The centralizers of semisimple elements of the Chevalley groups \(E_ 7\) and \(E_ 8\)
The purpose of this paper is to give in detailed tables all the centralizers and their orders of semisimple elements of the finite Chevalley groups \(E_ 7\) and \(E_ 8\). (Two tables on 21 pages.) Table 1: The structure and the orders of connected centralizers of semisimple elements in \(E_ 7\) (7 pp.). Table 2: The structure and the orders of the centralizers of semisimple elements in \(E_ 8\) (14 pp.). These tables are very useful since they give the character degrees of the semisimple irreducible complex representations constructed by Deligne and Lusztig for the finite Chevalley groups of adjoint type. These degrees can be obtained if we know what subgroups of the finite Chevalley groups of universal type are centralizers of semisimple elements [cf. \textit{P. Deligne} and \textit{G. Lusztig}, Ann. Math., II. Ser. 103, 103-161 (1976; Zbl 0336.20029)]. Similar tables giving these centralizers for the classical groups and for the groups \(G_ 2\), \(F_ 4\) and \(E_ 6\) have been obtained respectively by [\textit{R. W. Carter}, Proc. Lond. Math. Soc., III. Ser. 42, 1-41 (1981; Zbl 0455.20035)], \textit{B. Chang} and \textit{R. Ree}, Symp. Math. 13, 395-413 (1974; Zbl 0314.20034), \textit{T. Shoji}, J. Fac. Sci., Univ. Tokyo, Sect. IA 21, 1-17 (1974; Zbl 0279.20038), \textit{K. Mizuno}, ibid. 24, 525-563 (1977; Zbl 0399.20044). Cf. also \textit{K. Mizuno}, Tokyo J. Math. 3, 391-459 (1980; Zbl 0454.20046)].
connected centralizers of semisimple elements
semisimple irreducible complex representations
scientific article; zbMATH DE number 3847669