# Quasi-polynomial extensions of nonsymmetric Macdonald-Koornwinder
polynomials

Jasper Stokman  J. Stokman, KdV Institute for Mathematics, University of
Amsterdam, Science Park 105-107, 1098 XG Amsterdam, The Netherlands.

###### Abstract.

In a recent preprint of the author with Sahi and Venkateswaran families of
actions of the double affine Hecke algebra on spaces of quasi-polynomials were
introduced. These so-called quasi-polynomial representations led to the
introduction of quasi-polynomial extensions of the nonsymmetric Macdonald
polynomials which reduce to metaplectic Iwahori-Whittaker functions in the
$\mathfrak{p}$-adic limit. In this paper we extend the quasi-polynomial
representations to Sahi’s $5$-parameter double affine Hecke algebra and
introduce the quasi-polynomial extensions of the nonsymmetric Koornwinder
polynomials.

##  1\. Introduction

### 1.1.

The double affine Hecke algebra $\mathbb{H}$ with adjoint root data depends on
a deformation parameter $q$ and on a number of Hecke parameters. The Hecke
parameters are most conveniently encoded by a multiplicity function, which is
an affine Weyl group invariant function on the associated reduced affine root
system. Cherednik’s polynomial representation is a faithful representation of
$\mathbb{H}$ on Laurent polynomials in several variables, given explicitly in
terms of Demazure-Lusztig operators. Up to a multiplicative scalar, the
nonsymmetric Macdonald polynomials can be characterised as the simultaneous
eigenfunctions for the action of Bernstein’s [7] commuting elements
$Y^{\lambda}$ within the (double) affine Hecke algebra. See the monographs by
Cherednik [3] and Macdonald [9] for details and further references.

If the underlying finite root system is of type $\textup{C}_{r}$ then a
nonreduced extension of the Cherednik-Macdonald theory was developed in [11,
12, 15]. It depends on five Hecke parameters (four when $r=1$), which are
encoded by a multiplicity function on Macdonald’s [8] nonreduced affine root
system of type $\textup{C}^{\vee}\textup{C}_{r}$. The resulting simultaneous
polynomial eigenfunctions of the $Y^{\lambda}$ are Sahi’s [12] nonsymmetric
Koornwinder polynomials. Their symmetric versions are the celebrated
Koornwinder polynomials [6], which reduce to Askey-Wilson polynomials [1] for
$r=1$.

In a joint paper [14] with Sahi and Venkateswaran a quasi-polynomial extension
of the Cherednik-Macdonald theory was developed when the multiplicity function
on the reduced affine root system is invariant for the action of the extended
affine Weyl group (so it depends on one Hecke parameter if the underlying
finite root system has a single Weyl group orbit, and two otherwise). The role
of the polynomial representation is then replaced by explicit families of
$\mathbb{H}$-representations on spaces of quasi-polynomials, which are linear
combinations of monomials with possibly non-integral exponents, with the
action given in terms of truncated versions of Demazure-Lusztig operators. The
resulting quasi-polynomial extensions of the nonsymmetric Macdonald
polynomials are $q$-analogs of Iwahori-Whittaker functions on metaplectic
covers of reductive groups over non-Archimedean local fields.

In this paper we will construct the nonreduced extensions of the quasi-
polynomial representations and introduce the quasi-polynomial analogs of the
nonsymmetric Koornwinder polynomials. We will proceed by extending the
framework for the quasi-polynomial theory in such a way that it gives the
nonreduced quasi-polynomial theory when the underlying finite root system is
of type $\textup{C}_{r}$. For other types it will simply reduce to the quasi-
polynomial theory from [14]. The setup of the extended framework is modeled by
the treatment of the twisted polynomial theory with adjoint root datum from
[16].

In the remainder of the introduction we will explicitly state the main results
when the underlying root system is of type $\textup{C}_{r}$. In Section 2 we
will introduce affine root systems and the double affine Hecke algebra
$\mathbb{H}$ in the general, extended framework. Following [13], we start
Section 3 by introducing an $H^{X}$-action in terms of truncated Demazure-
Lusztig type operators on the space of all quasi-polynomials, where $H^{X}$ is
the copy of the affine Hecke algebra inside $\mathbb{H}$ that contains the
monomials. This $H^{X}$-representation is reducible, with subrepresentations
being naturally parametrised by affine Weyl group orbits in the ambient
Euclidean space of the root system. Following [14], we then give for each
subrepresentation a multiparameter extension of the $H^{X}$-action to an
action of the double affine Hecke algebra. It gives the quasi-polynomial
representations from [14, §4] as well as the new, nonreduced extensions when
the underlying root system is of type $\textup{C}_{r}$. In Section 4 we
introduce the quasi-polynomial extensions of the nonsymmetric Macdonald-
Koornwinder polynomials. Finally, in Section 5 we identify the quasi-
polynomial representations with $Y$-parabolically induced
$\mathbb{H}$-modules.

In [14, §6] various additional properties of the quasi-polynomial extensions
of the nonsymmetric Macdonald polynomials were derived, such as creation
formulas, (anti)symmetric versions of the quasi-polynomials, and orthogonality
relations. It is straightforward to derive the analogous results for the
nonreduced extension of the type $\textup{C}_{r}$ quasi-polynomials using the
intertwiners of Sahi’s [12] nonreduced extension of the double affine Hecke
algebra, but we will not discuss the details in this paper. We also do not
discuss the theory for extended lattices, which follows quite easily from the
theory for adjoint root data, cf. [14, §7].

### 1.2.

Let $\mathbf{F}$ be a field of characteristic zero. The algebra of quasi-
polynomials [14] in $r$ variables over $\mathbf{F}$ is the group algebra
$\mathbf{F}[\mathbb{R}^{r}]$ of the Euclidean space $\mathbb{R}^{r}$, viewed
as additive group. We write $x^{y}$ for the standard basis element in
$\mathbf{F}[\mathbb{R}^{r}]$ associated to the vector
$y=(y_{1},\ldots,y_{r})\in\mathbb{R}^{r}$, so that

$\mathbf{F}[\mathbb{R}^{r}]=\bigoplus_{y\in\mathbb{R}^{r}}\mathbf{F}x^{y},\qquad\quad
x^{y}x^{y^{\prime}}=x^{y+y^{\prime}},\quad x^{0}=1.$

Denote by $\\{\epsilon_{i}\\}_{i=1}^{r}$ the standard orthonormal basis of
$\mathbb{R}^{r}$. Then $x^{y}=x_{1}^{y_{1}}\cdots x_{r}^{y_{r}}$ with

$x_{i}^{\xi}:=x^{\xi\epsilon_{i}}\in\mathbf{F}[\mathbb{R}^{r}]\qquad\hbox{ for
}\,\xi\in\mathbb{R}.$

We call $x^{y}$ the quasi-monomial with quasi-exponent $y\in\mathbb{R}^{r}$.

