Spherical Harmonics and the Capelli Eigenvalue Problem for osp(1|2n)

Abstract

In this thesis, we define a dual action of sl₂(C) x osp(1|2n) on the space of superpolynomials P(C¹|²ⁿ) and thereby study the spherical harmonics for osp(1|2n). The harmonic polynomials are then used to give a decomposition of P(C¹|²ⁿ) into irreducible osp(1|2n)-modules. An action of gosp(1|2n) consistent with the action of osp(1|2n) on P(C¹|²ⁿ) decomposes P(C¹|²ⁿ) into a multiplicity-free decomposition and therefore defines Capelli operators. Lastly, we relate the surjectivity of the map Z(g) -> PD(V)ᵍ to the non-vanishing of certain determinants. These determinants are then given as polynomials in n along with a complete factorization with roots and their multiplicities. The new results are Theorem 4.3.3 where we give explicit formulas for the joint sl₂(C) x osp(1|2n)-highest weight vectors and Theorem 5.2.10 where we give the complete factorization of the aforementioned determinants.