Annihilators of highest weight $\frak{sl}(\infty)$-modules

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