BGG reciprocity for current algebras

Abstract

We study the category $\cal I_{\gr}$ of graded representations with finite--dimensional graded pieces for the current algebra $\lie g\otimes\bc[t]$ where $\lie g$ is a simple Lie algebra. This category has many similarities with the category $\cal O$ of modules for $\lie g$ and in this paper, we formulate and study an analogue of the famous BGG duality. We recall the definition of the projective and simple objects in $\cal I_{\gr}$ which are indexed by dominant integral weights. The role of the Verma modules is played by a family of modules called the global Weyl modules. We show that in the case when $\lie g$ is of type $\lie{sl}_2$, the projective module admits a flag in which the successive quotients are finite direct sums of global Weyl modules. The multiplicity with which a particular Weyl module occurs in the flag is determined by the multiplicity of a Jordan--Holder series for a closely associated family of modules, called the local Weyl modules. We conjecture that the result remains true for arbitrary simple Lie algebras. We also prove some combinatorial product--sum identities involving Kostka polynomials which arise as a consequence of our theorem.