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