Annihilators of highest weight $\mathfrak{s}\mathfrak{l}(\mathrm{\infty })$-modules

We give a criterion for the annihilator in U$(\mathfrak{s}\mathfrak{l}(\mathrm{\infty }))$ of a simple highest weight $\mathfrak{s}\mathfrak{l}(\mathrm{\infty })$-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$(\mathfrak{s}\mathfrak{l}(\mathrm{\infty }))$ of any simple highest weight $\mathfrak{s}\mathfrak{l}(\mathrm{\infty })$-module is integrable, i.e., coincides with the annihilator of an integrable $\mathfrak{s}\mathfrak{l}(\mathrm{\infty })$-module. Furthermore, we define the class of ideal Borel subalgebras of $\mathfrak{s}\mathfrak{l}(\mathrm{\infty })$, and prove that any prime integrable ideal in U$(\mathfrak{s}\mathfrak{l}(\mathrm{\infty }))$ is the annihilator of a simple ${\mathfrak{b}}^0$-highest weight module, where ${\mathfrak{b}}^0$ is any fixed ideal Borel subalgebra of $\mathfrak{s}\mathfrak{l}(\mathrm{\infty })$. This latter result is an analogue of the celebrated Duflo Theorem for primitive ideals.