Annales Henri Lebesgue 3 (2020) 341-380 MIC H A E L H E U S E N E R J O A N P O R TI H O L O M O R P HIC V O L U M E F O R M S O N R E P R E S E N TATIO N VA R IE TIE S O F S U R FA C E S W IT H B O U N D A R Y FORMES DE VOLUME HOLOMORPHES SUR L E S VA R IÉ T É S D E R E P R É S E N TATIO N S D E S S U R FA C E S À B O R D Abstract. — For closed and oriented hyperbolic surfaces, a formula of Witten establishes an equality between two volume forms on the space of representations of the surface in a semisimple Lie group. One of the forms is a Reidemeister torsion, the other one is the power of the Atiyah–Bott–Goldman–Narasimhan symplectic form. We introduce an holomorphic volume form on the space of representations of the circle, so that, for surfaces with boundary, it appears as peripheral term in the generalization of Witten’s formula. We compute explicit volume and symplectic forms for some simple surfaces and for the Lie group SLN (C). Résumé. — Pour les surfaces hyperboliques fermées et orientées, une formule de Witten établit une égalité entre deux formes de volume sur l’espace de représentations des groupes de surface dans un groupe de Lie semi-simple. Une de ces formes est une torsion de Reidemeister, l’autre est la forme de volume canoniquement associée à la forme symplectique d’Atiyah–Bott– Goldman–Narasimhan. Nous introduisons une forme de volume holomorphe sur l’espace des représentations du cercle, de sorte que, pour les surfaces à bord, elle apparaisse comme terme Keywords: representation varieties, volume forms. 2020 Mathematics Subject Classification: 53D30, 57M99. DOI: https://doi.org/10.5802/ahl.35 (*) Both authors partially supported by grant FEDER–Meic MTM2015–66165–P and by the Laboratoire de Mathématiques Blaise Pascal of the Université Clermont Auvergne. 342 M. HEUSENER & J. PORTI périphérique dans la généralisation de la formule de Witten. Pour certaines surfaces simples et pour le groupe de Lie SLN (C) nous calculons explicitement les formes volume et les formes symplectiques. 1. Introduction Along this paper S = Sg,b denotes a compact, oriented, connected surface with nonempty boundary, of genus g and with b > 1 boundary components. We assume that χ(S) = 2 − 2g − b < 0. The fundamental group π1 (S) is a free group Fk of rank k = 1 − χ(S) > 2. Let G be a connected, semisimple, complex, linear group with compact real form GR , e.g. G = SLN (C) and GR = SU(N ). We also assume that G is simply connected; notice that since π1 (S) is free, its representations lift to the universal covering of the Lie group. Fix a nondegenerate symmetric bilinear G–invariant form on the Lie algebra B: g × g → C, such that the restriction of B to gR , the Lie algebra of GR , is positive definite. This means that B is a negative multiple of the Killing form. Let R(S, G) denote the set of conjugacy classes of representations of π1 (S) ∼ = Fk into G. We are only interested in irreducible representations for which the centralizer coincides with the center of G. Following Johnson and Millson [JM87] we call such representations good (see Definition 2.3), and we use the notation R∗ (S, G) to denote the corresponding open subset of R(S, G). For a closed surface Σ, the bilinear form B induces two C–valued differential forms on R∗ (Σ, G), a holomorphic volume form ΩΣ defined as a Reidemeister torsion and the Atiyah–Bott–Goldman–Narasimhan (holomorphic) symplectic form ω. Witten has shown the following theorem for compact groups, here we state its complexifica- tion: Theorem 1.1 ([Wit91, Witten]). — If Σ is a closed, oriented and hyperbolic surface, then ωn ΩΣ = n! on R∗ (Σ, G), where n = 12 dim R∗ (Σ, G) = − 21 χ(Σ) dim(G). A proof of this theorem as well as an introduction to Reidemeister torsion and representation spaces of surface groups can be found in Labourie’s book [Lab13]. For surfaces with boundary S, we need to consider also R(S, ∂S, G)ρ0 , the relative set of conjugacy classes of representations (for each peripheral curve we require its image to be in a fixed conjugacy class), see Subsection 2.2. Let R∗ (S, ∂S, G)ρ0 denote the corresponding open subset of good representations. The holomorphic volume form ΩS is defined on R∗ (S, G) but the holomorphic symplectic form ω is defined on R∗ (S, ∂S, G)ρ0 . The geometric idea is to think of R(S, ∂S, G)ρ0 as the fibre of the restriction map R(S, G) → R(∂S, G). We aim to express the volume form (Reidemeister torsion) on ANNALES HENRI LEBESGUE Volume forms on representation varieties 343 R(S, G) as the product of a volume form on R(S, ∂S, G)ρ0 (induced by the symplectic form) and a volume form on R(∂S, G) that we construct in the next paragraph. To obtain a volume form on R(∂S, G), we deal with each component of ∂S, which is a circle. We identify the variety of representations of the circle S 1 with G, by mapping each representation to the image of a fixed generator of π1 (S 1 ). We restrict to regular representations namely, those which map the generators of π1 (S 1 ) to regular elements. Then Rreg (S 1 , G) ∼ = Greg /G. Using that G is simply connected (see Remark 2.13 when G is not simply connected), one of the consequences of Steinberg’s Theorem [Ste65] is that Rreg (S 1 , G) ∼ = Greg /G ∼ = Cr , where r = rank G, and that there is a natural isomorphism (Corollary 2.12): H 1 (S 1 ; Ad ρ) ∼ = T[ρ] Rreg (S 1 , G). Vr In Section 4.3 we show the existence of a form ν : H 1 (S 1 , Ad ρ) → C defined by the formula q (1.1) ν(∧v) = ± TOR(S 1 , Ad ρ, o, u, v) h∧v, ∧ui. Here u and v denote bases of H 0 (S 1 , Ad ρ) and H 1 (S 1 , Ad ρ) respectively and ∧u and ∧v their exterior product. Moreover, TOR denotes the Turaev’s sign refined tor- sion [Tur86, Section 3], o an homology orientation of H ∗ (S 1 ; R) (see Section 4.2), and h · , · i the duality pairing H 1 (S 1 , Ad ρ) × H 0 (S 1 , Ad ρ) → C. We prove in Lemma 4.5 that the value ν(∧v) ∈ C is independent of u. Steinberg’s Theorem ([Ste65], see also [Ste74, Pop11]) provides an isomorphism ∼ = (σ1 , . . . , σr ) : Greg /G −→ Cr , where σ1 , . . . , σr denotes a system of fundamental characters of G, which also proves the isomorphism G//G ∼ = Cr . When G = SLN (C), then r = N − 1 and (σ1 , . . . , σr ) are the coefficients of the characteristic polynomial. Proposition 1.2. — When G is simply connected, then ν = ±C dσ1 ∧ · · · ∧ dσr , for some constant C ∈ C∗ depending on G and B. In addition, for G = SLN (C) and B(X, Y ) = − tr(XY ) for X, Y ∈ SLN (C), √ C = ±(−1)(N −1)(N −2)/4 N . Let ρ0 ∈ R∗ (S, G) be ∂-regular, i.e. the image of each peripheral curve is a regular element of G (Definitions 2.8 and 2.9). We have an exact sequence (Corollary 2.12): b ∗ ∗ T[ρ(∂i )] Rreg (∂i , G) → 0, M 0 → T[ρ] R (S, ∂S, G)ρ0 → T[ρ] R (S, G) → i=1 where ∂S = ∂1 t · · · t ∂b denote the boundary components of S. For a ∂–regular rep- resentation ρ : π1 (S) → G we let νi denote the form corresponding to the restriction ρ|π1 (∂i ) : π1 (∂i ) → G as in (1.1) on ∂i ∼ = S 1. TOME 3 (2020) 344 M. HEUSENER & J. PORTI Set d = dim G, r = rank G, and b > 0 be the number of components of ∂S. The following generalizes Theorem 1.1 to surfaces with boundary, [Wit91] see also [BL99, Theorem 5.40] and Remark 4.9. Theorem 1.3. — Let ρ0 ∈ R(S, G) be a good, ∂–regular representation. Then on T[ρ0 ] R∗ (S, G) we have: ωn Ωπ1 (S) = ± ∧ ν1 ∧ · · · ∧ νb , n! where n = 12 dim R∗ (S, ∂S, G)ρ0 = 12 (−χ(S) d − b r). Notice that we write Ωπ1 (S) instead of ΩS , as the simple homotopy type of S only depends on π1 (S). Following Witten [Wit91] in the closed case, the proof of Theorem 1.3 is based on Franz–Milnor duality for Reidemeister torsion. The formula of Theorem 1.3 is homogeneous in the bilinear form B : g × g → C: if B is replaced by λ2 B for some λ ∈ C∗ , then ω is replaced by λ2 ω, νi by λr νi and Ωπ1 (S) by λ2 n+b r Ωπ1 (S) , as 2 n + b