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.

Non-Gatherable Triples for Non-Affine Root Systems

“…This paper is a continuation of the part of [2] devoted to non-gatherable triangle triples, NGT, in λ-sequences. The latter are the sequences of positive roots associated with reduced decompositions (words) in affine and non-affine Weyl groups.…”

“…[2]) for an explicit description of the irreducible representations of the affine (and double affine) Hecke algebras, complementary to the geometric theory of [7]. However, NGT are interesting in their own right.…”

“…For instance, α, β ∈ λ(w) ⇒ α + β ∈ λ(w) and the latter root must appear between α and β if this set is treated as a sequence. This property is necessary but not sufficient; here and below see [2] for a comprehensive discussion.…”

“…Generally, the admissibility condition from [2] is necessary and sufficient for the triple to be gatherable, which is formulated in terms of subsystems of R of types B 3 , C 3 or D 4 . We (re)establish this theorem in the non-affine case in this paper and make the proof very constructive.…”

“…We (re)establish this theorem in the non-affine case in this paper and make the proof very constructive. The proof presented in [2] was entirely algebraic, not quite complete for the system F 4 and sketchy in the D, E-cases.…”