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? -
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.