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

Abstract

Abstract. In this paper, we characterize unitary representations of π := π 1 (S 2 \{s 1 , . . . , s l }) whose generators u 1 , . . . , u l (lying in conjugacy classes fixed initially) can be decomposed as products of two Lagrangian involutions u j = σ j σ j+1 with σ l+1 = σ 1 . Our main result is that such representations are exactly the elements of the fixed-point set of an anti-symplectic involution defined on the moduli space M C := Hom C (π, U (n))/U (n). Consequently, as this fixed-point set is non-empty, it is a Lagrangian submanifold of M C . To prove this, we use the quasi-Hamiltonian description of the symplectic structure of M C and give conditions on an involution defined on a quasi-Hamiltonian U -space (M, ω, µ : M → U ) for it to induce an anti-symplectic involution on the reduced space M//U := µ −1 ({1})/U .

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

“…For further details on the structure of a quasi-Hamiltonian quotient M//U = µ −1 ({1})/U , we refer to [27]. Some of the results we are about to state also come from [29].…”

“…In the latter case, this notion has a simple geometric origin and was first introduced by Falbel and Wentworth in their study of representations of the fundamental group π g,0 of a punctured sphere into the unitary group U (n) (see [11]). Their approach (for the group π g,0 ) was generalized to arbitrary compact connected Lie groups (U, τ ) endowed with an involutive automorphism in [29]. After reviewing this in subsection 2.1, we will introduce the notion of decomposable representation for surface groups π g,l with g ≥ 1 in subsection 2.2.…”

“…As we shall see later, the key object in this paper is the involution β on (U × U ) g × C 1 × · · · × C l defined in definition 4.1, since it is this involution that will induce an anti-symplectic involutionβ on M g,l , whose fixed-point set F ix(β) will provide the promised Lagrangian submanifold. And this involution β was obtained by trying to generalize the expression found in the g = 0 case, for which we refer to [29]. But since in this g = 0 case, the involution β was obtained from the notion of decomposable representation, it seemed natural to work out a notion of decomposable representation in the g ≥ 1 case, that will later provide a nice geometric interpretation of the Lagrangian submanifold F ix(β) ⊂ M g,l .…”

“…As in section 2, the Lie group U is endowed with an involutive automorphism τ of maximal rank (see definition 2.1). The idea of characterizing decomposable representations in terms of an involution comes from the study of the special case U = U (n), as explained in detail in [29], a case where one can develop a more geometric intuition of the problem using Lagrangian subspaces of C n instead of symmetric elements of U (n). The case of the punctured sphere was treated in [29].…”

“…For details on the geometric origin of this definition, we refer to [29]: when U = U (n), the decomposition u j = w j w −1 j+1 is equivalent to saying that u j is a product u j = σ j σ j+1 of orthogonal reflections with respect to Lagrangian subspaces of C n (hence the name Lagrangian representation in [11]). Recall that…”