The big projective module as a nearby cycles sheaf

TL;DR

This paper presents a geometric construction of the big projective module in category O using nearby cycles of a family of Whittaker sheaves on the flag variety, linking representation theory with geometric methods.

Contribution

It introduces a novel geometric approach to realize the big projective module via nearby cycles of a degenerating family of Whittaker sheaves.

Findings

Unveiled a geometric construction of the big projective module.

Connected nearby cycles functor with representation-theoretic objects.

Provided a new perspective on category O modules through geometry.

Abstract

We give a new geometric construction of the big projective module in the principal block of the BGG category $\mathscr{O}$, or rather the corresponding $\mathscr{D}$-module on the flag variety. Namely, given a one-parameter family of nondegenerate additive characters of the unipotent radical of a Borel subgroup which degenerate to the trivial character, there is a corresponding one-parameter family of Whittaker sheaves. We show that the unipotent nearby cycles functor applied to this family yields the big projective $\mathscr{D}$-module.