Nearby Cycles of Whittaker Sheaves

Abstract

In this thesis we study the nearby cycles of a Whittaker sheaf as it degenerates to an object of the principal series category. In the case of a finite-dimensional flag variety, we prove that the nearby cycles sheaf corresponds under Beilinson-Bernstein localization to the big projective object of the BGG category O. We also introduce a resolution of singularities for Drinfeld's compactification, using it to prove a key local acyclicity statement for the intersection cohomology sheaf. This acyclicity is then applied in the study of nearby cycles of Whittaker sheaves on Drinfeld's compactification. Namely, we express the strata restrictions of the nearby cycles sheaf in Langlands dual terms, and also describe the subquotients of the monodromy filtration using the Picard-Lefschetz oscillators introduced by Schieder.