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

Schaffhauser, Florent

Canadian journal of mathematics, Tome 59 (2007) no. 4, pp. 845-879

Résumé

In this paper, we characterize unitary representations of $\pi \text{ }:=\text{ }\pi {{\text{ }}_{1}}({{S}^{2}}\backslash \{{{s}_{1}},\ldots ,{{s}_{l}}\})$ whose generators ${{u}_{1}},\,\ldots ,\,{{u}_{l}}$ (lying in conjugacy classes fixed initially) can be decomposed as products of two Lagrangian involutions ${{u}_{j}}\,=\,{{\sigma }_{j}}{{\sigma }_{j+1}}$ with ${{\sigma }_{l+1}}\,=\,{{\sigma }_{1}}$ . 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 ${{\mathcal{M}}_{C}}\,:\,=\,\text{Ho}{{\text{m}}_{C}}(\pi ,\,U(n))\,/\,U(n)$ . Consequently, as this fixed-point set is non-empty, it is a Lagrangian submanifold of ${{\mathcal{M}}_{C}}$ . To prove this, we use the quasi-Hamiltonian description of the symplectic structure of ${{\mathcal{M}}_{C}}$ and give conditions on an involution defined on a quasi-Hamiltonian $U$ -space $(M,\,\omega ,\,\mu :\,M\to \,U)$ for it to induce an anti-symplectic involution on the reduced space $M//U:=\,{{\mu }^{-1}}\,(\{1\})/U$ .

Détail

DOI : 10.4153/CJM-2007-036-9

Mots-clés : 53D20, 53D30, momentum maps, moduli spaces, Lagrangian submanifolds, anti-symplectic involutions, quasi-Hamiltonian

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