Some remarks on non-symmetric interpolation Macdonald polynomials

TL;DR

This paper establishes elementary identities and duality relations for non-symmetric interpolation Macdonald polynomials, enhancing understanding and providing new tools for their application in binomial formulas.

Contribution

It introduces new elementary identities and a duality for non-symmetric interpolation Macdonald polynomials, expanding theoretical understanding and potential applications.

Findings

Derived elementary identities relating different types of non-symmetric interpolation Macdonald polynomials

Established a duality property for these polynomials

Applied results to binomial formulas involving the polynomials

Abstract

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 for binomial formulas involving non-symmetric interpolation Macdonald polynomials.

Equations

$${\mathcal{P}}_n:=\left\{\lambda \in {\mathbb{Z}}^n|{\lambda }_1\ge {\lambda }_2\ge \cdots \ge {\lambda }_n\ge 0\right\}.$$

$$\overline{\mu }=\left(q^{{\mu }_1}{\tau }_1,\dots ,q^{{\mu }_n}{\tau }_n\right)\text{ where }\tau :=({\tau }_1,\dots ,{\tau }_n)\text{ with }{\tau }_i:=t^{1-i}.$$

$$R_{\lambda }(\overline{\mu })=0\text{ for }\mu \in {\mathcal{P}}_n\text{ such that }\left|\mu \right|\le \left|\lambda \right|,\mu \ne \lambda .$$

$$\tilde{\mu }=\left(q^{-{\mu }_n}{\tau }_1,\dots ,q^{-{\mu }_1}{\tau }_n\right),\mu \in {\mathcal{P}}_n$$

$$\frac{R_{\lambda }(a\tilde{\mu })}{R_{\lambda }(a\tau )}=\frac{R_{\mu }(a\tilde{\lambda })}{R_{\mu }(a\tau )}.$$

$$\overline{\beta }:=w_{\beta }(\overline{{\beta }_{+}})$$

$$G_{\alpha }(\overline{\beta })=0\text{ for }\beta \in {\mathcal{C}}_n\text{ such that }\left|\beta \right|\le \left|\alpha \right|,\beta \ne \alpha .$$

$$G_{\alpha }^{\prime }(\tilde{\beta })=0\text{ for }\left|\beta \right|<\left|\alpha \right|.$$

$$G_{\alpha }^{\prime }(t^{n-1}x;q^{-1},t^{-1})=t^{(n-1)|\alpha |-\text{I}(\alpha )}{w}_0{H}_{w_0}{G}_{\alpha }(x;q,t).$$

$$\frac{G_{\alpha }(a\tilde{\beta })}{G_{\alpha }(a\tau )}=\frac{G_{\beta }(a\tilde{\alpha })}{G_{\beta }(a\tau )}.$$

$$O_{\alpha }({\overline{\beta }}^{-1})=\frac{G_{\beta }(a\tilde{\alpha })}{G_{\beta }(a\tau )}\text{ for all }\beta .$$

$$O_{\alpha }\left(x\right)=\frac{G_{\alpha }(t^{1-n}a{w}_{0}x)}{G_{\alpha }(a\tau )}.$$

$$k_i(v):=\#\{k<i|v_k\ge v_i\}+\#\{k>i|v_k>v_i\}.$$

$$H_i=t{s}_i-\frac{(1-t)x_i}{x_i-x_{i+1}}(1-s_i)=t+\frac{x_i-t{x}_{i+1}}{x_i-x_{i+1}}(s_i-1)$$

$$\Delta f(x_1,\dots ,x_n)=f(q^{-1}{x}_n,x_1,\dots ,x_{n-1}).$$

$$\begin{array}{rl}(H_i-t)(H_i+1)& =0,\\ H_i{H}_j& =H_j{H}_i,|i-j|>1,\\ H_i{H}_{i+1}{H}_i& =H_{i+1}{H}_i{H}_{i+1},\\ \Delta H_{i+1}& =H_i\Delta ,\\ {\Delta }^2{H}_1& =H_{n-1}{\Delta }^2\end{array}$$

$${\xi }_i:=t^{1-n}{\overline{H}}_{i-1}\cdots {\overline{H}}_1{\Delta }^{-1}{H}_{n-1}\cdots H_i,1\le i\le n.$$

$${\xi }_i^{-1}={\overline{H}}_i\cdots {\overline{H}}_{n-1}\Delta H_1\cdots H_{i-1}.$$

$${\xi }_i^{-1}{E}_{\alpha }={\overline{\alpha }}_i{E}_{\alpha },i=1,\dots ,n$$

$$G_{\alpha}^{\circ}:=\iota\bigl{(}G_{\alpha}\bigr{)},\qquad E_{\alpha}^{\circ}:=\iota\bigl{(}E_{\alpha}\bigr{)}$$

$$\begin{array}{rl}& H_i^{\circ }=t^{-1}{s}_i-\frac{(1-t^{-1})x_i}{x_i-x_{i+1}}(1-s_i),\\ & {\Delta }^{\circ }f(x_1,\dots ,x_n)=f(q{x}_n,x_1,\dots ,x_{n-1}).\end{array}$$

$$G_{\alpha }^{\prime }(x)=t^{(1-n)|\alpha |+\text{I}(\alpha )}{w}_0{H}_{w_0}^{\circ }{G}_{\alpha }^{\circ }(t^{n-1}x)$$

$$G_{\alpha }^{\prime }(x;r)=(-1{)}^{|\alpha |}\sigma (w_0)w_0{G}_{\alpha }(-x-(n-1)r;r)$$

$$\sigma (s_i)=s_i+\frac{r}{x_i-x_{i+1}}(1-s_i),$$

$${\sigma }^{\circ }(s_i)=s_i-\frac{r}{x_i-x_{i+1}}(1-s_i)$$

$$t^{n-1}{w}_0\tilde{\beta }={\overline{\beta }}^{-1}$$

$$\bigl{(}w_{0}H_{w_{0}}^{\circ}G_{\alpha}^{\circ}(t^{n-1}x)\bigr{)}|_{x=\widetilde{\beta}}=\bigl{(}H_{w_{0}}^{\circ}G_{\alpha}^{\circ}(x)\bigr{)}|_{x=\overline{\beta}^{-1}}.$$

$$E_{\alpha }=t^{\text{I}(\alpha )}{w}_0{H}_{w_0}^{\circ }{E}_{\alpha }^{\circ }.$$

$$\begin{array}{rl}{H}_i\Psi & =t\Psi {\overline{H}}_i^{\circ },\\ \Delta \Psi & =t^{n-1}\Psi {\overline{H}}_{n-1}^{\circ }\cdots {\overline{H}}_1^{\circ }({\Delta }^{\circ }{)}^{-1}{H}_{n-1}^{\circ }\cdots H_1^{\circ }\end{array}$$

$$E_{\alpha }(x)=c_{\alpha }\Psi E_{\alpha }^{\circ }(x)$$

Full text

1. Introduction

The symmetric interpolation Macdonald polynomials $R_{\lambda}(x;q,t)=R_{\lambda}(x_{1},\ldots,x_{n};q,t)$ form a distinguished inhomogeneous basis for the algebra of symmetric polynomials in $n$ variables over the field $\mathbb{F}:=\mathbb{Q}(q,t)$. They were first introduced in [3, 11], building on joint work by one of the authors with F. Knop [4] and earlier work with B. Kostant [5, 6, 10]. These polynomials are indexed by the set of partitions with at most $n$ parts

