
Journal of Lie Theory 23 (2013), No. 1, 055083 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 coulembier@cage.ugent.be [Abstractpdf] \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(m2n)} We introduce the orthosymplectic superalgebra $\osp$ as the algebra of Killing vector fields on Riemannian superspace $\R^{m2n}$ which stabilize the origin. The Laplace operator and norm squared on $\R^{m2n}$, which generate $\s\l_2$, are orthosymplectically invariant, therefore we obtain the Howe dual pair $(\osp(m2n),\s\l_2)$. 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 $L_{(k,0,\cdots,0)}^{m2n}$. In particular we find branching rules with respect to $\osp(m12n)$. 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 [ Fulltextpdf (421 KB)] for subscribers only. 