UvA-DARE (Digital Academic Repository) A nonsymmetric version of Okounkov’s BC-type interpolation Macdonald polynomials

. Nonsymmetric interpolation Laurent polynomials in n variables are introduced, with the interpolation points depending on q and on a n -tuple of parameters (cid:28) = ( (cid:28) 1 ; : : : ; (cid:28) n ). When (cid:28) i = st n (cid:0) i , Okounkov’s 3-parameter BC n -type interpolation Macdonald polynomials are recovered from the nonsymmetric interpolation Laurent polynomials through Hecke algebra symmetrisation with respect to a type C n Hecke algebra action. In the Appendix we give some conjectures about extra vanishing, based on Mathematica computations in rank two.


Introduction
The goal of this paper is to introduce and solve a special class of Newton type interpolation problems for Laurent polynomials in several variables.An important special case leads to nonsymmetric analogs of Okounkov's [26], [27] BC n -type interpolation Macdonald polynomials.
Okounkov [27,Def. 4.4] calls a grid perfect when the extra vanishing property I + λ (Υ + (µ)) = 0 holds true for all λ, µ ∈ Λ + n such that λ µ, where ⊆ is the natural inclusion order on Λ + n .For generic q, s, t ∈ C * the grid is perfect, and all other perfect grids can be obtained from (1.1) by replacing the grid values Ω + (i, m) by aΩ + (i, m) + b where a ∈ C * and b ∈ C are independent of i and m (but they may depend on the parameters) and by taking limits (see [27]).
Macdonald and Koornwinder polynomials have natural nonsymmetric counterparts, see [2], [21], [37].They are the joint polynomial eigenfunctions of Cherednik's commuting q-difference reflection operators, which in turn constitute part of Cherednik's [3] polynomial representation of the double affine Hecke algebra in terms of Demazure-Lusztig operators.

Nonsymmetric interpolation Laurent polynomials
In this paper, we introduce nonsymmetric counterparts of the interpolation polynomials I + λ (y) for the grid (1.4) In the principal specialization τ i := st n−i , the grid (1.4) reduces to the perfect grid (1.1) underlying the BC n -type interpolation Macdonald polynomials.
In case of the grid (1.4), it is instrumental to look for nonsymmetric analogs of the symmetric interpolations within the space ] Wn of W n -invariant Laurent polynomials, where W n is the type C n Weyl group acting by permutations and inversions of the variables.Note that the interpolation points for I + λ (y) in terms of the x-variables are given by with Ω the type A n−1 grid defined by (1.3).The Weyl group W n acts on (C * ) n by permutations and inversions, and on Z n by permutations and sign changes.Write β + ∈ Λ + n for β ∈ Z n the unique partition in the W n -orbit of β, and w β ∈ W n for the element of smallest length such that Our main result is as follows (see Section 4).
Theorem 1.1.Assume q ∈ C * is not a root of unity, and denote by T n the generic set of parameters τ = (τ 1 , . . ., τ n ) defined by (3.3).For τ ∈ T n and α ∈ Z n there exists a unique Laurent polynomial where β := w β (Υ(β + )) (see (2.3) for the notation in bold ).Furthermore, with I + λ (y) the symmetric interpolation polynomial relative to the grid (1.4).The proof of the existence of the interpolation polynomials I + λ (y) and I α (y) are based on explicit recursion relations, which allow a direct proof by induction to the degree (see [27,Prop. 2.7] and [35,Cor. 4.4]).We revisit the proof for I + λ (y) with the grid (1.4) from the Laurent polynomial perspective in Section 3. It forms a convenient starting point for the much more elaborate inductive proof of Theorem 1.1, which is given in Section 4.
The nonsymmetric interpolation polynomials G α (x; q, s, t) (α ∈ Z n ) with parameters τ = (τ 1 , . . ., τ n ) specialized to the τ i := st n−i are nonsymmetric analogs of Okounkov's BC n -type interpolation Macdonald polynomials.In this case, the BC ntype interpolation Macdonald polynomials can alternatively be reobtained from the G α (x; q, s, t) by symmetrising with respect to a type C n Hecke algebra action on C[x ±1 ] in terms of Demazure-Lusztig type operators (see Section 5).It is a first indication that the G α (x; q, s, t) are amenable to the Hecke algebra techniques from [10], [36], [39].The missing ingredient from this perspective is the interpretation of the G α (x; q, s, t) as simultaneous eigenfunctions of commuting inhomogeneous Cherednik-type operators.This would allow one to involve double affine Hecke algebra techniques in deriving a binomial formula for nonsymmetric Koornwinder polynomials, and in deriving nonsymmetric analogs of extra vanishing and duality (compare with [10,39] for type A).We expect this to be the key step towards further applications of the nonsymmetric BC n -type interpolation polynomials in the theory of nonsymmetric Macdonald-Koornwinder polynomials, algebraic combinatorics, and exactly solvable models.