[TABLE]

For a partition $\mu\in\mathcal{P}_{n}$ we define $\left|\mu\right|=\mu_{1}+\cdots+\mu_{n}$ and write

[TABLE]

Then $R_{\lambda}(x)=R_{\lambda}(x;q,t)$ is, up to normalization, characterized as the unique nonzero symmetric polynomial of degree at most $\left|\lambda\right|$ satisfying the vanishing conditions

[TABLE]

The normalization is fixed by requiring that the coefficient of $x^{\lambda}:=x_{1}^{\lambda_{1}}\cdots x_{n}^{\lambda_{n}}$ in the monomial expansion of $R_{\lambda}(x)$ 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 $R_{\lambda}(x)$ is the Macdonald polynomial $P_{\lambda}(x)$ [8] and $R_{\lambda}(x)$ satisfies the extra vanishing property $R_{\lambda}(\overline{\mu})=0$ unless $\lambda\subseteq\mu$ as Ferrer diagrams. Other key properties of $R_{\lambda}(x)$, which were proven by A. Okounkov [9], include the binomial theorem, which gives an explicit expansion of $R_{\lambda}(ax)=R_{\lambda}(ax_{1},\ldots,ax_{n};q,t)$ in terms of the $R_{\mu}(x;q^{-1},t^{-1})$’s over the field $\mathbb{K}:=\mathbb{Q}(q,t,a),$ and the duality or evaluation symmetry, which involves the evaluation points

[TABLE]

and takes the form

[TABLE]

The interpolation polynomials have natural non-symmetric analogs $G_{\alpha}(x)=G_{\alpha}(x;q,t)$, which were also defined in [3, 11]. These are indexed by the set of compositions with at most $n$ parts, \mathcal{C}_{n}:=\bigl{(}\mathbb{Z}_{\geq 0}\bigr{)}^{n}. For a composition $\beta\in\mathcal{C}_{n}$ we define

[TABLE]

where $w_{\beta}$ is the shortest permutation such that $\beta_{+}=w_{\beta}^{-1}(\beta)$ is a partition. Then $G_{\alpha}(x)$ is, up to normalization, characterized as the unique polynomial of degree at most $\left|\alpha\right|:=\alpha_{1}+\cdots+\alpha_{n}$ satisfying the vanishing conditions

[TABLE]

The normalization is fixed by requiring that the coefficient of $x^{\alpha}:=x_{1}^{\alpha_{1}}\cdots x_{n}^{\alpha_{n}}$ in the monomial expansion of $G_{\alpha}(x)$ is $1$.

Many properties of the symmetric interpolation polynomials $R_{\lambda}(x)$ admit non-symmetric counterparts for the $G_{\alpha}(x)$. For instance, the top homogeneous part of $G_{\alpha}(x)$ is the non-symmetric Macdonald polynomial $E_{\alpha}(x)$ and $G_{\alpha}(x)$ 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 $G_{\alpha}(ax;q,t)$ in terms of a second family of interpolation polynomials $G_{\alpha}^{\prime}(x)=G_{\alpha}^{\prime}(x;q,t)$. These latter polynomials are characterized by having the same top homogeneous part as $G_{\alpha}(x)$, namely the non-symmetric polynomial $E_{\alpha}(x)$, and the following vanishing conditions at the evaluation points $\widetilde{\beta}:=\overline{(-w_{0}\beta)}$, with $w_{0}$ the longest element of the symmetric group $S_{n}$:

[TABLE]

The first result of the present paper is a Demazure-type formula for the primed interpolation polynomials $G_{\alpha}^{\prime}(x)$ in terms of $G_{\alpha}(x)$, which involves the symmetric group action on the algebra of polynomials in $n$ variables over $\mathbb{F}$ by permuting the variables, as well as the associated Hecke algebra action in terms of Demazure-Lusztig operators $H_{w}$ ($w\in S_{n}$) as described in the next section.

Theorem A. Write $\textup{I}(\alpha):=\#\left\{i<j\mid\alpha_{i}\geq\alpha_{j}\right\}$. Then we have

[TABLE]

This is restated and proved in Theorem 1 below.

The second result is the following duality theorem for $G_{\alpha}(x)$, which is the non-symmetric analog of Okounkov’s duality result.

Theorem B. For all compositions $\alpha,\beta\in\mathcal{C}_{n}$ we have

[TABLE]

This is a special case of Theorem 17 below.

We now recall the interpolation $O$-polynomials introduced in [12, Thm. 1.1]. Write $x^{-1}$ for $\left(x_{1}^{-1},\ldots,x_{n}^{-1}\right)$. Then it was shown in [12, Thm. 1.1] that there exists a unique polynomial $O_{\alpha}(x)=O_{\alpha}(x;q,t;a)$ of degree at most $\left|\alpha\right|$ with coefficients in the field $\mathbb{K}$ such that

[TABLE]

Our third result is a simple expression for the $O$-polynomials in terms of the interpolation polynomials $G_{\alpha}(x)$.

Theorem C. For all compositions $\alpha\in\mathcal{C}_{n}$ we have

[TABLE]

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 $G_{\alpha}^{\prime}(x)$ in terms of the $G_{\beta}(ax)$’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 non-symmetric 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).

2. Demazure-Lusztig operators and the primed interpolation polynomials

We use the notations from [12]. The correspondence with the notations from the other important references [3], [11] and [9] is listed in [12, §2] (directly after Lemma 2.8).

Let $S_{n}$ be the symmetric group in $n$ letters and $s_{i}\in S_{n}$ the permutation that swaps $i$ and $i+1$. The $s_{i}$ ($1\leq i<n$) are Coxeter generators for $S_{n}$. Let $\ell:S_{n}\rightarrow\mathbb{Z}_{\geq 0}$ be the associated length function. Let $S_{n}$ act on $\mathbb{Z}^{n}$ and $\mathbb{K}^{n}$ by $s_{i}v:=(\cdots,v_{i-1},v_{i+1},v_{i},v_{i+2},\ldots)$ for $v=(v_{1},\ldots,v_{n})$. Write $w_{0}\in S_{n}$ for the longest element, given explicitly by $i\rightarrow n+1-i$ for $i=1,\ldots,n$.

For $v=(v_{1},\ldots,v_{n})\in\mathbb{Z}^{n}$ define $\overline{v}=(\overline{v}_{1},\ldots,\overline{v}_{n})\in\mathbb{F}^{n}$ by $\overline{v}_{i}:=q^{v_{i}}t^{-k_{i}(v)}$ with

[TABLE]

If $v\in\mathbb{Z}^{n}$ has non-increasing entries $v_{1}\geq v_{2}\geq\cdots\geq v_{n}$, then $\overline{v}=(q^{v_{1}}\tau_{1},\ldots,q^{v_{n}}\tau_{n})$. For arbitrary $v\in\mathbb{Z}^{n}$ we have $\overline{v}=w_{v}(\overline{v_{+}})$ with $w_{v}\in S_{n}$ the shortest permutation such that $v_{+}:=w_{v}^{-1}(v)$ has nonincreasing entries, see [3, §2]. We write $\widetilde{v}:=\overline{-w_{0}v}$ for $v\in\mathbb{Z}^{n}$.

