ANNIHILATORS OF HIGHEST WEIGHT $\mathfrak{s}\mathfrak{l}$(∞)-MODULES

Abstract

We give a criterion for the annihilator in U($\mathfrak{s}\mathfrak{l}$(∞)) of a simple highest weight $\mathfrak{s}\mathfrak{l}$(∞)-module to be nonzero. As a consequence we show that, in contrast with the case of $\mathfrak{s}\mathfrak{l}$(n), the annihilator in U($\mathfrak{s}\mathfrak{l}$(∞)) of any simple highest weight $\mathfrak{s}\mathfrak{l}$(∞)-module is integrable, i.e., coincides with the annihilator of an integrable $\mathfrak{s}\mathfrak{l}$(∞)-module. Furthermore, we define the class of ideal Borel subalgebras of $\mathfrak{s}\mathfrak{l}$(∞), and prove that any prime integrable ideal in U($\mathfrak{s}\mathfrak{l}$(∞)) is the annihilator of a simple $\mathfrak{b}$ ⁰-highest weight module, where $\mathfrak{b}$ ⁰ is any fixed ideal Borel subalgebra of $\mathfrak{s}\mathfrak{l}$(∞). This latter result is an analogue of the celebrated Duoflo Theorem for primitive ideals.