The irreducible characters of a finite reductive group are partitioned into sets called Harish–Chandra series, describing their behavior under induction from and restriction to Levi subgroups. Broué, Malle, and Michel found a ‘cyclotomic’ generalization of this story, depending on geometric induction and restriction functors introduced by Deligne–Lusztig and Lusztig. I present some conjectures, joint with Ting Xue, that relate these cyclotomic Harish–Chandra series to the representation theory of certain Hecke algebras at roots of unity. These conjectures seem to be new even in type A. They suggest studying a certain virtual Hecke-algebra bimodule depending on two parameters. When these parameters are the same prime power, this virtual bimodule can be constructed from the cohomology of Deligne–Lusztig varieties. When they are certain roots of unity, I conjecture that it can be constructed from the cohomology of homogeneous affine Springer fibers. The motivation for this conjecture comes from work of Oblomkov–Yun on affine Springer fibers, my own work on braid varieties, and nonabelian Hodge theory.