Stable base loci of representations of algebraic groups (Q791596)

Let k be an algebraically closed field, H a connected linear algebraic group defined over k, and Y an affine k-variety on which H acts. Let R denote the coordinate ring of Y. This paper is concerned with the question of when the ring A of H-invariant elements of R is a finitely generated k-algebra. The prototypical case is when Y is a vector space on which H is represented by a representation \(\rho\). In this case the ring A is graded and \(I(\rho\),m) denotes the R-ideal generated by the m-th graded piece of A. The sum of these ideals is called the base locus ideal of \(\rho\). The base locus is stable if for all n sufficiently large \(I(\rho\),mn) and \(I(\rho\),m)n define the same sheaf of ideals on the projective space P(Y). For general affine Y, similar notions are defined via an H-equivariant embedding in a linear representation. The stability of the base locus is related to the finite generation of A. This is a complicated problem in general, and the author's results require a number of technical definitions to state properly, but the following theorem gives their flavor: In the case that Y is a vector space with linear H- action, if H has no characters and the base locus ideal is principal (hence stable) then A is finitely generated.