The orthosymplectic superalgebra in harmonic analysis
We introduce the orthosymplectic superalgebra osp(m vertical bar 2n) as the algebra of Killing vector fields on Riemannian superspace R-m vertical bar 2n which stabilize the origin. The Laplace operator and norm squared on R-m vertical bar 2n, which generate sl(2), are orthosymplectically invariant, therefore we obtain the Howe dual pair (osp(m vertical bar 2n), sl(2)). We study the osp (m vertical bar 2n)-representation structure of the kernel of the Laplace operator. This also yields the decomposition of the supersymmetric tensor powers of the fundamental osp(m vertical bar 2n)-representation under the action of sl(2) x osp(m vertical bar 2n). As a side result we obtain information about the irreducible osp(m vertical bar 2n)-representations L-(k, 0, ... , 0)(m vertical bar 2n). In particular we find branching rules with respect to osp(m - 1 vertical bar 2n). We also prove that integration over the supersphere is uniquely defined by its orthosymplectic invariance.