A positive-definite inner product for vector-valued Macdonald polynomials (original) (raw)

In a previous paper J.-G. Luque and the author {Sem. Loth. Combin. 2011) developed the theory of nonsymmetric Macdonald polynomials taking values in an irreducible module of the Hecke algebra of the symmetric group S N . The polynomials are parametrized by (q, t) and are simultaneous eigenfunctions of a commuting set of Cherednik operators, which were studied by Baker and Forrester (IMRN 1997). In the Dunkl-Luque paper there is a construction of a pairing between q -1 , t -1 polynomials and (q, t) polynomials, and for which the Macdonald polynomials form a biorthogonal set. The present work is a sequel with the purpose of constructing a symmetric bilinear form for which the Macdonald polynomials form an orthogonal basis and to determine the region of (q, t)-values for which the form is positive-definite. Irreducible representations of the Hecke algebra are characterized by partitions of N . The positivity region depends only on the maximum hook-length of the Ferrers diagram of the partition.