If G is either semisimple, a torus, or finite then a sufficient (and almost necessary) condition is that G acts on V unimodular, i.e., by transformations of determinant one. The case for general G is discussed in this paper. It turns out that the situation is much more complicated. The following result gives an impression which kind of conditions have to be taken into account.
Theorem: Let V be a representation of G and assume
1. the generic G-orbit in V is closed;
2. the set of all v in V with positive dimensional isotropy group
has at least codimension two;
3. the G-representation V\tensor(Lie G) is unimodular.
Then the ring R of G-invariants on V is Gorenstein.
The second main objective of the paper is to generalize bounds for the q-invariant of R. These are generalizations of results of an earlier paper in which the semisimple case is treated.
Appeared in: Journal of Algebra 127 (1989) 40-54
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