Parity Sheaves

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. 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. If moreover the resolution is semi-small, then the multiplicities of the indecomposable summands are encoded in certain intersection forms appearing in the work of de Cataldo and Migliorini. We give a criterion for the Decomposition Theorem to hold. Our framework applies in many situations arising in representation theory. We give examples in generalised flag varieties (in which case we recover a class of sheaves considered by Soergel), toric varieties, and nilpotent cones. Finally, we show that tilting modules and parity sheaves on the affine Grassmannian are related through the geometric Satake correspondence, when the characteristic is bigger than an explicit bound.