Category O over a deformation of the symplectic oscillator algebra

We discuss the representation theory of $H_f$, which is a deformation of the symplectic oscillator algebra $sp(2n)⋉h_n$, where $h_n$ 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 $\mathcal{O}$ is abelian, finite length, and self-dual. We decompose $\mathcal{O}$ as a direct sum of blocks $\text{calo}(\text{la})$, and show that each block is a highest weight category. In the second part, we focus on the case $H_f$ for $n=1$, where we prove all these assumptions, as well as the PBW theorem.