In the Appendix, we give a conjecture about the extra vanishing, based on Mathematica computations in rank two.
Noumi for sharing with us his insight on BC n -symmetric interpolation polynomials.A substantial part of Sections 3 and 4 is based on material in the master's thesis by the first author under supervision of the last two authors (University of Amsterdam, Faculty of Science, 2017).We thank the referees for valuable comments that led to significant improvements of the text.

Preliminaries
Throughout the paper, we assume that q ∈ C * := C \ {0} is not a root of unity.For a ∈ C, the q-shifted factorial is given by (a; q 2, . ..) and (a; q) 0 := 1.We also write (a 1 , . . ., a r ; q) k := (a When we write I = {i 1 , . . ., i k } ⊆ [1, n] for a subset of [1, n] of cardinality k, then we will always assume the ordering i 1 < i 2 < • • • < i k of its elements.We write I c for the complement of I in [1, n].
A Laurent polynomial f in the n complex variables x has the form with c α ∈ C and c α = 0 for only finitely many α.The degree of f in (2.1) is defined by Let W n = {±1} n S n be the Weyl group associated with the root system of type C n (and B n and BC n ).Then σ ∈ {±1} n and π ∈ S n act on Λ n (and C n ) by (σα) j := σ j α j , (πα Equivalently, if e 1 , . . ., e n is the standard basis of R n then σ(e j ) = σ j e j and π(e j ) = e π(j) .Since 2πiΛ n is invariant under the action of W n , we can exponentiate its action on C n to an action on (C * ) n .We will write this action in bold.Then (σx) j := x σj j , (πx The action of W n on (C * ) n induces an action of W n on Laurent polynomials (2.1) by for the n − 1 complex variables obtained from x by removing x n .Similarly, if τ = (τ 1 , . . ., τ n ) is a n-tuple of complex numbers, then we write τ := (τ 1 , . . ., τ n−1 ).
Sometimes we also need to remove an arbitrary complex variable x k from x.In that case we write A similar notation will be employed for n-tuples of complex numbers.Note that x = x (n) and τ = τ (n) .For the root system R of type C n we take β i := e i − e i+1 (i = 1, . . ., n − 1) and β n := 2e n as the simple roots.Then the set R + of positive roots consists of the vectors e i ± e j (i < j) and 2e i (i = 1, . . ., n).Write R − = −R + for the set of negative roots.Denote the simple reflections corresponding to the simple roots by s 1 , . . ., s n .Each w ∈ W n can be written as a product of simple reflections.The minimal number of factors in such a product representing w is called the length (w) of w.The length of w is also equal to the number of positive roots sent to negative roots by w; see [7,Lem. 10.3A] or [8,Cor. 1.7] r) and the r − k + 1 positive roots sent by w k to negative roots are precisely the positive roots The length (λ) ∈ {0, 1, . . ., n} of λ is the index such that λ i = 0 iff i > (λ).Denote the set of partitions of length at most n by Λ + n .For α ∈ Λ n there exists a unique partition α + ∈ Λ + n in the W n -orbit {wα} w∈Wn .If α ∈ Z n ≥0 then α + can also be characterized as the unique partition in the S n -orbit {πα} π∈Sn .For m ≤ n, we embed Λ + m → Λ + n by (λ 1 , . . ., λ m ) → (λ 1 , . . ., λ m , 0, . . ., 0).On Λ + n the dominance partial ordering ≤ and inclusion partial ordering ⊆ are defined by ) and π α ∈ S n is the unique permutation satisfying the following two properties: See also [37, p. 277] for a short formulation of what is essentially the description of π α in Lemma 2.2, and for an example.

Interpolation theorem for BC n -symmetric Laurent polynomials
Following [30], we call a Laurent polynomial (2.1) BC n -symmetric if it is invariant under the Weyl group W n .A basis of the linear space of BC n -symmetric Laurent polynomials is given by the symmetrized monomials The Laurent polynomial m λ has degree |λ|, and as identity in P as identity in P The result now follows from the fact that λ (j) < λ for j = 1, 2, . . ., .(b) From (3.1) we get 2) follows by induction on the weight of the partition.
