j (S ), for S near from S. We want to compute the monodromy of this connection on a fiber χ(S).
The fundamental group π 1 (C, [S]) is the pure braid group P 4 . It is generated by the 3 elements b 1 , b 2 , b 3 , where b i is the pure braid between s i and s i+1 , with the relation b 1 b 2 b 3 = id (note that the cross ratio induces an isomorphism from C on P 1 (C)\{0, 1, ∞}).
The generators b i induce an isomorphism from P 4 to the mapping class group of the disc punctured by 4 holes, with a base point on their boundaries. This interpretation allows us to construct an action from P 4 on the groupoid π 1 (X, S). We denote by h 1 , h 2 , h 3 the images of the braids in Aut 0 (π 1 (X, S)).
The automorphisms h i act on R by h i * : ρ → ρ • h i and an inner automorphism sends ρ on an equivalent representation. Therefore each [h i ] in Out * 0 (π 1 (X, S)) acts on χ(S). Looking at the picture of the groupoid, we immediately obtain: We have similar expressions for h 2 and h 3 by cyclic permutations of the indices. Now we compute the action h i * on χ(S) in three steps. Let [ρ] in χ(S) given by a normalized representation ρ, and therefore by 3 matrices M i up to a common conjugation. For each generator b i , 1) we compute ρ • h i on the generating morphisms γ i,j . The representation ρ • h i is not yet a normalized one; 2) we normalize ρ • h i in a new equivalent representation ρ , by changing the representation of the objects. Let M i be the matrices related to ρ ; For this last step, we will make use of an extended version of the Fricke lemma: We follow the notations of the Fricke Lemma 2.4. We have Proof . We only make use of the relation tr(AB) + tr(AB −1 ) = tr(A) tr(B): In these trace coordinates, h 1 * is given by We obtain h 2 * and h 3 * by a cyclic permutation indices (+1) of the indices 1, 2 and 3.
Proof . We have (M 1 M 2 ) −1 , and we normalize ρ • h 1 by setting: X 1 = X 1 , X 2 = X 2 and X 3 = X 3 · M 1 M 2 in order to obtain a representation ρ equivalent to ρ • b 1 which satisfies ρ (γ 2,3 ) = I. This representation ρ is characterized by the 3 matrices: The statement of the proposition is obtained from the extended Fricke Lemma 2.8.
By this way, we reach the same expressions of the dynamics as S. Cantat and F. Loray in [10]. The study of this dynamics allows them to give a new proof of the irreducibility of the Painlevé VI equation. This is an important motivation to extend the description of this dynamics to the non regular cases.
3 An irregular example (towards P V )

Standard facts about irregular singularities
We consider a rank n linear differential system at z = 0 where A(z) takes its values in gl(n, C). 3 The integer r is positive, and equal to 0 for a Fuchsian system. We suppose here that r > 0, and that the eigenvalues of A 0 = A(0) are non vanishing distincts complex numbers. We fix here Λ 0 = diag(λ i ) a normal form of A 0 in the Cartan subalgebra T 0 of the diagonal matrices, i.e., we choose an ordering of its eigenvalues. The formal local meromorphic classification is given by 1. Up to a local ramified formal meromorphic gauge equivalence, we have where z = t ν , Q (the "irregular type") = Λ 0 t r + · · · + Λ r−1 t , and the matrices Λ i and L (the residue matrix) are diagonal matrices. For a fixed Λ 0 , the pair (Q, L) is unique in T 0 × T 0 /T 0 (Z).

2.
Let F 0 be a conjugation between A 0 and Λ 0 : The system ∆ has a formal fundamental solution For a fixed Λ 0 , there are already two ambiguities in the above writing of X: -the choice of F 0 : we may change F 0 with F 0 D, and therefore X with XD, where D belongs to the centralizer of Λ 0 ; -the choice of a branch for the argument, and hence for log t and t L .
We suppose now that we are in the unramified case: ν = 1.

A regular sector
is an open sector S d of angle π/r bisected by d such that d is not a singular ray (or equivalently, such that its edges are not separating rays).
We can remark that: • If arg(z) = τ is a separating (resp. singular) ray then its opposite is also a separating (resp. singular) ray.
• A non singular ray is a ray on which the formal solution admits a unique sum, for the summation theory. 5 • A separating ray is a ray on which the asymptotic of a general solution (a linear combination of the columns of X) changes.
• The knowledge of the µ separating (resp. singular) rays in a regular sector generates the complete knowledge of all the separating (resp. singular) rays, by considering their opposites, and the ramification by z r .
