Complexity and rank of actions in invariant theory
Let \(G\) denote a connected reductive group \(G\) acting on an affine variety \(X.\) Let \(U \subset G\) be a maximal unipotent subgroup. By the result of Hadziev and Grosshans the covariant algebra \(k[X]^U\) is finitely generated. The author studies invariants of nonreductive subgroups of reductive groups. He uses a classification of actions by certain notions. One of that is the complexity, i.e. the minimal codimension of the orbits. The second one is the rank defined in terms of the semi-invariant functions of the function field \(k(X).\) One of the main results of the paper is a method that reduces the problem of finding rank and complexity to certain problems about actions of reductive groups. This method allows to find not only the rank of an action but the rank group and even the saturated rank semigroup for actions on affine varieties. Another theme of the paper is the investigation of the covariant algebra. One part is concerned with the reduction of the calculation of the covariant algebra to a certain `smaller' action. A second part is concerned with the `symmetry' of Poincaré series of the covariant algebra in case \(X\) is factorial and has rational singularities. As an application there is a classification of spherical nilpotent orbits (with respect to adjoint representation) in simple Lie algebras and a classification of affine homogeneous spaces of complexity 1 for simple algebraic groups.
scientific article; zbMATH DE number 1364727
scientific article