Then we define the parameter domain T n by and T n is invariant under scalar multiplication by q.For µ The map µ → µ is injective on Λ + n .Sometimes we write µ = µ(q, τ ) and µ i = µ i (q, τ ) if it is important to specify the dependence on q, τ .Remark 3.2.We will develop the theory of symmetric and nonsymmetric interpolation Laurent polynomials for parameters (q, τ ) with q ∈ C * not a root of unity and τ ∈ T n (see (3.3)).It is easy to check that the results also hold true with q and τ j rational indeterminates, and for (q, τ ) with 0 < |q| < 1 and 0 The latter case requires straightforward adjustments to the proofs of Corollary 4.3 and Lemma 4.9.
The following two properties of the interpolation points will play an important role in what follows: with µ(q, τ ) = (q µ1 τ 1 , . . ., q µn−1 τ n−1 ) for µ Proof.First note that both the space P Wn n,d of BC n -symmetric Laurent polynomials in x of degree at most d and the space of complex-valued functions on the set Therefore, surjectivity of the linear map which restricts a BC n -symmetric Laurent polynomial to the set of interpolation points {µ | µ ∈ Λ + n,d } implies injectivity, so that existence implies uniqueness.To prove existence we will use induction on n + d.If n + d = 1, so (n, d) = (1, 0), then Λ + 1,0 = {(0)} and 0 = τ 1 and there is nothing to prove (take f to be the appropriate constant function).Suppose that the existence of f , with (n, d) replaced by ( n, d), is true for n + d < n + d for all possible parameters in T n and all possible maps and τ ∈ T n and let µ be µ(q, τ ).To establish the induction step, we need to prove the existence of f ∈ P Wn n,d satisfying f (µ) = f (µ) for all µ ∈ Λ + n,d .We first construct g ∈ P Wn n,d satisfying the partial interpolation property First assume n > 1.By induction, there exists a BC n−1 -symmetric Laurent polynomial g in x of degree at most d such that By Lemma 3.1(c) there exists a BC n -symmetric Laurent polynomial g in x of degree at most d such that g(x , τ n ) = g(x ).Then ).Now the first formula of (3.5) gives (3.6).If n = 1 then put g(x) := f (0), where g has degree d ≥ 0.Then, in particular, g(τ 1 ) = f (0).This concludes the proof of (3.6) in all cases.
Note that (3.6) already concludes the proof of the induction step when n > d.Indeed, in this case we can simply take f = g since µ n = 0 for all µ ∈ Λ + n,d .To complete the induction step we thus may and will assume from now on that d ≥ n.We make the Ansatz that the symmetric interpolation Laurent polynomial f we are searching for is of the form with g as constructed above and with µ n = 0.The identity f (µ) = f (µ) will also hold for Note that, since τ ∈ T n , no factors in the above denominator vanish.Hence what remains to show is the existence of a BC n -symmetric Laurent polynomial h ∈ P Wn n,d−n satisfying (3.9).Note that we have a bijection given by µ → µ − 1 := (µ 1 − 1, . . ., µ n − 1).By the induction hypothesis, there exists a h ∈ P Wn n,d−n such that By the second formula of (3.5) we have ), hence we conclude that h ∈ P Wn n,d−n satisfies the desired interpolation property (3.9).This concludes the proof of the induction step.
In view of Proposition 3.3 we can give the following definition.Definition 3.4.Fix τ ∈ T n .The BC n -symmetric interpolation Laurent polynomial of degree λ ∈ Λ + n is the unique BC n -symmetric Laurent polynomial R λ (x; q, τ ) in n variables x of degree at most |λ| such that R λ (λ; q, τ ) = 1 and where µ := µ(q, τ ).
It follows from Proposition 3.3 that {R λ (x; q, τ ) with on the left-hand side the interpolation Laurent polynomial in n variables and on the right-hand side the interpolation Laurent polynomial in for the partitions of length at most n and weight d.
The following important property is less immediate.
We prove this by induction on n + d.For n = 1, the result follows from Example 3.7.To prove the induction step we need to consider two cases. If in G(P Wn n ) by Proposition 3.5(b).The result now immediately follows from the induction hypothesis.