r = −χ(S)d = dim R(S, G). We focus now on G = SLN (C), which is simply connected and has rank r = N − 1. We fix a bilinear form on the Lie algebra: Convention 1.4. — Along this paper, when G = SLN (C) we always assume B(X, Y ) = − tr(XY ) for X, Y ∈ slN (C). We compute explicit volume forms for spaces of representations of free groups in SL2 (C) and SL3 (C). We start with a pair of pants S0,3 . The fundamental group π1 (S0,3 ) ∼ = F2 is free on two generators γ1 and γ2 . By Fricke–Klein–Vogt Theorem [Fri96, FK97, Vog89] (see [Gol09, Mag80] for a modern treatment), X(F2 , SL2 (C)) ∼ = C3 and the coordinates are precisely the traces of the peripheral elements γ1 , γ2 , and γ1 γ2 , denoted by t1 , t2 , and t12 respectively. In this case the relative character variety is just a point, and√the symplectic form is trivial. Thus, by applying Theorem 1.3 and equality ν = ± 2 d trγ (Proposition 1.2), we have √ ΩF2 = Ωπ1 (S0,3 ) = ±2 2 dt1 ∧ dt2 ∧ dt12 , on R∗ (F2 , SL2 (C)) (see also [Mar16, 4.3.1]). By [GAMA93], for k > 3, the 3k−3 trace functions t1 , t2 , t12 , t3 , t13 , t23 , . . . , tk , t1k , t2k define a local parameterization T : R∗ (Fk , SL2 (C)) \ crit(T ) → C3k−3 , where crit(T ) = i>3 {t12i = t21i } ∪ {t121̄2̄ = 2}. Here, ti1 ...il : R∗ (Fk , SL2 (C)) → C S stands for the trace function trγ if γ = γi1 . . . γil with the convention γ ī = γi−1 . Theorem 1.5. — The holomorphic volume form on R∗ (Fk , SL2 (C)) \ crit(T ) is SL (C) ΩFk 2 = ±T ∗ Ω, where √ ^k √ dti ∧ dt1i ∧ dt2i Ω = ±2 2 dt1 ∧ dt2 ∧ dt12 2 . i=3 t12i − t21i Next we deal with SL3 (C). To avoid confusion with SL2 (C), the trace functions in SL3 (C) are denoted by τi1 ...ik ; notice that τī 6= τi . Lawton obtains in [Law07] an ANNALES HENRI LEBESGUE Volume forms on representation varieties 345 explicit description of the variety of characters X(F2 , SL3 (C)). It follows from his result that T := (τ1 , τ1̄ , τ2 , τ2̄ , τ12 , τ1̄2̄ , τ12̄ , τ1̄2 ) : R∗ (F2 , SL3 (C)) \ {τ121̄2̄ = τ212̄1̄ } → C8 defines a local parameterization. Using the computation of the symplectic form in [Law09], we prove in Proposition 5.7 that, on R∗ (F2 , SL3 (C)) \ {τ212̄1̄ = τ121̄2̄ } the SL (C) volume form is ΩF2 3 = T ∗ Ω, where √ 3 −3 Ω=± dτ1 ∧ dτ1̄ ∧ dτ2 ∧ dτ2̄ ∧ dτ12 ∧ dτ1̄2̄ ∧ dτ12̄ ∧ dτ1̄2 . τ212̄1̄ − τ121̄2̄ This is generalized to a free group of arbitrary rank. The next proposition is a special case of a theorem of Lawton [Law10, Theorem 6]: Proposition 1.6. — For k > 3, the 8k − 8 trace functions T = (τ1 , τ1̄ , τ2 , τ2̄ , . . . , τk , τk̄ , τ12 , τ1̄2̄ , τ13 , τ1̄3̄ , . . . , τ1k , τ1̄k̄ , τ23 , τ2̄3̄ , . . . , τ2k , τ2̄k̄ , τ12̄ , τ1̄2 , τ13̄ , τ1̄3 , . . . , τ1k̄ , τ1̄k ) define a local parameterization T : R∗ (Fk , SL3 (C)) \ crit(T ) → C8k−8 , with {∆12i = 0}, [ [ crit(T ) = {τ1i1̄ī = τi1ī1̄ } ∪ i>2 i>3 ∆12i = (τ12i − τ1i2 )(τ1̄2̄ī − τ1̄ī2̄ ) − (τ12̄ī − τ1ī2̄ )(τ1̄2i − τ1̄i2 ). Next we provide the holomorphic volume form: SL (C) Theorem 1.7. — The volume form on R∗ (Fk , SL3 (C)) \ crit(T ) is ΩFk 3 = ∗ ±T Ω, for k ^ ω1i ∧ νi ∧ ν1i ∧ ν2i Ω = ω12 ∧ ν1 ∧ ν2 ∧ ν12 i=3 3∆12i where √ √ 1 νi = −3 dτi ∧ dτī , ν`i = −3 dτ`i ∧ dτ`¯ī , ω1i = dτ ∧ dτ1̄i , τ1i1̄ī − τi1ī1̄ 1ī and ∆12i is as in Proposition 1.6. Remark 1.8. — Notice that all the formulas for the volume and symplectic forms are rational. Rationality of the volume forms coming from Reidemeister torsion has been addressed by several authors [Bén17, Bén16, DG16, Mar16, Por97], and rationality of the symplectic form can be deduced from Goldman’s works [Gol86, Gol04]. The paper is organized as follows. In Section 2 we review the results on spaces of representations that we need, in particular we describe the relative variety of representations. In Section 3 we recall the tools of Reidemeister torsion, including the duality formula, on which Theorem 1.3 is based. In Section 4 we describe all forms and we prove Theorem 1.3. Section 5 is devoted to formulas for SLN (C), the form ν and as well as the volume form for the free groups of rank 2 in SL2 (C) and SL3 (C). In Section 6 we use Goldman’s formula for the Poisson bracket to give the TOME 3 (2020) 346 M. HEUSENER & J. PORTI symplectic form in terms of trace functions for the relative varieties of representations of S0,4 and S1,1 in SL2 (C). Finally, in Section 7 we compute volume forms on spaces of representations of free groups of higher rank in SL2 (C) and SL3 (C). Acknowledgments We are indebted to Simon Riche for helpful discussions and for pointing out Steinberg’s results to us, and to Sean Lawton for useful references. We thank the referees for suggestions that improved the paper, and for letting us know about a lecture by Narasimhan where he presented the formula of the symplectic form. 2. Varieties of representations Throughout this article G denotes a simply-connected semisimple complex linear Lie group. We let d denote the dimension of G, and r its rank. Also recall that along this paper S = Sg,b denotes a compact, oriented, connected surface with nonempty boundary, of genus g and with b > 1 boundary components, ∂S = ∂1 t · · · t ∂b . We assume that χ(S) = 2 − 2g − b < 0. The fundamental group of S is a free group Fk of rank k = 1 − χ(S) > 2. 2.1. The variety of good representations The set of representations of π1 (S) ∼ = Fk in G is R(S, G) = hom(π1 (S), G) ∼ = Gk . It follows from [OV90, Chapter 4, Section 1.2] that G is algebraic, and hence R(S, G) is an affine algebraic set (it has a natural algebraic structure independent of the choice of the isomorphism π1 (S) ∼= Fk ). The group G acts on R(S, G) by conjugation and we are interested in the quotient R(S, G) = R(S, G)/G . This is not a Hausdorff space, so we need to restrict to representations with some regularity properties. Following [JM87], we define: Definition 2.1. — A representation ρ ∈ R(S, G) is irreducible if its image is not contained in a proper parabolic subgroup of G. For ρ ∈ R(S, G), its centralizer is Z(ρ) = {g ∈ G | gρ(γ) = ρ(γ)g, ∀ γ ∈ π1 (S)} . Proposition 2.2 ([JM87, Proposition 1.1]). — The representation ρ ∈ R(S, G) is irreducible if and only if the orbit O(ρ) is closed in R(S, G) and Z(ρ) is finite. Definition 2.3. — A representation ρ ∈ R(S, G) is good if it is irreducible and its centralizer Z(ρ) is the center of the group G, i.e. Z(ρ) = Z(G). ANNALES HENRI LEBESGUE Volume forms on representation varieties 347 The set of good representations is denoted by R∗ (S, G), and its orbit space by R∗ (S, G) = R∗ (S, G)/G. Proposition 2.4 ([JM87, Propositions 1.2 and 1.3]). — The set of good repre- sentations R∗ (S, G) is a Zariski open subset of R(S, G). Furthermore the action of G on R∗ (S, G) is proper. The variety of characters is the quotient in the algebraic category: X(S, G) = R(S, G)//G . Namely, it is an algebraic affine set defined by its ring of polynomial functions, as the ring of functions on R(S, G) invariant by conjugation. The projection R(S, G) → X(S, G) factors through a surjective map R(S, G) → X(S, G). For good representations we have: Proposition 2.5. — The natural map restricts to an injection R∗ (S, G) ,→ X(S, G) whose image is a Zariski open subset and a smooth complex manifold. For the proof, see for instance [JM87, Section 1], or [New78, Proposition 3.8] for injectivity, as irreducibility is equivalent to stability in GIT [JM87, Section 1]. For smoothness see [Gol84]. Given a representation ρ ∈ R(S, G), the Lie algebra g turns into an π1 (S)–module via Ad ◦ρ. If there is no ambiguity this module is denoted just by g, and the coeffi- cients in cohomology are denoted by Ad ρ. Proposition 2.6. — Let ρ ∈ R∗ (S, G) be a good representation. Then there is a natural isomorphism T[ρ] R∗ (S, G) ∼ = H 1 (S; Ad ρ). In particular the dimension of R∗ (S, G) is −χ(S) d. This proposition can be found for instance in [Sik12, Corollary 50], but we