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Parity sheaves

Authors: Daniel Juteau, Carl Mautner and Geordie Williamson
Journal: J. Amer. Math. Soc. 27 (2014), 1169-1212
MSC (2010): Primary 55N33, 20C20
Published electronically: May 21, 2014
MathSciNet review: 3230821
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Abstract: Given a stratified variety $ X$ with strata satisfying a cohomological parity-vanishing condition, we define and show the uniqueness of ``parity sheaves,'' which are objects in the constructible derived category of sheaves with coefficients in an arbitrary field or complete discrete valuation ring. This construction depends on the choice of a parity function on the strata.

If $ X$ admits a resolution also satisfying a parity condition, then the direct image of the constant sheaf decomposes as a direct sum of parity sheaves, and the multiplicities of the indecomposable summands are encoded in certain refined intersection forms appearing in the work of de Cataldo and Migliorini. We give a criterion for the Decomposition Theorem to hold in the semi-small case.

Our framework applies to many stratified varieties arising in representation theory such as generalised flag varieties, toric varieties, and nilpotent cones. Moreover, parity sheaves often correspond to interesting objects in representation theory. For example, on flag varieties we recover in a unified way several well-known complexes of sheaves. For one choice of parity function we obtain the indecomposable tilting perverse sheaves. For another, when using coefficients of characteristic zero, we recover the intersection cohomology sheaves and in arbitrary characteristic the special sheaves of Soergel, which are used by Fiebig in his proof of Lusztig's conjecture.

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Additional Information

Daniel Juteau
Affiliation: LMNO, Université de Caen Basse-Normandie, CNRS, BP 5186, 14032 Caen, France

Carl Mautner
Affiliation: Max-Planck-Institut für Mathematik, Vivatsgasse 7, 53111 Bonn, Germany

Geordie Williamson
Affiliation: Max-Planck-Institut für Mathematik, Vivatsgasse 7, 53111 Bonn, Germany

Received by editor(s): November 5, 2012
Received by editor(s) in revised form: October 21, 2013
Published electronically: May 21, 2014
Additional Notes: The first author was supported by ANR Grant No. ANR-09-JCJC-0102-01.
The second author was supported by an NSF postdoctoral fellowship.
Article copyright: © Copyright 2014 American Mathematical Society