Consider the hyperoctahedral group $S_{r}\ltimes(\pm 1)^{r}$. It is a Coxeter
group with Coxeter system $\\{s_{1},\ldots,s_{r}\\}$ given by the simple
neighboring transpositions $s_{i}=(i,i+1)$ for $1\leq i<r$ and
$s_{r}=(1,\ldots,1,-1)$ (we identify $S_{r}$ and $(\pm 1)^{r}$ with the
corresponding subgroups in $S_{r}\ltimes(\pm 1)^{r}$). The Coxeter generators
satisfy the type $\textup{C}_{r}$ braid relations

(1.1) $\begin{split}s_{i}s_{i+1}s_{i}&=s_{i+1}s_{i}s_{i+1}\qquad(1\leq
i<r-1),\\\ s_{r-1}s_{r}s_{r-1}s_{r}&=s_{r}s_{r-1}s_{r}s_{r-1},\\\
s_{i}s_{i^{\prime}}&=s_{i^{\prime}}s_{i}\qquad\qquad\,\,\,\hbox{ if
}\,|i-i^{\prime}|>1.\end{split}$

The formulas

(1.2)
$\begin{split}s_{i}y&:=(y_{1},\ldots,y_{i-1},y_{i+1},y_{i},y_{i+2},\ldots,y_{r})\qquad(1\leq
i<r),\\\ s_{r}y&:=(y_{1},\ldots,y_{r-1},-y_{r})\end{split}$

define a linear action of the hyperoctahedral group $S_{r}\ltimes(\pm 1)^{r}$
on $\mathbb{R}^{r}$. The action (1.2) naturally gives rise to a
$S_{r}\ltimes(\pm 1)^{r}$-action by algebra automorphisms on
$\mathbf{F}[\mathbb{R}^{r}]$ by letting $s_{j}$ act on the quasi-exponents of
the quasi-monomials $x^{y}$ according to (1.2). The subalgebra
$\mathbf{F}[x^{\pm 1}]$ of $\mathbf{F}[\mathbb{R}^{r}]$ spanned $x^{\mu}$
($\mu\in\mathbb{Z}^{r}$) is the algebra of Laurent polynomials in the
variables $x_{i}:=x_{i}^{1}=x^{\epsilon_{i}}$ ($1\leq i\leq r$), which
inherits a $S_{r}\ltimes(\pm 1)^{r}$-action from $\mathbf{F}[\mathbb{R}^{r}]$.

For $\xi\in\mathbb{R}$ write

  1. $\lfloor\xi\rfloor$ for the largest integer $\leq\xi$,

  2. $\lfloor\xi\rfloor_{e}$ for the largest even integer $\leq\xi$,

  3. $\lfloor\xi\rfloor_{o}$ for the largest odd integer $\leq\xi$.

Definition A. For $1\leq i<r$ let $\nabla_{i},\nabla_{r}^{e},\nabla_{r}^{o}$
be the linear operators on $\mathbf{F}[\mathbb{R}^{r}]$ defined by

$\begin{split}\nabla_{i}(x^{y})&:=\left(\frac{1-(x_{i+1}/x_{i})^{\lfloor
y_{i}-y_{i+1}\rfloor}}{1-x_{i}/x_{i+1}}\right)x^{y},\\\
\nabla_{r}^{e}(x^{y})&:=\left(\frac{1-x_{r}^{-\lfloor
2y_{r}\rfloor_{e}}}{1-x_{r}^{2}}\right)x^{y},\qquad\,\,\,\nabla_{r}^{o}(x^{y}):=\left(\frac{x_{r}-x_{r}^{-\lfloor
2y_{r}\rfloor_{o}}}{1-x_{r}^{2}}\right)x^{y}\end{split}$

for $y\in\mathbb{R}^{r}$.

Note that the $\nabla_{i}$ are well defined because of the truncation by floor
functions of the exponents in the numerator. For $y=\mu\in\mathbb{Z}^{r}$ the
$\nabla_{i}$ reduce to divided difference operators,

(1.3)
$\nabla_{i}(x^{\mu})=\frac{x^{\mu}-x^{s_{i}\mu}}{1-x_{i}/x_{i+1}},\qquad\quad\nabla_{r}^{e}(x^{\mu})=\frac{x^{\mu}-x^{s_{r}\mu}}{1-x_{r}^{2}}=x_{r}^{-1}\nabla_{r}^{o}(x^{\mu}).$

For a subset $B$ in a set $X$, we denote by

$\chi_{B}:X\rightarrow\\{0,1\\}$

the indicator function of $B$. We use the shorthand notations $\chi_{e}$ and
$\chi_{o}$ for the indicator function of the even and odd integers inside
$\mathbb{R}$.

Theorem A. For $k,k_{r},u_{r}\in\mathbf{F}^{\times}$ the linear operators
$\mathcal{T}_{1},\ldots,\mathcal{T}_{r}$ on $\mathbf{F}[\mathbb{R}^{r}]$
defined by

(1.4)
$\begin{split}\mathcal{T}_{i}(x^{y})&:=k^{\chi_{\mathbb{Z}}(y_{i}-y_{i+1})}x^{s_{i}y}+(k-k^{-1})\nabla_{i}(x^{y})\qquad\hbox{
for }\,\,1\leq i<r,\\\
\mathcal{T}_{r}(x^{y})&:=k_{r}^{\chi_{e}(2y_{r})}u_{r}^{\chi_{o}(2y_{r})}x^{s_{r}y}+(k_{r}-k_{r}^{-1})\nabla_{r}^{e}(x^{y})+(u_{r}-u_{r}^{-1})\nabla_{r}^{o}(x^{y})\end{split}$

satisfy the type $\textup{C}_{r}$ braid relations (1.1) and the quadratic
Hecke relations

(1.5) $\begin{split}(\mathcal{T}_{i}-k)(\mathcal{T}_{i}+k^{-1})&=0\qquad\hbox{
for }\,\,1\leq i<r,\\\
(\mathcal{T}_{r}-k_{r})(\mathcal{T}_{r}+k_{r}^{-1})&=0.\end{split}$

Theorem A thus provides a representation of the $2$-parameter Hecke algebra
$H_{0}=H_{0}(k,k_{r})$ of type $\textup{C}_{r}$ on
$\mathbf{F}[\mathbb{R}^{r}]$. Using the natural $\mathbf{F}[x^{\pm 1}]$-module
structure on $\mathbf{F}[\mathbb{R}^{r}]$ the $H_{0}$-action on
$\mathbf{F}[\mathbb{R}^{r}]$ extends to a representation of the $3$-parameter
affine Hecke algebra $\widetilde{H}=H(u_{r},k,k_{r})$ of type $\textup{C}_{r}$
(this uses the Bernstein presentation of $\widetilde{H}$). Note that the
$\mathcal{T}_{j}$ preserve $\mathbb{C}[x^{\pm 1}]$, in which case they reduce
to the operators

