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1. Rational Representations: Induction, Cohomology Vanishing


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      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?


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          Problem 1.2.

          [Williamson] What representations of $G(\F_q)$ arise by restricting a tilting module of $G$?


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              Problem 1.3.

              [Scott] What rational representations of $B$ arise by restricting a tilting module of $G$?


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                  Problem 1.4.

                  [Achar] What objects of the derived category of representations of $B$ correspond to tilting modules of $G$?


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                      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.