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Super dual pairs and highest weight modules of orthosymplectic algebras. (English) Zbl 0802.17002
For a symplectic Lie group $$\text{Sp} (2n,\mathbb{R})$$, all unitary lowest (or highest) weight representations can be realized as a subrepresentation in some power of tensor products of the Weil representation (also known as the oscillator representation). In the present paper this result is extended to the orthosymplectic Lie superalgebra $$\text{osp} (2n, m; \mathbb{R})$$. A complete result is obtained only for $$\text{osp} (2,2; \mathbb{R})$$, but also for the general case interesting results are proved. The method used in this paper is based upon the notion of super dual pairs, following Lie algebra dual pairs of R. Howe [Bull. Am. Math. Soc., New Ser. 3, 821-843 (1980; Zbl 0442.43002)].

##### MSC:
 17A70 Superalgebras 17B10 Representations of Lie algebras and Lie superalgebras, algebraic theory (weights)
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