A nonsymmetric version of Okounkov's BC-type interpolation Macdonald polynomials

Symmetric and nonsymmetric interpolation Laurent polynomials are introduced with the interpolation points depending on $q$ and a $n$-tuple of parameters $\tau=(\tau_1,\ldots,\tau_n)$. For the principal specialization $\tau_i=st^{n-i}$ the symmetric interpolation Laurent polynomials reduce to Okounkov's $BC$-type interpolation Macdonald polynomials and the nonsymmetric interpolation Laurent polynomials become their nonsymmetric variants. We expand the symmetric interpolation Laurent polynomials in the nonsymmetric ones. We show that Okounkov's $BC$-type interpolation Macdonald polynomials can also be obtained from their nonsymmetric versions using a one-parameter family of actions of the finite Hecke algebra of type $B_n$ in terms of Demazure-Lusztig operators. In the Appendix we give some experimental results and conjectures about extra vanishing.Comment: 30 pages, 9 figures; v4: experimental results and conjectures added about extra vanishin