The orthosymplectic superalgebra in harmonic analysis

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.

partial overlap with arXiv:1202.0668

Cited by in corpus (8)

• Superspace conformal field theory
• A minimal representation of the orthosymplectic Lie supergroup
• The Capelli eigenvalue problem for Lie superalgebras
• Fischer decomposition for spinor valued polynomials in several variables
• Polynomial realisations of Lie (super)algebras and Bessel operators
• A Fock Model and the Segal-Bargmann Transform for the Minimal Representation of the Orthosymplectic Lie Superalgebra $\mathfrak{osp}(m,2|2n)$
• Fischer decomposition for polynomials on superspace
• Explicit formulas for Eigenvalues of Capelli operators for the Lie superalgebra $\mathfrak{osp}(1|2n)$