Spaces of coinvariants and fusion product, I: From equivalence theorem to Kostka polynomials

Abstract

The fusion rule gives the dimensions of spaces of conformal blocks in Wess-Zumino-Witten (WZW) theory. We prove a dimension formula similar to the fusion rule for spaces of coinvariants of affine Lie algebras $\widehat{\mathfrak{g}}$. An equivalence of filtered spaces is established between spaces of coinvariants of two objects: highest weight $\widehat{\mathfrak{g}}$-modules and tensor products of finite-dimensional evaluation representations of $\mathfrak{g}\otimes \mathbb{C}[t]$.

In the $\widehat{\mathfrak{sl}}$₂-case we prove that their associated graded spaces are isomorphic to the spaces of coinvariants of fusion products and that their Hilbert polynomials are the level-restricted Kostka polynomials.