Category O over a deformation of the symplectic oscillator algebra

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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 setup, involving an 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 (), 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.