A positive-definite inner product for vector-valued Macdonald polynomials

Abstract

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 $\mathcal{S}_{N}$. The polynomials are parametrized by $\left( q,t\right) $ 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 $\left( q^{-1},t^{-1}\right) $ polynomials and $\left( q,t\right) $ 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 $\left( q,t\right) $-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.