Generation of global sections for arbitrary reductive groups

Establish that the generation-of-zeroth-cohomology assertion for homogeneous vector bundles on cotangent bundles of partial flag varieties extends to every reductive group G whenever the Springer map T^*(G/P) to the associated Richardson orbit closure is birational and that Richardson orbit closure is normal.

Background

The paper proves, for G = SL_n and an arbitrary parabolic subgroup P, that the global sections of the pullback to T^*(G/P) of the homogeneous vector bundle associated with an irreducible Levi representation are generated over C[g^*] by their degree-zero global sections. The relevant surjection is established using the moment map and the Borel–Weil theorem.

The authors explicitly conjecture that this generation statement remains valid for arbitrary reductive groups under two geometric hypotheses: the Springer map from T^*(G/P) to the corresponding Richardson orbit closure must be birational, and the Richardson orbit closure must be normal. They note that these conditions are necessary because the degree-zero case already fails when the Springer map has degree greater than one or when the Richardson orbit closure is non-normal.