VECTOR INVARIANT IDEALS OF ABELIAN GROUP ALGEBRAS UNDER THE ACTION OF THE SYMPLECTIC GROUPS

Let F be a finite field and let Sp(2v) (F) be the symplectic group over F. If Sp(2v) (F) acts on the F-vector space F-2v, then it can induce an action on the vector space F-2v circle plus F-2v, defined by (x, y)(A) = (xA, yA), for all x, y is an element of F-2v, A is an element of Sp(2v) (F). If K is a field with char K not equal char F, then Sp(2v) (F) also acts on the group algebra K[F-2v circle plus F-2v]. In this paper, we determine the structures of Sp(2v) (F)-stable ideals of the group algebra K[F-2v circle plus F-2v] by augmentation ideals, and describe the relations between the invariant ideals of K[F-2v] and the vector invariant ideals of K[F-2v circle plus F-2v].