Lusztig Induction, Unipotent Supports, and Character Bounds

Recently, a strong exponential character bound has been established in [3] for all elements \(g \in \mathbf{G}^F\) of a finite reductive group \(\mathbf{G}^F\) which satisfy the condition that the centraliser \(C_{\mathbf{G}}(g)\) is contained in a \((\mathbf{G},F)\)-split Levi subgroup \(\mathbf{M}\) of \(\mathbf{G}\) and that \(\mathbf{G}\) 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 \(\mathbf{M}\) is split. This assumption is known to hold whenever \(Z(\mathbf{G})\) is connected or when \(\mathbf{G}\) is a special linear or symplectic group and \(\mathbf{G}\) is defined over a sufficiently large finite field.