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# Representations of the Fundamental Group of an $L$-Punctured Sphere Generated by Products of Lagrangian Involutions

Published:2007-08-01
Printed: Aug 2007
• Florent Schaffhauser
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## Abstract

In this paper, we characterize unitary representations of $\pi:=\piS$ whose generators $u_1, \dots, u_l$ (lying in conjugacy classes fixed initially) can be decomposed as products of two Lagrangian involutions $u_j=\s_j\s_{j+1}$ with $\s_{l+1}=\s_1$. 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 $\Mod:=\Hom_{\mathcal C}(\pi,U(n))/U(n)$. Consequently, as this fixed-point set is non-empty, it is a Lagrangian submanifold of $\Mod$. To prove this, we use the quasi-Hamiltonian description of the symplectic structure of $\Mod$ and give conditions on an involution defined on a quasi-Hamiltonian $U$-space $(M, \w, \mu\from M \to U)$ for it to induce an anti-symplectic involution on the reduced space $M/\!/U := \mu^{-1}(\{1\})/U$.
 Keywords: momentum maps, moduli spaces, Lagrangian submanifolds, anti-symplectic involutions, quasi-Hamiltonian
 MSC Classifications: 53D20 - Momentum maps; symplectic reduction 53D30 - Symplectic structures of moduli spaces