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 $x_1\cdots x_lv_0$, 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 $L^{\mathrm{gr}}$ with respect to the PBW filtration. The “top-down” description deals with a structure of $L^{\mathrm{gr}}$ 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 $L^{\mathrm{gr}}$ as a representation of the current algebra $\mathfrak g\otimes\mathbb C[t]$. We prove that each quotient $F_m/F_{m-1}$ can be filtered by graded deformations of the tensor products of $m$ copies of $\mathfrak g$.