Journal of Lie Theory 26 (2016), No. 1, 097--115
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 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.
Keywords: Special linear Lie algebra, symplectic Lie algebra, oscillator representation, irreducible module, polynomial algebra, exponential-polynomial function.
MSC: 17B10; 17B20
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