Orbits that always have affine stable neighbourhoods
Let \(G\) be a connected linear algebraic reductive group over an algebraically closed ground field. The main theorem characterizes the closed subgroups \(H\) of \(G\) which are not contained in a proper parabolic subgroup of \(G\). It is shown that \(H\) has this property if and only if the following is true: For every algebraic variety \(X\) with a regular \(G\)-action every normal point of \(X\) which is fixed by \(H\) has a \(G\)- stable affine neighborhood in \(X\). This is well known for \(H=G\) and is proved here mainly by amplifying methods of \textit{G. R. Kempf} [Ann. Math., II. Ser. 108, 299-316 (1978; Zbl 0406.14031)] and using structure theory of linear algebraic groups. The above condition for \(H\) is also equivalent to the condition that \(H\) is reductive and \(C_ G(H)^ 0=\text{Rad} G\). The theorem on p. 54 is not correctly stated. Condition (4) has to be read as follows: ``\(H\) is reductive and the centralizer of \(H\) in \(G\) is finite''.
closed subgroups of linear algebraic reductive group
scientific article; zbMATH DE number 57109