Irreducibility of Perfect Representations of Double Affine Hecke Algebras

In the paper, we prove that the quotient of the polynomial representation of the double affine Hecke algebra (DAHA) by the radical of the duality pairing is always irreducible assuming that it is finite dimensional, which is always the case at roots of unity. We also find necessary and sufficient conditions for the radical to be zero, a q-generalization of Opdam’s formula for the singular k-parameters with the multiple zero-eigenvalues of the corresponding Dunkl operators. Concerning the terminology, perfect representations in the paper posses a non-degenerate (=perfect) pairing, which induces the canonical duality anti-involution of DAHA. In the polynomial representation, it is given in terms of the evaluation at qk . We assume that they are spherical representations, i.e., quotients of the polynomial representation of DAHA, but do not impose the semisimplicity in contrast to [C3]. The irreducibility theorem from this paper is stronger and at the same time the proof is simpler than those in [C3]. The polynomial representation, denoted by V in the paper, is irreducible and semisimple for generic values of the DAHA-parameters q, t.