If λ n = 0 then first consider R λ (x ; q, τ ).By the induction hypothesis, m λ (x ) occurs with nonzero coefficient in the linear expansion of R λ (x ; q, τ ) in the basis . By the Proof of Lemma 3.1(c), there exists g ∈ P Wn n,d such that g(x , τ n ) = R λ (x ; q, τ ) and such that m λ (x) occurs with nonzero coefficient in the linear expansion of g(x) in the basis Fixing this choice of g, there exists by (the proof of) Proposition 3.3 a unique h ∈ P Wn n,d−n such that ) should be read as R λ (x; q, τ ) = g(x) and the proof below goes through with the obvious adjustments).Hence , and the result follows from the fact that the linear expansion of Example 3.7.If n = 1 then the interpolation parameter τ ∈ T 1 is a complex number s ∈ C * satisfying s 2 ∈ q Z .We denote the corresponding symmetric interpolation Laurent polynomial R ( ) (x; q, τ ) in one variable x by R (x; q, s) for ∈ Z ≥0 .Then R (x; q, s) = (sx, sx −1 ; q) (q s 2 , q − ; q) (3.12) and the coefficient of m (x) in the linear expansion of R (x; q, s) with respect to the basis q s 2 , q − ; q .
and they admit explicit binomial, combinatorial and integral formulas; see [26], [27].The combinatorial formula [26,Thm. 5.2] allows to obtain more precise information on the expansion components of R λ (x; q, s, t) in symmetric monomials, while the binomial formula [26,Thm. 7.1] provides the explicit expansion of Koornwinder polynomials in terms of BC n -type interpolation Macdonald polynomials.
Remark 3.9.The interpolation grid for the BC n -type interpolation Macdonald polynomials naturally appears in the theory of Koornwinder polynomials in the following way.Koornwinder polynomials are the BC n -symmetric Laurent polynomial eigenfunctions of the commuting Koornwinder-van Diejen q-difference operators [6,13], depending on five parameters a, b, c, d, t.These operators generate a commutative algebra isomorphic to P Wn n through the Harish-Chandra isomorphism (cf.[15, §2]).Through this isomorphism, the eigenvalues of the Koornwinder-van Diejen q-difference operators are described by the evaluation morphisms P Wn n → C, p → p(q λ st ρ ) (λ ∈ Λ + n ), where s = q −1 abcd.

Interpolation theorem for nonsymmetric Laurent polynomials
We extend definition (3.4) of the interpolation points µ from µ ∈ Λ + n to µ ∈ Λ n as follows.Put τ ∈ T n , with T n defined by (3.3) Here π α is as in Lemma 2.2.We write α = α(q, τ ) and α i = α i (q, τ ) if we need to emphasize the dependence of α on the parameters.Recall the actions (2.2), (2.3), (2.4) of W n on Λ n , (C * ) n and P n , respectively.The resulting action of W n on the interpolation points α ∈ (C * ) n can be described as follows.Proof of Lemma 4.1.First we prove (a).For j = n, this reduces by (4.1) and the assumption s n α = α to showing that π −1 snα = π −1 α .It follows immediately from Lemma 2.2 that these two permutations are equal.For j < n, the statement of (a) reduces by (4.1) to showing that π −1 sj α = π −1 α • s j if s j α = α.Also the equality of these two permutations under the given condition follows immediately from Lemma 2.2.
(b) This follows from the explicit expression (4.1) using the fact that q is not a root of unity and that τ ∈ T n (see (3.3)).
(c) By part (b) of the Corollary and Lemma 4.1(a) we have for s j α = α that s j α = s j α = α.Hence α β j = α j /α j+1 = 1 for j < n and α (a) For every map f : R(n, d, I) → C there exists a Laurent polynomial f ∈ P n such that For every map f : T (n, d, I) → C there exists a Laurent polynomial f ∈ P n such that Note that sgn(β i ) = sgn(α i ) and, using (2.6), π β = π α .Hence, by (4.1), α(q, τ ) = β(q, qτ ).By the induction hypothesis, statement (b) with τ replaced by qτ is valid for the function The first step is to prove the existence of a Laurent polynomial g ∈ P n such that For n = 1, we have d ≥ 1 and In this case, we can take g(x) to be the constant polynomial f (−1).Assume that n > 1.In this case, we solve the interpolation problem (4.6) by rewriting it as an interpolation problem for a function on with γ := α (i1) = (α 1 , . . ., α i1−1 , α i1+1 , . . ., α n ).In other words, The interpolation points behave under this bijection in the following manner.By the explicit description of π α (see Lemma 2.2) we have π −1 α (i 1 ) = n and π −1 Then, by (4.1) we have Consider the function g ∨ : R(n − 1, d − 1, J) → C, defined by g ∨ (γ) := f (α).Since |J| < k the induction hypothesis (either the one on the sum of the number of variables and the weight, or the one on the size of the subset) implies that statement (a) is true for g ∨ and τ ∈ T n−1 .Hence there exists a Laurent polynomial g ), then it follows that g satisfies (4.6).This completes the first step.
