On the Converse to a Theorem of Atiyah and Bott

Throughout this paper, C denotes a smooth projective curve of genus at least one, G denotes a reductive linear algebraic group over C, and ξ0 is a C ∞ principal G-bundle over C. The space of all (0, 1)-connections on ξ0 is an affine space A = A(ξ0) associated to the infinite dimensional complex vector space H0,1(C; ad ξ0). Following Shatz [6] for the case G = GL(n), Atiyah and Bott [1] defined a natural stratification of this space. If h is a Cartan subalgebra for G, then the strata are indexed by the orbits under the Weyl group of a certain discrete set of points in hR, the split real form of h, which we call points of Atiyah-Bott type for ξ0. Fix a set ∆ of simple roots for G with respect to h. Since every Weyl orbit in hR has a unique representative in the positive Weyl chamber C0 associated to ∆, it is natural to index the strata by points μ of Atiyah-Bott type for ξ0 which lie in C0. We denote by Cμ the stratum corresponding to μ. A point μ of Atiyah-Bott type for ξ0 determines a parabolic subgroup P (μ) of G, together with a C ∞ P (μ)-bundle η0(μ) over P (μ) such that η0(μ)×P (μ)G is C∞ isomorphic to ξ0. The condition that μ ∈ C0 is just the condition that P (μ) is a standard parabolic subgroup. Recall that an unstable holomorphic bundle ξ whose underlying C∞-bundle is C∞ isomorphic to ξ0 has a canonical reduction to a parabolic subgroup, called the Harder-Narasimhan reduction. A (0, 1)-connection lies in Cμ if and only if the parabolic subgroup of its Harder-Narasimhan reduction is conjugate to P (μ) in such a way that the corresponding holomorphic P (μ)-bundle is C∞ isomorphic to η0(μ). The strata Cμ are of finite codimension in A and are invariant under the action of the group of C∞-changes of gauge. We define Cμ′ r Cμ if the closure of Cμ′ meets Cμ, and define the relation on the strata Cμ by taking the unique extension of r to a transitive relation. More concretely, Cμ′ Cμ if and only if there is a sequence Cμ′ = Cμ0 , Cμ1 , . . . , Cμn = Cμ such that for each i , 0 ≤ i ≤ n− 1 the closure of Cμi meets Cμi+1. For Atiyah and Bott, the stratification arises as follows. Let K be a compact Lie group whose complexification is G, and let ξK be a C ∞ principal K-bundle such that ξK ×K G is C∞ isomorphic to ξ0. One can then identify K-connections on ξK with (0, 1)-connections The first author was partially supported by NSF grant DMS-99-70437. The second author was partially supported by NSF grant DMS-97-04507.