Extensions of the conformal representations for orthogonal Lie algebras

Abstract

The conformal transformations with respect to the metric defining o(n,C) give rise to an inhomogeneous polynomial representation of o(n+2,C). Using Shenʼs technique of mixed product, we generalize the above representation to an inhomogeneous representation of o(n+2,C) on the tensor space of any finite-dimensional irreducible o(n,C)-module with the polynomial space, where a hidden central transformation is involved. Moreover, we find a condition on the constant value taken by the central transformation such that the generalized representation is irreducible. In our approach, Pieriʼs formulas, invariant operators and the idea of Kostantʼs characteristic identities play key roles. The result could be useful in understanding higher-dimensional conformal field theory with the constant value taken by the central transformation as the central charge. Our representations virtually provide natural extensions of the conformal transformations on a Riemannian manifold to its vector bundles.