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 g. The m-th space Fm of the PBW filtration on L is a linear span of vectors of the form x₁ ···x_lv₀, where l ≤ m, x_i ∈ g and v0 is a highest weight vector of L. In this paper we give two descriptions of the associated graded space L^gr with respect to the PBW filtration. The "top-down" description deals with a structure of L^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 eθ(z)², which corresponds to the longest root θ. The bottom-up description deals with the structure of L^gr as a representation of the current algebra g ⊗ 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 g.