Note that $\overline{\alpha}_{n}=t^{1-n}$ if $\alpha\in\mathcal{C}_{n}$ with $\alpha_{n}=0$.

For a field $F$ we write $F[x]:=F[x_{1},\ldots,x_{n}]$, $F[x^{\pm 1}]:=F[x_{1}^{\pm 1},\ldots,x_{n}^{\pm 1}]$ and $F(x)$ for the quotient field of $F[x]$. The symmetric group acts by algebra automorphisms on $\mathbb{F}[x]$ and $\mathbb{F}(x)$, with the action of $s_{i}$ by interchanging $x_{i}$ and $x_{i+1}$ for $1\leq i<n$. Consider the $\mathbb{F}$-linear operators

[TABLE]

on $\mathbb{F}(x)$ ($1\leq i<n$) called Demazure-Lusztig operators, and the automorphism $\Delta$ of $\mathbb{F}(x)$ defined by

[TABLE]

Note that $H_{i}$ ($1\leq i<n$) and $\Delta$ preserve $\mathbb{F}[x^{\pm 1}]$ and $\mathbb{F}[x]$. Cherednik [1, 2] showed that the operators $H_{i}$ ($1\leq i<n$) and $\Delta$ satisfy the defining relations of the type A extended affine Hecke algebra,

[TABLE]

for all the indices such that both sides of the equation make sense (see also [3, §3]). For $w\in S_{n}$ we write $H_{w}:=H_{i_{1}}H_{i_{2}}\cdots H_{i_{\ell}}$ with $w=s_{i_{1}}s_{i_{2}}\cdots s_{i_{\ell}}$ a reduced expression for $w\in S_{n}$. It is well defined because of the braid relations for the $H_{i}$’s. Write $\overline{H}_{i}:=H_{i}+1-t=tH_{i}^{-1}$ and set

[TABLE]

The operators $\xi_{i}$’s are pairwise commuting invertible operators, with inverses

[TABLE]

The $\xi_{i}^{-1}$ ($1\leq i\leq n$) are the Cherednik operators [2, 3].

The monic non-symmetric Macdonald polynomial $E_{\alpha}\in\mathbb{F}[x]$ of degree $\alpha\in\mathcal{C}_{n}$ is the unique polynomial satisfying

[TABLE]

and normalized such that the coefficient of $x^{\alpha}$ in $E_{\alpha}$ is $1$.

Let $\iota$ be the field automorphism of $\mathbb{K}$ inverting $q$, $t$ and $a$. It restricts to a field automorphism of $\mathbb{F}$, inverting $q$ and $t$. We extend $\iota$ to a $\mathbb{Q}$-algebra automorphism of $\mathbb{K}[x]$ and $\mathbb{F}[x]$ by letting $\iota$ act on the coefficients of the polynomial. Write

[TABLE]

for $\alpha\in\mathcal{C}_{n}$. Note that $\overline{v}^{-1}=(\iota(\overline{v}_{1}),\ldots,\iota(\overline{v}_{n}))$.

Put $H_{i}^{\circ}$, $H_{w}^{\circ}$, $\overline{H}_{i}^{\circ}$, $\Delta^{\circ}$ and $\xi_{i}^{\circ}$ for the operators $H_{i}$, $H_{w}$, $\overline{H}_{i}$, $\Delta$ and $\xi_{i}$ with $q,t$ replaced by their inverses. For instance,

[TABLE]

We then have $\xi_{i}^{\circ}E_{\alpha}^{\circ}=\overline{\alpha}_{i}E_{\alpha}^{\circ}$ for $i=1,\ldots,n$, which characterizes $E_{\alpha}^{\circ}$ up to a scalar factor.

Theorem 1**.**

For $\alpha\in\mathcal{C}_{n}$ we have

[TABLE]

with $\textup{I}(\alpha):=\#\{i<j\,|\,\alpha_{i}\geq\alpha_{j}\}$.

Remark. Formally set $t=q^{r}$, replace $x$ by $1+(q-1)x$, divide both sides of (2) by $(q-1)^{|\alpha|}$ and take the limit $q\rightarrow 1$. Then

[TABLE]

for the non-symmetric interpolation Jack polynomial $G_{\alpha}(\cdot\,;r)$ and its primed version (see [12]). Here $\sigma$ denotes the action of the symmetric group with $\sigma(s_{i})$ the rational degeneration of the Demazure-Lusztig operators $H_{i}$, given explicitly by

[TABLE]

see [12, §1]. To establish the formal limit (3) one uses that $\sigma(w_{0})w_{0}=w_{0}\sigma^{\circ}(w_{0})$ with $\sigma^{\circ}$ the action of the symmetric group defined in terms of the rational degeneration

[TABLE]

of $H_{i}^{\circ}$. Formula (3) was obtained before in [12, Thm. 1.10].

Proof.

We show that the right hand side of (2) satisfies the defining properties of $G_{\alpha}^{\prime}$. For the vanishing property, note that

[TABLE]

(this is the $q$-analog of [12, Lem. 6.1(2)]), hence

[TABLE]

This expression is zero for $|\beta|<|\alpha|$ since it is a linear combination of the evaluated interpolation polynomials $G_{\alpha}^{\circ}(\overline{w\beta}^{-1})$ ($w\in S_{n}$) by [12, Lem. 2.1(2)].

It remains to show that the top homogeneous terms of both sides of (2) are the same, i.e. that

[TABLE]

Note that $\Psi:=w_{0}H_{w_{0}}^{\circ}$ satisfies the intertwining properties

[TABLE]

for $1\leq i<n$ (use e.g. [2, Prop. 3.2.2]). It follows that $\xi_{i}^{-1}\Psi=\Psi\xi_{i}^{\circ}$ for $i=1,\ldots,n$. Therefore

[TABLE]

for some constant $c_{\alpha}\in\mathbb{F}$. But the coefficient of $x^{\alpha}$ in $\Psi x^{\alpha}$ is $t^{-\textup{I}(\alpha)}$, hence $c_{\alpha}=t^{\textup{I}(\alpha)}$. ∎

Consider the Demazure operators $H_{i}$ and the Cherednik operators $\xi_{j}^{-1}$ as operators on the space $\mathbb{F}[x^{\pm 1}]$ of Laurent polynomials. For an integral vector $u\in\mathbb{Z}^{n}$, let $E_{u}\in\mathbb{F}[x^{\pm 1}]$ be the common eigenfunction of the Cherednik operators $\xi_{j}^{-1}$ with eigenvalues $\overline{u}_{j}$ ($1\leq j\leq n$), normalized such that the coefficient of $x^{u}:=x_{1}^{u_{1}}\cdots x_{n}^{u_{n}}$ in $E_{u}$ is $1$. For $u=\alpha\in\mathcal{C}_{n}$ this definition reproduces the non-symmetric Macdonald polynomial $E_{\alpha}\in\mathbb{F}[x]$ as defined before. Note that

[TABLE]

It is now easy to check that formula (5) is valid with $\alpha$ replaced by an arbitrary integral vector $u$,

[TABLE]

with $E_{u}^{\circ}:=\iota(E_{u})$. Furthermore, one can show in the same vein as the proof of (5) that

[TABLE]

for an integral vector $u$, where $p(x^{-1})$ stands for inverting all the parameters $x_{1},\ldots,x_{n}$ in the Laurent polynomial $p(x)\in\mathbb{F}[x^{\pm 1}]$. Combining this equality with (7) yields

[TABLE]

which is a special case of a known identity for non-symmetric Macdonald polynomials (see [2, Prop. 3.3.3]).