As a second step we add an appropriate term to g to obtain the desired interpolation properties for the full set R(n, d, I).Note that Laurent polynomials f of the form Note that the right-hand side is well defined, since the conditions (3.3) on the parameters together with (4.1) and the fact that α i1 = −1 imply that the denominator is nonzero.Due to the induction hypothesis, we are allowed to apply statement (a) to the function This gives a Laurent polynomial h ∈ P n fulfilling (4.9) and satisfying the degree Then f given by (4.8) satisfies f (α(q, τ )) = f (α) for all α ∈ R(n, d, I).Furthermore, by the degree properties of g and h the degree conditions deg(f (x)x J ) ≤ d − n + k for all J ⊆ I are satisfied.Hence f satisfies (4.3) as desired.
Proof of statement (b).The proof is along the same lines as the proof of statement (a), but there are subtle differences in the combinatorics.We may assume that d ≥ n − k.If I = ∅ then the statement is correct due to Remark 4.5.
Let k > 0 and assume that statement (b) is true for all f ∧ : R(n We claim that there exists g ∈ P n such that For n = 1, we have d ≥ 1 and I = {1}, hence T ∧ (1, d, {1}) = {0} and we can take g(x) to be the constant polynomial equal to f (0).For n > 1, consider the bijection with δ := α (i k ) .In other words, As in the proof of statement (a) one then shows that for α ∈ T ∧ (n, d, I).By the induction hypothesis (either the induction hypothesis on the sum of the number of variables and the weight, or the induction hypothesis on the size of the subset), there exists g ∧ ∈ P n such that g ∧ (δ(q, τ )) = f (α) for δ ∈ T (n − 1, d, I \ {i k }) and satisfying the degree conditions deg g . Then g ∈ P n , defined by g(x) := g ∧ (x (i k ) ), satisfies (4.10).Now define h : Note that the right-hand side is well defined, since the conditions (3.3) on the parameters together with (4.1) and the fact that α i k = 0 imply that the denominator is nonzero.By the induction hypothesis, there exists h ∈ P n such that h(α(q, τ )) = h(α) for all α ∈ T (n, d, I\{i k }) which satisfies the degree conditions deg h(x)x K ≤ d for all K ⊆ I c ∪ {i k }.Furthermore, with this choice of h and (4. satisfies the desired interpolation property f (α(q, τ )) = f (α) for all α ∈ T (n, d, I).
By the degree conditions on g(x) and h(x), we have deg f (x)x J ≤ d for all J ⊆ I c , which completes the proof of statement (b).
Proof.Denote by F q,τ n,d the space of complex-valued functions on Then, by Corollary 4.3(b), the space F q,τ n,d has dimension |Λ n,d |.Define the linear map φ q,τ n,d : P n,d → F q,τ n,d by φ q,τ n,d (f ) := f | S q,τ n,d .Proposition 4.4(b) with I = [1, n] implies that φ q,τ n,d is surjective.Then φ q,τ n,d is also injective, since both vector spaces P n,d and F q,τ n,d are of dimension |Λ n,d |.Hence φ q,τ n,d is a linear isomorphism, which implies the theorem.
In the remainder of this section, we fix τ ∈ T n and write α := α(q, τ ) for α ∈ Λ n .In view of Theorem 4.6, we can give the following definition.
As in Section 3, one concludes from Theorem 4.
