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The big projective module as a nearby cycles sheaf

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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.

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Acknowledgments

First and foremost I thank my advisor Dennis Gaitsgory, without whom this paper would not exist. I am grateful to Sam Raskin for many helpful conversations, as well as feedback which substantially simplified the proof of the theorem. I also thank the anonymous referee who pointed out several errors in an earlier draft.

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  1. Cambridge, MA, USA

    Justin Campbell

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  1. Justin Campbell
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Correspondence to Justin Campbell.

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Campbell, J. The big projective module as a nearby cycles sheaf. Sel. Math. New Ser. 23, 721–726 (2017). https://doi.org/10.1007/s00029-016-0257-7

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  • DOI: https://doi.org/10.1007/s00029-016-0257-7

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