Conformal oscillator representations of orthosymplectic Lie superalgebras

Abstract

In this paper, we generalize the one-parameter ( c ) family of inhomogeneous first-order differential operator representations of the orthogonal Lie algebras arising from conformal transformations to those of orthosymplectic Lie superalgebras , and determine the irreducible condition. Letting these supersymmetric differential operators act on the space of supersymmetric exponential-polynomial functions that depend on a parametric vector a → ∈ C n , we prove that the space forms an irreducible o s p ( n + 2 | 2 m ) -module for any c ∈ C if a → is not on a certain hypersurface . By partially swapping differential operators and multiplication operators, we obtain more general supersymetric differential operator representations of o s p ( n + 2 | 2 m ) on the polynomial algebra C in n + m supersymetric variables. Moreover, we prove that C forms an infinite-dimensional irreducible weight o s p ( n + 2 | 2 m ) -module with finite-dimensional weight subspaces if c ∉ Z / 2 .