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Decomposing oscillator representations of $${\mathfrak osp}(2n/n;{\mathbb{R}{}})$$ by a super dual pair $${\mathfrak osp}(2/1;{\mathbb{R}{}}) \times{} {\mathfrak so}(n)$$. (English) Zbl 0741.17002
The oscillator representation of the Lie superalgebra $$\text{osp}(2n/n;\mathbb{R})$$ is introduced, and is shown to be a unitary representation. It is shown that $$\text{osp}(2n/n;\mathbb{R})$$ contains a super dual pair $$\text{osp}(2/1;\mathbb{R})\times\text{so}(n)$$. The decomposition of the oscillator representation into irreducible representations of $$\text{osp}(2/1;\mathbb{R})\times\text{so}(n)$$ is solved, and some interesting byproducts in superanalysis are obtained.

##### MSC:
 17A70 Superalgebras 17B70 Graded Lie (super)algebras 22E70 Applications of Lie groups to the sciences; explicit representations
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##### References:
 [1] A. Bohm , M. Kmiecik , and L.J. Boya. Representation theory of superconformal quantum mechanics . J. Math. Phys., 29:1163-1170, 1988. · Zbl 0654.17013 [2] N. Bourbaki. Algèbre, Chap. 9 . Hermann, 1959. · Zbl 0102.25503 [3] C. Chevalley. Theory of Lie groups I . Princeton Univ. Press, 1946. · Zbl 0063.00842 [4] L. Corwin , Y. Ne’Eman , and S. Sternberg. Graded Lie algebras in mathematics and physics (bose-Fermi symmetry) . Reviews of Modern Phys., 47:573-603, 1975. · Zbl 0557.17004 [5] C. Fronsdal, editor. Essays on Supersymmetry . Reidel, 1986. [6] H. Furutsu and T. Hirai. Representations of Lie super algebras, I. Extensions of representations of the even part . J. Math. Kyoto Univ., 28:695-749, 1988. · Zbl 0674.17010 [7] R. Howe. On the role of Heisenberg group in harmonic analysis . Bull. AMS, 3:821-843, 1980. · Zbl 0442.43002 [8] R. Howe. Remarks on classical invariant theory . Trans. AMS, 313:539-570, 1989. · Zbl 0674.15021 [9] M. Kashiwara and M. Vergne. On the Segal-Shale-Weil representations and harmonic polynomials . Inv. Math., 44:1-47, 1978. · Zbl 0375.22009 [10] K. Nishiyama. Oscillator representations for orthosymplectic algebras . J. Alg., 129:231-262, 1990. · Zbl 0688.17002 [11] K. Nishiyama. Super dual pairs and unitary highest weight modules of orthosymplectic algebras . To appear in Adv. in Math. · Zbl 0802.17002 [12] R. Parthasarathy. Dirac operator and the discrete series . Ann. Math., 96:1-30, 1972. · Zbl 0249.22003 [13] J.H. Schwartz. Dual resonance theory . Phys. Rep., 8C:269-335, 1973. [14] S. Sternberg and J.A. Wolf. Hermitian Lie algebras and metaplectic representations. I . Trans. AMS, 238:1-43, 1978. · Zbl 0386.22010 [15] H. Tilgner. Graded generalizations of Weyl- and Clifford algebras . J. Pure and Appl. Alg., 10:163-168, 1977. · Zbl 0386.17003 [16] A. Weil. Sur certain groupes d’operateurs unitairs . Acta Math., 111:143-211, 1964. · Zbl 0203.03305
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