The orthosymplectic superalgebra in harmonic analysis

摘要

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 × osp(m|2n). As a side result we obtain information about the irreducible osp(m|2n)-representations L m|2n (k;0;...;0). In particular we find branching rules with respect to osp(m - 1|2n). We also prove that integration over the supersphere is uniquely defined by its orthosymplectic invariance.