Reduction algebras are known by many names in the literature, including step
algebras, Mickelsson algebras, Zhelobenko algebras, and transvector algebras,
to name a few. These algebras, realized by raising and lowering operators,
allow for the calculation of Clebsch-Gordan coefficients, branching rules, and
intertwining operators; and have connections to extremal equations and
dynamical R-matrices in integrable face models.
In this paper we continue the study of the diagonal reduction superalgebra
A of the orthosymplectic Lie superalgebra osp(1∣2). We construct
a Harish-Chandra homomorphism, Verma modules, and study the Shapovalov form on
each Verma module. Using these results, we prove that the ghost center (center
plus anti-center) of A is generated by two central elements and one
anti-central element (analogous to the Scasimir due to Le\'{s}niewski for
osp(1∣2)). As another application, we classify all
finite-dimensional irreducible representations of A. Lastly, we calculate an
infinite-dimensional tensor product decomposition explicitly.Comment: 27 pages; updated introduction: references and motivation;
readability; comments welcomed