Irreducibility of perfect representations of double affine Hecke algebras

It is proved that the quotient of the polynomial representation of the double affine Hecke algebra by the radical of the duality pairing is always irreducible apart from the roots of unity provided that it is finite dimensional. We also find necessary and Sufficient conditions for the radical to be zero, a generalization of Opdam's formula for the singular parameters such that the corresponding Dunkl operators have multiple zero-eigenvalues.