
Journal of Lie Theory 26 (2016), No. 1, 097115 Copyright Heldermann Verlag 2016 Projective Oscillator Representations of sl(n+1) and sp(2m+2) Xiaoping Xu Mathematical Institute, Academy of Mathematics and Systems Sciences, Chinese Academy of Sciences, Beijing 100190, P. R. China xiaoping@math.ac.cn The ndimensional projective group gives rise to a oneparameter family of inhomogeneous firstorder 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 exponentialpolynomial functions, we construct new multiparameter families of explicit infinitedimensional 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 infinitedimensional multiplicityfree irreducible representations for sl(n). They can also be used to study free bosonic field irreducible representations of the corresponding affine KacMoody algebras. Keywords: Special linear Lie algebra, symplectic Lie algebra, oscillator representation, irreducible module, polynomial algebra, exponentialpolynomial function. MSC: 17B10; 17B20 [ Fulltextpdf (310 KB)] for subscribers only. 