(1.6)
$\begin{split}\mathcal{T}_{i}(x^{\mu})&=kx^{s_{i}\mu}+(k-k^{-1})\Bigl{(}\frac{x^{\mu}-x^{s_{i}\mu}}{1-x_{i}/x_{i+1}}\Bigr{)}\\\
&=kx^{\mu}+k^{-1}\frac{1-k^{2}x_{i}/x_{i+1}}{1-x_{i}/x_{i+1}}\bigl{(}x^{s_{i}\mu}-x^{\mu}\bigr{)},\\\
\mathcal{T}_{r}(x^{\mu})&=k_{r}x^{s_{r}\mu}+\bigl{(}(k_{r}-k_{r}^{-1})+(u_{r}-u_{r}^{-1})x_{r}\bigr{)}\bigl{(}\frac{x^{\mu}-x^{s_{r}\mu}}{1-x_{r}^{2}}\bigr{)}\\\
&=k_{r}x^{r}+k_{r}^{-1}\frac{(1-ax_{r})(1-bx_{r})}{1-x_{r}^{2}}\bigl{(}x^{s_{r}\mu}-x^{\mu}\bigr{)}\end{split}$

for $\mu\in\mathbb{Z}^{r}$ and $1\leq i<r$ by (1.3), with
$\\{a,b\\}=\\{k_{r}u_{r},-k_{r}u_{r}^{-1}\\}$. These Demazure-Lusztig type
operators on $\mathbb{C}[x^{\pm 1}]$ are used in the definition of the
polynomial representation of the double affine Hecke algebra of type
$\textup{C}^{\vee}\textup{C}_{r}$, see [11, 12].

Theorem A will be proven in Subsection 3.1 (it is the special case of Theorem
3.3 when the underlying finite root system is of type $\textup{C}_{r}$). When
$k_{r}=u_{r}$ Theorem A was proven before in [13, 14].

### 1.3.

The affine extension of Theorem A depends on various additional parameters.
First of all it depends on a dilation parameter
$q^{\frac{1}{2}}\in\mathbf{F}^{\times}$, which naturally appears in the
following affine extension of the $S_{r}\ltimes(\pm 1)^{r}$-action on
$\mathbf{F}[x^{\pm 1}]$.

The affine Weyl group of type $\textup{C}_{r}$ is $W:=(S_{r}\ltimes(\pm
1)^{r})\ltimes\mathbb{Z}^{r}$, with the rightmost semidirect product defined
in terms of the action (1.2) restricted to $\mathbb{Z}^{r}$. It acts by
algebra automorphisms on $\mathbf{F}[x^{\pm 1}]$ by

(1.7)
$\begin{split}(s_{i}p)(x)&=p(x_{1},\ldots,x_{i-1},x_{i+1},x_{i},x_{i+2},\ldots,x_{r})\qquad(1\leq
i<r),\\\ (s_{r}p)(x)&=p(x_{1},\ldots,x_{r-1},x_{r}^{-1}),\\\
(\tau(\lambda)p)(x)&=p(q^{-\lambda_{1}}x_{1},\ldots,q^{-\lambda_{r}}x_{r})\qquad\qquad\qquad\qquad\,(\lambda\in\mathbb{Z}^{r})\end{split}$

for $p(x)=p(x_{1},\ldots,x_{r})\in\mathbf{F}[x^{\pm 1}]$, with $\tau(\lambda)$
the affine Weyl group element corresponding to $\lambda\in\mathbb{Z}^{r}$.
Regarding $\mathbf{F}[x^{\pm 1}]$ as the algebra of regular functions on the
$F$-torus

$\mathbf{T}:=(\mathbf{F}^{\times})^{r},$

we may view (1.7) as the $W$-action on $\mathbf{F}[x^{\pm 1}]$ contragredient
to the left $W$-action

$\begin{split}s_{i}t&:=(t_{1},\ldots,t_{i-1},t_{i+1},t_{i},t_{i+2},\ldots,t_{r})\qquad(1\leq
i<r),\\\ s_{r}t&:=(t_{1},\ldots,t_{r-1},t_{r}^{-1}),\\\
\tau(\lambda)t&:=(q^{\lambda_{1}}t_{1},\ldots,q^{\lambda_{r}}t_{r})\end{split}$

on $t=(t_{1},\ldots,t_{r})\in\mathbf{T}$.

The affine Weyl group $W$ is a Coxeter group with Coxeter system
$\\{s_{0},s_{1},\ldots,s_{r}\\}$ containing the extra simple reflection

$s_{0}:=\tau(\epsilon_{1})s_{\epsilon_{1}},\qquad
s_{\epsilon_{1}}:=s_{1}\cdots s_{r-1}s_{r}s_{r-1}\cdots s_{1}.$

The simple reflection $s_{0}$ acts on $\mathbf{F}[x^{\pm 1}]$ by

$(s_{0}p)(x)=p(qx_{1}^{-1},x_{2},\ldots,x_{r}).$

The braid relations involving $s_{0}$ are

(1.8) $\begin{split}s_{0}s_{1}s_{0}s_{1}&=s_{1}s_{0}s_{1}s_{0},\\\
s_{0}s_{i}&=s_{i}s_{0}\qquad\quad(1<i\leq r).\end{split}$

The linear $S_{r}\ltimes(\pm 1)^{r}$-action (1.2) on $\mathbb{R}^{r}$ extends
to an affine linear $W$-action with $\tau(\lambda)$
($\lambda\in\mathbb{Z}^{r}$) acting on $\mathbb{R}^{r}$ as translation
operators,

$\tau(\lambda)y:=y+\lambda\qquad(y\in\mathbb{R}^{r}).$

The alcove

$C_{+}:=\\{\,y\in\mathbb{R}^{r}\,\,\,|\,\,\,0<y_{r}<y_{r-1}<\cdots<y_{1}<\tfrac{1}{2}\,\\}$

is the intersection of the half-spaces
$\\{y\in\mathbb{R}^{r}\,\,|\,\,\alpha_{j}(y)>0\\}$ ($0\leq j\leq r$), where
the affine linear functionals $\alpha_{j}:\mathbb{R}^{r}\rightarrow\mathbb{R}$
are defined by

$\alpha_{0}(y):=1-2y_{1},\qquad\alpha_{i}(y):=y_{i}-y_{i+1}\,\,\,\,(1\leq
i<r),\qquad\alpha_{r}(y):=2y_{r}.$

The closure $\overline{C_{+}}$ of $C_{+}$ in $\mathbb{R}^{r}$ is a fundamental
domain for the $W$-action on $\mathbb{R}^{r}$. For a $W$-orbit $\mathcal{O}$
in $\mathbb{R}^{r}$ we denote by $c^{\mathcal{O}}$ the unique vector in
$\mathcal{O}\cap\overline{C_{+}}$. Note that $\mathbb{Z}^{r}$ is a $W$-orbit
in $\mathbb{R}^{r}$, and $c^{\mathbb{Z}^{r}}=0$.

