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Representations of the Fundamental Group of an
$L$–Punctured Sphere Generated by Products of Lagrangian Involutions
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- Journal:
- Canadian Journal of Mathematics / Volume 59 / Issue 4 / 01 August 2007
- Published online by Cambridge University Press:
- 20 November 2018, pp. 845-879
- Print publication:
- 01 August 2007
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- Article
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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 
$M//U:=\,{{\mu }^{-1}}\,(\{1\})/U$.
