Abstract
The Joseph ideal in the universal enveloping algebra Graphic is the annihilator ideal of the Graphic-representation on the harmonic functions on Graphic. The Joseph ideal for Graphic 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 Graphic for Graphic equal to Graphic or Graphic. In this paper, we construct two analogous ideals in Graphic and Graphic for Graphic the orthosymplectic Lie super-algebra Graphic 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 Graphic-representation on the spherical harmonics on Graphic and a generalization of the metaplectic representation to Graphic. 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 Graphic and the algebra of symmetries of the super-conformal Laplace operator, regarded as an intertwining operator between principal series representations for Graphic. 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].