3. Evaluation formulas

In [12, Thm. 1.1] the following combinatorial evaluation formula

[TABLE]

was obtained, with $a(s)$, $l(s)$, $a^{\prime}(s)$ and $l^{\prime}(s)$ the arm, leg, coarm and coleg of $s=(i,j)\in\alpha$, defined by

[TABLE]

By (8) we have

[TABLE]

which is the well known evaluation formula [1, 2] for the non-symmetric Macdonald polynomials. Note that for $\alpha\in\mathcal{C}_{n}$,

[TABLE]

Lemma 2**.**

For $\alpha\in\mathcal{C}_{n}$ we have

[TABLE]

Proof.

Since $t^{n-1}w_{0}\tau=\tau^{-1}=\overline{0}^{-1}$ we have by Theorem 1,

[TABLE]

where we have used [12, Lem. 2.1(2)] for the second equality. ∎

We now derive a relation between the evaluation formulas for $G_{\alpha}(x)$ and $G_{\alpha}^{\circ}(x)$. To formulate this we write, following [7],

[TABLE]

Note that $n^{\prime}(\alpha)=\sum_{i=1}^{n}{{\alpha_{i}}\choose{2}}$, hence it only depends on the $S_{n}$-orbit of $\alpha$, while

[TABLE]

The following lemma is a non-symmetric version of the first displayed formula on [9, Page 537].

Lemma 3**.**

For $\alpha\in\mathcal{C}_{n}$ we have

[TABLE]

Proof.

This follows from the explicit evaluation formula (8) for the non-symmetric interpolation Macdonald polynomial $G_{\alpha}$. ∎

Following [7, (3.9)] we define $\tau_{\alpha}\in\mathbb{F}$ ($\alpha\in\mathcal{C}_{n}$) by

[TABLE]

It only depends on the $S_{n}$-orbit of $\alpha$.

Corollary 4**.**

For $\alpha\in\mathcal{C}_{n}$ we have

[TABLE]

Proof.

Use Lemma 2, Lemma 3 and (9). ∎

4. Normalized interpolation Macdonald polynomials

We need the basic representation of the (double) affine Hecke algebra on the space of $\mathbb{K}$-valued functions on $\mathbb{Z}^{n}$, which is constructed as follows.

For $v\in\mathbb{Z}^{n}$ and $y\in\mathbb{K}^{n}$ write $v^{\natural}:=(v_{2},\ldots,v_{n},v_{1}+1)$ and $y^{\natural}:=(y_{2},\ldots,y_{n},qy_{1})$. Denote the inverse of ♮ by ♯, so $v^{\sharp}=(v_{n}-1,v_{1},\ldots,v_{n-1})$ and $y^{\sharp}=(y_{n}/q,y_{1},\ldots,y_{n-1})$. We have the following lemma (cf. [3, 12, 11]).

Lemma 5**.**

Let $v\in\mathbb{Z}^{n}$ and $1\leq i<n$. Then we have

1. 1.

$s_{i}(\overline{v})=\overline{s_{i}v}$* if $v_{i}\not=v_{i+1}$.* 2. 2.

$\overline{v}_{i}=t\overline{v}_{i+1}$* if $v_{i}=v_{i+1}$.* 3. 3.

$\overline{v}^{\natural}=\overline{v^{\natural}}$.

Let $\mathbb{H}$ be the double affine Hecke algebra over $\mathbb{K}$. It is isomorphic to the subalgebra of $\textup{End}(\mathbb{K}[x^{\pm 1}])$ generated by the operators $H_{i}$ ($1\leq i<n$), $\Delta^{\pm 1}$, and the multiplication operators $x_{j}^{\pm 1}$ ($1\leq j\leq n$).

For a unital $\mathbb{K}$-algebra $A$ we write $\mathcal{F}_{A}$ for the space of $A$-valued functions $f:\mathbb{Z}^{n}\rightarrow A$ on $\mathbb{Z}^{n}$.

Corollary 6**.**

Let $A$ be a unital $\mathbb{K}$-algebra. Consider the $A$-linear operators $\widehat{H}_{i}$ ($1\leq i<n$), $\widehat{\Delta}$ and $\widehat{x}_{j}$ ($1\leq j\leq n$) on $\mathcal{F}_{A}$ defined by

[TABLE]

for $f\in\mathcal{F}_{A}$ and $v\in\mathbb{Z}^{n}$. Then $H_{i}\mapsto\widehat{H}_{i}$ ($1\leq i<n$), $\Delta\mapsto\widehat{\Delta}$ and $x_{j}\mapsto\widehat{x}_{j}$ ($1\leq j\leq n$) defines a representation $\mathbb{H}\rightarrow\textup{End}_{A}(\mathcal{F}_{A})$, $X\mapsto\widehat{X}$ ($X\in\mathbb{H}$) of the double affine Hecke algebra $\mathbb{H}$ on $\mathcal{F}_{A}$.

Proof.

Let $\mathcal{O}\subset\mathbb{K}^{n}$ be the smallest $S_{n}$-invariant and $\natural$-invariant subset which contains $\{a\overline{v}\,\,|\,\,v\in\mathbb{Z}^{n}\}$. Note that $\mathcal{O}$ is contained in $\{y\in\mathbb{K}^{n}\,\,|\,\,y_{i}\not=y_{j}\hbox{ if }i\not=j\}$. The Demazure-Lusztig operators $H_{i}$ ($1\leq i<n$), $\Delta^{\pm 1}$ and the coordinate multiplication operators $x_{j}$ ($1\leq j\leq n$) act $A$-linearly on the space $F_{A}^{\mathcal{O}}$ of $A$-valued functions on $\mathcal{O}$, and hence turns $F_{A}^{\mathcal{O}}$ into a $\mathbb{H}$-module. Define the surjective $A$-linear map

[TABLE]

by $\textup{pr}(g)(v):=g(a\overline{v})$ ($v\in\mathbb{Z}^{n}$).