For a $W$-orbit $\mathcal{O}$ in $\mathbb{R}^{r}$, consider the free
$\mathbf{F}[x^{\pm 1}]$-submodule

$\mathbf{F}[\mathcal{O}]:=\bigoplus_{y\in\mathcal{O}}\mathbf{F}x^{y}$

of $\mathbf{F}[\mathbb{R}^{r}]$ of finite rank. The operators
$\mathcal{T}_{i}$ ($1\leq i\leq r$) preserve $\mathbf{F}[\mathcal{O}]$, and we
write

$\mathcal{T}_{i}^{\mathcal{O}}:=\mathcal{T}_{i}|_{\mathbf{F}[\mathcal{O}]}$

for the resulting linear operators on $\mathbf{F}[\mathcal{O}]$. We now define
a linear operator $\mathcal{T}_{0}^{\mathcal{O}}$ on $\mathbf{F}[\mathcal{O}]$
that will provide the local affine extension of Theorem A (’local’ in the
sense that the operators should be restricted to $\mathbf{F}[\mathcal{O}]$).
The operator $\mathcal{T}_{0}^{\mathcal{O}}$ will depend on additional
parameters that lie in an $\mathcal{O}$-dependent affine subtorus
$\mathbf{T}_{\mathcal{O}}$ of $\mathbf{T}$. We define the affine subtorus
$\mathbf{T}_{\mathcal{O}}$ now first.

Consider the simple co-roots
$\alpha_{j}^{\vee}\in\mathbb{Z}^{r}\times\frac{1}{2}\mathbb{Z}$, defined by

$\alpha_{0}^{\vee}=(-\epsilon_{1},\tfrac{1}{2}),\qquad\alpha_{i}^{\vee}=(\epsilon_{i}-\epsilon_{i+1},0)\quad(1\leq
i<r),\qquad\alpha_{r}^{\vee}:=(\epsilon_{r},0).$

The (affine) subtorus $\mathbf{T}_{\mathcal{O}}$ is then given by

$\mathbf{T}_{\mathcal{O}}:=\\{\,t\in\mathbf{T}\,\,\,|\,\,\,t^{\alpha_{j}^{\vee}}=1\,\hbox{
for }\,j\in\\{0,\ldots,r\\}\,\hbox{ satisfying
}\,\alpha_{j}(c^{\mathcal{O}})=0\,\\}$

where

$t^{(\mu,\ell)}:=q^{\ell}t^{\mu}=q^{\ell}t_{1}^{\mu_{1}}\cdots
t_{r}^{\mu_{r}}\qquad\textup{ for
}\,\,(\mu,\ell)\in\mathbb{Z}^{r}\times\tfrac{1}{2}\mathbb{Z}.$

Note that $\mathbf{T}_{\mathcal{O}}=\mathbf{T}$ if $\mathcal{O}$ is a regular
$W$-orbit, while $\mathbf{T}_{\mathbb{Z}^{r}}=\\{1_{\mathbf{T}}\\}$.

For $y\in\mathbb{R}^{r}$ let $\mathbf{g}_{y}\in W$ be the unique element of
minimal length such that $\mathbf{g}_{y}^{-1}y\in\overline{C_{+}}$.

Definition B. Let $\mathcal{O}$ be a $W$-orbit in $\mathbb{R}^{r}$. For
$k_{0},u_{0}\in\mathbf{F}^{\times}$ and $t\in\mathbf{T}_{\mathcal{O}}$ let
$\mathcal{T}_{0}^{\mathcal{O}}$ be the linear operator on
$\mathbf{F}[\mathcal{O}]$ defined by

(1.9)
$\mathcal{T}_{0}^{\mathcal{O}}(x^{y}):=k_{0}^{\chi_{e}(2y_{1})}u_{0}^{\chi_{o}(2y_{1})}(\mathbf{g}_{y}t)^{\epsilon_{1}}s_{\epsilon_{1}}(x^{y})+(k_{0}-k_{0}^{-1})\nabla_{0}^{e}(x^{y})+(u_{0}-u_{0}^{-1})\nabla_{0}^{o}(x^{y})$

for $y\in\mathcal{O}$, where $\nabla_{0}^{e},\nabla_{0}^{o}$ are the linear
operators on $\mathbf{F}[\mathbb{R}^{r}]$ defined by

$\nabla_{0}^{e}(x^{y}):=\left(\frac{1-(q^{-\frac{1}{2}}x_{1})^{\lfloor-2y_{1}\rfloor_{e}}}{1-qx_{1}^{-2}}\right)x^{y},\qquad\nabla_{0}^{o}(x^{y}):=\left(\frac{q^{\frac{1}{2}}x_{1}^{-1}-(q^{-\frac{1}{2}}x_{1})^{\lfloor-2y_{1}\rfloor_{o}}}{1-qx_{1}^{-2}}\right)x^{y}$

for $y\in\mathbb{R}^{r}$.

It is straightforward to check that $\mathcal{T}_{0}$ is a well-defined linear
operator on $\mathbf{F}[\mathcal{O}]$. Furthermore, in case of the $W$-orbit
$\mathcal{O}=\mathbb{Z}^{r}$ we have
$\mathbf{g}_{\mu}1_{T}=(q^{\mu_{1}},\ldots,q^{\mu_{r}})$ for
$\mu\in\mathbb{Z}^{r}$, and hence $\mathcal{T}_{0}^{\mathbb{Z}^{r}}$ reduces
to

(1.10)
$\begin{split}\mathcal{T}_{0}^{\mathbb{Z}^{r}}(x^{\mu})&=k_{0}s_{0}(x^{\mu})+\bigl{(}(k_{0}-k_{0}^{-1})+(u_{0}-u_{0}^{-1})q^{\frac{1}{2}}x_{1}^{-1}\bigr{)}\Bigl{(}\frac{x^{\mu}-s_{0}(x^{\mu})}{1-qx_{1}^{-2}}\Bigr{)}\\\
&=k_{0}x^{\mu}+k_{0}^{-1}\frac{(1-cx_{1}^{-1})(1-dx_{1}^{-1})}{1-qx_{1}^{-2}}\bigl{(}s_{0}(x^{\mu})-x^{\mu}\bigr{)}\end{split}$

for $\mu\in\mathbb{Z}^{r}$ with
$\\{c,d\\}=\\{q^{\frac{1}{2}}k_{0}u_{0},-q^{\frac{1}{2}}k_{0}u_{0}^{-1}\\}$,
which is the Demazure-Lusztig operator associated to the affine simple
reflection $s_{0}$ appearing in the polynomial representation of the double
affine Hecke algebra of type $\textup{C}^{\vee}\textup{C}_{r}$, see [11, 12].

