UvA-DARE (Digital Academic Repository) Some Remarks on Non-Symmetric Interpolation Macdonald Polynomials

We provide elementary identities relating the three known types of non-symmetric interpolation Macdonald polynomials. In addition we derive a duality for non-symmetric interpolation Macdonald polynomials. We consider some applications of these results, in particular to binomial formulas involving non-symmetric interpolation Macdonald polynomials

The normalization is fixed by requiring that the coefficient of x λ := x λ 1 1 • • • x λn n in the monomial expansion of R λ (x) is 1.In spite of their deceptively simple definition, these polynomials possess some truly remarkable properties.For instance, as shown in [3,11], the top homogeneous part of R λ (x) is the Macdonald polynomial P λ (x) [8] and R λ (x) satisfies the extra vanishing property R λ (µ) = 0 unless λ ⊆ µ as Ferrer diagrams.Other key properties of R λ (x), which were proven by A. Okounkov [9], include the binomial theorem, which gives an explicit expansion of R λ (ax) = R λ (ax 1 , . . ., ax n ; q, t) in terms of the R µ (x; q −1 , t −1 )'s over the field K := Q(q, t, a), and the duality or evaluation symmetry, which involves the evaluation points µ = q −µn τ 1 , . . ., q −µ 1 τ n , µ ∈ P n and takes the form R λ (a µ) R λ (aτ ) = R µ (a λ) R µ (aτ ) .
The interpolation polynomials have natural non-symmetric analogs G α (x) = G α (x; q, t), which were also defined in [3,11].These are indexed by the set of compositions with at most n parts, C n := Z ≥0 n .For a composition β ∈ C n we define where w β is the shortest permutation such that β + = w −1 β (β) is a partition.Then G α (x) is, up to normalization, characterized as the unique polynomial of degree at most |α| := α 1 + • • • + α n satisfying the vanishing conditions The normalization is fixed by requiring that the coefficient of Many properties of the symmetric interpolation polynomials R λ (x) admit non-symmetric counterparts for the G α (x).For instance, the top homogeneous part of G α (x) is the nonsymmetric Macdonald polynomial E α (x) and G α (x) satisfies an extra vanishing property [3].An analog of the binomial theorem, proved by one of us in [12,Thm. 1.1], gives an explicit expansion of G α (ax; q, t) in terms of a second family of interpolation polynomials G ′ α (x) = G ′ α (x; q, t).These latter polynomials are characterized by having the same top homogeneous part as G α (x), namely the non-symmetric polynomial E α (x), and the following vanishing conditions at the evaluation points β := (−w 0 β), with w 0 the longest element of the symmetric group S n : The first result of the present paper is a Demazure-type formula for the primed interpolation polynomials G ′ α (x) in terms of G α (x), which involves the symmetric group action on the algebra of polynomials in n variables over F by permuting the variables, as well as the associated Hecke algebra action in terms of Demazure-Lusztig operators H w (w ∈ S n ) as described in the next section.
This is restated and proved in Theorem 1 below.The second result is the following duality theorem for G α (x), which is the non-symmetric analog of Okounkov's duality result.
This is a special case of Theorem 17 below.We now recall the interpolation O-polynomials introduced in [12, Thm.
Our third result is a simple expression for the O-polynomials in terms of the interpolation polynomials G α (x).
Theorem C. For all compositions α ∈ C n we have This is deduced in Proposition 22 below as a direct consequence of non-symmetric duality.We also obtain new proofs of Okounkov's [9] duality theorem, as well as the dual binomial theorem of A. Lascoux, E. Rains and O. Warnaar [7], which gives an expansion of the primed-interpolation polynomials G ′ α (x) in terms of the G β (ax)'s.
