An analogue of the Duistermaat-van der Kallen theorem for group algebras

Let G be a group, R an integral domain, and V _G the R-subspace of the group algebra RG] consisting of all the elements of RG] whose coefficient of the identity element 1 _G of G is equal to zero. Motivated by the Mathieu conjecture Mathieu O., Some conjectures about invariant theory and their applications, In: Algèbre non Commutative, Groupes Quantiques et Invariants, Reims, June 26–30, 1995, Sémin. Congr., 2, Société Mathématique de France, Paris, 1997, 263–279], the Duistermaat-van der Kallen theorem Duistermaat J.J., van der Kallen W., Constant terms in powers of a Laurent polynomial, Indag. Math., 1998, 9(2), 221–231], and also by recent studies on the notion of Mathieu subspaces, we show that for finite groups G, V _G also forms a Mathieu subspace of the group algebra RG] when certain conditions on the base ring R are met. We also show that for the free abelian groups G = ℤ ⁿ , n ≥ 1, and any integral domain R of positive characteristic, V _G fails to be a Mathieu subspace of RG], which is equivalent to saying that the Duistermaat-van der Kallen theorem cannot be generalized to any field or integral domain of positive characteristic.