Koszul algebras of invariants

Abstract

The notion of a Koszul algebra is recalled in Section 1. The cases of O(n), SO(n), and U(n/2) are dealt with in [S, 91. For these groups A and its quadratic dual have an interesting interpretation in terms of the representation theory of Spin(n, 1) and SU(n, 1). Moreover A appears in a natural way in differential geometry. In the orthogonal (resp. unitary) case A is the universal algebra of invariant differential operators acting on differential forms on a riemannian (resp. kahlerian) manifold, and was implicitly presented in [16] (resp. [17]). This aspect is developed in [9]. The last section is dedicated to the proof of the following fact. Suppose G is an arbitrary subgroup of GL(n, C), and let A = Oka0 Ak denote the grading of A = (Xn@ End A C”)‘. Here and in the sequel @ = @Jc, End = End,, and Horn = Horn,. Also let d(A) and d(A, A’) be, respectively, the (absolute) global dimension of A and the A’-relative global dimension of A in the sense of [lo] (see Sect. 1).