Spaces of coinvariants and fusion product II. character formulas in terms of Kostka polynomials

In this paper, we continue our study of the Hilbert polynomials of coinvariants begun in our previous work [B. Feigin et al., math.QA/0205324, 2002]. We describe the fusion products for symmetric tensor representations following the method of [B. Feigin, E. Feigin, math.QA/0201111, 2002], and show that their Hilbert polynomials are An−1-supernomials. We identify the fusion product of arbitrary irreducible -modules with the fusion product of their restriction to . Then using the equivalence theorem from [B. Feigin et al., math.QA/0205324, 2002] and the results above for we give a fermionic formula for the Hilbert polynomials of a class of coinvariants in terms of the level-restricted Kostka polynomials. The coinvariants under consideration are a generalization of the coinvariants studied in [B. Feigin et al., Transfom. Groups 6 (2001) 25–52; math.QA/0009198, 2000; math.QA/0012190, 2000]. Our formula differs from the fermionic formula established in [B. Feigin et al., Transfom. Groups 6 (2001) 25–52; math.QA/0009198, 2000; math.QA/0012190, 2000] and implies the alternating sum formula conjectured in [B. Feigin, S. Loktev, math.QA/9812093, 1998; Amer. Math. Sci. Transl. 194 (1999) 61–79] for this case.