Theorem B. For
$q^{\frac{1}{2}},k_{0},u_{0},k,k_{r},u_{r}\in\mathbf{F}^{\times}$ and
$t\in\mathbf{T}_{\mathcal{O}}$, the operators
$\mathcal{T}_{0}^{\mathcal{O}},\ldots,\mathcal{T}_{r}^{\mathcal{O}}$ satisfy
the affine type $\textup{C}_{r}$ braid relations (1.1)&(1.8) and the Hecke
relations

$(\mathcal{T}_{0}^{\mathcal{O}}-k_{0})(\mathcal{T}_{0}^{\mathcal{O}}+k_{0}^{-1})=0,\qquad(\mathcal{T}_{i}^{\mathcal{O}}-k)(\mathcal{T}_{i}^{\mathcal{O}}+k^{-1})=0,\qquad(\mathcal{T}_{r}^{\mathcal{O}}-k_{r})(\mathcal{T}_{r}^{\mathcal{O}}+k_{r}^{-1})=0$

for $1\leq i<r$.

Theorem B thus gives rise to a representation of the $3$-parameter Hecke
algebra $H:=H(k_{0},k,k_{r})$ of type $\textup{C}_{r}$ on
$\mathbf{F}[\mathcal{O}]$ (using now the Coxeter presentation of the affine
Hecke algebra $H$). Adding the action of $\mathbf{F}[x^{\pm 1}]$ on
$\mathbf{F}[\mathcal{O}]$ by multiplication operators yields a representation
of Sahi’s [12] double affine Hecke algebra
$\mathbb{H}=\mathbb{H}(k_{0},u_{0},k,k_{r},u_{r};q^{\frac{1}{2}})$ of type
$\textup{C}^{\vee}\textup{C}_{r}$ on $\mathbf{F}[\mathcal{O}]$, depending on
$t\in\mathbf{T}_{\mathcal{O}}$. We call it the quasi-polynomial representation
of $\mathbb{H}$. By (1.6) and (1.10) the quasi-polynomial representation for
$\mathcal{O}=\mathbb{Z}^{r}$ is the polynomial representation of $\mathbb{H}$
from [11, 12] governing the Koornwinder polynomials.

Theorem B will be proven in Subsection 3.3 (it is the special case of Theorem
3.12 when the underlying finite root system is of type $\textup{C}_{r}$). In
Section 5 we will also show that the quasi-polynomial representation is
isomorphic to a $Y$-parabolically induced $\mathbb{H}$-module. These results
were obtained before in [14] when $k_{0}=u_{0}=k_{r}=u_{r}$.

When $k_{0}=u_{0}=k_{r}=u_{r}$ a particular reparametrisation of the extra
parameters $t\in\mathbf{T}_{\mathcal{O}}$ in terms of so-called $g$-parameters
allows to glue the quasi-polynomial representations into a family of
$\mathbb{H}$-representations on $\mathbf{F}[\mathbb{R}^{r}]$ with the Coxeter
type generators of $H$ acting by global $g$-dependent truncated Demazure-
Lusztig type operators [14]. This is an important intermediate step in
establishing the link to representation theory of metaplectic covers of
reductive groups over non-Archimedean local fields when taking the
$\mathfrak{p}$-adic limit $q\rightarrow\infty$ (the $g$-parameters are then
given in terms of Gauss sums).

### 1.4.

The quasi-polynomial extensions of the monic nonsymmetric Koornwinder
polynomials are defined as follows. For $y^{\prime},y\in\mathbb{R}^{r}$ write

$y^{\prime}\leq y$

if $y^{\prime}$ and $y$ lie in the same $W$-orbit and
$\mathbf{g}_{y^{\prime}}\leq_{B}\mathbf{g}_{y}$, where $\leq_{B}$ is the
Bruhat order on $W$. For a $W$-orbit $\mathcal{O}$ and an element
$y\in\mathcal{O}$ the subset
$\\{y^{\prime}\in\mathbb{R}^{r}\,\,|\,\,y^{\prime}\leq y\\}$ of $\mathcal{O}$
is finite, and $c^{\mathcal{O}}$ is its unique minimal element.

We say that $p(x)\in\mathbf{F}[\mathbb{R}^{r}]$ is a quasi-polynomial of
degree $y\in\mathbb{R}^{r}$ if

$p(x)-dx^{y}\in\bigoplus_{y^{\prime}<y}\mathbf{F}x^{y^{\prime}}$

for some $d\in\mathbf{F}^{\times}$. We then say that $d$ is the leading term
of $p(x)$, and $p(x)$ is said to be monic if $d=1$. If $p(x)$ is of degree $y$
with $y$ lying in the $W$-orbit $\mathcal{O}$, then
$p(x)\in\mathbf{F}[\mathcal{O}]$.

Fix a $W$-orbit $\mathcal{O}$ in $\mathbb{R}^{r}$ and fix
$t\in\mathbf{T}_{\mathcal{O}}$. Consider the invertible linear operators

(1.11)
$\mathcal{Y}_{i}^{\mathcal{O}}:=(\mathcal{T}_{i-1}^{\mathcal{O}})^{-1}\cdots(\mathcal{T}_{1}^{\mathcal{O}})^{-1}\mathcal{T}_{0}^{\mathcal{O}}\mathcal{T}_{1}^{\mathcal{O}}\cdots\mathcal{T}_{r-1}^{\mathcal{O}}\mathcal{T}_{r}^{\mathcal{O}}\mathcal{T}_{r-1}^{\mathcal{O}}\cdots\mathcal{T}_{i}^{\mathcal{O}}\qquad(1\leq
i\leq r)$

on $\mathbf{F}[\mathcal{O}]$. These operators are the images under
$\pi^{\mathcal{O}}$ of the commuting elements $Y^{\epsilon_{i}}\in H$ in the
Bernstein presentation of $H$ (cf. Subsection 2.4). In particular,

$[\mathcal{Y}_{i}^{\mathcal{O}},\mathcal{Y}_{j}^{\mathcal{O}}]=0\quad\,\,\,\hbox{
for }\,1\leq i,j\leq r.$

Theorem C. Fix a $W$-orbit $\mathcal{O}$ and fix generic parameters
$q^{\frac{1}{2}},k_{0},u_{0},k,k_{r},u_{r}\in\mathbf{F}^{\times}$ and
$t\in\mathbf{T}_{\mathcal{O}}$. For each $y\in\mathcal{O}$ there exists a
unique quasi-polynomial

$E_{y}^{\mathcal{O}}(x)=E_{y}^{\mathcal{O}}(x;k_{0},u_{0},k,k_{r},u_{r},t;q^{\frac{1}{2}})\in\mathbf{F}[\mathcal{O}]$

satisfying the following two properties:

  1. (1)

$E_{y}^{\mathcal{O}}(x)$ is a monic quasi-polynomial of degree $y$.