sketch the proof as it may be useful for the relative case. Proof. — Let Z 1 = Z 1 (S; Ad ρ) denote the space of crossed morphisms from π1 (S) to g, i.e. maps d : π1 (S) → g satisfying d(γµ) = d(γ) + Adρ(γ) d(µ), ∀ γ, µ ∈ π1 (S). Let B 1 = B 1 (S; Ad ρ) denote the subspace of inner crossed morphisms: for a ∈ g the corresponding inner morphism maps γ ∈ π1 (S) to Adρ(γ) (a) − a. Weil’s construction identifies Z 1 with Tρ R(S, G) (usually Z 1 is the Zariski tangent space of a scheme, possibly non-reduced, but as π1 (S) is free, R(S, G) is a smooth algebraic variety). The subspace B 1 corresponds to the tangent space of the orbit AdG (ρ). Then, in order to identify the tangent space of R∗ (S, G) with the cohomology group H 1 (S; Ad ρ) = Z 1 /B 1 , we use a slice, for instance an étale slice provided by Luna’s Theorem [PS94, Theorem 6.1], or an analytic slice (cf. [JM87]). In the setting of a good representation ρ, a slice is a subvariety S ⊂ R(S, G) containing ρ, invariant by Z(ρ) = Z(G), such that the conjugation map G/Z(G) × S → R(S, G) TOME 3 (2020) 348 M. HEUSENER & J. PORTI is locally bi-analytic at (e, ρ) and the projection S → X(S, G) is also bi-analytic at ρ. (If ρ was not good, we should take care of the action of Z(ρ)/Z(G). In addition, for Γ not a free group the description is more involved). Then the assertion follows easily from the properties of the slice.  2.2. The relative variety of representations Let ∂S = ∂1 t · · · t ∂b denote the decomposition in connected components. By abuse of notation, we also let ∂i denote an element of the fundamental group represented by the corresponding oriented peripheral curve. This is well-defined only up to conjugacy in π1 (S), but our constructions do not depend on the representative in the conjugacy class. Definition 2.7 ([Kap01, Section 4.3]). — For ρ0 ∈ R(S, G), the relative variety of representations is R(S, ∂S, G)ρ0 = {[ρ] ∈ R(S, G) | ρ(∂i ) ∈ O(ρ0 (∂i )), i = 1, . . . , b} . Here O(ρ0 (∂i )) denotes the conjugacy class of ρ0 (∂i ). We also denote R∗ (S, ∂S, G)ρ0 = R(S, ∂S, G)ρ0 ∩ R∗ (S, G) . Besides considering good representations, we restrict our attention to representa- tions which map peripheral elements to regular elements of G. Definition 2.8 ([Ste74, Section 3.5]). — An element g ∈ G is called regular if its centralizer Z(g) has minimal dimension among centralizers of elements of G. Equivalently, its conjugacy class O(g) has maximal dimension. This minimal dimension is r the rank of G [Ste74, Section 3.5, Proposition 1]. In SLN (C), a diagonal matrix is regular if and only if all eigenvalues are different. More generally, g ∈ SLN (C) is regular if and only if its minimal polynomial is of degree N [Ste74, Section 3.5, Proposition 2]. In particular the companion matrix of a monic polynomial is regular. Definition 2.9. — A representation ρ ∈ R(S, G) is called ∂–regular if the ele- ments ρ(∂1 ), . . . , ρ(∂b ) are regular. Proposition 2.10. — Let ρ0 ∈ R∗ (S, G) be a good, ∂–regular representation. (a) R∗ (S, ∂S, G)ρ0 is a complex smooth manifold of dimension d0 = −d χ(S) − b r = d(2g(S) − 2) + b(d − r). (b) For [ρ] ∈ R∗ (S, ∂S, G)ρ0 , there is a natural isomorphism: T[ρ] R∗ (S, ∂S, G)ρ ∼   0= ker H 1 (S; Ad ρ) → H 1 (∂S; Ad ρ) . Proof. — We first show that the map H 1 (S; Ad ρ) → H 1 (∂S; Ad ρ) is a surjection. By Poincaré duality H 2 (S, ∂S; Ad ρ) ∼ = H 0 (S; Ad ρ) ∼ = gAd ρ(π1 (S)) , that vanishes because Z(ρ) is finite. Thus, by the long exact sequence of the pair (S, ∂S), the map H 1 (S; Ad ρ) → H 1 (∂S; Ad ρ) is a surjection. ANNALES HENRI LEBESGUE Volume forms on representation varieties 349 We use a slice at ρ0 , S ⊂ R(M ) as in the proof of Proposition 2.6. The fact that H 1 (S; Ad ρ) → H 1 (∂S; Ad ρ) is a surjection means that the restriction map b R(∂i , G) = Gb Y res |S : S → R(∂S, G) = i=1 is transverse to the products of orbits by conjugation b Y O= O(ρ(∂i )) . i=1 Namely, (res |S )∗ (Tρ S)+Tres(ρ) O = Tres(ρ) Gb . It follows from the rank Theorem [Łoj91, Chapter 4.1] that O(ρ(∂i )) ⊂ G is a complex analytic subvariety of dimension d − r because ρ is ∂–regular. Thus (res |S )−1 (O) is an analytic C–submanifold of codimension b   dim Gb − dim O = X dim G − dim O(ρ(∂i )) = b r . i=1 Now the proposition follows from the properties of the slice.  2.3. Steinberg map In order to understand the space of conjugacy classes of regular representations of Z we identify each representation with the image of its generator, so that Rreg (Z, G) = Greg and Rreg (Z, G) = Greg /G. Consider the Steinberg map (2.1) (σ1 , . . . , σr ) : G → Cr where σ1 , . . . , σr denote the characters corresponding to a system of fundamental representations (for SLN (C) those are the coefficients of the characteristic polyno- mial). Theorem 2.11 ([Ste65]). — If G is simply connected, then the map (2.1) is a surjection and has a section s : Cr → Greg so that s(Cr ) is a subvariety that intersects each orbit by conjugation in Greg precisely once. For instance, when G = SLN (C) the section in Theorem 2.11 can be chosen to be the companion matrix (see [Ste74, p. 120] and [Hum95, Section 4.15]). Corollary 2.12. — If G is simply connected, then: (i) The map (2.1) induces natural isomorphisms between the space of regular orbits by conjugation, the variety of characters, and Cr : Rreg (S 1 , G) ∼ = X(S 1 , G) ∼ = Cr . (ii) The Steinberg map induces a natural isomorphism H 1 (S 1 , Ad ρ) ∼ = T[ρ] Rreg (S 1 , G) ∼ = Cr . TOME 3 (2020) 350 M. HEUSENER & J. PORTI Moreover, for each good, ∂-regular representation ρ0 ∈ R∗ (S, G) and [ρ] ∈ R∗ (S, ∂S, G)ρ0 there is an exact sequence b ∗ ∗ T[ρ(∂i )] Rreg (∂i , G) → 0. M 0 → T[ρ] R (S, ∂S, G)ρ0 → T[ρ] R (S, G) → i=1 Proof. (i) Notice that what we aim to prove is the isomorphism Greg /G ∼ = G//G ∼ = Cr ; which is straightforward from the existence of the section in Theorem 2.11. (ii) By the existence of the section we also know that the differential of Steinberg’s map Z 1 (Z, Ad ρ) ∼ = g → Cr is surjective whenever ρ is regular [Hum95, Section 4.19]. In addition it maps B 1 (Z, Ad ρ) to 0, because Steinberg map is constant on orbits by conjugation. Thus we have a well-defined surjection H 1 (S 1 , Ad ρ) → Cr , which is an isomorphism because of the dimension. The exact sequence follows from the long exact sequence in cohomology of the pair (S, ∂S) and the identification of cohomology groups with tangent spaces, cf. Proposition 2.10.  Remark 2.13. — When G is not simply connected, then the universal covering G e → G is finite and π1 (G) can be identified with a (finite) central subgroup Z of G. e The center of G acts on the quotient G//G and we obtain a commutative diagram e e e e −−−→ G/ G e /G e      ϕ y y G −−−→ G//G where (G//G, ϕ) is a quotient for the action of Z on G/ e (see [Pop11, Lemma 2.5]). e /G Notice that ϕ is a finite branched covering. Then part (ii) of Corollary 2.12 can be easily adapted for those [g] ∈ G//G which are outside the branch set of ϕ. 3. Reidemeister torsion Let ρ ∈ R(S, G) be a representation; recall that we consider the action of π1 (S) on g via the adjoint of ρ. Most of the results in this section apply not only to g but to its real form gR , provided that the image of the representation is contained in GR . Recall also that we assume that B restricted to the compact real form gR is positive definite. Consider a cell decomposition K of S. If C∗ (K; f Z) denotes the cellular chain complex on the universal covering, one defines (3.1) C ∗ (K; Ad ρ) = homπ1 (S) (C∗ (K; f Z), g). We consider the so called geometric basis. Start with a B–orthonormal C–basis {m1 , . . . , md } of g. For each i–cell eij of K we choose a lift eeij to the universal covering f then K, ci = {(eeij )∗ ⊗ mk }jk ANNALES HENRI LEBESGUE Volume forms on representation varieties 351 is a basis of C i (K; Ad ρ), called the