Abstract

Abstract. 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. Many of the constructions in the classical BGG case go through here as well; in particular, assuming the PBW theorem and finite length of all Vermas, the category O is abelian, finite length, and self-dual. We decompose O as a direct sum of blocks O(λ), and show that enough projectives exist in each block, so that each O(λ) is equivalent to a finite-dimensional algebra. Each block is also a highest weight category, so that we have BGG reciprocity. We focus on the case Hf for n = 1, where we show all these assumptions to hold,

Citations