• The generic case is the situation in which there exists exactly one pair of eigenvalues (λ j (ν), λ k (ν)) defining each separating ray τ ν . In this case, we have µ = n(n − 1)/2 separating rays in a regular sector, and m = n(n − 1)r = 2rµ separating (resp. singular) rays in S 1 .
Theorem 3.3. On a regular sector S d containing the µ separating rays τ ν , . . . , τ ν+µ−1 , there exists a unique holomorphic fundamental system of solutions X d admitting the asymptotic expansion X. Furthermore X d can be extended to a solution (with the same asymptotic) on the sector S ν delimited by the two nearest separating rays τ ν−1 and τ ν+µ outside S d .
There exists two proofs of this fact using either the asymptotic theory (see [2]), or the summation theory (see [26]).
For ν = 1 to m, Theorem 3.3 gives us a unique solution X ν on the large sector delimited by τ ν−1 and τ ν+µ admitting X as asymptotic expansion on S ν .
with a m-periodic indexation.
In the generic case (the support of each singular ray reduces to a unique pair of eigenvalues), the constant matrices U ν , ν = 1, . . . , m − 1 are transvection matrices: the diagonal entries are equal to 1, and the unique non vanishing coefficient off the diagonal is the coefficient in position (j(ν), k(ν)) where the separating ray τ ν (the only ray in S ν \S ν ∩ S ν+1 ) is defined by the pair (λ j , λ k ). The diagonal of the matrix U m is exp(2iπL), where L is the residue coefficient of A(x) after diagonalisation.
In order to construct the wild fundamental groupoid (for a fixed Λ 0 ), we first use a real blowing up at each singularity in P 1 and we obtain a variety X with 3 exceptional divisors D 0 , D 1 and D ∞ (circles in dotted lines in Fig. 3). As in the previous classical case, we choose a base point s 0 , s 1 and s ∞ on each of them, and we consider the morphism (path up to homotopy) γ i,j joining s i to s j . The paths γ i,i are homotopic in X to the curves D i . We choose s ∞ such that it corresponds to a non singular ray.
Since we also have to consider the continuation of X (the formal monodromy along an arc is induced by the substitution z → ze iθ ), we introduce a second copy D ∞ of D ∞ inside the first one, with a base point τ 1 which is the separating ray between σ 2 and σ 1 . For each singular direction σ i (denoted in the picture below by a ray with a cross in the annulus between D ∞ and D ∞ ), we add two loops delimited by two rays r − i and r + i which are non singular and non separating, and two arcs α i on D ∞ and α i and D ∞ of opening strictly lower than π, bisected by the singular rays. Let σ − i and σ − i the two points on D ∞ and D ∞ joined by r − i , and σ + i and σ + i joined by r + i . Finally we put a ray r ∞ from τ 1 to s ∞ .
Remark 3.5. In the picture above, we put arbitrarily the two base points s ∞ on D ∞ and τ 1 on D ∞ in the same direction: it is our initial configuration. Nevertheless, in the dynamical study of the next section, s ∞ will remain fixed, while the separating ray τ 1 (and all the other data related to Λ 0 ) will move on D ∞ .
Definition 3.6. The wild fundamental groupoid π 1 (X, S(Λ 0 )) is the groupoid defined by • the objects S(Λ 0 ): the three points s 0 , s 1 and s ∞ , the points τ i (separating rays), σ i ± on D ∞ around the singular rays σ i (denoted on the figure by a ray with a cross), and the corresponding points σ i ± on D ∞ .
• the morphisms: they are generated by the paths γ i,j (up to homotopy) between s 0 , s 1 , and s ∞ in X, the rays r ∞ , r ± i , the arcs α i on D ∞ , and all the arcs on D ∞ : α i from σ i − to σ i + , and the connecting arcs β i ± as indicated on the figure.
We still have two relations r int and r ext between the generating morphisms: and a new one (the wild relation): A representation ρ of this groupoid induced by the differential system (∆) is defined in the following way. We first choose a "compatible" representation of the objects: • We choose analytic fundamental systems X(s 0 ), X(s 1 ), at s 0 , s 1 , as in the regular case (i.e., we choose a logarithmic branch in the corresponding direction); • We choose a formal fundamental system X ∞ given by Proposition 3.1. For each object σ i ± , τ i on D ∞ , we choose a formal fundamental system X( σ i ± ), X( τ i ) by choosing a determination in the corresponding direction of the formal fundamental solution X ∞ .
• For each object σ i ± on D ∞ , we choose an actual sectorial solutions X(σ i ± ), given by Theorem 3.3, whose asymptotic expansion is some determinacy of the same formal fundamental solution X ∞ (this is the compatibility condition).
