Perverse sheaves on the semi-infinite flag variety and representations of the Frobenius kernel
Emilien Zabeth
TL;DR
This work extends the geometric framework linking modular perverse sheaves and Frobenius-kernel representations to the semi-infinite flag setting. By globalizing local problems through Drinfeld compactifications and Zastava spaces, it builds a regular perverse sheaf \mathcal{R} and an exact Conv^{Hecke} functor from the category mod^Y_{I_u}(\mathcal{R}) to the Iwahori-equivariant semi-infinite flag category Perv_{I_u}(Fl^{∞/2}), preserving Verdier duality and aligning simple objects via IC-sheaves. Leveraging the modular adaptations of the geometric Satake equivalence, Finkelberg-Mirković conjecture, and Achar–Riche’s model, the paper proves an exact, faithful functor with a bijection on simples and outlines conditions under which this yields an equivalence; SL_n is known to satisfy the stalk-independence conjecture, illustrating the approach. The construction provides a concrete geometric model for representations of the Frobenius kernel in the extended principal block, and paves the way toward a full characteristic-free equivalence via the Conv^{Hecke} program, with potential applications to character formulas and modular Langlands-type correspondences.
Abstract
We study a category of Iwahori-equivariant modular perverse sheaves on some avatar of the semi-infinite flag variety, by adapting the work of Arkhipov-Bezrukavnikov-Braverman-Gaitsgory-Mirković. We then construct a functor between the latter category and the category of graded representations of the Frobenius kernel, and conjecture that this functor is an equivalence.