We claim that $\textup{Ker}(\textup{pr})$ is a $\mathbb{H}$-submodule of $F^{\mathcal{O}}_{A}$. Clearly $\textup{Ker}(\textup{pr})$ is $x_{j}$-invariant for $j=1,\ldots,n$. Let $g\in\textup{Ker}(\textup{pr})$. Part 3 of Lemma 5 implies that $\Delta g\in\textup{Ker}(\textup{pr})$. To show that $H_{i}g\in\textup{Ker}(\textup{pr})$ we consider two cases. If $v_{i}\not=v_{i+1}$ then $s_{i}\overline{v}=\overline{s_{i}v}$ by part 1 of Lemma 5. Hence

[TABLE]

If $v_{i}=v_{i+1}$ then $\overline{v}_{i}=t\overline{v}_{i+1}$ by part 2 of Lemma 5. Hence

[TABLE]

Hence $\mathcal{F}_{A}$ inherits the $\mathbb{H}$-module structure of $F^{\mathcal{O}}_{A}/\textup{Ker}(\textup{pr})$. It is a straightforward computation, using Lemma 5 again, to show that the resulting action of $H_{i}$ ($1\leq i<n$), $\Delta$ and $x_{j}$ ($1\leq j\leq n$) on $\mathcal{F}_{A}$ is by the operators $\widehat{H}_{i}$ ($1\leq i<n$), $\widehat{\Delta}$ and $\widehat{x}_{j}$ ($1\leq j\leq n$). ∎

Remark 7*.*

With the notations from (the proof of) Corollary 6, let $\widetilde{g}\in F_{A}^{\mathcal{O}}$ and set $g:=\textup{pr}(\widetilde{g})\in\mathcal{F}_{A}$. In other words, $g(v):=\widetilde{g}(a\overline{v})$ for all $v\in\mathbb{Z}^{n}$. Then

[TABLE]

for $X=H_{i},\Delta^{\pm 1},x_{j}$.

Remark 8*.*

Let $\mathcal{F}^{+}_{A}$ be the space of $A$-valued functions on $\mathcal{C}_{n}$. We sometimes will consider $\widehat{H}_{i}$ ($1\leq i<n$), $\widehat{\Delta}^{-1}$ and $\widehat{x}_{j}$ ($1\leq j\leq n$), defined by the formulas (11), as linear operators on $\mathcal{F}^{+}_{A}$.

Definition 9**.**

We call

[TABLE]

the normalized non-symmetric interpolation Macdonald polynomial of degree $\alpha$.

We frequently use the shorthand notation $K_{\alpha}(x):=K_{\alpha}(x;q,t;a)$. 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 $G_{\alpha}(\tau)=G_{\alpha}(\overline{0})=0$ if $0\not=\alpha\in\mathcal{C}_{n}$. Note furthermore that

[TABLE]

since $\lim_{a\rightarrow\infty}a^{-|\alpha|}G_{\alpha}(ax)=E_{\alpha}(x)$.

Recall from [3] the operator $\Phi=(x_{n}-t^{1-n})\Delta\in\mathbb{H}$ and the inhomogeneous Cherednik operators

[TABLE]

The operators $H_{i}$, $\Xi_{j}$ and $\Phi$ preserve $\mathbb{K}[x]$ (see [3]), hence they give rise to $\mathbb{K}$-linear operators on $\mathcal{F}^{+}_{\mathbb{K}[x]}$ (e.g., $(H_{i}f)(\alpha):=H_{i}(f(\alpha))$ for $\alpha\in\mathcal{C}_{n}$). Note that the operators $H_{i},\Xi_{j}$ and $\Phi$ on $\mathcal{F}_{\mathbb{K}[x]}^{+}$ commute with the hat-operators $\widehat{H}_{i}$, $\widehat{x}_{j}$ and $\widehat{\Delta}^{-1}$ on $\mathcal{F}_{\mathbb{K}[x]}^{+}$ (cf. Remark 8). The same remarks hold true for the space $\mathcal{F}_{\mathbb{K}(x)}$ of $\mathbb{K}(x)$-valued functions on $\mathbb{Z}^{n}$ (in fact, in this case the hat-operators define a $\mathbb{H}$-action on $\mathcal{F}_{\mathbb{K}(x)}$).

Let $K\in\mathcal{F}^{+}_{\mathbb{K}[x]}$ be the map $\alpha\mapsto K_{\alpha}(\cdot)$ ($\alpha\in\mathcal{C}_{n}$).

Lemma 10**.**

For $1\leq i<n$ and $1\leq j\leq n$ we have in $\mathcal{F}_{\mathbb{K}[x]}^{+}$,

1. 1.

$H_{i}K=\widehat{H}_{i}K$. 2. 2.

$\Xi_{j}K=a\widehat{x}_{j}^{-1}K$. 3. 3.

$\Phi K=t^{1-n}(a^{2}\widehat{x}_{1}^{-1}-1)\widehat{\Delta}^{-1}K$.

Proof.

1. To derive the formula we need to expand $H_{i}K_{\alpha}$ as a linear combination of the $K_{\beta}$’s. As a first step we expand $H_{i}G_{\alpha}$ as linear combination of the $G_{\beta}$’s.

If $\alpha\in\mathcal{C}_{n}$ satisfies $\alpha_{i}<\alpha_{i+1}$ then

[TABLE]

by [12, 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

[TABLE]

if $\alpha\in\mathcal{C}_{n}$ satisfies $\alpha_{i}>\alpha_{i+1}$. Finally, $H_{i}G_{\alpha}(x)=tG_{\alpha}(x)$ if $\alpha\in\mathcal{C}_{n}$ satisfies $\alpha_{i}=\alpha_{i+1}$ by [3, Cor. 3.4].

An explicit expansion of $H_{i}K_{\alpha}$ as linear combination of the $K_{\beta}$’s can now be obtained using the formula

[TABLE]

for $\alpha\in\mathcal{C}_{n}$ satisfying $\alpha_{i}>\alpha_{i+1}$, cf. the proof of [12, Lem 3.1]. By a direct computation the resulting expansion formula can be written as $H_{i}K=\widehat{H}_{i}K$.

2. See [3, Thm. 2.6].

3. Let $\alpha\in\mathcal{C}_{n}$. By [12, Lem. 2.2 (1)],

[TABLE]

By the evaluation formula (8) we have

[TABLE]

Hence

[TABLE]

∎

Remark 11*.*

Note that

[TABLE]

for $\alpha\in\mathcal{C}_{n}$ since $\overline{\alpha}^{-1}=t^{n-1}w_{0}\widetilde{\alpha}$.

5. Interpolation Macdonald polynomials with negative degrees

In this section we give the natural extension of the interpolation Macdonald polynomials $G_{\alpha}(x)$ and $K_{\alpha}(x)$ to $\alpha\in\mathbb{Z}^{n}$. It will be the unique extension of $K\in\mathcal{F}_{\mathbb{K}[x]}^{+}$ to a map $K\in\mathcal{F}_{\mathbb{K}(x)}$ such that Lemma 10 remains valid.

Lemma 12**.**

For $\alpha\in\mathcal{C}_{n}$ we have

[TABLE]

Proof.

