Conformal oscillator representations of orthogonal Lie algebras

Abstract

The conformal transformations with respect to the metric defining the orthogonal Lie algebra o(n, C) give rise to a one-parameter (c) family of inhomogeneous first-order differential operator representations of the orthogonal Lie algebra o(n+2, C). Letting these operators act on the space of exponential-polynomial functions that depend on a parametric vector $\vec a \in \mathbb{C}^n $ , we prove that the space forms an irreducible o(n+2, C)-module for any c ∈ C; if $\vec a$ is not on a certain hypersurface. By partially swapping differential operators and multiplication operators, we obtain more general differential operator representations of o(n+2, C) on the polynomial algebra C in n variables. Moreover, we prove that C forms an infinite-dimensional irreducible weight o(n+2, C)-module with finite-dimensional weight subspaces if c ∉ Z/2.