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 \(G_2\).


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


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