Lusztig induction, unipotent supports, and character bounds

Recently, a strong exponential character bound was established in [Acta Math. 221 (2018), pp. 1–57] for all elements g ∈ G ^F of a finite reductive group G ^F which satisfy the condition that the centralizer C _G(g) is contained in a (G,F)-split Levi subgroup M of G and that G is defined over a field of good characteristic. In this paper, assuming a weak version of Lusztig’s conjecture relating irreducible characters and characteristic functions of character sheaves holds, we considerably generalize this result by removing the condition that M is split. This assumption is known to hold whenever Z(G) is connected or when G is a special linear or symplectic group and G is defined over a sufficiently large finite field.