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

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.