Branching laws for spherical harmonics on superspaces in exceptional cases

Abstract

It turns out that harmonic analysis on the superspace R^{m|2n} is quite parallel to the classical theory on the Euclidean space R^{m} unless the superdimension M:=m-2n is even and non-positive. The underlying symmetry is given by the orthosymplectic superalgebra osp(m|2n). In this paper, when the symmetry is reduced to osp(m-1|2n) we describe explicitly the corresponding branching laws for spherical harmonics on R^{m|2n} also in exceptional cases. In unexceptional cases, these branching laws are well-known and quite analogous as in the Euclidean framework.

Figures (3)

• Figure 1: The Fischer decomposition for $M\not\in -2\mathbb{N}_0$.
• Figure 2: The Fischer decomposition for $M\in -2\mathbb{N}_0$ and $m=0$.
• Figure 3: The Fischer decomposition for $M=-4$ and $m\not=0$.

Theorems & Definitions (6)

• Theorem 1
• proof
• Theorem 2
• proof
• Theorem 3
• proof