
## 5. Parity Sheaves and Torsion in IC Sheaves

1. #### Problem 5.1.

[Achar–Williamson]
1. For which primes $p$ is it true that all indecomposable $B$-constructible parity $\F_p$ sheaves on $G/B$ are perverse?
2. Same question for $I$-constructible parity $\F_p$-sheaves on $Gr$.

(Expect this to be true for large $p$.)
• #### Problem 5.2.

[Mautner] For which primes $p$ do the IC sheaves on $\operatorname{Perv}_I(Gr, \Z_p)$ have torsion-free stalks?

(Related to Problem 10.1 part 2, and to Ext-vanishing between reduced standard and costandard modules.)
• #### Problem 5.3.

[Williamson] Give nontrivial sufficient conditions for indecomposable parity sheaves on $G/B$ or $Gr$ or $Fl$ to be simple perverse sheaves.
• #### Problem 5.4.

[Achar] Is there a faster algorithm to compute indecomposable parity sheaves on $\operatorname{Perv}_{\mathrm{sph}}(Gr, \F_p)$ than by computing the $p$-canonical basis?

(Motivated by A. Broer’s algorithm in characteristic 0.)

Cite this as: AimPL: Sheaves and modular representations of reductive groups, available at http://aimpl.org/sheavemodular.