geometric basis. Here, (eeij )∗ ⊗ mk : C∗ (K; f Z) → g is the unique π1 (S)–homomorphism given by (eeij )∗ ⊗ mk (eeil ) = δjl mk . On the other hand, if B i = Im(δ : C i−1 (K; Ad ρ) → C i (K; Ad ρ)) is the space of coboundaries, choose bi a basis for B i ⊂ C i and choose lifts b fi to C i−1 of the coboundary map. For a basis hi of H i (K; Ad ρ), consider also representatives fi ∈ C i (K; Ad ρ). Then the disjoint union h fi t bi i+1 t h bg is also a basis for C i (K; Ad ρ). Notice that we are interested in the case where the zero and second cohomology groups vanish, so we assume that h f2 = ∅. f0 = h Reidemeister torsion is defined as f1 t b1 : c1 ] f2 t h [b (3.2) 1 tor(S, Ad ρ, h ) = f ∈ C∗ /{±1} 1 0 2 2 [b : c ][b : c ] Here, for two bases a and b of a vector space, [a : b] denotes the determinant the matrix whose colons are the coefficients of the vectors of a as linear combination of b. Remark 3.1. — The choice of the bilinear form B is relevant, as we use a B– orthonormal basis for g and χ(S) 6= 0. Namely, if we replace B by λ2 B, then the orthonormal basis will be λ1 {m1 , . . . , md } and the torsion will be multiplied by a factor λ−χ(S)d = λdim R(S,G) . For an ordered basis a = {a1 , . . . , am } of a vector space, denote ∧a = a1 ∧ · · · ∧ am . Since ∧a = [a : b](∧b), the notation [a : b] = ∧a /∧b is often used in the literature (cf. [Mil62]). 3.1. The holomorphic volume form The tangent space of R∗ (S, G) at [ρ] is identified with H 1 (S; Ad ρ), by Proposi- tion 2.6. There is a natural holomorphic volume form on H 1 (S; Ad ρ): ΩS (∧h) = ± tor(S, Ad ρ, h) 1 where h is a basis for H (S; Ad ρ). The surface S has the simple homotopy type of a graph. Moreover, graphs that are homotopy equivalent are also simple-homotopy equivalent, thus this volume form depends only on the fundamental group Ωπ1 (S) = ΩS . The bilinear form B defines a bi-invariant volume form θG on the Lie group G in the usual way. Hence (θG )k is a volume form on R(π1 (S), G) ∼ = Gk . For a good representation ρ the form θG induces also a form θO(ρ) on the orbit O(ρ) by push-forward: the orbit map fρ : G → R(π1 (S), G), fρ (g) = Adg ◦ρ factors TOME 3 (2020) 352 M. HEUSENER & J. PORTI through the quotient G/Z(G). The quotient map G → G/Z(G) is a Lie group covering and we get an isomorphism f¯ρ : G/Z(G) → O(ρ), and hence (3.3) θO(ρ) = (fρ )∗ (θG ) . The next lemma justifies why Reidemeister torsion is the natural choice of volume form on the variety of representations up to conjugation. Lemma 3.2. — Let π : R∗ (S, G) → R∗ (S, G) denote the projection. Then at ρ ∈ R∗ (S, G) we have: (θG )k = ±θO(ρ) ∧ π ∗ ΩS . Proof. — We use a graph G with one vertex and k edges to compute the torsion of S. The Reidemeister torsion of this graph is tor(G, Ad ρ, h) = ±[b1 t h e : c1 ]/[b f1 : c0 ]. f1 = c0 , which is a basis for g, then If we make the choice b   tor(G, Ad ρ, h) = [δc0 t h e : c1 ] = ∧ δc0 ∧ h e /∧c1 . We identify the 1–cells with the generators of Fk , so that every element in c1 is viewed as a tangent vector of the variety of representations, and c1 has volume one, (θG )k (∧c1 ) = 1 because we started with an B–orthonormal basis for g. Thus (3.4) (θG )k (∧δc0 ∧ h) e = ± tor(G, Ad ρ, h) = ±Ω (∧h). S As δc0 is a basis of the tangent space of the orbitπ∗ (δc0 ) = 0. Moreover, using π∗ (h) e = h: (3.5) (θO(ρ) ∧ π ∗ ΩS )(∧δc0 ∧ h) e =θ 0 O(ρ) (∧δc )ΩS (∧h). By (3.4) and (3.5), to conclude the proof of the Lemma 3.2 we claim that θO(ρ) (∧δc0 ) = 1. For that purpose, we use the canonical identification Tρ O(ρ) ∼ = B 1 (π1 (S); Ad ρ). Using this identification, the tangent map of the orbit map fρ : G → O(ρ) at e ∈ G, dfρ (e) : g → Tρ O(ρ), corresponds to dfρ (e)(X) = −δ(X), where δ : g → B 1 (π1 (S); Ad ρ) denotes the coboundary operator δ(X)(γ) = Adρ(γ) (X) − X, for γ ∈ π1 (S). Therefore for the basis δc0 of B 1 (π1 (S); Ad ρ) we obtain by (3.3): θO (ρ)(∧δc0 ) = θO (ρ)(∧dfρ (e)c0 ) = ±θG (∧c0 ) = 1. This concludes the proof of the claim and the Lemma 3.2.  ANNALES HENRI LEBESGUE Volume forms on representation varieties 353 3.2. The nondegenerate pairing Consider K 0 the cell decomposition dual to K: for each i–dimensional cell eij of K there exists a dual (2 − i)–dimensional cell fj2−i of the dual complex (K 0 , ∂K 0 ). The complex C ∗ (K 0 , ∂K 0 ; Z) yields the relative cohomology of the pair (S, ∂S). This can be generalized to cohomology with coefficients. If C∗ (K; f Z) denotes the simplicial chain complex on the universal covering, recall from (3.1) that C ∗ (K; Ad ρ) = homπ1 (S) (C∗ (K; f Z), g), and we similarly define C ∗ (K 0 , ∂K 0 ; Ad ρ) = homπ1 (S) (C∗ (K f0 , ∂ K f0 ; Z), g) where π1 (S) acts on g by the adjoint representation. Following Milnor [Mil62], there is a paring [ · , · ] : Ci (K; f Z) × C (K f0 , ∂ K f0 ; Z) → Zπ (S) 2−i 1 defined by [c, c0 ] := (c · γc0 ) γ. X γ∈π1 (S) Here “·” denotes the intersection number in the universal covering. The main prop- erties of this paring are that for η ∈ Zπ1 (S) we have: (3.6) [ηc, c0 ] = η[c, c0 ], [c, ηc0 ] = [c, c0 ]η̄ and [∂c, c0 ] = ±[c, ∂c0 ]. Here the bar · : Zπ1 (S) → Zπ1 (S) denotes the anti-involution that extends Z–linearly the anti-morphism of π1 (S) that maps γ ∈ π1 (S) to γ −1 . Notice that the sign ± in equation (3.6) depends only on the dimension of the chains. For each i–dimensional cell eij we fix a lift eeij to K. f Also, we choose a (2 − i)– dimensional cell fe2−i j which projects to fj2−i . By replacing fe2−i j by a translate, we can assume that eeij · fe2−i k = δjk . f Z) and each (2−i)-chain c0 ∈ C (K We obtain, for each i–chain c ∈ Ci (K; f0 , ∂ K f0 ; Z) 2−i that c = [c, fe2−i eij and c0 = [eeij , c0 ] fe2−i X X j ]e j . j j i 0 2−i−1 0 0 Given α ∈ C (K; Ad ρ) and α ∈ C (K , ∂K ; Ad ρ) the formula   (α, α0 ) 7→ B α(eeij ), α0 (fej2−i ) X j defines a nondegenerate pairing (3.7) h · , · i : C i (K; Ad ρ) × C 2−i (K 0 , ∂K 0 ; Ad ρ) → C . By using equation (3.6), it is easy to see that this pairing satisfies (3.8) hδα, α0 i = ±hα, δα0 i , and therefore it induces a non-singular pairing in cohomology (3.9) h · , · i : H 1 (S; Ad ρ) × H 1 (S, ∂S; Ad ρ) → C. TOME 3 (2020) 354 M. HEUSENER & J. PORTI Given a basis h = {hi }i of H 1 (S; Ad ρ) and h0 = {h0i }i a basis of H 1 (S, ∂S; Ad ρ), we introduce the notation   (3.10) h∧h, ∧h0 i := det hhi , h0j iij which is the natural extension of the pairing (3.9) to d d H 1 (S; Ad ρ) H 1 (S, ∂S; Ad ρ) → C , ^ O^ where d = −χ(S) dim G. 3.3. The duality formula Let ρ ∈ R(π1 (S), G) be a representation. Proposition 3.3 (Duality formula). — Let h = {hi }i be a basis for H 1 (S; Ad ρ), and let h0 = {h0i }i be a basis for H 1 (S, ∂S; Ad ρ). Assume that the cohomology groups H k (S; Ad ρ) and H k (S, ∂S; Ad ρ) vanish in dimension k = 0, 2. Then tor(S, Ad ρ, h) tor(S, ∂S, Ad ρ, h0 ) = ±h∧h, ∧h0 i This is E. Witten’s generalization of the duality formula of W. Franz and J. Milnor. We reproduce the proof for completeness. In Witten’s article [Wit91] the proof of this particular formula is only given in the closed case, and Milnor [Mil62] and Franz [Fra36] consider only the acyclic case. Proof. — We choose the geometric basis of C i (K; Ad ρ) and C 2−i (K 0 , ∂K 0 ; Ad ρ) to be dual to each other, by choosing dual lifts of the cells and a B–orthonormal basis of the Lie algebra g. In this way, the matrix of the intersection form (3.7) with respect the geometric basis is the identity, in particular its determinant is 1: h∧ci , ∧(c2−i )0 i = 1. Thus we view the product of torsions in the statement of the proposition as three changes of basis, one for each intersection form: (3.11) tor(S, Ad ρ, h) tor(S, ∂S, Ad ρ, h0 ) 0 h∧c1 , ∧c1 )0 i = ± tor(S, Ad ρ, h) tor(S, ∂S, Ad ρ, h ) h∧c0 , ∧(c2 )0 ih∧c2 , ∧(c0 )0 i [bf2 t b1 t h e : c1 ] [(b ] 2 )0 t (b1 )0 t hf0 : (c1 )0 ] h∧c1 , ∧(c1 )0 i =± f f1 )0 : (c0 )0 ][(b2 )0 : (c2 )0 ] h∧c0 , ∧(c2 )0 ih∧c2 , ∧(c0 )0 i [b1 : c0 ][b2 : c2 ] [(b f2 ∧ b1 ∧ h, h∧b ] e ∧(b 2 )0 ∧ (b1 )0 ∧ h f0 i =± f1 , ∧(b2 )0 ih∧b2 , ∧(b f1 )0 i . h∧b Next, following Witten, we may choose the lift of the coboundaries to be orthogonal to the lift of the cohomology of the other complex: hh 2 0 e2 f0 j ) i = hbi , hj i = 0. f, (b g i In addition, by direct application of (3.8): hb1i , (b1j )0 i = hb1i , hj f, (b1 )0 i = 0. f0 i = hh i j ANNALES HENRI LEBESGUE Volume forms on representation varieties 355 Thus the numerator in (3.11) is the determinant of a matrix with some vanishing blocks, and (3.11) becomes: f2 , ∧(b1 )0 ih∧b1 , ∧(b h∧b ] 2 )0 ih∧h, ∧h0 i (3.12) ± f1 , ∧(b2 )0 ih∧b2 , ∧(b f1 )0 i . h∧b Finally, since δ bfi = bi and δ (b gi )0 = (bi )0 , h∧b f2 , ∧b1 )0 i = ±h∧b2 , ∧(b f1 )0 i and h∧b1 , ∧bg f1 , ∧(b2)0 i, by (3.8). Hence (3.12) equals ±h∧h, ∧h0 i, concluding 2 )0 i = ±h∧b the proof of Proposition 3.3.  