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An infinitesimal study of the moduli of Hitchin pairs. (English) Zbl 0819.58007

Let \(G\) be an algebraic group, \(\rho\) a representation of \(G\), \(V\) a vector bundle on a smooth projective curve. The authors consider deformations of pairs \((P,\phi)\) consisting of a principal \(G\)-bundle \(P\) and a section \(\phi\) of \(\rho P\otimes V\). Section 2 contains the infinitesimal computation, while in section 3 the question is considered whether the moduli functor is formally smooth.
Then \(\rho\) is taken to be the coadjoint representation and \(V=K\) and one defines a symplectic structure on the moduli space. Confining to the pairs where \(P\) is stable, the moduli space can be identified with the cotangent bundle of the moduli space of \(G\)-bundles and the symplectic structure can be identified with the Hamiltonian structure. In the case when \(G=\text{SL}(n)\), Hitchin considered the global analogue of the map, which maps the Lie algebra of \(G\) into \(\mathbb{C}^{n+1}\), given by the coefficients of the characteristic polynomial. The authors look at the analogue of the Kostant map of the Lie algebra of \(G\) into \(\mathbb{C}^ \ell\) for all semisimple groups and show that the fibres are Lagrangian at the smooth points of the fibre and also that the symplectic form vanishes on any smooth variety in the fibre over 0. In section 6, they extend these results to pairs with parabolic structures.

MSC:

37J99 Dynamical aspects of finite-dimensional Hamiltonian and Lagrangian systems
14D20 Algebraic moduli problems, moduli of vector bundles
22E46 Semisimple Lie groups and their representations
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