Projective Oscillator Representations of sl(n+1) and sp (2m+2)

摘要

The n-dimensional projective group gives rise to a one-parameter family of inhomogeneous first-order differential operator representations of sl (n + 1). By partially swapping differential operators and multiplication operators, we obtain more general differential operator representations of sl (n + 1). Letting these differential operators act on the corresponding polynomial algebra and the space of exponential-polynomial functions, we construct new multi-parameter families of explicit infinite-dimensional irreducible representations for sl (n + 1) and sp (2m + 2) when n = 2m + 1. Our results can be viewed as extensions of Howe's oscillator construction of infinite-dimensional multiplicity-free irreducible representations for sl (n). They can also be used to study free bosonic field irreducible representations of the corresponding affine Kac-Moody algebras.