I'm surprised this hasn't been posted yet:

Lusztig's conjecture in modular representation theory, which would describe the characters of simple $G_1$-modules for $p$ larger than the Coxeter number, was generally believed to be true for a long time. Here $G$ is a reductive algebraic group in characteristic $p$ and $G_1$ is its Frobenius kernel. The conjecture in turn leads to character formulae (or at least, algorithms for calculating characters) for simple representations of $G$.

They were proven for large $p$ by Andersen-Jantzen-Soergel, with later contributions by others, including an explicit bound by Fiebig. But recently Geordie Williamson proved that the original condition on $p$ is not enough.

See What to do now that Lusztig's and James' conjectures have been shown to be false? for more details.

Lusztig's conjecture in modular representation theory, which would describe the characters of simple $G_1$-modules for $p$ larger than the Coxeter number, was generally believed to be true for a long time. Here $G$ is a reductive algebraic group in characteristic $p$ and $G_1$ is its Frobenius kernel. The conjecture in turn leads to character formulae (or at least, algorithms for calculating characters) for simple representations of $G$.

They were proven for large $p$ by Andersen-Jantzen-Soergel, with later contributions by others, including an explicit bound by Fiebig. But recently Geordie Williamson proved that the original condition on $p$ is not enough.

See What to do now that Lusztig's and James' conjectures have been shown to be false? for more details.