The orthosymplectic supergroup in harmonic analysis

Abstract

The orthosymplectic supergroup OSp(m|2n) is introduced as the supergroup of isometries of flat Riemannian superspace R^{m|2n} which stabilize the origin. It also corresponds to the supergroup of isometries of the supersphere S^{m-1|2n}. 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)). This Howe dual pair solves the problems of the dual pair (SO(m)xSp(2n),sl(2)), considered in previous papers. In particular we characterize the invariant functions on flat Riemannian superspace and show that the integration over the supersphere is uniquely defined by its orthosymplectic invariance. The supersphere manifold is also introduced in a mathematically rigorous way. Finally we study the representations of osp(m|2n) on spherical harmonics. This corresponds to the decomposition of the supersymmetric tensor space of the m|2n-dimensional super vectorspace under the action of sl(2)xosp(m|2n). As a side result we obtain information about the irreducible osp(m|2n)-representations L_{(k,0,...,0)}^{m|2n}. In particular we find branching rules with respect to osp(m-1|2n).