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

Published online by Cambridge University Press:  20 November 2018

Florent Schaffhauser
Florent Schaffhauser*
Affiliation:
Institut de Mathématiques, Université Pierre et Marie Curie-Paris 6, 4, place Jussieu F-75252 Paris Cedex 05 email: [email protected]
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Abstract

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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 $M//U:=\,{{\mu }^{-1}}\,(\{1\})/U$.

Keywords

53D2053D30momentum mapsmoduli spacesLagrangian submanifoldsanti-symplectic involutionsquasi-Hamiltonian
Type
Research Article
Information
Canadian Journal of Mathematics , Volume 59 , Issue 4 , 01 August 2007 , pp. 845 - 879
Copyright
Copyright © Canadian Mathematical Society 2007

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