1202.0668 (Kevin Coulembier)

The orthosymplectic supergroup in harmonic analysis    [PDF]

Kevin Coulembier

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).

View original: http://arxiv.org/abs/1202.0668