Remark 3.4. — Notice that the proof generalizes in any dimension, after changing the product by a quotient in the odd dimensional case, and taking care of the intersection product in all cohomology groups. 4. Symplectic form and volume forms 4.1. The symplectic form on the relative variety of representations For a good and ∂–regular representation ρ0 , the tangent space of R∗ (S, ∂S, G)ρ0 is the kernel of the map i : H 1 (S; Ad ρ) → H 1 (∂S; Ad ρ) induced by inclusion (Propo- sition 2.10). The long exact sequence in cohomology of the pair is: β j i 0 → H 0 (∂S; Ad ρ) → H 1 (S, ∂S; Ad ρ) → H 1 (S; Ad ρ) → H 1 (∂S; Ad ρ) → 0 For a, b ∈ ker(i), we define (4.1) ω(a, b) = hae, bi = ha, ebi where ae, eb ∈ H 1 (S, ∂S; Ad ρ) satisfy j(ae) = a, j(eb) = b. This form is well-defined (in- dependent of the lift), because i and β are dual maps with respect to the pairing (3.9), that is hβ( · ), · i = h · , i( · )i. Moreover we have: Theorem 4.1 ([Gol84, GHJW97, Law09]). — Assume that ρ0 is a good and ∂–regular. Then the form ω is symplectic on R∗ (S, ∂S, G)ρ0 . The fact that ω is bilinear and alternating is clear from construction, non-degen- eracy follows from Poincaré duality, and the deep result is to prove dω = 0. When S is closed this was proved by Goldman in [Gol84]. When ∂S 6= ∅, the result with real coefficients is due to Guruprasad, Huebschmann, Jeffrey, and Weinstein [GHJW97], and in [Law09] Lawton explains why it applies also in the complex case. 4.2. Sign refined Reidemeister torsion for the circle Let V be a finite dimensional real or complex vector space, and ϕ : π1 (S 1 ) → SL(V ) be a representation. In what follows we use the refined torsion with sign due to Turaev, that we denote TOR(S 1 , ϕ, o, u, v) [Tur86, Section 3]. This torsion depends TOME 3 (2020) 356 M. HEUSENER & J. PORTI on the choice of an orientation o in cohomology with constant coefficients of S 1 and the choice of respective basis u for H 0 (S 1 ; ϕ) and v for H 1 (S 1 ; ϕ). For a circle S 1 , the choice of an orientation determines a fundamental class, hence an orientation in homology. We start with a cell decomposition K of S 1 , with i–cells ei1 , . . . , eia , i = 0, 1, and a (real or complex) basis {m1 , . . . , mk } for the vector space V . The geometric basis for C i (K; ϕ) is then ci = {(eei1 )∗ ⊗ m1 , (eei1 )∗ ⊗ m2 , . . . , (eeia )∗ ⊗ mk }. As before, let B 1 = Im(δ : C 0 (K; ϕ) → C 1 (K; ϕ)) denote the coboundary space and choose b1 as basis for B 1 and lift it to b f1 in C 0 (K; ϕ). Consider also v e ⊂ C 1 (K; ϕ) a representative of v and similarly u e ⊂ C 0 (K; ϕ) for u. Then we define the torsion: e t b1 : c1 ] [v tor(S 1 , ϕ, u, v, c0 , c1 ) = f ∈ C∗ . [b t u 1 0 e :c ] Notice that there is no sign indeterminacy, because we include ci in the notation. In fact sign indeterminacy comes from changing the order or the orientation of the cells of K. The sign is not affected by the choice of a basis for V , because χ(S 1 ) = 0. Following [Tur86, Section 3] we consider i i dim C l (K; ϕ), dim H l (S 1 ; ϕ) and N = X X X αi = βi = αi βi . l=0 l=0 i>0 We define Tor(S 1 , ϕ, u, v, c0 , c1 ) = (−1)N tor(S 1 , ϕ, u, v, c0 , c1 ). This quantity is invariant under subdivision of the cells of K, but it still depends on their ordering and orientation. To make this quantity invariant, Turaev introduces the notion of cohomology orientation, i.e. an orientation of the R–vector space H 0 (S 1 ; R) ⊕ H 1 (S 1 ; R). We consider a geometric basis of the complex with trivial coefficients C i (K; R), ci = {(ei1 )∗ , . . . , (eia )∗ }, with the same ordering and orientation of cells. We choose any basis hi of H i (S 1 ; R) that yield the orientation o. Definition 4.2. — The sign determined torsion is  dim ϕ TOR(S 1 , ϕ, o, u, v) = Tor(S 1 , ϕ, u, v, c0 , c1 ) · sgn Tor(S 1 , 1, h0 , h1 , c0 , c1 ) Let −o denote the homology orientation opposite to o. It is straightforward from construction that (4.2) TOR(S 1 , ϕ, −o, u, v) = (−1)dim ϕ TOR(S 1 , ϕ, o, u, v) In particular, we do not need the homology orientation when dim ϕ is even. For a circle S 1 , the choice of an orientation determines a fundamental class, hence an orientation in cohomology. Let ϕi : π1 (S 1 ) → SL(Vi ) be representations into finite dimensional vector spaces, for i = 1, 2. Then H ∗ (S 1 ; ϕ1 ⊕ ϕ2 ) ∼ = H ∗ (S 1 ; ϕ1 ) ⊕ H ∗ (S 1 ; ϕ2 ). Let ui and vi denote 0 1 1 1 bases for H (S ; ϕi ) and H (S ; ϕi ) respectively. The following lemma reduces to an elementary calculation: ANNALES HENRI LEBESGUE Volume forms on representation varieties 357 Lemma 4.3. — Let ϕi : π1 (S 1 ) → SL(Vi ) be representations into finite dimen- sional vector spaces, for i = 1, 2. Then TOR(S 1 , ϕ1 ⊕ ϕ2 , o, u1 × {0} t {0} × u2 , v1 × {0} t {0} × v2 ) = TOR(S 1 , ϕ1 , o, u1 , v1 ) · TOR(S 1 , ϕ2 o, u2 , v2 ). 4.3. An holomorphic volume form on Rreg (S 1 , G) As in the introduction we let G denote a simply-connected, semisimple, complex, linear Lie group, d = dim G, and r = rk G. Definition 4.4. — We call a representation ρ : π1 (S 1 ) → G regular if the image of the generator of π1 (S 1 ) is a regular element g ∈ G. The set of conjugacy classes of regular representations is denoted by Rreg (S 1 , G). Let ρ : π1 (S 1 ) → G be regular. Then dim H 0 (S 1 ; Ad ρ) = r, because H 0 (S 1 ; Ad ρ) ∼ = gAd ρ . As the Euler characteristic of S 1 vanishes, dim H 1 (S 1 ; Ad ρ) = r. Furthermore, since G is simply connected, we have H 1 (S 1 ; Ad ρ) ∼ = T[ρ] Rreg (S 1 , G) (Corollary 2.12). By Poincaré duality, the pairing h · , · i : H 0 (S 1 ; Ad ρ) × H 1 (S 1 ; Ad ρ) → H 1 (S 1 ; C) ∼=C is non degenerate. In the next lemma we use the refined torsion with sign due to Turaev (see Sec- tion 4.2). By (4.2) changing the orientation of S 1 changes the torsion TOR(S 1 , Ad ρ, o, u, v) by a factor (−1)d = (−1)r , as well as h∧v, ∧ui by the same factor. Let GR denote the compact real form of the semisimple complex linear group G. We will assume that the restriction of the nondegenerate symmetric bilinear G–invariant form B on the Lie algebra to gR is positive definite. This means that B is a negative multiple of the Killing form. In what follows we will denote by AdR : GR → Aut(gR ) the restriction of Ad to the real form GR . Lemma 4.5. — If ρ : π1 (S 1 ) → G is a regular representation, and if u and v are bases of H 0 (S 1 ; Ad ρ) and H 1 (S 1 ; Ad ρ) respectively, then the product TOR(S 1 , Ad ρ, o, u, v) h∧v, ∧ui is independent of u. Lemma 4.6. — If ρ : π1 (S 1 ) → GR is a regular representation and if u and v are bases of H 0 (S 1 ; AdR ρ), and H 1 (S 1 ; AdR ρ) respectively, then TOR(S 1 , AdR ρ, o, u, v) h∧v, ∧ui > 0 . Proof of Lemma 4.5. — Let u and u0 be bases for H 0 (S 1 ; Ad ρ), and v and v0 , for H 1 (S 1 ; Ad ρ). We change bases by means of the following formulas: [v0 : v] TOR(S 1 , Ad ρ, o, u0 , v0 ) = TOR(S 1 , Ad ρ, o, u, v) 0 [u : u] and h∧v0 , ∧u0 i = h∧v, ∧ui[v0 : v][u0 : u] . TOME 3 (2020) 358 M. HEUSENER & J. PORTI Hence (4.3) TOR(S 1 , Ad ρ, o, u0 , v0 )h∧v0 , ∧u0 i = TOR(S 1 , Ad ρ, o, u, v)h∧v, ∧ui[v0 : v]2 . This proves independence of u.  