1. Rational Representations: Induction, Cohomology Vanishing

Problem 1.1.
[Williamson] What does the induction theorem of Achar–Riche/Hodge–Karuppuchamy–Scott imply about the relationship between representations of $G(\F_q)$ and $B(\F_q)$ in defining characteristic? 
Problem 1.2.
[Williamson] What representations of $G(\F_q)$ arise by restricting a tilting module of $G$? 
Problem 1.3.
[Scott] What rational representations of $B$ arise by restricting a tilting module of $G$? 
Problem 1.4.
[Achar] What objects of the derived category of representations of $B$ correspond to tilting modules of $G$? 
Problem 1.5.
[Nakano] Let $P$ be a parabolic subgroup of $G$. Is it true in characteristic $p$ that $R^i\operatorname{Ind}_P^G\operatorname{Sym}(\mathfrak{n}_P^\bullet) = 0$ for all $i > 0$? (Known for $P = B$ and for certain other parabolics. Possible connection with normality of nilpotent orbit closures.)
Cite this as: AimPL: Sheaves and modular representations of reductive groups, available at http://aimpl.org/sheavemodular.