Category O over a Deformation of the Symplectic Oscillator Algebra

We discuss the representation theory of Hf , which is a deformation of the symplectic oscillator algebra sp(2n) ⋉ hn, where hn is the ((2n+1)-dimensional) Heisenberg algebra. We first look at a more general algebra with a triangular decomposition. Assuming the PBW theorem, and one other hypothesis, we show that the BGG category O is abelian, finite length, and self-dual. We decompose O as a direct sum of blocks O(λ), and show that each block is a highest weight category. In the second part, we focus on the case Hf for n = 1, where we prove all these assumptions, as well as the PBW theorem.