Nonsemisimple Macdonald polynomials

Abstract.

The paper is mainly devoted to the irreducibility of the polynomial representation of the double affine Hecke algebra for an arbitrary reduced root system and generic “central charge” q. The technique of intertwiners in the nonsemisimple variant is the main tool. We introduce the Macdonald nonsemisimple polynomials and use them to analyze the reducibility of the polynomial representation in terms of the affine exponents, counterparts of the classical Coxeter exponents. The focus is on principal aspects of the technique of intertwiners, including related problems of the theory of reduced decomposition in affine Weyl groups and semisimple submodules of the polynomial representation.