  2. (2)

$E_{y}^{\mathcal{O}}(x)$ is a joint eigenfunction of the commuting operators
$\mathcal{Y}_{i}^{\mathcal{O}}$ ($1\leq i\leq r$).

We will prove Theorem C in Section 4 (it is the special case of Theorem 4.9
when the underlying finite root system is of type $\textup{C}_{r}$). It was
derived before in [14] when $k_{0}=u_{0}=k_{r}=u_{r}$. The main step in
proving Theorem C is showing that the $\mathcal{Y}_{i}^{\mathcal{O}}$ ($1\leq
i\leq r$) are triangular operators relative to the partially ordered quasi-
monomial basis $\\{x^{y}\\}_{y\in\mathcal{O}}$ of $\mathbf{F}[\mathcal{O}]$,
with the partial order on $\\{x^{y}\\}_{y\in\mathcal{O}}$ induced from the
partial order $\leq$ on the corresponding set $\mathcal{O}$ of quasi-
exponents. Concretely, we will show that
$\mathcal{Y}_{i}^{\mathcal{O}}(x^{y})\in\mathbf{F}[\mathcal{O}]$ is a quasi-
polynomial of degree $y$ with leading term
$\gamma_{i}^{\mathcal{O}}(y)\in\mathbf{F}^{\times}$ given explicitly by

$\begin{split}\gamma_{i}^{\mathcal{O}}(y)=&\bigl{(}\mathbf{g}_{y}(\mathfrak{s}^{\mathcal{O}}t)\bigr{)}_{i}^{-1},\\\
\mathfrak{s}^{\mathcal{O}}:=&\bigl{(}(k_{0}k_{r})^{-\chi_{e}(2c_{1}^{\mathcal{O}})}(u_{0}u_{r})^{\chi_{o}(2c_{1}^{\mathcal{O}})}k^{n_{1}^{\mathcal{O}}},\ldots,(k_{0}k_{r})^{-\chi_{e}(2c_{r}^{\mathcal{O}})}(u_{0}u_{r})^{\chi_{o}(2c_{r}^{\mathcal{O}})}k^{n_{r}^{\mathcal{O}}}\bigr{)},\end{split}$

where

$n_{i}^{\mathcal{O}}:=\sum_{j=i+1}^{r}\bigl{(}\eta(c_{i}^{\mathcal{O}}-c_{j}^{\mathcal{O}})+\eta(c_{i}^{\mathcal{O}}+c_{j}^{\mathcal{O}})\bigr{)}+\sum_{j=1}^{i-1}\bigl{(}\eta(c_{j}^{\mathcal{O}}+c_{i}^{\mathcal{O}})-\eta(c_{j}^{\mathcal{O}}-c_{i}^{\mathcal{O}})\bigr{)}$

and $\eta:=\chi_{\mathbb{Z}_{>0}}-\chi_{\mathbb{Z}_{\leq 0}}$. Then

$\mathcal{Y}_{i}^{\mathcal{O}}(E_{y}(x))=\gamma_{i}^{\mathcal{O}}(y)E_{y}(x)\qquad\hbox{
for }\,\,1\leq i\leq r\,\,\,\,\hbox{ and }\,\,\,y\in\mathcal{O},$

and the generic conditions on the parameters in Theorem C boil down to the
requirement that the map

$\mathcal{O}\rightarrow\mathbf{T},\qquad
y\mapsto(\gamma_{1}^{\mathcal{O}}(y),\ldots,\gamma_{r}^{\mathcal{O}}(y))$

is an embedding.

The $E_{\lambda}^{\mathbb{Z}^{r}}(x)\in\mathbf{F}[x^{\pm 1}]$
($\lambda\in\mathbb{Z}^{r}$) are Sahi’s [12] monic nonsymmetric Koornwinder
polynomials (recall that in this case necessarily $t=1_{\mathbf{T}}$). Hence
the $E_{y}^{\mathcal{O}}(x)$ ($y\in\mathcal{O}$) may be viewed as quasi-
polynomial generalisations of the nonsymmetric Koornwinder polynomials,
depending on the extra parameters $t\in\mathbf{T}_{\mathcal{O}}$. If
$\mathcal{O}$ and $\mathcal{O}^{\prime}$ are two $W$-orbits in
$\mathbb{R}^{r}$ intersecting $\overline{C_{+}}$ in the same face, then the
corresponding families of quasi-polynomials are essentially the same (cf. [14,
Thm. 6.2(4)]).

##  2\. Preliminaries

###  2.1. Reduced affine root systems

Let $(E,\langle\cdot,\cdot\rangle)$ be an Euclidean space, $\Phi_{0}$ an
irreducible reduced root system in $E^{*}$ with Weyl group $W_{0}$, and
$\Phi_{0}^{\vee}=\\{\alpha^{\vee}\\}_{\alpha\in\Phi_{0}}$ the dual root system
in $E$. The co-root $\alpha^{\vee}$ is the unique vector in $E$ such that

(2.1) $\langle y,\alpha^{\vee}\rangle=\frac{2\alpha(y)}{\|\alpha\|^{2}}$

for all $y\in E$.

Consider the corresponding reduced affine root system

$\Phi:=\Phi_{0}\times\mathbb{Z}\subset E^{*}\times\mathbb{R}.$

We will view an element $(\phi,\xi)\in E^{*}\times\mathbb{R}$ in the ambient
space as an affine linear functional on $E$ by $y\mapsto\phi(y)+\xi$ for $y\in
E$.

The projection $E^{*}\times\mathbb{R}\rightarrow E^{*}$ on the first component
will be denoted by

$f\mapsto\overline{f}.$

It restricts to a surjective map $\Phi\twoheadrightarrow\Phi_{0}$.
Furthermore, we have $a=(\overline{a},a(0))$ for $a\in\Phi$. Throughout the
paper we will identify a root $\alpha\in\Phi_{0}$ with $(\alpha,0)\in\Phi$.

For $a\in\Phi$ let $s_{a}:E\rightarrow E$ be the orthogonal reflection in the
affine root hyperplane $a^{-1}(0)\subset E$. Then

(2.2) $s_{a}(y)=y-a(y)\overline{a}^{\vee}$

for $y\in E$. The affine Weyl group $W$ of $\Phi$ is the subgroup of affine
linear transformations of $E$ generated by the orthogonal reflections $s_{a}$
($a\in\Phi$). The finite Weyl group $W_{0}$ is the subgroup generated by
$s_{\alpha}$ ($\alpha\in\Phi_{0}$).

