The PBW Filtration, Demazure Modules and Toroidal Current Algebras

Let $L$ be the basic (level one vacuum) representation of the affine Kac-Moody Lie algebra $\widehat{\mathfrak{g}}$. The $m$-th space $F_m$ of the PBW filtration on $L$ is a linear span of vectors of the form $x_1\mathrm{.}\mathrm{.}\mathrm{.}{x}_l{v}_0$, where $l\le m$, $x_i\in \widehat{\mathfrak{g}}$ and $v_0$ is a highest weight vector of $L$. In this paper we give two descriptions of the associated graded space $L^{\mathrm{g}\mathrm{r}}$ with respect to the PBW filtration. The “top-down” description deals with a structure of $L^{\mathrm{g}\mathrm{r}}$ as a representation of the abelianized algebra of generating operators. We prove that the ideal of relations is generated by the coefficients of the squared field $e_{\theta }(z{)}^2$, which corresponds to the longest root $\theta$. The “bottom-up” description deals with the structure of $L^{\mathrm{g}\mathrm{r}}$ as a representation of the current algebra $\mathfrak{g}\otimes \mathbb{C}[t]$. We prove that each quotient $F_m\mathrm{/}{F}_{m-1}$ can be filtered by graded deformations of the tensor products of $m$ copies of $\mathfrak{g}$.