×

On the centralizer of a regular, semi-simple, stable conjugacy class. (English) Zbl 1107.20034

Let \(k\) be a field and \(G\) be a semi-simple simply-connected algebraic group that is quasi-split over \(k\). Let \(s\) be a regular semi-simple stable conjugacy class in \(G\). Then the centralizer \(G_\gamma\), where \(\gamma\) is a semi-simple element representing the conjugacy class \(s\), is a maximal torus \(T_s\) over \(k\).
The author gives an abstract description of the character group \(X(T_s)\) as an integral representation of the Galois group of \(k\). He also describes \(T_s\) concretely for the linear unitary and symplectic groups, and for the group \(G_2\).

MSC:

20G15 Linear algebraic groups over arbitrary fields
20E45 Conjugacy classes for groups
PDFBibTeX XMLCite

References:

[1] Benedict H. Gross and Curtis T. McMullen, Automorphisms of even unimodular lattices and unramified Salem numbers, J. Algebra 257 (2002), no. 2, 265 – 290. · Zbl 1022.11016
[2] Benedict H. Gross and David Pollack, On the Euler characteristic of the discrete spectrum, J. Number Theory 110 (2005), no. 1, 136 – 163. · Zbl 1080.11041
[3] Robert E. Kottwitz, Rational conjugacy classes in reductive groups, Duke Math. J. 49 (1982), no. 4, 785 – 806. · Zbl 0506.20017
[4] Jean-Pierre Serre, Cohomologie galoisienne, 5th ed., Lecture Notes in Mathematics, vol. 5, Springer-Verlag, Berlin, 1994 (French). · Zbl 0143.05901
[5] Jean-Pierre Serre, L’invariant de Witt de la forme \?\?(\?²), Comment. Math. Helv. 59 (1984), no. 4, 651 – 676 (French). · Zbl 0565.12014
[6] T. A. Springer and R. Steinberg, Conjugacy classes, Seminar on Algebraic Groups and Related Finite Groups (The Institute for Advanced Study, Princeton, N.J., 1968/69) Lecture Notes in Mathematics, Vol. 131, Springer, Berlin, 1970, pp. 167 – 266. · Zbl 0249.20024
[7] Robert Steinberg, Regular elements of semisimple algebraic groups, Inst. Hautes √Čtudes Sci. Publ. Math. 25 (1965), 49 – 80.
This reference list is based on information provided by the publisher or from digital mathematics libraries. Its items are heuristically matched to zbMATH identifiers and may contain data conversion errors. In some cases that data have been complemented/enhanced by data from zbMATH Open. This attempts to reflect the references listed in the original paper as accurately as possible without claiming completeness or a perfect matching.