Level-Rank Dualities from $\Phi$-Cuspidal Pairs and Affine Springer Fibers

We propose a generalization of the level-rank dualities arising from Uglov's work on higher-level Fock spaces. The statements use Hecke algebras defined by Brou\'{e}-Malle, which conjecturally describe the endomorphisms of Lusztig induction modules, and a generalization of Harish-Chandra theory due to Brou\'{e}-Malle-Michel. For any generic finite reductive group $\mathbb{G}$ and integers $e, m > 0$, we conjecture that: (1) the intersection of a $\Phi_e$-Harish-Chandra series and a $\Phi_m$-Harish-Chandra series is parametrized by a union of blocks of the Hecke algebra of the $\Phi_e$-cuspidal pair at an $m$th root of unity, and similarly for the Hecke algebra of the $\Phi_m$-cuspidal pair at an $e$th root of unity; (2) these parametrizations match the blocks on the two sides; (3) when two blocks match, the bijection between them lifts to a derived equivalence between associated blocks of rational DAHAs. Surprisingly, these structures also appear in bimodules formed from the cohomology of affine Springer fibers studied by Oblomkov-Yun. When $\mathbb{G} = \mathbb{GL}_n$ and $e, m$ are coprime, we show that (1)-(3) hold, and that (3) recovers the level-rank dualities conjectured by Chuang-Miyachi and later proved through the work of several other people. Finally, we verify for many cases where $\mathbb{G}$ is exceptional that Brou\'{e}-Malle's parameters are numerically compatible with our conjectures.