On the separation property of orbits in representation spaces.
The starting point of the paper under review was the following question asked by Jens-Carsten Jantzen in relation with some work of \textit{A. Premet} [see: J. Lond. Math. Soc., II. Ser. 55, No. 2, 236-250 (1997; Zbl 0889.17019)]: Question. Let \(K\) be an algebraically closed field. Is it true that the minimal nilpotent conjugacy class \(C_{\min}\) in the Lie algebra \(\mathfrak g\) of a simple affine algebraic \(K\)-group has the following ``Separation Property'': (SP) For any pair of linearly independent linear functions \((\alpha,\beta)\) on \(\mathfrak g\) there is an element \(x\in C_{\min}\) such that \(\alpha(x)=0\) and \(\beta(x)\neq 0\)? In the paper under review the authors show that the answer is ``yes'' except for \(\mathfrak{sl}_2\) and the symplectic Lie algebras \(\mathfrak{sp}_{2n}\) (\(n>2\)) where they give explicit counterexamples. As a consequence, some results in the above mentioned paper of Premet do not hold in the given generality. (In the mean-time the authors have been informed that the basic results of Premet are still correct and have been proved along different lines.) Nevertheless, the separation property is an interesting concept in itself and the authors study this property in the paper. In Section 2, they first discuss the case of matrices and show that the set of nilpotent rank 1 matrices in \(\mathfrak{sl}_n(K)\subset M_n(K)\) has the separation property (SP) in case \(n\geqslant 3\). In Section 3, the authors discuss the separation property in general and introduce in addition the ``strong'' and the ``weak'' separation property, (SSP) and (WSP). A number of examples illuminate these concepts. In Section 4 and 5, they study the separation property for orbits in representation spaces of algebraic groups and give a classification of those representations of reductive groups where the minimal orbit \(O_{\min}\) satisfies the various separation properties. For example, the authors discover the fact that in an irreducible representation of a connected semisimple group every linear hyperplane meets every orbit, and they show that a generic orbit always has the separation property.
Lie algebras of simple affine algebraic groups
scientific article; zbMATH DE number 1868087