Acknowledgements: We thank Eric Rains for sharing with us his unpublished results with Alain Lascoux and Ole Warnaar on a one-parameter rational extension of the nonsymmetric interpolation Macdonald polynomials.It leads to a different proof of the duality of the non-symmetric interpolation Macdonald polynomials (Theorem B).We thank an anonymous referee for detailed comments.The research of S. Sahi was partially supported by a Simons Foundation grant (509766).
Let S n be the symmetric group in n letters and s i ∈ S n the permutation that swaps i and i + 1.The s i (1 ≤ i < n) are Coxeter generators for S n .Let ℓ : S n → Z ≥0 be the associated length function.Let S n act on Z n and K n by 1 , . . ., x ±1 n ] and F (x) for the quotient field of F [x].The symmetric group acts by algebra automorphisms on F[x] and F(x), with the action of s i by interchanging x i and x i+1 for 1 ≤ i < n.Consider the F-linear operators on F(x) (1 ≤ i < n) called Demazure-Lusztig operators, and the automorphism ∆ of F(x) defined by ∆f (x 1 , . . ., x n ) = f (q −1 x n , x 1 , . . ., x n−1 ).Note that H i (1 ≤ i < n) and ∆ preserve F[x ±1 ] and F[x].Cherednik [1,2] showed that the operators H i (1 ≤ i < n) and ∆ satisfy the defining relations of the type A extended affine Hecke algebra, (H i − t)(H i + 1) = 0, for all the indices such that both sides of the equation make sense (see also [3, §3]).For w ∈ S n we write It is well defined because of the braid relations for the H i 's.Write and set (1) The operators ξ i 's are pairwise commuting invertible operators, with inverses and normalized such that the coefficient of x α in E α is 1.
Let ι be the field automorphism of K inverting q, t and a.It restricts to a field automorphism of F, inverting q and t.We extend ι to a Q-algebra automorphism of K[x] and F[x] by letting ι act on the coefficients of the polynomial.Write and ξ i with q, t replaced by their inverses.For instance, We then have ξ Remark.Formally set t = q r , replace x by 1 + (q − 1)x, divide both sides of ( 2) by (q − 1) |α| and take the limit q → 1.Then for the non-symmetric interpolation Jack polynomial G α (• ; r) and its primed version (see [12]).Here σ denotes the action of the symmetric group with σ(s i ) the rational degeneration of the Demazure-Lusztig operators H i , given explicitly by see [12, §1].To establish the formal limit (3) one uses that σ(w 0 )w 0 = w 0 σ • (w 0 ) with σ • the action of the symmetric group defined in terms of the rational degeneration 3) was obtained before in [12, Thm.1.10].Proof.We show that the right hand side of (2) satisfies the defining properties of G ′ α .For the vanishing property, note that (4) (this is the q-analog of [12, Lem.6.1(2)]), hence It remains to show that the top homogeneous terms of both sides of (2) are the same, i.e. that ( 5) Consider the Demazure operators H i and the Cherednik operators ξ −1 j as operators on the space F[x ±1 ] of Laurent polynomials.For an integral vector u ∈ Z n , let E u ∈ F[x ±1 ] be the common eigenfunction of the Cherednik operators ξ −1 j with eigenvalues u j (1 ≤ j ≤ n), normalized such that the coefficient of It is now easy to check that formula ( 5) is valid with α replaced by an arbitrary integral vector u, ).Furthermore, one can show in the same vein as the proof of ( 5) that for an integral vector u, where p(x −1 ) stands for inverting all the parameters x 1 , . . ., x n in the Laurent polynomial p(x) ∈ F[x ±1 ].Combining this equality with ( 7) yields , which is a special case of a known identity for non-symmetric Macdonald polynomials (see [2,Prop. 3.3.3]).

