A de Rham weight part of Serre's conjecture and generalized mod $p$ BGG decompositions

We propose the use of de Rham cohomology of special fibers of Shimura varieties to formulate a geometric version of the weight part of Serre's conjecture. We conjecture that this formulation is equivalent to the one using Serre weights and the \'etale cohomology of Shimura varieties. We prove this equivalence for generic weights and generic non-Eisenstein eigensystems for a compact $U(2,1)$ Shimura variety such that $G_{\mathbb{Q}_p}=GL_3$. We do this by proving a generic concentration in middle degree of mod $p$ de Rham cohomology with coefficients. In turn, we prove this generic concentration by constructing generalized mod $p$ BGG decompositions for de Rham cohomology. After applying the results from our companion paper, this reduces to computing some BGG-like resolutions in a certain mod $p$ version of category $\mathcal{O}$, which is the main content of the article. In the $GSp_4$ case we also compute some explicit BGG decompositions, and assuming the generic concentration in middle degree of de Rham cohomology we obtain an improvement on the main result of arxiv:2410.09602.