Dennis Gaitsgory's 2016 preprint, From Geometric to Function-Theoretic Langlands (or How to Invent Shtukas) includes in the third section a very compressed but suggestive discussion of the representation theory of Chevalley groups, using a categorified version of the Grothendieck's faisceaux-functions correspondence. Particularly, if $G$ is a reductive algebraic group (say, $GL_2$), their setup allows making sense of the notion that the category $\textrm{Rep}_{G(\mathbb{F}_q)}$ is the categorified 'trace of Frobenius' on the 2-category of categorified representations of $G$, defined as some kind of module categories over the monoidal category of sheaves on $G$ under convolution. I am suppressing some difficulties here, for example all of the categories under consideration are derived, and the flavour of sheaf theory is a bit difficult to pin down.
At the end of a few pages, he is able to derive a seemingly nontrivial observation connecting an object of Springer's theory to Deligne-Lusztig representations. I don't understand it.
I'm led to understand there has been some serious progress on at least the formal aspect, for example the 2020 preprint of Gaitsgory-Kazhdan-Rozenblyum-Varshavsky, A toy model for the Drinfeld-Lafforgue shtuka construction offers a much expanded discussion of the formal setup, but without the application to Chevalley groups.
At last, here is a question: has anyone developed this approach to the representation theory of finite groups beyond the very compressed discussion in the 2016 preprint? Of course this theorem is not the target of either paper and people have their eyes set on bigger game, but I would love to read something about this example. Has anyone written about it at length in the last 5 years?
- Gaitsgory-2016 (section 3): https://arxiv.org/abs/1606.09608
- Gaitsgory-Kazhdan-Rozenblyum-Varshavsky-2020 (section 3 and possibly 4): https://arxiv.org/abs/1908.05420
