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Journal of Lie Theory 23 (2013), No. 1, 055--083
Copyright Heldermann Verlag 2013

The Orthosymplectic Superalgebra in Harmonic Analysis

Kevin Coulembier
Dept. of Mathematical Analysis, Ghent University, Krijgslaan 281, 9000 Gent, Belgium
and: School of Mathematics and Statistics, University of Sydney, Sydney, Australia


\def\l{{\frak l}} \def\o{{\frak o}} \def\p{{\frak p}} \def\s{{\frak s}} \def\R{{\Bbb R}} \def\osp{\o\s\p(m|2n)} We introduce the orthosymplectic superalgebra $\osp$ as the algebra of Killing vector fields on Riemannian superspace $\R^{m|2n}$ which stabilize the origin. The Laplace operator and norm squared on $\R^{m|2n}$, which generate $\s\l_2$, are orthosymplectically invariant, therefore we obtain the Howe dual pair $(\osp(m|2n),\s\l_2)$. We study the $\osp$-representation structure of the kernel of the Laplace operator. This also yields the decomposition of the supersymmetric tensor powers of the fundamental $\osp$-representation under the action of $\s\l_2\times\osp$. As a side result we obtain information about the irreducible $\osp$-representations $L_{(k,0,\cdots,0)}^{m|2n}$. In particular we find branching rules with respect to $\osp(m-1|2n)$. We also prove that integration over the supersphere is uniquely defined by its orthosymplectic invariance.

Keywords: Howe dual pair, orthosymplectic superalgebra, not completely reducible representations, supersymmetric tensor product.

MSC: 17B10, 58C50, 17B15

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