A note on the complete integrability of the Hitchin system

Abstract.

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 Ad G-invariant polynomials on the Lie algebra of G.