A local-global principle for unipotent characters (Q6656598)

The so-called Dade's Conjecture ([\textit{E. C. Dade}, Invent. Math. 109, No. 1, 187--210 (1992; Zbl 0738.20011)]), asserts that, given a prime number \(\ell\), there exists a precise formula for counting the number of irreducible characters of a finite group, with a given \(\ell\)-defect and belonging to a given Brauer \(\ell\)-block, in terms of the \(\ell\)-local structure of the group itself. \textit{B. Späth}, in [J. Eur. Math. Soc. (JEMS) 19, No. 4, 1071--1126 (2017; Zbl 1459.20005)], introduces the Character Triple Conjecture and proves that Dade's conjecture holds for all finite groups if the Character Triple Conjecture holds for all quasisimple groups.\N\NIn the paper under review, the author adapts and proves the two conjectures mentioned above in the case of unipotent characters of finite reductive groups. Building on ideas introduced by the author in [Adv. Math. 436, Article ID 109403, 61 p. (2024; Zbl 1535.20067)], this approach offers further evidence for certain conjectures that have been shown to imply Dade's Conjecture and the Character Triple Conjecture for all finite reductive groups in nondefining characteristic. In particular, the results by the author (see [\textit{D. Rossi}, ``The Brown complex in non-defining characteristic and applications'', Preprint, \url{arXiv:2303.13973}]) enable the replacement of \(\ell\)-local structures with more suitable \(e\)-local structures, derived from the geometry of the underlying algebraic group and aligned with the framework of Deligne-Lusztig theory.