Joseph Ideals and Harmonic Analysis for $\mathfrak {osp}(m|2n)$

The Joseph ideal in the universal enveloping algebra $\mathcal U(\mathfrak so(m))$ is the annihilator ideal of the $\mathfrak {so}(m)$-representation on the harmonic functions on $\mathbb R^{m-2}$. The Joseph ideal for $\mathfrak {sp}(2n)$ 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 $\otimes \mathfrak g$ for $\mathfrak g$ equal to $\mathfrak {so}(m)$ or $\mathfrak {sp}(2n)$. In this paper, we construct two analogous ideals in $\otimes \mathfrak g$ and $\mathcal U(\mathfrak g)$ for $\mathfrak g$ the orthosymplectic Lie super-algebra $\mathfrak {osp}(m|2n)=\mathfrak {spo}(2n|m)$ 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 $\mathfrak {osp}(m|2n)$-representation on the spherical harmonics on $\mathbb R^{m-2|2n}$ and a generalization of the metaplectic representation to $\mathfrak {spo}(2n|m)$. This proves that these ideals are reasonable candidates to establish the theory of Joseph-like ideals for Lie super-algebras. We also discuss the relation between the Joseph ideal of $\mathfrak {spo}(2n|m)$ and the algebra of symmetries of the super-conformal Laplace operator, regarded as an intertwining operator between principal series representations for $\mathfrak {spo}(2n|m)$. As a side result, we obtain the proof of a conjecture of Eastwood about the Cartan product of irreducible representations of semisimple Lie algebras made in [10].