(Co)stalks of IC_{\overline{Bun}_B} and the Gaitsgory sheaf in bad characteristic

Determine the stalk and costalk cohomology of the intersection cohomology complex IC_{\overline{Bun}_B} on Drinfeld’s compactification of Bun_B and of the Gaitsgory sheaf Ga on the affine Grassmannian (in the Iwahori-equivariant derived category) when the coefficient field has bad characteristic for the connected reductive group G. In particular, ascertain whether the good-characteristic formulas and independence-of-coefficients results for the dimensions of (co)stalk cohomology extend to bad characteristic.

Background

The paper proves that, for coefficients in a field of good characteristic for G, the cohomology dimensions of stalks and costalks of IC_{\overline{Bun}_B} and of the Gaitsgory sheaf Ga are described by the q-analogue of Kostant’s partition function, and are independent of the coefficient field. This relies on the Mirković–Vilonen conjecture holding in good characteristic.

However, the Mirković–Vilonen conjecture is known to fail in bad characteristic, and the authors do not extend their results to this setting. Thus, the behavior of (co)stalk cohomology for IC_{\overline{Bun}_B} and Ga when the coefficient field has bad characteristic remains unsettled.