Note that for $f\in\mathbb{K}[x]$,

[TABLE]

The first formula then follows by iteration of [12, Lem. 2.2(1)] and the second formula from part 3 of Lemma 10. ∎

For $m\in\mathbb{Z}_{\geq 0}$ we define $A_{m}(x;v)\in\mathbb{K}(x)$ by

[TABLE]

with \bigl{(}y;q\bigr{)}_{m}:=\prod_{j=0}^{m-1}(1-q^{j}y) the $q$-shifted factorial.

Definition 13**.**

Let $v\in\mathbb{Z}^{n}$ and write $|v|:=v_{1}+\cdots+v_{n}$. Define $G_{v}(x)=G_{v}(x;q,t)\in\mathbb{F}(x)$ and $K_{v}(x)=K_{v}(x;q,t;a)\in\mathbb{K}(x)$ by

[TABLE]

where $m$ is a nonnegative integer such that $v+(m^{n})\in\mathcal{C}_{n}$ (note that $G_{v}$ and $K_{v}$ are well defined by Lemma 12).

Example 14*.*

If $n=1$ then for $m\in\mathbb{Z}_{\geq 0}$,

[TABLE]

Lemma 15**.**

For all $v\in\mathbb{Z}^{n}$,

[TABLE]

Proof.

Let $v\in\mathbb{Z}^{n}$. Clearly $G_{v}(x)$ and $K_{v}(x)$ only differ by a multiplicative constant, so it suffices to show that $K_{v}(a\tau)=1$. Fix $m\in\mathbb{Z}_{\geq 0}$ such that $v+(m^{n})\in\mathcal{C}_{n}$. Then

[TABLE]

where the last formula follows from a direct computation using the evaluation formula (8). ∎

We extend the map $K:\mathcal{C}_{n}\rightarrow\mathbb{K}[x]$ to a map

[TABLE]

by setting $v\mapsto K_{v}(x)$ for all $v\in\mathbb{Z}^{n}$. Lemma 10 now extends as follows.

Proposition 16**.**

We have, as identities in $\mathcal{F}_{\mathbb{K}(x)}$,

1. 1.

$H_{i}K=\widehat{H}_{i}K$. 2. 2.

$\Xi_{j}K=a\widehat{x}_{j}^{-1}K$. 3. 3.

$\Phi K=t^{1-n}(a^{2}\widehat{x}_{1}^{-1}-1)\widehat{\Delta}^{-1}K$.

Proof.

Write $A_{m}\in\mathcal{F}_{\mathbb{K}(x)}$ for the map $v\mapsto A_{m}(x;v)$ for $v\in\mathbb{Z}^{n}$. Consider the linear operator on $\mathcal{F}_{\mathbb{K}(x)}$ defined by $(A_{m}f)(v):=A_{m}(x;v)f(v)$ for $v\in\mathbb{Z}^{n}$ and $f\in\mathcal{F}_{\mathbb{K}(x)}$. For $1\leq i<n$ we have $[H_{i},A_{m}]=0$ as linear operators on $\mathcal{F}_{\mathbb{K}(x)}$, since $A_{m}(x;v)$ is a symmetric rational function in $x_{1},\ldots,x_{n}$. Furthermore, for $v\in\mathbb{Z}^{n}$ and $f\in\mathcal{F}_{\mathbb{K}(x)}$,

[TABLE]

by part 2 of Lemma 5 and the fact that $A_{m}(x;v)$ is symmetric in $\overline{v}_{1},\ldots,\overline{v}_{n}$. Fix $v\in\mathbb{Z}^{n}$ and choose $m\in\mathbb{Z}_{\geq 0}$ such that $v+(m^{n})\in\mathcal{C}_{n}$. Since

[TABLE]

we obtain from $[H_{i},A_{m}]=0$ and (15) that $(H_{i}K)(v)=(\widehat{H}_{i}K)(v)$ if $v_{i}\not=v_{i+1}$. This also holds true if $v_{i}=v_{i+1}$ since then $(\widehat{H}_{i}K)(v)=tK_{v}$ and $H_{i}K_{v+(m^{n})}(q^{m}x)=tK_{v+(m^{n})}(q^{m}x)$. This proves part 1 of the proposition.

Note that $\Phi K_{v}(x)=t^{1-n}(a\overline{v}_{1}^{-1}-1)K_{v^{\natural}}(x)$ for arbitrary $v\in\mathbb{Z}^{n}$ by Lemma 10 and the commutation relation

[TABLE]

where $\Phi^{(q^{m})}:=(q^{m}x_{n}-t^{1-n})\Delta$. This proves part 3 of the proposition.

Finally we have $\Xi_{j}K_{v}(x)=\overline{v}_{j}^{-1}K_{v}(x)$ for all $v\in\mathbb{Z}^{n}$ by $[H_{i},A_{m}]=0$, (16) and Lemma 10. This proves part 2 of the proposition. ∎

6. Duality of the non-symmetric interpolation Macdonald polynomials

Recall the notation $\widetilde{v}=\overline{-w_{0}v}$ for $v\in\mathbb{Z}^{n}$.

Theorem 17** (Duality).**

For all $u,v\in\mathbb{Z}^{n}$ we have

[TABLE]

Example 18*.*

If $n=1$ and $m,r\in\mathbb{Z}_{\geq 0}$ then

[TABLE]

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.

Step 1. If $K_{u}(a\widetilde{v})=K_{v}(a\widetilde{u})$ for all $v\in\mathbb{Z}^{n}$ then $K_{s_{i}u}(a\widetilde{v})=K_{v}(a\widetilde{s_{i}u})$ for $v\in\mathbb{Z}^{n}$ and $1\leq i<n$.

Proof of step 1. Writing out the formula from part 1 of Proposition 16 gives

[TABLE]

Replacing in (19) the role of $u$ and $v$ and replacing $i$ by $n-i$ we get

[TABLE]

Suppose that $s_{n-i}v=v$. Then $\overline{v}_{n-i}=t\overline{v}_{n+1-i}$ by the second part of Lemma 5. Since $\widetilde{v}=t^{1-n}w_{0}\overline{v}^{-1}$, i.e. $\widetilde{v}_{i}=t^{1-n}\overline{v}_{n+1-i}^{-1}$, we then also have $\widetilde{v}_{i}=t\widetilde{v}_{i+1}$. It then follows by a direct computation that (19) reduces to $K_{s_{i}u}(a\widetilde{v})=K_{u}(a\widetilde{v})$ and (20) to $K_{v}(a\widetilde{s_{i}u})=K_{v}(a\widetilde{u})$ if $s_{n-i}v=v$.