For $y\in E$ let $\tau(y):E\rightarrow E$ be the translation map $z\mapsto
z+y$. Then

(2.3) $s_{a}=s_{\overline{a}}\,\tau(a(0)\overline{a}^{\vee})$

for $a\in\Phi$. Consequently $W\simeq W_{0}\ltimes Q^{\vee}$ with
$Q^{\vee}=\mathbb{Z}\Phi_{0}^{\vee}$ the co-root lattice of $\Phi_{0}$.

The linear, contragredient $W$-action on the space $E^{*}\times\mathbb{R}$ of
affine linear functionals on $E$ restricts to a $W$-action on $\Phi$. It
satisfies

(2.4)
$\begin{split}s_{a}(b)&=b-\overline{b}(\overline{a}^{\vee})a=(s_{\overline{a}}(\overline{b}),b(0)-a(0)\overline{b}(\overline{a}^{\vee})),\\\
\tau(\lambda)b&=(\overline{b},b(0)-\overline{b}(\lambda))\end{split}$

for $a,b\in\Phi$ and $\lambda\in Q^{\vee}$.

We fix an ordered basis $\Delta_{0}=\\{\alpha_{1},\ldots,\alpha_{r}\\}$ of the
root system $\Phi_{0}$ once and for all. We will use the convention that the
simple root $\alpha_{r}$ is always a long root (if all the roots in $\Phi_{0}$
have the same root length, then all roots are considered to be long as well as
short). We denote by $\Phi_{0}^{+}$ the set of positive roots in $\Phi_{0}$
relative to $\Delta_{0}$. The corresponding set of negative roots is denoted
by $\Phi_{0}^{-}:=-\Phi_{0}^{+}$.

The Weyl group $W_{0}$ is a Coxeter group with Coxeter generators
$\\{s_{1},\ldots,s_{r}\\}$ given by the simple reflections
$s_{i}:=s_{\alpha_{i}}$ ($1\leq i\leq r$). The closure of the positive Weyl
chamber

(2.5) $E_{+}:=\\{y\in
E\,\,\,|\,\,\,\alpha(y)>0\quad\forall\,\alpha\in\Phi_{0}^{+}\\}$

is a fundamental domain for the $W_{0}$-action on $E$.

The ordered basis $\Delta_{0}$ of $\Phi_{0}$ extends to an ordered basis
$\Delta=\\{\alpha_{0},\ldots,\alpha_{r}\\}$ of $\Phi$ with the additional
affine simple root

$\alpha_{0}=(-\varphi,1)$

where $\varphi$ is the highest root of $\Phi_{0}$ relative to $\Delta_{0}$.
The corresponding sets of positive and negative roots are denoted by
$\Phi^{+}$ and $\Phi^{-}$, respectively. The affine Weyl group $W$ is a
Coxeter group with Coxeter generators $\\{s_{0},\ldots,s_{r}\\}$ the simple
reflections $s_{j}:=s_{\alpha_{j}}$ ($0\leq j\leq r$). By (2.3) we have

$s_{0}=s_{\varphi}\tau(-\varphi^{\vee})=\tau(\varphi^{\vee})s_{\varphi}.$

The closure $\overline{C_{+}}$ of the fundamental alcove

(2.6) $C_{+}:=\\{y\in E_{+}\,\,\,|\,\,\,\alpha_{0}(y)>0\\}$

is a fundamental domain for the action of $W\simeq W_{0}\ltimes Q^{\vee}$ on
$E$ by reflections and translations. Since $\alpha_{0}=(-\varphi,1)$, we have
the following alternative description

$C_{+}=\\{y\in
E\,\,\,|\,\,\,0<\alpha(y)<1\quad\forall\,\alpha\in\Phi_{0}^{+}\\}$

of the fundamental alcove. Note furthermore that

(2.7) $\bigcup_{w\in W_{0}}w(\overline{C_{+}})=\\{y\in
E\,\,\,\,\,|\,\,\,\,\,|\alpha(y)|\leq 1\quad\forall\,\alpha\in\Phi_{0}\\}.$

###  2.2. Nonreduced extensions and multiplicity functions

For $a\in\Phi$ such that $\overline{a}(Q^{\vee})=\mathbb{Z}$ we have

$Wa=W_{0}\,\overline{a}\times\mathbb{Z}$

by (2.4). A case by case inspection of the Dynkin diagrams shows that
$\alpha(Q^{\vee})=\mathbb{Z}$ for $\alpha\in\Phi_{0}$ unless
$\alpha\in\Phi_{0}$ is long and $\Phi_{0}$ is of type $\textup{C}_{r}$ ($r\geq
1$), in which case $\alpha(Q^{\vee})=2\mathbb{Z}$ (note that
$\textup{C}_{1}=\textup{A}_{1}$ and $\textup{C}_{2}=\textup{B}_{2}$). Recall
our concention that the simple root $\alpha_{r}$ is always a long root. If
$\Phi_{0}$ is of type $\textup{C}_{r}$ ($r\geq 1$) then $\alpha_{r}$ is the
only long simple root in $\Delta_{0}$.

The $W$-orbits in $\Phi$ can now be described as follows.

  1. (1)

If all the roots in $\Phi_{0}$ has the same root length but $\Phi_{0}$ is not
of type $\textup{A}_{1}$, then $W$ acts transitively on $\Phi$.

  2. (2)

If $\Phi_{0}$ is of type $\textup{C}_{1}=\textup{A}_{1}$, then
$\Phi=W\alpha_{0}\sqcup W\alpha_{1}$ and

$W\alpha_{0}=\Phi_{0}\times\mathbb{Z}_{o},\qquad
W\alpha_{1}=\Phi_{0}\times\mathbb{Z}_{e}$

with $\mathbb{Z}_{o}$ (resp. $\mathbb{Z}_{e}$) the set of odd (resp. even)
integers.

  3. (3)

If $\Phi_{0}$ is of type $\textup{C}_{r}$ ($r\geq 2$) then
$\Phi=W\alpha_{0}\sqcup W\alpha_{1}\sqcup W\alpha_{r}$ and $\alpha_{i}\in
W\alpha_{1}$ for all $1\leq i<r$. Furthermore,

$W\alpha_{0}=\Phi_{0}^{\ell}\times\mathbb{Z}_{o},\qquad
W\alpha_{1}=\Phi_{0}^{s}\times\mathbb{Z},\qquad
W\alpha_{r}=\Phi_{0}^{\ell}\times\mathbb{Z}_{e}$

with $\Phi_{0}^{\ell}$ (resp. $\Phi_{0}^{s}$) the long (resp. short) roots in
$\Phi_{0}$.

