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

X. Xu

Journal of Lie theory, Tome 26 (2016) no. 1, pp. 97-115

Résumé

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.

Détail

Classification : 17B10, 17B20
Mots-clés : Special linear Lie algebra, symplectic Lie algebra, oscillator representation, irreducible module, polynomial algebra, exponential-polynomial function