Category O over Skew Group Rings

We study the BGG Category O over a skew group ring, involving a finite group acting on a regular triangular algebra. We relate the representation theory of the algebra to Clifford theory for the skew group ring, and obtain results on block decomposition, semisimplicity, and enough projectives. O is also shown to be a highest weight category; the BGG Reciprocity formula is slightly different because the duality functor need not preserve each simple module. Next, we turn to tensor products of such skew group rings. Such a product is also a skew group ring; we explore its representation theory in relation to that of the components. We apply these results to the wreath product of symplectic oscillator algebras, and show that the PBW property does not hold if we deform certain relations. Part 1 : Preliminaries 0.