Symplectic Differential Reduction Algebras and Generalized Weyl Algebras

Given a map $\Xi\colon U(\mathfrak{g})\rightarrow A$ of associative algebras, with $U(\mathfrak{g})$ the universal enveloping algebra of a (complex) finite-dimensional reductive Lie algebra $\mathfrak{g}$, the restriction functor from $A$-modules to $U(\mathfrak{g})$-modules is intimately tied to the representation theory of an $A$-subquotient known as the reduction algebra with respect to $(A,\mathfrak{g},\Xi)$. Herlemont and Ogievetsky described differential reduction algebras for the general linear Lie algebra $\mathfrak{gl}(n)$ as algebras of deformed differential operators. Their map $\Xi$ is a realization of $\mathfrak{gl}(n)$ in the $N$-fold tensor product of the $n$-th Weyl algebra tensored with $U(\mathfrak{gl}(n))$. In this paper, we further the study of differential reduction algebras by finding a presentation in the case when $\mathfrak{g}$ is the symplectic Lie algebra of rank two and $\Xi$ is a canonical realization of $\mathfrak{g}$ inside the second Weyl algebra tensor the universal enveloping algebra of $\mathfrak{g}$, suitably localized. Furthermore, we prove that this differential reduction algebra is a generalized Weyl algebra (GWA), in the sense of Bavula, of a new type we term skew-affine. It is believed that symplectic differential reduction algebras are all skew-affine GWAs; then their irreducible weight modules could be obtained from standard GWA techniques.