Then, we construct the representations of the generating morphisms in the following way: • We use analytic continuation to represent the morphisms between s 0 , s 1 , and s ∞ , as in the singular regular case (see (2.1)).
• In the same way, we use the analytic continuation of the formal solutions to represent the morphisms β i ± , and α i on D ∞ between the formal objects: this formal monodromy is defined by the substitution z → ze iθ in the formal expressions.
• We also use analytic continuation preserving the same asymptotic, to represent the arcs α i on D ∞ . Note that the regular sectors given by Theorem 3.3 centered on r i − and on r i + allow us to define this continuation along α i and α −1 i : indeed the intersection of these two sectors is a sector of opening π delimited by the 2 separating rays, and therefore contains α i .
• We represent r i ± and r ∞ by using Theorem 3.3: starting from the representation X( σ i ± ) of the formal objects σ i ± , this theorem gives us an actual solution in this direction, which is denoted by X( σ i ± ). The comparison with the representation X(σ i ± ) of the final object σ i ± defines ρ(r i ± ): The representation of the inverse paths (r i ± ) −1 are obtained in the following way: starting from the representation X(σ i ± ) of σ i ± on D ∞ we use its asymptotic expansion X(σ i ± ) and compare it with the representation X( σ i ± ) of the final object σ i ± in order to define ρ((r i ± ) −1 ).
Definition 3.7. Two representations of π 1 (X, S(Λ 0 )) are equivalent if they are obtained by different compatible representations of the objects. The wild character variety χ(Λ 0 ) is the set of the representations of π 1 (X, S(Λ 0 )) up to the above equivalence relation. The local wild character variety χ loc (Λ 0 ) is the set of the representations of π loc 1 (X, S(Λ 0 )) up to the above equivalence relation, and we have a natural fibration π : χ(Λ 0 ) −→ χ loc (Λ 0 ).
Then we choose X( τ i ), X( σ i ± ) and X(σ i ± ) such that There remains five matrices Remark 3.8.
• For such a normalized representation, we also have U i = ρ(st i ), U i is the representation of the Stokes loops. From the above definition of the representation of the paths r ± i , the matrices U i are the Stokes multipliers introduced in the previous section. In particular, they are unipotent matrices • We also have M = ρ( γ 1,1 ). Therefore this matrix is a representation of the formal loop. It is a diagonal matrix • If we change the choice of the representation of the initial object X( τ 1 ) setting X ( τ 1 ) = X( τ 1 ) · D α , D α in C(T 0 ), the 4-uple (M 0 , U 1 , U 2 , M ) changes by the common conjugacy with D α .
Therefore, for a given Λ 0 , a representation ρ is characterized by a 4-uple (M 0 , U 1 , U 2 , M ) up to the conjugation by C(T 0 ). According to the previous description, the character variety is Its dimension is (3 + 1 + 1 + 1) − 1 = 5. The character variety of the local datas is where [M 0 ] ∼ is the conjugation class of M 0 , [M 1 ] ∼ is the (independent) conjugation class of M 1 , and M is diagonal. If M 0 and M 1 are semi-simple matrices, it is a 3-dimensional variety, and the fiber of χ(Λ 0 ) → χ loc (Λ 0 ) is a 2-dimensional variety.
Changing the choice of Λ 0 . Let W := {id, w} be the group of permutations of two objects. W is isomorphic to the quotient of the subgroup {I, P w } of SL 2 by {±I}, where The conjugation by P w on sl 2 only depends on the class of P w in the quotient. Therefore, we denote c w (M ) = P −1 w M P w . Let w · Λ 0 := c w (Λ 0 ) = −Λ 0 . We consider the fundamental groupoid π 1 (X, S(w · Λ 0 )) obtained by a new indexation of the singular rays. The objects are Proof . Clearly, these quantities are invariant. Suppose now that they are equal. We choose α such that From the other relations we obtain: We consider the 6 coordinates They are invariant by the action of the centralizer C(Λ 0 ) and therefore they induce a map T : χ(Λ 0 ) → C 6 . The Fricke lemma applied on the triple (M 0 , U 1 U 2 , M ) defines a codimension 1 Fricke variety F given by t 2 1 − P t 1 + Q = 0, with P = t 0 λ −1 − λ + λs + sy + λ + λ −1 x, i.e., This is a family of cubics parametrized by (t 0 , t 1 , λ). Proof . Clearly, from the Fricke lemma, T takes its values in F . In order to check that this map is invertible, we have to solve: T (M 0 , U 1 , U 2 , M ) = (λ, t 0 , t 1 , s, x, y). This equation is equivalent to the system If λ = ±1, we obtain from equations (3.1a) and (3.1e) a unique solution for a 0 and d 0 For Λ 0 = −Λ 0 , we have a similar description of χ(Λ 0 ) as an affine algebraic variety in the coordinates (λ , t 0 , t 1 , s , x , y ).
