  We introduce the orthosymplectic superalgebra osp(m|2n) 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
sl(2), are orthosymplectically invariant, therefore we obtain the Howe dual
pair (osp(m|2n),sl(2)). We study the osp(m|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|2n)-representation under
the action of sl(2) x osp(m|2n). As a side result we obtain information about
the irreducible osp(m|2n)-representations L_(k,0,...,0). In particular we find
branching rules with respect to osp(m-1|2n) and an interesting formula for the
Cartan product inside the tensor powers of the natural representation of
osp(m|2n). We also prove that integration over the supersphere is uniquely
defined by its orthosymplectic invariance.
