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.
Categories: math.RT math.AG