Joseph-like ideals and harmonic analysis for osp(m|2n)

TL;DR

This paper extends the concept of Joseph ideals to the orthosymplectic Lie superalgebra osp(m|2n), constructing and characterizing new ideals that relate to harmonic analysis and symmetries in supergeometry.

Contribution

It introduces and characterizes Joseph-like ideals for osp(m|2n), unifying classical and superalgebra cases, and links these ideals to harmonic functions and symmetry operators.

Findings

Constructed two analogues of Joseph ideals for osp(m|2n).

Proved these ideals are the annihilators of specific osp(m|2n) representations.

Connected the ideals to symmetries of the super conformal Laplace operator.

Abstract

The Joseph ideal in the universal enveloping algebra U(so(m)) is the annihilator ideal of the so(m)-representation on the harmonic functions on R^{m-2}. The Joseph ideal for sp(2n) is the annihilator ideal of the Segal-Shale-Weil (metaplectic) representation. Both ideals can be constructed in a unified way from a quadratic relation in the tensor algebra of g for g equal to so(m) or sp(2n). In this paper we construct two analogous ideals in the tensor algebra of g and U(g) for g the orthosymplectic Lie superalgebra osp(m|2n)=spo(2n|m) and prove that they have unique characterizations that naturally extend the classical case. Then we show that these two ideals are the annihilator ideals of respectively the osp(m|2n)-representation on the spherical harmonics on R^{m-2|2n} and a generalization of the metaplectic representation to spo(2n|m). This proves that these ideals are reasonable candidates to establish the theory of Joseph-like ideals for Lie superalgebras. We also discuss the relation between the Joseph ideal of osp(m|2n) and the algebra of symmetries of the super conformal Laplace operator, regarded as an intertwining operator between principal series representations for osp(m|2n).