We now use these observations to prove step 1. Assume that $K_{u}(a\widetilde{v})=K_{v}(a\widetilde{u})$ for all $v$. We have to show that $K_{s_{i}u}(a\widetilde{v})=K_{v}(a\widetilde{s_{i}u})$ for all $v$. It is trivially true if $s_{i}u=u$, so we may assume that $s_{i}u\not=u$. Suppose that $v$ satisfies $s_{n-i}v=v$. Then it follows from the previous paragraph that

[TABLE]

If $s_{n-i}v\not=v$ then (19) and the induction hypothesis can be used to write $K_{s_{i}u}(a\widetilde{v})$ as an explicit linear combination of $K_{v}(a\widetilde{u})$ and $K_{s_{n-i}v}(a\widetilde{u})$. Then (20) can be used to rewrite the term involving $K_{s_{n-i}v}(a\widetilde{u})$ as an explicit linear combination of $K_{v}(a\widetilde{u})$ and $K_{v}(a\widetilde{s_{i}u})$. Hence we obtain an explicit expression of $K_{s_{i}u}(a\widetilde{v})$ as linear combination of $K_{v}(a\widetilde{u})$ and $K_{v}(a\widetilde{s_{i}u})$, which turns out to reduce to $K_{s_{i}u}(a\widetilde{v})=K_{v}(a\widetilde{s_{i}u})$ after a direct computation.

Step 2. $K_{0}(a\widetilde{v})=1=K_{v}(a\widetilde{0})$ for all $v\in\mathbb{Z}^{n}$.

Proof of step 2. Clearly $K_{0}(x)=1$ and $K_{v}(a\widetilde{0})=K_{v}(a\tau)=1$ for $v\in\mathbb{Z}^{n}$ by Lemma 15.

Step 3. $K_{\alpha}(a\widetilde{v})=K_{v}(a\widetilde{\alpha})$ for $v\in\mathbb{Z}^{n}$ and $\alpha\in\mathcal{C}_{n}$.

Proof of step 3. We prove it by induction. It is true for $\alpha=0$ by step 2. Let $m\in\mathbb{Z}_{>0}$ and suppose that $K_{\gamma}(a\widetilde{v})=K_{v}(a\widetilde{\gamma})$ for $v\in\mathbb{Z}^{n}$ and $\gamma\in\mathcal{C}_{n}$ with $|\gamma|<m$. Let $\alpha\in\mathcal{C}_{n}$ with $|\alpha|=m$.

We need to show that $K_{\alpha}(a\widetilde{v})=K_{v}(a\widetilde{\alpha})$ for all $v\in\mathbb{Z}^{n}$. By step 1 we may assume without loss of generality that $\alpha_{n}>0$. Then $\gamma:=\alpha^{\sharp}\in\mathcal{C}_{n}$ satisfies $|\gamma|=m-1$, and $\alpha=\gamma^{\natural}$. Furthermore, note that we have the formula

[TABLE]

for all $u,v\in\mathbb{Z}^{n}$, which follows by writing out the formula from part 3 of Lemma 16. Hence we obtain

[TABLE]

where we used the induction hypothesis for the third equality and (21) for the second and fourth equality. This proves the induction step.

Step 4. $K_{u}(a\widetilde{v})=K_{v}(a\widetilde{u})$ for all $u,v\in\mathbb{Z}^{n}$.

Proof of step 4. Fix $u,v\in\mathbb{Z}^{n}$. Let $m\in\mathbb{Z}_{\geq 0}$ such that $u+(m^{n})\in\mathcal{C}_{n}$. Note that $q^{m}\widetilde{v}=\widetilde{v-(m^{n})}$ and $q^{-m}\widetilde{u}=\widetilde{u+(m^{n})}$. Then

[TABLE]

where we used step 3 in the third equality. The result now follows from the fact that

[TABLE]

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 $E_{\alpha}(x)$ of degree $\alpha$ is the top homogeneous component of $G_{\alpha}(x)$, i.e.

[TABLE]

The normalized non-symmetric Macdonald polynomials are

[TABLE]

We write $\overline{K}\in\mathcal{F}^{+}_{\mathbb{F}[x]}$ for the resulting map $\alpha\mapsto\overline{K}_{\alpha}$. Taking limits in Lemma 10 we get

Lemma 19**.**

We have for $1\leq i<n$ and $1\leq j\leq n$,

1. 1.

$H_{i}\overline{K}=\widehat{H}_{i}\overline{K}$. 2. 2.

$\xi_{j}\overline{K}=\widehat{x}_{j}^{-1}\overline{K}$. 3. 3.

$x_{n}\Delta\overline{K}=t^{1-n}\widehat{x}_{1}^{-1}\widehat{\Delta}^{-1}\overline{K}$.

Note that

[TABLE]

Then repeated application of part 3 of Lemma 19 shows that for $\alpha\in\mathcal{C}_{n}$,

[TABLE]

As is well known and already noted in Section 2, the first equality allows to relate the non-symmetric Macdonald polynomials $E_{v}(x):=E_{v}(x;q,t)\in\mathbb{F}[x^{\pm 1}]$ for arbitrary $v\in\mathbb{Z}^{n}$ to those labeled by compositions through the formula

[TABLE]

The second formula of (22) can now be used to explicitly define the normalized non-symmetric Macdonald polynomials for degrees $v\in\mathbb{Z}^{n}$.

Definition 20**.**

Let $v\in\mathbb{Z}^{n}$ and $m\in\mathbb{Z}_{\geq 0}$ such that $v+(m^{n})\in\mathcal{C}_{n}$. Then $\overline{K}_{v}(x):=\overline{K}_{v}(x;q,t)\in\mathbb{F}[x^{\pm 1}]$ is defined by

[TABLE]

Using

[TABLE]

and the definitions of $G_{v}(x)$ and $K_{v}(x)$ it follows that

[TABLE]

for all $v\in\mathbb{Z}^{n}$, so in particular

[TABLE]

Lemma 19 holds true for the extension of $\overline{K}$ to the map $\overline{K}\in\mathcal{F}_{\mathbb{F}[x^{\pm 1}]}$ defined by $v\mapsto\overline{K}_{v}$ ($v\in\mathbb{Z}^{n}$). 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 $u,v\in\mathbb{Z}^{n}$,

[TABLE]

7.2. $O$-polynomials

We now show that the duality of the non-symmetric interpolation Macdonald polynomials (Theorem 17) directly implies the existence of the $O$-polynomials $O_{\alpha}$ (which is the nontrivial part of the proof of [12, Thm. 1.2]), and that it provides an explicit expression for $O_{\alpha}$ in terms of the non-symmetric interpolation Macdonald polynomial $K_{\alpha}$.

Proposition 22**.**

For all $\alpha\in\mathcal{C}_{n}$ we have

[TABLE]

Proof.

The polynomial $\widetilde{O}_{\alpha}(x):=K_{\alpha}(t^{1-n}aw_{0}x)$ is of degree at most $|\alpha|$ and

[TABLE]

for all $\beta\in\mathcal{C}_{n}$ by (4) and Theorem 17. Hence $\widetilde{O}_{\alpha}=O_{\alpha}$. ∎

7.3. Okounkov’s duality

Write $F[x]^{S_{n}}$ for the symmetric polynomials in $x_{1},\ldots,x_{n}$ with coefficients in a field $F$. Write $C_{+}:=\sum_{w\in S_{n}}H_{w}$. The symmetric interpolation Macdonald polynomial $R_{\lambda}(x)\in\mathbb{F}[x]^{S_{n}}$ is the multiple of $C_{+}G_{\lambda}$ such that the coefficient of $x^{\lambda}$ is one (see, e.g., [11]). We write

[TABLE]

for the normalized symmetric interpolation Macdonald polynomial. Then

[TABLE]

for $\alpha\in\mathcal{C}_{n}$. Okounkov’s [9, §2] duality result now reads as follows.