Evaluation formulas
In [12, Thm.1.1] the following combinatorial evaluation formula was obtained, with a(s), l(s), a ′ (s) and l ′ (s) the arm, leg, coarm and coleg of s = (i, j) ∈ α, defined by By (8) we have which is the well known evaluation formula [1,2] for the non-symmetric Macdonald polynomials.Note that for α ∈ C n , Lemma 2. For α ∈ C n we have where we have used [12, Lem.2.1(2)] for the second equality.
We now derive a relation between the evaluation formulas for G α (x) and G • α (x).To formulate this we write, following [7], 2 , hence it only depends on the S n -orbit of α, while ( 9) The following lemma is a non-symmetric version of the first displayed formula on [9, Page 537].
Lemma 3.For α ∈ C n we have . Proof.This follows from the explicit evaluation formula (8) for the non-symmetric interpolation Macdonald polynomial G α .
It only depends on the S n -orbit of α.

Normalized interpolation Macdonald polynomials
We need the basic representation of the (double) affine Hecke algebra on the space of K-valued functions on Z n , which is constructed as follows.
Lemma 5. Let v ∈ Z n and 1 ≤ i < n.Then we have 1.
Let H be the double affine Hecke algebra over K.It is isomorphic to the subalgebra of End(K[x ±1 ]) generated by the operators H i (1 ≤ i < n), ∆ ±1 , and the multiplication operators x ±1 j (1 ≤ j ≤ n).For a unital K-algebra A we write F A for the space of A-valued functions f : Z n → A on Z n .Corollary 6.Let A be a unital K-algebra.Consider the A-linear operators H i (1 ≤ i < n), ∆ and x j (1 ≤ j ≤ n) on F A defined by

and hence turns F O
A into a H-module.Define the surjective A-linear map pr : We claim that Ker(pr) is a H-submodule of F O A .Clearly Ker(pr) is x j -invariant for j = 1, . . ., n.Let g ∈ Ker(pr).Part 3 of Lemma 5 implies that ∆g ∈ Ker(pr).To show that H i g ∈ Ker(pr) we consider two cases.
Hence F A inherits the H-module structure of F O A /Ker(pr).It is a straightforward computation, using Lemma 5 again, to show that the resulting action of

Remark 7. With the notations from (the proof of) Corollary 6, let g ∈ F O
A and set g := pr( g) ∈ F A .In other words, g(v) := g(av) for all v ∈ Z n .Then Remark 8. Let F + A be the space of A-valued functions on C n .We sometimes will consider H i (1 ≤ i < n), ∆ −1 and x j (1 ≤ j ≤ n), defined by the formulas (11), as linear operators on F + A .
We frequently use the shorthand notation K α (x) := K α (x; q, t; a).We will see in a moment that formulas for non-symmetric interpolation Macdonald polynomials take the nicest form in this particular normalization.
Note that a cannot be specialized to 1 in (12) since Recall from [3] the operator Φ = (x n − t 1−n )∆ ∈ H and the inhomogeneous Cherednik operators The operators H i , Ξ j and Φ preserve K[x] (see [3]), hence they give rise to K-linear operators on F + K[x] (e.g., (H i f )(α) := H i (f (α)) for α ∈ C n ).Note that the operators H i , Ξ j and Φ on F + K[x] commute with the hat-operators H i , x j and ∆ −1 on F + K[x] (cf.Remark 8).The same remarks hold true for the space F K(x) of K(x)-valued functions on Z n (in fact, in this case the hat-operators define a H-action on F K(x) ). Let Lemma 10.For 1 ≤ i < n and 1 ≤ j ≤ n we have in 1.
To derive the formula we need to expand H i K α as a linear combination of the K β 's.As a first step we expand H i G α as linear combination of the G β 's.
If α ∈ C n satisfies α i < α i+1 then  [3,Cor. 3.4].An explicit expansion of H i K α as linear combination of the K β 's can now be obtained using the formula for α ∈ C n satisfying α i > α i+1 , cf. the proof of [12,Lem 3.1].By a direct computation the resulting expansion formula can be written as By the evaluation formula (8) we have for α ∈ C n since α −1 = t n−1 w 0 α.

Interpolation Macdonald polynomials with negative degrees
In this section we give the natural extension of the interpolation Macdonald polynomials G α (x) and K α (x) to α ∈ Z n .It will be the unique extension of K ∈ F + K[x] to a map K ∈ F K(x) such that Lemma 10 remains valid.Lemma 12.For α ∈ C n we have The first formula then follows by iteration of [12, Lem.2.2(1)] and the second formula from part 3 of Lemma 10.
For m ∈ Z ≥0 we define A m (x; v) ∈ K(x) by ( 14) with y; q m := m−1 j=0 (1 − q j y) the q-shifted factorial.Definition 13.Let v ∈ Z n and write |v| where m is a nonnegative integer such that v + (m n ) ∈ C n (note that G v and K v are well defined by Lemma 12).
Example 14.If n = 1 then for m ∈ Z ≥0 , Proof.Let v ∈ Z n .Clearly G v (x) and K v (x) only differ by a multiplicative constant, so it suffices to show that where the last formula follows from a direct computation using the evaluation formula (8).
We extend the map K : by setting v → K v (x) for all v ∈ Z n .Lemma 10 now extends as follows.
Proposition 16.We have, as identities in F K(x) , 1. ( by part 2 of Lemma 5 and the fact that This proves part 1 of the proposition.
for arbitrary v ∈ Z n by Lemma 10 and the commutation relation where Φ (q m ) := (q m x n − t 1−n )∆.This proves part 3 of the proposition.Finally we have 16) and Lemma 10.This proves part 2 of the proposition.

