Archiv der Mathematik

, Volume 73, Issue 1, pp 50–55 | Cite as

A note on the complete integrability of the Hitchin system

  • Renata Scognamillo


We consider the Hitchin map \( {\cal H}\!: T^{*}{\cal M}_{G}\longrightarrow {\cal K},\) defined on the cotangent bundle of the moduli space of stable principal G-bundles over a compact Riemann surface, G being any reductive complex Lie group. In [5] Hitchin showed that the \(m=\dim{\cal M}_{G}\) components of \(\cal H\) are analytic functions that commute with respect to the natural Poisson structure on \(T^{*}{\cal M}_{G}\). In order to do this, Hitchin used a description of \(T^{*}{\cal M}_{G}\) as a Marsden-Weinstein quotient \(\mu ^{-1}(0)/{\cal G}, \mu \) being the moment map for the action of the Gauge group \(\cal G\) on the infinite dimensional space of stable holomorphic structures on a fixed principal G-bundle. In this note we obtain the same result as an application of a few elementary properties satisfied by the homogeneous AdG-invariant polynomials on the Lie algebra of G.


Modulus Space Gauge Group Riemann Surface Poisson Structure Cotangent Bundle 


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Copyright information

© Birkhäuser Verlag, Basel 1999

Authors and Affiliations

  • Renata Scognamillo
    • 1
  1. 1.Scuola Normale Superiore, Piazza dei Cavalieri 7, I-56126 Pisa, ItalyItaly

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