  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.