Recall the notation
Theorem 17 (Duality).For all u, v ∈ Z n we have Example 18.If n = 1 and m, r ∈ Z ≥0 then (18) K m (aq −r ) = q −mr (a −1 ; q) m+r (a −1 ; q) m (a −1 ; q) r by the explicit expression for K m (x) from Example 14.The right hand side of (18) is manifestly invariant under the interchange of m and r.
Proof.We divide the proof of the theorem in several steps.
for all u, v ∈ Z n , which follows by writing out the formula from part 3 of Lemma 16.Hence we obtain where we used the induction hypothesis for the third equality and (21) for the second and fourth equality.This proves the induction step.
where we used step 3 in the third equality.The result now follows from the fact that which follows by a straightforward computation using (4).
7. Some applications of duality 7.1.Non-symmetric Macdonald polynomials.Recall that the (monic) non-symmetric Macdonald polynomial E α (x) of degree α is the top homogeneous component of G α (x), i.e.
The normalized non-symmetric Macdonald polynomials are We write K ∈ F + F[x] for the resulting map α → K α .Taking limits in Lemma 10 we get Lemma 19.We have for 1 ≤ i < n and 1 ≤ j ≤ n, 1.
Then repeated application of part 3 of Lemma 19 shows that for α ∈ C n , As is well known and already noted in Section 2, the first equality allows to relate the non-symmetric Macdonald polynomials E v (x) := E v (x; q, t) ∈ F[x ±1 ] for arbitrary v ∈ Z n to those labeled by compositions through the formula The second formula of ( 22) can now be used to explicitly define the normalized nonsymmetric Macdonald polynomials for degrees v ∈ Z n .
] is defined by and the definitions of G v (x) and K v (x) it follows that for all v ∈ Z n , so in particular Lemma 19 holds true for the extension of K to the map Taking the limit in Theorem 17 we obtain the well known duality [1] of the Laurent polynomial versions of the normalized non-symmetric Macdonald polynomials.
Corollary 21.For all u, v ∈ Z n , Sn is the multiple of C + G λ such that the coefficient of x λ is one (see, e.g., [11]).We write for the normalized symmetric interpolation Macdonald polynomial.Then (23) for α ∈ C n .Okounkov's [9, §2] duality result now reads as follows.

7.4.
A primed version of duality.We first derive the following twisted version of the duality of the non-symmetric interpolation Macdonald polynomials (Theorem 17).
Lemma 24.For u, v ∈ Z n we have Proof.We proceed as in the previous subsection.Set 4), Remark 7 implies that Now H w 0 Jw 0 = Jw 0 H w 0 by (26), hence which completes the proof.
Proposition 26.For all u, v ∈ Z n we have Proof.Note that by (4).By (27) the right hand side is invariant under the interchange of u and v. Applying the automorphism ι of F to (29) we get ) .
Theorem 27.For α, β ∈ C n we have the binomial formula .
Now use (28) to complete the proof of (32).
[12,2, Lem.2.2].Using part 1 of Lemma 5 and the fact that H i satisfies the quadratic relation (H i − t)(H i + 1) = 0, it follows that [12,O-polynomials.We now show that the duality of the non-symmetric interpolation Macdonald polynomials (Theorem 17) directly implies the existence of the O-polynomials O α (which is the nontrivial part of the proof of[12, Thm.1.2]),and that it provides an explicit expression for O α in terms of the non-symmetric interpolation Macdonald polynomial K α .
α) for all β ∈ C n by (4) and Theorem 17. Hence O α = O α .7.3.Okounkov's duality.Write F [x] Sn for the symmetric polynomials in x 1 , . . ., x n with coefficients in a field F .Write C + := w∈Sn H w .The symmetric interpolation Macdonald polynomial R λ