Joseph Ideals and Harmonic Analysis for osp(m vertical bar 2n)

Author:

Coulembier, Kevin  Somberg, Petr  Soucek, Vladimir  

Journal:

INTERNATIONAL MATHEMATICS RESEARCH NOTICES

Issue Date:

2014

Abstract(summary):

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 Rm-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 circle times g for g equal to so(m) or sp(2n). In this paper, we construct two analogous ideals in circle times g and U(g) for g the orthosymplectic Lie super-algebra osp(m vertical bar 2n)=spo(2n vertical bar 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 vertical bar 2n)-representation on the spherical harmonics on Rm-2 vertical bar 2n and a generalization of the metaplectic representation to spo(2n vertical bar 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 osp(m vertical bar 2n) and the algebra of symmetries of the super-conformal Laplace operator, regarded as an intertwining operator between principal series representations for osp(m vertical bar 2n). 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].

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