Proof of Lemma 4.6. — We are assuming that the image of ρ is contained in the compact real form, ρ(π1 (S 1 )) ⊂ GR . By (4.3) in the proof of Lemma 4.5, the sign is independent of v. By regularity, the image ρ(π1 (S 1 )) is contained in an unique maximal torus T , and H 0 (S 1 ; AdR ρ) ⊂ gR is the corresponding Cartan subalgebra h. Recall that B restricted to h is positive definite. Hence we may choose an R– basis of gR compatible with the orthogonal decomposition gR = h ⊕ h⊥ . This is also a decomposition of π1 (S 1 )–modules, and by Lemma 4.3 the torsion decomposes accordingly as a product of torsions. We compute the torsion of h first. Since the adjoint action of T on h is trivial, we have natural isomorphisms (4.4) H 1 (S 1 ; h) ∼ = H 1 (S 1 , R) ⊗ h and H 0 (S 1 ; h) ∼ = H 0 (S 1 , R) ⊗ h. We choose a cell decomposition of S 1 with a single (positively oriented) cell in each dimension. In particular, as the adjoint action of T on h is trivial, the boundary operator δ : C 0 (K; h) → C 1 (K; h) vanishes. Choose a B–orthonormal basis for h; this provides geometric basis c1 and c0 , and since δ = 0, those are also representatives of basis in cohomology. By choosing those bases (u = c0 and v = v1 ), tor(S 1 , Ad ρ|h , c0 , c1 , c0 , c1 ) = 1. Following the construction in Section 4.2, we compute α0 = β0 = r and α1 = β1 = 2r ≡ 0 mod 2. Thus N ≡ r2 ≡ r mod 2 and Tor(S 1 , Ad ρ|h , c0 , c1 , c0 , c1 ) = (−1)r . As the torsion for the trivial representation corresponds to the case r = 1, the torsion “Tor” for the trivial representation is −1 and (4.5) TOR(S 1 , Ad ρ|h , o, c1 , c0 ) = (−1)r · sgn(−1)r = 1. Also, by construction, h∧c1 , ∧c0 i = 1. Next we compute the torsion of h⊥ . We have H ∗ (S 1 ; h⊥ ) = 0 and, since dim h⊥ is even, TOR(S 1 , Ad ρ|h⊥ , o) = tor(S 1 , Ad ρ|h⊥ , c0 , c1 ) = det(AdR (g) − Id)|h⊥ , where g ∈ G is the image of a generator of π1 (S 1 ). Notice that, as dim h⊥ is even, the sign is independent of the cohomology orientation. Let ∆G be the Weyl function [GW09]. Then det(Ad(g) − Id)|h⊥ = ∆G (g)∆G (g −1 ) = |∆G (g)|2 > 0 (see [GW09, (7.47)] for details). This finishes the proof of the Lemma 4.6.  ANNALES HENRI LEBESGUE Volume forms on representation varieties 359 Definition 4.7. — Let ρ : π1 (S 1 ) → G be a regular representation. The form r H 1 (S 1 ; Ad ρ) → C ^ ν: is defined by the formula q (4.6) ν(∧u) = ± TOR(S 1 , Ad ρ, o, u, v) h∧v, ∧ui for any basis u of H 1 (S 1 ; Ad ρ). (By Lemma 4.5, it is independent of u.) We are interested in understanding ν as a differential form on Rreg (S 1 , G) for G simply connected. Recall from Section 2.3 that when G is simply connected, the Steinberg map has coordinates the fundamental characters (σ1 , . . . , σr ) : G → Cr . Proposition 4.8. — For G simply connected, there exists a constant C ∈ C∗ and a choice of sign for ν such that ν = C dσ1 ∧ · · · ∧ dσr . Proof. — Using Steinberg’s section s : Cr → Greg (Theorem 2.11), consider for each p ∈ Cr the subalgebra gAd s(p) of elements fixed by Ad s(p). By the constant rank theorem this defines an algebraic vector bundle gAd ◦s → E(s) → Cr . Since algebraic vector bundles over Cr are trivial [Qui76, Sus76], there is a trivial- ization u = (u1 , . . . , ur ) : Cr → E(s), so that {u1 (p), . . . , ur (p)} is a basis for gAd s(p) , for each p ∈ Cr . By the identifications, T[s(p)] Rreg (S 1 , G) ∼ = H 1 (S 1 , Ad s(p)) (Corol- lary 2.12), and the identification gAd s(p) ∼ = H 0 (S 1 , Ad s(p)), we have two (r, 0)–forms r on C : (4.7) hs∗ (−), ∧ui and TOR(S 1 , Ad s, o, u, s∗ (−)). We claim that these forms are both algebraic. Assuming the claim, they are a polynomial multiple of dz1 ∧ · · · ∧ dzr , for (z1 , . . . , zr ) the standard coordinate system for Cr . Since they vanish nowhere in Cr , both forms in (4.7) are a constant multiple of dz1 ∧ · · · ∧ dzr . Viewed as forms on Rreg (S 1 , G), they are both a constant multiple of dσ1 ∧ · · · ∧ dσr and the proposition follows, once we have shown the claim. To prove that the forms in (4.7) are algebraic, use a CW-decomposition K of S 1 with a 1 and a 0-cell, so that the groups of cochains C i (K, Ad s(p)), for i = 0, 1, are naturally identified with g. We also have a natural isomorphism (Rs(p)−1 )∗ : Ts(p) G → g, which is precisely the tangent map of the right multiplication by s(p)−1 . This identification maps s∗ (∂zi ) at p ∈ Cn to !   ∂s vi (p) = Rs(p)−1 (p) ∈ g, ∗ ∂zi which is a map algebraic on p ∈ Cr . Hence the intersection product is hs∗ (∂z1 ∧ · · · ∧ ∂zr ), ∧ui = det(hs∗ (∂zi ), uj iij ) = det(B(vi , uj )ij ), which is polynomial on p ∈ Cr . To show that the torsion is algebraic, using again triviality of algebraic bundles on Cr , complete u to a section of the trivial bundle (u1 , . . . , ur , . . . , ud ) : Cr → g. TOME 3 (2020) 360 M. HEUSENER & J. PORTI e 1 = {u e1 Setting b r+1 , . . . , ud }, then u(p) t b (p) is a basis for g, for each p ∈ C. We view u(p) t be 1 (p) as a basis for C 0 (K, Ad s(p)), so that u(p) projects to a basis for H (S , Ad s(p)), for every p ∈ Cr . Fix c0 = c1 a basis for g. By construction: 0 1 [v t ∂ be 1 : c1 ] TOR(S 1 , Ad s, o, u, s∗ (∂z1 ∧ · · · ∧ ∂zr )) = ± , [u t be 1 : c0 ] where the sign depends on the orientation in homology, but it is constant on p. Thus this is a quotient of algebraic polynomial functions on Cr , but since it is defined everywhere, it is polynomial.  4.4. Witten’s formula Let ρ : π1 (S) → G be a good ∂-regular representation. Let νi denote the peripheral form of the i−th component of ∂S (Definition 4.7), and let ω denote the symplectic form of the relative character variety (4.1). We aim to prove Theorem 1.3, namely, that Ωπ1 (S) = ± n!1 ω n ∧ ν1 ∧ · · · ∧ νb . Proof of Theorem 1.3. — We apply the duality formula (Proposition 3.3) and the formula of the torsion for the long exact sequence of the pair, Equation (4.9) below. For this purpose we discuss the bases in cohomology. Start with u a basis for H 0 (∂S; Ad ρ). If β denotes the connecting map of the long exact sequence, then complete β(u) to a basis for H 1 (S, ∂S; Ad ρ): β(u) t h. e Next we choose v a basis for H 1 (∂S; Ad ρ) that we lift to v e = h, then h t v e by i, and if we set j(h) e is a basis 1 for H (S; Ad ρ) (and h is a basis for ker(i) = Im(j)). The bases are organized as follows: β j i (4.8) 0 → H 0 (∂S; Ad ρ) → H 1 (S, ∂S; Ad ρ) → H 1 (S; Ad ρ) → H 1 (∂S; Ad ρ) → 0 u hte v v h β(u)te As the bases have been chosen compatible with the maps of the long exact sequence, the product formula for the torsion [Mil66] gives: (4.9) tor(S, Ad ρ, h t v e ) = ± tor(S, ∂S, Ad ρ, β(u) t h) e tor(∂S, Ad ρ, u, v). We shall combine (4.9) with the duality formula (Proposition 3.3): (4.10) tor(S, Ad ρ, h t v e ) tor(S, ∂S, Ad ρ, β(u) t h) e = ± h∧(h t v e ), ∧(β(u) t h)i e . We next decompose the right hand side in (4.10). By naturality of the intersection form, hhi , β(uj )i = hi(hi ), uj i = hi(j(h f)), u i = 0 . i j Hence the right hand side in (4.10) becomes: h∧(h t v e ), ∧(β(u) t h)i e = h∧h, ∧hi e · h∧v e , ∧β(u)i. Again by naturality h∧v e , ∧β(u)i = h∧i(v e ), ∧ui = h∧v, ∧ui. In addition, by definition h∧h, ∧hi e = ω(∧h, ∧h). ANNALES HENRI LEBESGUE Volume forms on representation varieties 361 Thus (4.11) h∧(h t v e ), ∧(β(u) t h)i e = ±h∧u, ∧vi ω(∧h, ∧h). Hence by (4.9), (4.10), and (4.11): e )2 = ±ω(∧h, ∧h) TOR(∂S, Ad ρ, u, v)hu, vi . tor(S, Ad ρ, h t v Notice that on the right hand side we use Turaev’s sign refined torsion. Next we claim that the sign of this formula is + and not −. It suffices to determine the sign in the compact case. Then the formula will follow in the complex case by a connectedness argument (the variety of characters of a free group is connected and irreducible, and ∂–regularity and being good are Zariski open properties, hence they fail in a set of real codimension > 2). We show that the sign is + in the compact case by showing that all terms are positive. Since TOR(∂S, Ad ρ, u, v)hu, vi is positive by Lemma 4.6, the sign will follow from the equality 2 1 n   (4.12) ω(∧h, ∧h) = ω (∧h) , n! that will also complete the proof of the Theorem 1.3. We give a self-contained proof of (4.12) for completeness. By Darboux’s Theorem there are local coordinates so that ω = dx1 ∧ dx2 + · · · + dx2n−1 ∧ dx2n . Let A be a matrix of size 2n×2n whose colons are the components of the vectors of h in this coordinate system. Then, if J denotes the matrix of the standard symplectic form, ω(∧h, ∧h) = det (ω(hi , hj )ij ) = det(At JA) = (det A)2 . On the other hand ω n = n! dx1 ∧ dx2 ∧ · · · ∧ dx2n , hence 1 n ω (∧h) = det A n! and we are done.  