Double affine Hecke algebras, conformal coinvariants and Kostka polynomials

Abstract: We study a class of representations called ``calibrated representations'' of the degenerate double affine Hecke algebra and those of the rational Cherednik algebra of type ${\mathrm{GL}}_n$. We give a realization of calibrated irreducible modules as spaces of coinvariants constructed from integrable modules over the affine Lie algebra $\hat{\mathfrak{gl}}_m$. Moreover, we give a character formula of these irreducible modules in terms of a level-restricted Kostka polynomials. These results were conjectured by Arakawa, Suzuki and Tsuchiya based on the conformal field theory. The proofs using recent results on the representation theory of the double affine Hecke algebras will be presented in the forthcoming papers.