Recall from Proposition 3.6 that the coefficient of m λ (x) in the linear expansion of the symmetric interpolation Laurent polynomial R λ (x; q, τ ) in symmetric monomials m µ (x) (µ ∈ Λ + n ) is nonzero.For the nonsymmetric interpolation Laurent polynomial G α (x; q, τ ) we have the following result.Lemma 4.9.Let α ∈ Λ n .The coefficients c α γ (q, τ ) ∈ C in the linear expansion are rational functions in the variables q, τ 1 , . . ., τ n .The rational function representing c α α (q, τ ) for q not a root of unity and τ ∈ T n , is nonzero.Proof.We will prove that the c α γ (q, τ ) are rational in q, τ by considering them for fixed α as solutions of a linear system with rational coefficients.We use the notations introduced in the proof of Theorem 4.6.Let d ∈ Z ≥0 and set m := |Λ n,d |.Identify Λ n,d with [1, m] by fixing an enumeration of the elements in Λ n,d .This provides vector space identifications The linear isomorphism φ q,τ n,d : P n,d ∼ −→ F q,τ n,d is then represented by the invertible matrix A(q, τ ) := β(q, τ ) γ β,γ ∈ GL m (C).
In the following section and in the Appendix, we present first steps towards answering the question whether the nonsymmetric BC n -type interpolation Macdonald polynomials satisfy extra vanishing properties and admit explicit binomial formulas.For Knop's [10] type A n−1 nonsymmetric interpolation Macdonald polynomials, extra vanishing and explicit binomial formulas were derived in [10], [35], [39].Their proofs lean on a generalization of Cherednik's action of the double affine Hecke algebra on polynomials in n variables for which the type A n−1 nonsymmetric interpolation Macdonald polynomials are common eigenfunctions of the resulting Y -operators.
It is not known whether the nonsymmetric BC n -type interpolation Macdonald polynomials G α (x; q, s, t) satisfy extra vanishing properties.In the Appendix, we will present the outcome of computer algebra computations describing extra vanishing for G α (x; q, s, t) when n = 2 and |α| = 4.

The action of Demazure-Lusztig operators
We introduce an action of the type C n Hecke algebra on the space of Laurent polynomials in n variables, defined in terms of Demazure-Lusztig operators.We explicitly compute its action on nonsymmetric BC n -type interpolation Macdonald polynomials.Similar to the type A n−1 case in [10], [35], the Hecke algebra techniques in this section can only be applied when taking the principal specialization τ = st ρ .
Recall our notations associated with root system C n in Section 2.
Definition 5.1 (Hecke algebra of type B n or C n ).Let H n (t, t n ) be the complex unital associative algebra with generators T 1 , . . ., T n , parameters t, t n ∈ C * , and defining relations with t i := t for i ∈ [1, n).
Remark 5.2.The relations (T i − t i )(T i + 1) = 0 are related to the usual Hecke relations ( [37, §2.3] has Hecke relations as in [18] with with a, b ∈ C such that ab = −t n .As we shall see in Proposition 5.5, the specialization of the Hecke parameters t, t n and the representation parameters a, b that is needed for the application to nonsymmetric BC n -type interpolation Macdonald polynomials is t n = −1 and a = s, b = s −1 with s ∈ C * .Noumi's representation then takes the following form. define a one-parameter family of representations π s : H n (t, −1) → End(P n ) on P n .
In the following lemma, we show that π s preserves the degree-filtration on P n .
)-submodule of P n with respect to the action π s .
Proof.We have to prove that H A straightforward computation gives that Using Proposition 5.5, the coefficient e λ λ− in the expansion is the same as the coefficient of G λ− in the expansion of π s (C λ + )G λ in nonsymmetric BC n -type interpolation Macdonald polynomials.Hence it suffices to show that where the third equality follows from the well-known description of the set of positive roots mapped by w λ 0 to negative roots in terms of the reduced expression w λ 0 = s i1 • • • s ir (see Section 2).
The meaning of the colours of the dots in the pictures is as follows: • cross: α = (α 1 , α 2 ).
Remark A.2.If α ∈ Λ + n (i.e., α is a partition) then for all β ∈ W n α, Thus Conjecture A.1 implies that, for α a partition, Z α ∩ Λ + n consists of all partitions µ which do not include the partition α.Compare with the case of Okounkov's BC n -type interpolation Macdonald polynomials; see (3.14).
