Decomposable representations and Lagrangian submanifolds of moduli spaces associated to surface groups

Abstract

Abstract. The importance of explicit examples of Lagrangian submanifolds of moduli spaces is revealed by papers such as [9,25]: given a 3-manifold M with boundary ∂M = Σ, Dostoglou and Salamon use such examples to obtain a proof of the Atiyah-Floer conjecture relating the symplectic Floer homology of the representation space Hom(π 1 (Σ = ∂M ), U )/U (associated to an explicit pair of Lagrangian submanifolds of this representation space) and the instanton homology of the 3-manifold M . In the present paper, we construct a Lagrangian submanifold of the space of representations M g,l := Hom C (π g,l , U )/U of the fundamental group π g,l of a punctured Riemann surface Σ g,l into an arbitrary compact connected Lie group U . This Lagrangian submanifold is obtained as the fixed-point set of an anti-symplectic involutionβ defined on M g,l . We show that the involutionβ is induced by a form-reversing involution β defined on the quasi-Hamiltonian space (The fact thatβ has a non-empty fixed-point set is a consequence of the real convexity theorem for group-valued momentum maps proved in [28]. The notion of decomposable representation provides a geometric interpretation of the Lagrangian submanifold thus obtained.

Traces, Cross-Ratios and 2-Generator Subgroups of SU(2, 1)

“…Following remark 4, the isometries A and B admit the lifts to SU(2,1) a=⇒b If the lifts A and B have real trace, then A and B have real eigenvalues, as seen in the proof of proposition 4. Thus, taking the imaginary part in the two relations (40) and (41) yields: The same kind of results has been obtained in a different frame by Schaffhauser in [26] and [27]. If A and B are two loxodromic isometries such that [A, B] is unipotent, lifts of A and B may be chosen such that their commutator has trace 3.…”

“…It is a direct consequence of the relation (26) that for any σ ∈ V 4 , Ω σ = f σ (Ω). The proof of the following proposition is done by repeated use of relation (26).…”

Representations of the Fundamental Group of an L–Punctured Sphere Generated by Products of Lagrangian Involutions

“…Remark (Addendum -26.07.06). As a matter of fact, corollary 6.11 does hold without any assumption on the conjugacy classes C j and a proof of this is availablle in [Sch05].…”

“…All one has to do is then define β as in (1) in subsection 6.5 (that is, replace u t by τ (u −1 ) in the definition of β given in subsection 6.6) : the σ 0 -decomposable representations are exactly the elements of the fixed-point set of β, a given representation u is decomposable if and only if β(u) is equivalent to u, and the set of equivalence classes of decomposable representations is a Lagrangian submanifold of Hom C (π, U )/U , obtained as the fixed-point set of an antisymplectic involutionβ. We refer to [Sch05] for further details in that direction.…”

“…The fact that there always exist Lagrangian representations was first proved in [FW06], where the dimension of the submanifold of (equivalence classes of) Lagrangian representations was shown to be half the dimension of the moduli space. For a proof of the non-emptiness using ideas from (quasi-) Hamiltonian geometry, we refer to [Sch05] or to the forthcoming paper [Sch] (see also theorem 5.3). For now, we will use the quasi-Hamiltonian description of the symplectic structure of the moduli space to prove theorem 2.…”