Theorem 23**.**

For partitions $\lambda,\mu\in\mathcal{P}_{n}$ we have

[TABLE]

Let us derive Theorem 23 as consequence of Theorem 17. Write $\widehat{C}_{+}=\sum_{w\in S_{n}}\widehat{H}_{w}$, with $\widehat{H}_{w}:=\widehat{H}_{i_{1}}\cdots\widehat{H}_{i_{r}}$ for a reduced expression $w=s_{i_{1}}\cdots s_{i_{r}}$. Write $f_{\mu}\in\mathcal{F}_{\mathbb{K}}$ for the function $f_{\mu}(u):=K_{u}(a\widetilde{\mu})$ ($u\in\mathbb{Z}^{n}$). Then

[TABLE]

by part 1 of Proposition 16. The duality (17) of $K_{u}$ and (4) imply that

[TABLE]

with $(Jf)(x):=f(x_{1}^{-1},\ldots,x_{n}^{-1})$ for $f\in\mathbb{K}(x)$. A direct computation shows that

[TABLE]

for $1\leq i<n$. In particular, $Jw_{0}C_{+}=C_{+}Jw_{0}$. Combined with Remark 7 we conclude that

[TABLE]

By (23) and (4) this simplifies to

[TABLE]

Returning to (24) we conclude that $K_{\lambda}^{+}(a\widetilde{\mu})=K_{\mu}^{+}(a\widetilde{\lambda})$. Since $K_{\lambda}^{+}$ is symmetric we obtain from (4) that

[TABLE]

which is Okounkov’s duality result.

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 $u,v\in\mathbb{Z}^{n}$ we have

[TABLE]

Proof.

We proceed as in the previous subsection. Set $f_{v}(u):=K_{u}(a\widetilde{v})$ for $u,v\in\mathbb{Z}^{n}$. By part 1 of Proposition 16,

[TABLE]

Since f_{v}(u)=\bigl{(}Iw_{0}K_{v}\bigr{)}(a^{-1}t^{n-1}\overline{u}) by (4), Remark 7 implies that

[TABLE]

Now $H_{w_{0}}Jw_{0}=Jw_{0}H_{w_{0}}$ by (26), hence

[TABLE]

which completes the proof. ∎

Recall from Theorem 1 that

[TABLE]

with $\Psi:=w_{0}H_{w_{0}}^{\circ}$. We define normalized versions by

[TABLE]

with $K_{v}^{\circ}:=\iota(K_{v})$ for $v\in\mathbb{Z}^{n}$ (the second formula follows from Lemma 2). More generally, we define for $v\in\mathbb{Z}^{n}$,

[TABLE]

We write $K^{\prime}:\mathbb{Z}^{n}\rightarrow\mathbb{K}(x)$ for the map $v\mapsto K_{v}^{\prime}$ ($v\in\mathbb{Z}^{n}$). Since $H_{i}\Psi=\Psi H_{i}^{\circ}$, part 1 of Proposition 16 gives $H_{i}K^{\prime}=\widehat{H}_{i}^{\circ}K^{\prime}$. Considering the action of $((x_{n}-1)\Delta^{\circ})^{n}$ on $K_{\beta}^{\prime}(x)$ we get, using the fact that $((x_{n}-1)\Delta^{\circ})^{n}$ commutes with $\Psi$ and part 3 of Proposition 16,

[TABLE]

in particular

[TABLE]

Example 25*.*

For $n=1$ we have $K_{v}^{\prime}(x)=K_{v}^{\circ}(x)$ for $v\in\mathbb{Z}$, hence

[TABLE]

for $m\in\mathbb{Z}_{\geq 0}$ by Example 14.

Proposition 26**.**

For all $u,v\in\mathbb{Z}^{n}$ we have

[TABLE]

Proof.

Note that

[TABLE]

by (4). By (27) the right hand side is invariant under the interchange of $u$ and $v$. ∎

7.5. Binomial formula and dual binomial formula

In [12] the existence and uniqueness of $O_{\alpha}$ was used to prove the following binomial theorem [12, Thm. 1.3]. Define for $\alpha,\beta\in\mathcal{C}_{n}$ the generalized binomial coefficient by

[TABLE]

Applying the automorphism $\iota$ of $\mathbb{F}$ to (29) we get

[TABLE]

Theorem 27**.**

For $\alpha,\beta\in\mathcal{C}_{n}$ we have the binomial formula

[TABLE]

Remark 28*.*

1. Note that the sum in (30) is finite, since the generalized binomial coefficient (29) is zero unless $\beta\subseteq\alpha$, with $\beta\subseteq\alpha$ meaning $\beta_{i}\leq\alpha_{i}$ for $i=1,\ldots,n$.

2. By Corollary 4 and (28) the binomial formula (30) can be alternatively written as

[TABLE]

with $\Psi=w_{0}H_{w_{0}}^{\circ}$ (note that the dependence on $a$ in the right hand side of (31) is through the normalization factors of the interpolation polynomials $K_{\beta}^{\circ}(x)$ and $K_{\beta}^{\prime}(x)$).

3. The binomial formula (30) and Theorem 1 imply the twisted duality (27) of $K_{\alpha}$ as follows. By the identity $H_{w_{0}}\Psi=w_{0}$ the binomial formula (31) implies the finite expansion

[TABLE]

Substituting $x=\widetilde{\gamma}$ and using (4) we obtain

[TABLE]

The right hand side is manifestly invariant under interchanging $\alpha$ and $\gamma$, which is equivalent to twisted duality (27).

In [7, §4] it is remarked that an explicit identity relating $G_{\alpha}^{\prime}$ and $G_{\alpha}$ is needed to provide a proof of the dual binomial formula [7, Thm. 4.4] as a direct consequence of the binomial formula (30). We show here that Theorem 1 is providing the required identity. Instead of Theorem 1 we use its normalized version, encoded by (28).

The dual binomial formula [7, Thm. 4.4] in our notations reads as follows.

Theorem 29**.**

For all $\alpha\in\mathcal{C}_{n}$ we have

[TABLE]

The starting point of the alternative proof of (32) is the binomial formula in the form

[TABLE]

see (31). Replace $(a,x,q,t)$ by $(a^{-1},at^{n-1}x,q^{-1},t^{-1})$ and act by $w_{0}H_{w_{0}}$ on both sides. Since $w_{0}H_{w_{0}}\Psi=\textup{Id}$ we obtain

[TABLE]

Now use (28) to complete the proof of (32).

Remark 30*.*

It follows from this proof of (32) that the dual binomial formula (32) can be rewritten as

[TABLE]

As observed in [7, (4.11)], the binomial and dual binomial formula directly imply the orthogonality relations

[TABLE]

Since $\left[\begin{matrix}\delta\\ \epsilon\end{matrix}\right]_{q,t}=0$ unless $\delta\supseteq\epsilon$, the terms in the sum are zero unless $\gamma\subseteq\beta\subseteq\alpha$.