Lusztig induction, unipotent supports, and character bounds

Taylor, Jay; Tiep, Pham

doi:10.1090/tran/8188

Lusztig induction, unipotent supports, and character bounds
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by Jay Taylor and Pham Huu Tiep PDF

Trans. Amer. Math. Soc. 373 (2020), 8637-8676 Request permission

Abstract:

Recently, a strong exponential character bound was established in [Acta Math. 221 (2018), pp. 1–57] for all elements $g \in \mathbf {G}^F$ of a finite reductive group $\mathbf {G}^F$ which satisfy the condition that the centralizer $C_\mathbf {G}(g)$ 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.

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