Vol. 104, No. 1, 1983

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Centralizers of irregular elements in reductive algebraic groups

John Francis Kurtzke, Jr.

Vol. 104 (1983), No. 1, 133–154

Let G be a reductive linear algebraic group defined over an algebraically closed field K. An element x G is called regular if dimZG(x) is the rank of G—which is the smallest possible dimension for a centralizer—otherwise x is called irregular. T. A. Springer has shown that if x G is regular, then ZG(x)0 is abelian. In this paper, we show that when char K is good for G, the converse is also true—if x G is irregular, then ZG(x)0 is nonabelian. In the course of the proof of this, we show that if G is a classical group or G2 (char K good), then dimZ(ZG(x)0) is at most rank G. Further, if G = Sp2n(K)(char K2), then Z(ZG(x)0) consists of polynomials in x.

Mathematical Subject Classification 2000
Primary: 20G15
Secondary: 14L10
Received: 25 January 1980
Revised: 30 November 1981
Published: 1 January 1983
John Francis Kurtzke, Jr.