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- Title
Joseph Ideals and Harmonic Analysis for.
- Authors
Coulembier, Kevin; Somberg, Petr; Souček, Vladimír
- Abstract
The Joseph ideal in the universal enveloping algebra is the annihilator ideal of the -representation on the harmonic functions on . The Joseph ideal for 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 for equal to or . In this paper, we construct two analogous ideals in and for the orthosymplectic Lie super-algebra 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 -representation on the spherical harmonics on and a generalization of the metaplectic representation to . 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 and the algebra of symmetries of the super-conformal Laplace operator, regarded as an intertwining operator between principal series representations for . 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].
- Subjects
IDEALS (Algebra); HARMONIC analysis (Mathematics); TENSOR algebra; SPHERICAL harmonics; LIE superalgebras; LAPLACE distribution
- Publication
IMRN: International Mathematics Research Notices, 2014, Vol 2014, Issue 15, p4291
- ISSN
1073-7928
- Publication type
Article