Remark 4.9. — Theorem 1.3 is a version for complex groups of [BL99, Theo- rem 5.40], which holds for compact groups. In particular, in [BL99, Definition 5.38] peripheral elements are required to be semisimple, through we only assume that the peripheral elements are regular (but perhaps not semisimple). By taking real forms, we recover [BL99, Theorem 5.40] for compact Lie groups, as we illustrate for SU(N ) (see Proposition 5.4), but also a formula for other real forms, like SLN (R) (see Remark 5.5). 5. Formulas for the group SLN (C) If G = SLN (C) we can give explicit formulas for several volume forms. TOME 3 (2020) 362 M. HEUSENER & J. PORTI 5.1. The form ν for SLN (C) We know that ν is a constant multiple of dσ1 ∧ · · · ∧ dσr and we shall determine the constant, completing the proof of Proposition 1.2. Recall that we have fixed the C–bilinear form on slN (C) to be B(X, Y ) = − tr(X Y ) ∀ X, Y ∈ SLN (C). In SLN (C) the invariant functions are the symmetric functions on the spectrum: if the eigenvalues of A ∈ SLN (C) are λ1 , . . . , λN , then X X X 1 σ1 (A) = λi , σ2 (A) = λi λj , . . . , σN −1 (A) = . i ij which is the product ∆G (g)∆G (g −1 ) of Weyl functions [GW09, Section 7]. Thus √ N (N −1)/2 Y (5.1) ν = ± −1 (eui − euj ) θH , i>j where q (5.2) θH (∧v) = TOR(S 1 , h, o, ∧v, ∧u)h∧v, ∧ui. We use coordinates for the Cartan algebra via the entries of the diagonal matrices: ∼ h = {(u1 , . . . , uN ) ∈ CN | ui = 0}, P Lemma 5.3. — The form θH is the restriction to {(u1 , . . . , uN ) ∈ CN | P ui = 0} of the form √ (N −1) 1 X N ± −1 √ (−1)N −i du1 ∧ · · · ∧ du d ∧ · · · ∧ du , i N N i=1 or, equivalently, of √ (N −1) √ ± −1 N du1 ∧ · · · ∧ duN −1 . Proof. — In order to compute TOR(S 1 , h, o, ∧v, ∧u) we proceed as in the proof of Lemma 4.6. In particular we choose a cell decomposition of S 1 with a single (positively oriented) cell in each dimension, and bases in homology represented by the geometric bases. With this choice of u and v, by (4.5), TOR(S 1 , h, o, v, u) = 1. Next we compute h∧v, ∧ui. The bases u and v are constructed from dual basis in H ∗ (S 1 ; Z) tensorized by a basis of h. We choose a basis for the Cartan subalgebra, e = {e1 , . . . , eN −1 }:       1 0 0  0   1   ..  e1 =   , e2 =   , . . . , eN −1 =  .  .  ...  ...   1             −1 −1 −1 Since the cells of S 1 are positively oriented, h∧v, ∧ui = det(B(ei , ej ))i,j ). In addition, as B(ei , ei ) = −2 and B(ei , ej ) = −1 for i = 6 j, det(B(ei , ej ))i,j ) = (−1)N −1 N . Thus q (5.3) θH (∧v) = ± (−1)N −1 N On the other hand, direct computation yields: N (−1)N −i du1 ∧ · · · ∧ du X d ∧ · · · ∧ du (e ∧ · · · ∧ e i N 1 N −1 ) = N . i=1 TOME 3 (2020) 364 M. HEUSENER & J. PORTI By the natural identification of H 1 (S 1 ; h) with the Cartan algebra h we get the Lemma 5.3.  We conclude the proof of Proposition 5.1. By (5.1) and Lemma 5.3, √ (N ) √ Y (5.4) ν ∧ (du1 + · · · + duN ) = −1 N (eui − euj ) du1 ∧ · · · ∧ duN . i>j Next we use Newton’s identities: σ1 = eu1 + · · · + euN σ2 = − 21 (e2u1 + · · · + e2uN − Pol(σ1 )) .. . σj = (−1)j+1 1j (eju1 + · · · + ejuN − Pol(σ1 , . . . , σj−1 )) were Pol(σ1 , . . . , σj−1 ) denotes a polynomial expression on σ1 , . . . , σj−1 , whose precise value is not relevant here. From them we deduce 1 d( euj ) ∧ d( e2uj ) ∧ · · · ∧ d( e(N −1)uj ) X X X dσ1 ∧ · · · ∧ dσN −1 = ± (N − 1)! Since eiuj ) = i eiuj duj , X X d( for i = 1, . . . , m − 1, Vandermonde determinant yields (eui − euj )du1 ∧ · · · ∧ duN . Y (5.5) (du1 + · · · + duN ) ∧ dσ1 ∧ · · · ∧ dσN −1 = ± i>j Then combine (5.4) and (5.5) to prove the theorem, knowing that our tangent space is the kernel of du1 + · · · + duN .  5.2. The form ν for SU(N ) An element in SU(N ) is conjugate to a diagonal element   eiθ1   ...     eiθN with θi ∈ 2πZ. A matrix is regular if and only if eiθj 6= eiθk for j 6= k. P By identifying Rreg (S 1 , SU(N )) with the image of the generator (or its conjugacy class), functions on θ1 , . . . , θN invariant under permutations are well-defined on Rreg (S 1 ,P SU(N )). Also the form dθ1 ∧ · · · ∧ dθN −1 is well-defined up to sign by the relation θi ∈ 2πZ. Proposition 5.4. — On Rreg (S 1 , SU(N )) (for B(X, Y ) = − tr(X Y )), √ Y ! N (N −1)/2 θi − θj ν = ±2 N sin dθ1 ∧ · · · ∧ dθN −1 . i 2. Therefore dim(Eλ ∩ B(Eλ )) > 1. Choose a nonzero vector v ∈ Eλ ∩ B(Eλ ), by construction B −1 (v) ∈ Eλ and (ABA−1 B −1 )(v) = v. This yields that 1 is an eigenvalue of the commutator ABA−1 B −1 , therefore it has the same eigenvalues as its inverse, which implies that tr(ABA−1 B −1 ) = tr(BAB −1 A−1 ). TOME 3 (2020) 374 M. HEUSENER & J. PORTI (ii) By contradiction, assume that L ⊂ C3 is a proper subspace invariant by both A and B. If dim L = 1, then this is and eigenspace of ABA−1 B −1 with eigen- value 1, and if dim L = 2, by looking at the action on C3 /L we also deduce that 1 is an eigenvalue of ABA−1 B −1 . Therefore, by the discussion on the previous item, this contradicts the hypothesis tr(ABA−1 B −1 ) 6= tr(BAB −1 A−1 ).  Lemma 7.3. — Let A ∈ SL3 (C). If A is regular then the AdA -invariant subspace of sl3 (C) is −1 sl3 (C)AdA = hA − tr(A) 3 Id, A−1 − tr(A3 ) Idi. −1 Proof. — It is clear from construction that both A− tr(A) 3 Id and A−1 − tr(A3 ) Id are AdA -invariant. All we need to show is that those elements are linearly independent, −1 as by regularity dim sl3 (C)AdA = 2. If A − tr(A) 3 Id and A−1 − tr(A3 ) Id were linearly dependent, then A, Id, and A−1 would satisfy a nontrivial linear relation. Multiplying it by A, the same relation would be satisfied by A2 , A and Id, and hence A would have an eigenspace of dimension at least 2, contradicting regularity.  Remark 7.4. — It follows from Schur’s Lemma [FH91] that every irreducible representation ρ : Γ → SLN (C) is good, that is the centralizer of ρ(Γ) coincides with the center of SLN (C). Proof of Proposition 1.6. — Assume k = 3, the general case follows from an induction argument as in the proof of Theorem 1.5. We choose generators F3 = hγ1 , γ2 , γ3 |−i and we identify F3 with π1 (S0,4 ). We represent S0,4 as the union of two pairs of pants P 0 and P 00 , so that P 0 ∩ P 00 is a circle. Choose the generators of the fundamental group so that π1 (P 0 ) = hγ1 , γ2 i, π1 (P 00 ) = hγ1 , γ3 i, and γ1 is the generator of π1 (P 0 ∩ P 00 ). Then the peripheral elements of P 0 are γ1 , γ2 , and γ1 γ2 , and those of P 00 , γ1 , γ3 , and γ1 γ3 . The peripheral elements of S are γ2 , γ3 , γ1 γ2 , and γ1 γ3 . Let [ρ] ∈ R∗ (F3 , SL3 (C)) \ {τ121̄2̄ = τ212̄1̄ } ∪ {τ131̄3̄ = τ313̄1̄ } ∪ {∆123 = 0} be a representation where (7.14) ∆123 = (τ123 − τ132 )(τ1̄2̄3̄ − τ1̄3̄2̄ ) − (τ12̄3̄ − τ13̄2̄ )(τ1̄23 − τ1̄32 ). We have to show that the 16 functions (τ1 , τ1̄ , τ2 , τ2̄ , τ3 , τ3̄ , τ12 , τ1̄2̄ , τ13 , τ1̄3̄ , τ23 , τ2̄3̄ , τ12̄ , τ1̄2 , τ13̄ , τ1̄3 ) define a local parameterization at [ρ]. The hypothesis tr(ρ([γ1 , γi ])]) 6= tr(ρ([γi , γ1 ])) for i = 2, 3 implies that ρ(γj ), j = 1, 2, 3, are regular elements (Lemma 7.2). It follows also that ρ(γ1 γ2 ) and ρ(γ1 γ3 ) are regular since tr(ρ([γ1 γi , γ1 ])) = tr(ρ([γi , γ1 ])) and tr(ρ([γ1 , γ1 γi ])) = tr(ρ([γ1 , γi ])) for i = 2, 3. The Mayer–Vietoris long exact sequence is: β j (7.15) 0 → H 0 (γ1 , Ad ρ) → − H 1 (P 0 , Ad ρ) ⊕ H 1 (P 00 , Ad ρ) − H 1 (S, Ad ρ) → ∆ → H 1 (γ1 , Ad ρ) → 0 . − Choose u a basis for H 0 (γ1 , Ad ρ). We will proceed as in the proof of Proposition 6.5. Viewing the cohomology groups as tangent spaces, the proposition will follow from ANNALES HENRI LEBESGUE Volume forms on representation varieties 375 the local parameterizations for the representation space of P 0 and P 00 , and from (7.15), provided we show that dτ23 ∧ dτ2̄3̄ (∧β(u)) 6= 0. We prove below in Lemma 7.5 that dτ23 ∧ dτ2̄3̄ (∧β(u)) = ±∆123 , which is nonzero by hypothesis.  