The orthosymplectic superalgebra in harmonic analysis

TL;DR

This paper explores the orthosymplectic superalgebra osp(m|2n) within harmonic analysis, establishing its invariance properties, representation structures, and decompositions relevant to supersymmetric spaces and tensor powers.

Contribution

It introduces the osp(m|2n) superalgebra as Killing vector fields on superspace, analyzes its representation structure, and derives new branching rules and invariance properties in harmonic analysis.

Findings

Established osp(m|2n) as the algebra of Killing vector fields stabilizing the origin.

Derived the decomposition of supersymmetric tensor powers under osp(m|2n) and sl(2).

Proved the uniqueness of integration over the supersphere based on orthosymplectic invariance.

Abstract

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.