Proposition 3.12. The change of variables between these two charts describing the character variety χ is given by Proof . Since from Remark 3.9 we have we immediately obtain λ = λ −1 , s = s and y = y. The only non trivial computation is for x = tr(M 0 U 2 U 1 ). The Fricke lemma applied on the three matrices U 1 , U 2 , M 0 gives us x + x = t 0 s + 2b 0 u 2 + 2c 0 u 1 .
From the system (3.1) written in the proof of Proposition 3.11, we have By substitution in the above expression of x , we obtain the result.
We will see in the next section that the natural parameter here is not Λ 0 but the conjugacy class [A 0 ] of A 0 (see also [7,Remarks (8.5) and (10.6)]). Therefore the character variety χ([A 0 ]) appears as a gluing of the two affine algebraic charts χ(Λ 0 ) and χ(−Λ 0 ) which turns out to be a scheme on our example, from the above proposition. We can conjecture that it is the general case.

The dynamics on the wild character variety
This part is rather an experimental one. The way used here to encode all the information of irregular representations is not completely well justified. We need other experiments (for example in the P II context) and a very precise presentation of the background to present a definitive version. Nevertheless, we think that a lot of tools are present here to deal with the program presented in the first section.
In the classical previous case, the configuration space C was the set of the positions of the 4 singularities, identified to P 1 \{0, 1, ∞} through the cross ratio. In the present context we can't move the three singularities in P 1 . We can only move Λ 0 in T * 0 , which is the usual configuration space considered for example in [17]. Ph. Boalch in [5] has introduced a coordinate independent version of this space "through the notion of irregular curve". Its fundamental group is again a pure braid group.
Nevertheless, the most natural configuration space C is not the set of the Λ 0 's but Definition 3.13. The configuration space C is the set of the initial terms A 0 up to a conjugation.
Indeed, a gauge equivalence on ∆ acts by conjugacy on the initial coefficient A 0 . In the same way, Ph. Boalch has introduced a "bare irregular curve" in [7]. We begin with a description of C in a general framework.
Let G be a reductive algebraic group, with Lie-algebra G. Let Let T 0 be a fixed Cartan subalgebra (for sl 2 , D is the set of the diagonalisable matrices, and T 0 the Cartan subalgebra of the diagonal matrices). Let T reg be the subset of T of matrices with distinct eigenvalues. Recall that all the Cartan subalgebras are conjugated by some element g in G. This element is not unique: if g and g are two conjugations between T 0 and T , g −1 g keeps (globally) invariant T 0 and therefore belongs to the normalizer N (T 0 ) of the Cartan torus T 0 . We obtain a fibration which sends T 0 on the identity element I. Note that the quotient space G/N (T 0 ) of left classes modulo N (T 0 ) is not a group.
Lemma 3.14. If the algebraic group G is connected, and simply connected (which is the case for SL n ), the fundamental group π 1 (G/N (T 0 ), I) is the Weyl group W := N (T 0 )/T 0 .
Remark that the fibration D −→ G/N (T 0 ) has natural local trivialisations: we can lift a path g(s) in G/N (T 0 ) from A 0 by using the conjugation g(s)A 0 g(s) −1 . The conjugation class [A 0 ] of A 0 in D cuts the fiber T 0 in a discrete set which is the orbit of Λ 0 under the action of W by the monodromy of the above fibration. Therefore, an equivalent definition of the configuration space is Definition 3.15. If the algebraic group G is connected, and simply connected, the configuration space C is the set of the orbits of the Weyl group W acting on T reg 0 .
It induces a loop w in G/N (T 0 ) since P belongs to N (T 0 ). This loop can be lifted in a path δ w in D joining Λ 0 until −Λ 0 δ w (s) = w(s)Λ 0 w(s) −1 .
This path permutes the eigenvalues of Λ 0 without moving the eigenvalues themselves.
Proof . Clearly the pure braid group P n = π 1 (T reg 0 , Λ 0 ) acts on χ(Λ 0 ), keeping invariant each element of the orbit W · Λ 0 . Now let B n be the whole braid group. From the exact sequence 0 −→ P n −→ B n p −→ W −→ 0, a braid b sends Λ 0 on p(b) · Λ 0 . Then we can lift the Weyl loop w = p(b) in order to come back to Λ 0 .