Lemma 7.5. — dτ23 ∧ dτ2̄3̄ (β(∧u)) = ±∆123 , where ∆123 is as in (7.14). Proof. — Set A1 = ρ(γ1 ). By Lemma 7.3, the elements tr A1 tr A−1 1 x = A1 − Id and y = A−1 1 − Id 3 3 form a basis of the invariant subspace sl3 (C)Ad ρ(γ1 ) . We choose u = {x, y} via the isomorphism H 0 (γ1 , Ad ρ) ∼ = sl3 (C)AdA1 . Then β(x) is the tangent vector of the infinitesimal bending: γ1 7→ A, γ2 7→ B, γ3 7→ (Id +εx)C(Id −εx) = C + ε(xC − Cx) in C[ε]/ε2 and similarly for β(y). To compute dτ23 and dτ2̄3̄ on β(x) and β(y), we must evaluate the infinitesimal deformations on γ2 γ3 and γ̄2 γ̄3 . Thus the path corresponding to β(x) evaluated at γ2 γ3 is (7.16) γ2 γ3 7→ BC + ε(BxC − BCx) = BC + ε(BAC − BCA). Therefore, taking traces we get: (7.17) dτ23 (β(x)) = τ213 − τ231 = τ132 − τ123 . The same argument for y instead of x gives: (7.18) dτ23 (β(y)) = τ21̄3 − τ231̄ = τ1̄32 − τ1̄23 . To evaluate dτ2̄3̄ = dτ3̄2̄ , we take inverses in (7.16) (7.19) (γ2 γ3 )−1 7→ C −1 B −1 +ε(xC −1 B −1 −C −1 xB −1 ) = C −1 B −1 +ε(AC −1 B −1 −C −1 AB −1 ) and taking traces we get: (7.20) dτ2̄3̄ (β(x)) = τ13̄2̄ − τ3̄12̄ = τ13̄2̄ − τ12̄3̄ . Again the same argument for y instead of x gives: (7.21) dτ2̄3̄ (β(y)) = τ1̄3̄2̄ − τ3̄1̄2̄ = τ1̄3̄2̄ − τ1̄2̄3̄ . Hence dτ23 ∧ dτ2̄3̄ (β(x) ∧ β(y)) = ±(τ123 − τ132 )(τ1̄2̄3̄ − τ1̄3̄2̄ ) − (τ12̄3̄ − τ13̄2̄ )(τ1̄23 − τ1̄32 ) = ±∆123 , which concludes the proof of the Lemma 7.5.  Proof of Theorem 1.7. — We assume again that k = 3. The general case follows with the same argument as in Theorem 1.5. As in the proof of Proposition 1.6 we decompose S = S0,4 = P 0 ∪ P 00 , γ1 = P 0 ∩ P 00 . Also, we choose generators of π1 (P 0 ), π1 (P 00 ), and π1 (P 0 ∩ P 00 ) as in the proof of Proposition 1.6. The peripheral elements of S are γ2 , γ3 , γ1 γ2 , and γ1 γ3 . TOME 3 (2020) 376 M. HEUSENER & J. PORTI For a representation ρ : π1 (S) → SL3 (C) we let ρ0 : π1 (P 0 ) → SL3 (C) and ρ : π1 (P 00 ) → SL3 (C) denote the restriction of ρ to π1 (P 0 ) and π1 (P 00 ) respectively. 00 Let [ρ] ∈ R(S, SL3 (C)) \ {τ121̄2̄ = τ212̄1̄ } ∪ {τ131̄3̄ = τ313̄1̄ } ∪ {∆123 = 0}. It follows from Lemma 7.2 and Remark 7.4 that ρ0 and ρ00 are good, ∂–regular representations. In what follows we let ω12 and ω13 denote the pullback of the symplectic form ωP 0 on R∗ (P 0 , ∂P 0 , SL3 (C))ρ0 and ωP 00 on R∗ (P 00 , ∂P 00 , SL3 (C))ρ00 respectively. Given a basis v for H 1 (γ1 , Ad ρ) we can choose lifts v0 ⊂ H 1 (P 0 , Ad ρ), and v00 ⊂ H 1 (P 00 , Ad ρ) which map to v. By exactness there exists v e ⊂ H 1 (S, Ad ρ) which maps to (v0 , −v00 ). Lemma 7.6. — Let v a basis for H 0 (γ1 , Ad ρ) and v a basis for H 1 (γ1 , Ad ρ). Then h∧u, ∧vi (7.22) ΩS = ± ω12 ∧ ω13 ∧ ν2 ∧ ν3 ∧ ν12 ∧ ν13 ∧ ν1 ∧ ν23 . (ν1 ∧ ν23 )(∧v ∧ β(u)) Proof of Lemma 7.6. — Choose a0 a basis of ker(H 1 (P 0 , Ad ρ) → H 1 (P 0 ∩P 00 , Ad ρ)) and a00 a basis of ker(H 1 (P 00 , Ad ρ) → H 1 (P 0 ∩ P 00 , Ad ρ)). Moreover, we can choose lifts (a g 0 ), (a g 00 ) ⊂ H 1 (S, Ad ρ) which map under j : H 1 (S, Ad ρ) → H 1 (P 0 , Ad ρ) ⊕ H 1 (P 00 , Ad ρ) to (a0 , 0) and (0, a00 ) respectively. Then, by using (7.15), a0 t v0 is a basis for H 1 (P 0 , Ad ρ), a00 t v00 is a basis for H 1 (P 00 , Ad ρ) and (a g 0 ) t (a g e is a basis for H 1 (S, Ad ρ). 00 ) t β(u) t v The product formula applied to (7.15) yields: 0 ) ∧ (a 00 ) ∧ β(u) ∧ v ΩP 0 (∧a0 ∧ v0 ) ΩP 00 (∧a00 ∧ v00 ) ΩS (∧(a g g e) = ± tor(P 0 ∩ P 00 , Ad ρ, u, v) ν1 (∧v)2 = ±(ω12 ∧ ν2 ∧ ν12 )(∧a0 )(ω13 ∧ ν3 ∧ ν13 )(∧a00 ) . tor(P 0 ∩ P 00 , Ad ρ, u, v) The last equality follows since dτ2 , dτ2̄ , dτ12 , dτ1̄2̄ , dτ12̄ , dτ1̄2 vanish on each cocycle vi0 of v0 = (v10 , v20 ), and dτ3 , dτ3̄ , dτ13 , dτ1̄3̄ , dτ13̄ , dτ1̄3 vanish on each cocycle vi00 of v00 = (v100 , v200 ). By Definition 4.7, ν1 (v)2 /tor(P 0 ∩ P 00 , Ad ρ, u, v) = ±h∧u, ∧vi, hence ΩS (∧(a g 0 ) ∧ (a g 00 ) ∧ β(u) ∧ v e) = ±(ω12 ∧ ν2 ∧ ν12 )(∧a0 )(ω13 ∧ ν3 ∧ ν13 )(∧a00 )h∧u, ∧vi (ν1 ∧ ν23 )(∧v ∧ β(u)) = ±(ω12 ∧ ν2 ∧ ν12 )(∧a0 )(ω13 ∧ ν3 ∧ ν13 )(∧a00 )h∧u, ∧vi (ν1 ∧ ν23 )(∧v ∧ β(u)) h∧u, ∧vi =± ω12 ∧ ω13 ∧ ν2 ∧ ν3 ∧ ν12 ∧ ν13 ∧ ν1 ∧ ν23 (ν1 ∧ ν23 )(∧v ∧ β(u))   ∧(a g 0 ) ∧ (a g 00 ) ∧ β(u) ∧ v e . The last equality follows since β(u) is an infinitesimal bending that vanish on ν1 , and β(u) is in the kernel of j (see (7.15)). Moreover, the bases (a g 0 ) and (a g 00 ) map to (a0 , 0) and (0, a00 ) respectively.  ANNALES HENRI LEBESGUE Volume forms on representation varieties 377 To conclude the proof of Theorem 1.7 we need to compute the quotient h∧u, ∧vi (ν1 ∧ ν23 )(∧v ∧ β(u)) As β(u) consist of infinitesimal bendings that vanish on dτ1 and dτ1̄ , (ν1 ∧ ν23 )(∧v ∧ β(u)) = ν1 (∧v)ν23 (∧β(u)). Write A = ρ(γ1 ), B = ρ(γ2 ), and C = ρ(γ3 ), and tr(A) tr(A−1 ) x=A− Id and y = A−1 − Id . 3 3 Hence x, y ∈ sl3 (C) generate the A–invariant subspace by Lemma 7.3. By the natural identification H 0 (γ1 , Ad ρ) ∼ = sl3 (C)AdA , we choose u = {x, y}. To finish the proof of Theorem 1.7, we assume semi-simplicity, so that H 1(γ1 ,Adρ) ∼ = H 1 (γ1 , R) ⊗R sl3 (C)AdA and we may choose v to be {x, y} times the fundamental class. Therefore ! tr(x2 ) tr(xy) (7.23) h∧u, ∧vi = det . tr(xy) tr(y 2 ) Next we compute ν(∧v). Write v = {vx , vy }, where vx and vy are the infinitesimal deformations corresponding to x and y respectively. Namely, the tangent vector of the infinitesimal paths (7.24) γ1 7→ (Id +εx)A = A + εx A and γ1 7→ (Id +εy)A = A + εy A in C[ε]/ε2 . These infinitesimal deformations evaluated at γ1−1 are, respectively, (7.25) γ1−1 7→ A−1 (Id −εx) = A−1 − εA−1 x and γ1−1 7→ A−1 (Id −εy) = A−1 − εA−1 y in C[ε]/ε2 ]. Thus, dτ1 (vx ) = tr(x A), and as tr(x) = 0, tr(x A) = tr(x A − τ31 x) = tr(x x). By the very same argument, tr(y A) = tr(A−1 x) = tr(x y) and tr(A−1 y) = tr(y 2 ), and (7.24) and (7.25) yield dτ1 (vx ) = tr(x2 ) , dτ1 (vy ) = tr(x y) , (7.26) dτ1̄ (vx ) = − tr(x y) , dτ1̄ (vy ) = − tr(y 2 ). From (7.23) and (7.26) we have (7.27) dτ1 ∧ dτ1̄ (∧v) = ±h∧u, ∧vi. In addition, by Lemma 7.5 √ √ (7.28) ν23 (β(u)) = −3 dτ23 ∧ dτ2̄3̄ (β(u)) = ± −3∆23 . √ Hence, as ν1 = −3 dτ1 ∧ dτ1̄ , by (7.27) and (7.28): h∧u, ∧vi h∧u, ∧vi 1 = =± 1 . (ν1 ∧ ν23 )(∧v ∧ β(u)) ν1 (∧v)ν23 (∧β(u)) 3∆23 TOME 3 (2020) 378 M. HEUSENER & J. PORTI Now the volume formula follows from Lemma 7.6, the last equation, and the expres- sion of the symplectic forms ω12 and ω13 in (5.8).  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Phys. 141 (1991), no. 1, 153–209. ↑342, 344, 354 Manuscript received on 18th April 2018, revised on 2nd May 2019, accepted on 21st May 2019. Recommended by Editor A. Wienhard. Published under license CC BY 4.0. This journal is a member of Centre Mersenne. Michael HEUSENER Université Clermont Auvergne, CNRS, Laboratoire de Mathématiques Blaise Pascal, 63000 Clermont-Ferrand (France) michael.heusener@uca.fr Joan PORTI Departament de Matemàtiques, Universitat Autònoma de Barcelona, 08193 Cerdanyola del Vallès (Spain) and Barcelona Graduate School of Mathematics (BGSMath) porti@mat.uab.cat ANNALES HENRI LEBESGUE