 
Summary: PARITY SHEAVES
DANIEL JUTEAU, CARL MAUTNER, AND GEORDIE WILLIAMSON
Abstract. Given a stratified variety X with strata satisfying
a cohomological parityvanishing condition, we define and show
the uniqueness of "parity sheaves", which are objects in the con
structible derived category of sheaves with coefficients in an arbi
trary 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 semismall, 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 represen
tation 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 character
