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

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 ̂g. An equivalence of filtered spaces is established between spaces of coinvariants of two objects: highest weights-modules and tensor products of finite-dimensional evaluation representations g ⊗ ℂ[t]. In the ̂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.