Abstract

Let X be an irreducible smooth projective curve over an algebraically closed field k of characteristic p, with p > 5. Let G be a connected reductive algebraic group over k. Let H be a Levi factor of some parabolic subgroup of G and χ a character of H. Given a reduction E_H of the structure group of a G-bundle E_G to H, let E_χ be the line bundle over X associated to E_H for the character χ. If G does not contain any SL(n) ∕ Z as a simple factor, where Z is a subgroup of the center of SL(n), we prove that a G-bundle E_G over X admits a connection if and only if for every such triple (H,χ,E_H), the degree of the line bundle E_χ is a multiple of p. If G has a factor of the form SL(n) ∕ Z, then this result is valid if n is not a multiple of p. If G is a classical group but not of the form SL(n) ∕ Z, then this criterion for the existence of connection remains valid even if p ≥ 3.

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