In [10, §4] Knop introduced a new partial order on Z n ≥0 to describe the extra vanishing of the type A n−1 nonsymmetric interpolation Macdonald polynomials.Knop's order relation between two elements α, β ∈ Z n ≥0 can be described in terms of inequalities of the entries of the corresponding partitions α + , β + ∈ Λ + n , with the strictness or non-strictness of the inequalities depending on the defining permutation π = u α u −1 β (here u α , u β ∈ S n are the permutations of shortest lengths such that u α (α + ) = α and u β (β + ) = β).One may wonder whether the extra vanishing of the nonsymmetric BC n -type interpolation Macdonald polynomials G α (x; q, s, t) can be formulated in terms of a hyperoctahedral version of Knop's partial order, with the strictness or nonstrictness of the entries of the associated partitions α + , β + ∈ Λ + n now described in terms of w α w −1 β ∈ W n for α, β ∈ Λ n .

. 4 ) 4 . 5 .
Remark Note that statement (b) for I = ∅ is statement (a) for I = [1, n].Proof of Proposition 4.4.If f is a map on an empty set then choose f identically zero.Thus statement (a) holds trivially when d < n and statement (b) holds trivially when d < n − k.If (n, d) = (1, 0) then statement (a), and statement (b) for I = ∅, are true by the remark in the previous paragraph.For statement (b) with I = {1} note that T (1, 0, {1}) = {0}, hence we can take f (x) to be the constant polynomial f (0).Now let n + d ≥ 2. Suppose that all the statements of the Proposition, with (n, d) replaced by ( n, d), are true for all subsets I ⊂ [1, n] and all τ ∈ T n when n + d < n + d.We will then successively prove statements (a) and (b) by induction on the cardinality |I| of the subset I ⊆ [1, n].Proof of statement (a).We may assume that d ≥ n.First consider the case I = ∅.Note that R(n, d, ∅) = {α ∈ Λ n,d | α j = 0, −1 for all j}.Fix a map f : R(n, d, ∅) → C and τ ∈ T n .We prove the existence of a Laurent polynomial f ∈ P n,d−n such that f (α(q, τ )) = f (α) for all α ∈ R(n, d, ∅) by solving a related interpolation problem on the set T (n, d − n, [1, n]) = Λ n,d−n , using statement (b) with shifted parameters qτ ∈ T n .Consider for this the bijection α → β : R(n, d, ∅) which completes the proof of statement (a) for I = ∅.Let k > 0 and assume that statement (a) is true for all f ∨ : R(n ∨ , d ∨ , I ∨ ) → C when n ∨ + d ∨ ≤ n + d, τ ∨ ∈ T n ∨ and with I ∨ ⊆ [1, n ∨ ] of cardinality < k. (Note that for n ∨ + d ∨ < n + d this assumption already holds by our earlier induction hypothesis.)Let I = {i 1 , . . ., i k } ⊆ [1, n] be a set of cardinality k, τ ∈ T n , and consider a function f : R(n, d, I) → C. We prove the existence of an interpolation Laurent polynomial f ∈ P n satisfying (4.3) by splitting the interpolation problem in two pieces.For this we use the disjoint union R(n, d, I) = R ∨ (n, d, I) R(n, d, I\{i 1 }) with R ∨ (n, d, I) := {α ∈ R(n, d, I) | α i1 = −1}.
a set of cardinality < k. (Note that for n ∨ + d ∨ < n + d this assumption already holds by our earlier induction hypothesis.)Let I = {i 1 , . . ., i k } ⊆ [1, n] be a set of cardinality k and consider a function f : T (n, d, I) → C. We have to prove the existence of an interpolation Laurent polynomial f ∈ P n satisfying (4.4).Consider this time the decomposition

j
(x α ) (j ∈ [1, n)) are Laurent polynomials of degree at most |α|.Clearly, it is sufficient to prove the first claim for n = 1 and the second claim for n = 2 and j = 1.

Figure 7 .Figure 9 .
Figure 7. α = (−2, 2) Figure 8. α = (−3, 1) wα if f (x) = x α .Note that |wα| = |α| for w ∈ W n and α ∈ Λ n .In particular, deg(wf ) = deg(f ) for w ∈ W n and f ∈ P n .Hence the W n -action on P n is an action by filtered algebra automorphisms and induces a W n -action by graded algebra automorphisms on the associated graded algebra G(P n ).We write P Wn n , P Wn n,d and G Wn n,d for the subspaces of W n -invariant elements in P n , P n,d and G n,d , respectively.By construction the associated graded algebra G(P This is independent of the choice of the reduced expression; see [9, Prop.1.15].Define the Hecke symmetrizer of H n (t, t n ) by ] introduced a one-parameter family of representations of H n (t, t n ) on P n in terms of Demazure-Lusztig type operators [16, Prop.3.6].Concretely, it is given by