Dimensions of reductive automorphism groups (Q1176553)

It is known that the dimension of a reductive algebraic group \(G\) acting regularly and effectively on a connected reduced algebraic variety of dimension \(n\) over an algebraically closed field of characteristic 0 is at most \(n^ 2+n\) [ \textit{H. Carayol}, Bull. Sci. Math., II. Ser. 99, 135-143 (1975; Zbl 0333.20031)]. The author reproves this result using some standard facts from the theory of semi-simple Lie algebras. In fact, he improves the estimate by showing that \(\dim(G)\leq(n-c)^ 2+2(n-c)+c\), where \(c\) is the dimension of the center of \(G\).