[For the entire collection see Zbl 0665.00004.]
An interesting overview is given of what is known about the canonical module of the ring \(A=R^ G\) of invariants of a linearly reductive group G acting linearly on the polynomial ring \(R=K[X_ 1,...,X_ n]\), K an algebraically closed field. The extensive first section on background material makes the paper almost self-contained. The central theme is the \(conjecture\quad G\) which says that, with the notations above and assuming that A is Cohen-Macaulay, A is Gorenstein if the action of G factors through SL(V), where V denotes the space \(RX_ 1+...+RX_ n\), or, in other words, if the determinant of any \(g\in G\) acting on V is the trivial character. Cases where the conjecture is known to be true include finite G such that the order of G is invertible in K, or G an algebraic torus, or a connected semi-simple group over a field of characteristic zero. Results of K. Watanabe and of R. Stanley, based on Molien’s formula (for a finite group G) for the Hilbert-Poincaré series are discussed. The condition for A to be Gorenstein can be translated in terms of the canonical module \(\omega_ A\) and this fact implies a functional equation for the Hilbert-Poincaré series of A if and only if A is Gorenstein.
In the third section the canonical module \(\omega_ A\), \(A=R^ G\), is calculated under the assumption that \(conjecture\quad G\) holds. A useful, auxiliary action of G on R[z], z a new indeterminate, is introduced by letting \(g\in G\) act on z by multiplication with the inverse of its determinant and \(\omega_ A\) turns out to be the module of semi- invariants for the inverse of the determinant for many good actions of G, e.g. when for the auxiliary action of G on R[z], \(R[z]^ G\) is Gorenstein. Explicit descriptions of \(\omega_ A\) are obtained for finite G, in which case one recovers Watanabe’s result, and for G an algebraic torus.
In the fourth section a special situation where the Gorenstein property of \(A=R^ G\) holds is discussed. The notion of excellent action of the connected linear algebraic (not necessarily reductive) group G on the ring R is defined (in terms of characters) and the main result which says that for an excellent action and under an additional condition on the normalizer of G in GL(V), \(A=R^ G\) (assumed to be Cohen-Macaulay) is Gorenstein.
The last section treats several examples in some detail, in particular, certain determinantal rings are studied.
In a footnote on the first page the author remarks that he recently had become aware of a result of F. Knop that says that \(conjecture\quad G\) is not true in general.