  4. (4)

If $\Phi_{0}$ is of type $\textup{B}_{r}$ ($r\geq 3$), $F_{4}$ or $G_{2}$ and
if $\alpha_{i}\in\Delta_{0}$ ($1\leq i<r$) is a short simple root, then
$\Phi=W\alpha_{0}\sqcup W\alpha_{i}$ and

$W\alpha_{0}=\Phi_{0}^{\ell}\times\mathbb{Z}=W\alpha_{r},\qquad
W\alpha_{i}=\Phi_{0}^{s}\times\mathbb{Z}.$

The set

$\Phi^{\textup{nr}}:=\Phi\sqcup\left\\{a/2\,\,\,|\,\,\,a\in\Phi\,\,\textup{such
that}\,\,\overline{a}(Q^{\vee})=\mathbb{Z}_{e}\right\\}$

forms an affine root system in the affine space $E^{*}\times\mathbb{R}$ (see
[8]). If $\Phi_{0}$ is of type $C_{r}$ ($r\geq 1$) then $\Phi^{\textup{nr}}$
is the nonreduced irreducible affine root system of type
$\textup{C}^{\vee}\textup{C}_{r}$ (see [8]). In this case $\Phi^{\textup{nr}}$
has five $W$-orbits
$W\alpha_{0},W\frac{\alpha_{0}}{2},W\alpha_{1},W\alpha_{r},W\frac{\alpha_{r}}{2}$
when $r\geq 2$, and four $W$-orbits
$W\alpha_{0},W\frac{\alpha_{0}}{2},W\alpha_{1},W\frac{\alpha_{1}}{2}$ when
$r=1$. If $\Phi_{0}$ is not of type $\textup{C}_{r}$ ($r\geq 1$) then
$\Phi^{\textup{nr}}=\Phi$.

Let $\mathbf{F}$ be a field of characteristic zero. We call a $W$-invariant
function

$\mathbf{k}:\Phi^{\textup{nr}}\rightarrow\mathbf{F}^{\times},\qquad
a\mapsto\mathbf{k}_{a}$

a multiplicity function. Denote by $\mathcal{K}$ the set of multiplicity
functions. In order to obtain uniform notations, we extend a multiplicity
function $\mathbf{k}:\Phi^{\textup{nr}}\rightarrow\mathbf{F}^{\times}$ to a
$W$-invariant function

$\mathbf{k}:\Phi\sqcup\tfrac{1}{2}\Phi\rightarrow\mathbf{F}^{\times}$

by declaring $\mathbf{k}_{\frac{a}{2}}:=\mathbf{k}_{a}$ when
$\frac{a}{2}\not\in\Phi^{\textup{nr}}$. Note that for any root
$\alpha\in\Phi_{0}$,

(2.8)
$\mathbf{k}_{\alpha}=\mathbf{k}_{(\alpha,1)}=\mathbf{k}_{\frac{\alpha}{2}}=\mathbf{k}_{(\frac{\alpha}{2},\frac{1}{2})}\quad\,\,\textup{
if $\Phi_{0}$ is not of type $\textup{C}_{r}$ ($r\geq 1$)}.$

For the value of a multiplicity function $\mathbf{k}$ at a simple root
$\alpha_{j}$ and at $\frac{\alpha_{j}}{2}$ we will use the shorthand notations

$k_{j}:=\mathbf{k}_{\alpha_{j}},\qquad\qquad
u_{j}:=\mathbf{k}_{\frac{\alpha_{j}}{2}}.$

If $\Phi_{0}$ has rank $r=1$ then the multiplicity function $\mathbf{k}$ is
determined by the four parameters $k_{0},u_{0},k_{1},u_{1}$, which can be
chosen arbitrarily. If $\Phi_{0}$ has rank $r>1$ then $\mathbf{k}$ is
determined by the five parameters $k_{0},u_{0},k:=k_{i},k_{r},u_{r}$, where
$1\leq i<r$ is such that $\alpha_{i}$ is a short root. These parameters can be
chosen arbitrarily when $\Phi_{0}$ is of type $\textup{C}_{r}$ ($r\geq 2$). If
$\Phi_{0}$ is not of type $\textup{C}_{r}$ ($r\geq 1$) then
$k_{0}=u_{0}=k_{r}=u_{r}$ by (2.8), hence
$\mathcal{K}\simeq\mathbf{F}^{\times}$ if all the roots in $\Phi_{0}$ have the
same length and $\mathcal{K}\simeq(\mathbf{F}^{\times})^{2}$ otherwise (i.e.,
if $\Phi_{0}$ is of type $\textup{B}_{r}$ ($r\geq 3$), $F_{4}$ or $G_{2}$).

The extended affine Weyl group is the subgroup

$W^{\textup{ext}}:=W_{0}\tau(P^{\vee})$

of the group of affine linear transformations of $E$, with $P^{\vee}$ the co-
weight lattice of $\Phi_{0}$. The linear, contragredient
$W^{\textup{ext}}$-action on the space $E^{*}\times\mathbb{R}$ of affine
linear functionals on $E$ restricts to a $W^{\textup{ext}}$-action on $\Phi$.
The explicit formulas for this action are again given by (2.4), now with
$\lambda\in P^{\vee}$ in the second formula.

Write $\mathcal{K}^{\textup{res}}\subseteq\mathcal{K}$ for the subset of
multiplicity functions $\mathbf{k}$ satisfying the following two additional
conditions:

  1. (1)

$\mathbf{k}$ is $W^{\textup{ext}}$-invariant,

  2. (2)

$\mathbf{k}_{\frac{a}{2}}=\mathbf{k}_{a}$ for all $a\in\Phi$.

By (2) a restricted multiplicity function
$\mathbf{k}\in\mathcal{K}^{\textup{res}}$ is uniquely determined by its values
on $\Phi$, and $\mathbf{k}_{a}=\mathbf{k}_{\overline{a}}$ for $a\in\Phi$. Its
value $\mathbf{k}_{\alpha}$ at $\alpha\in\Phi_{0}$ only depends on the length
of $\alpha$. Hence $\mathcal{K}^{\textup{res}}\simeq\mathbf{F}^{\times}$ if
all roots in $\Phi_{0}$ have the same length and
$\mathcal{K}^{\textup{res}}\simeq(\mathbf{F}^{\times})^{2}$ otherwise, which
implies that

$\mathcal{K}^{\textup{res}}=\mathcal{K}\quad\textup{ if }\,\Phi_{0}\,\textup{
is not of type }\,\textup{C}_{r}\,\,(r\geq 1).$

In particular, for $\mathbf{k}\in\mathcal{K}^{\textup{res}}$ formula (2.8)
holds true for root systems $\Phi_{0}$ of any type,

$k_{0}=u_{0}=k_{r}=u_{r}\quad\,\hbox{ if
}\,\,\,\mathbf{k}\in\mathcal{K}^{\textup{res}}.$

Define an involution

(2.9)
$\mathcal{K}\overset{\sim}{\longrightarrow}\mathcal{K},\qquad\quad\mathbf{k}\mapsto\widetilde{\mathbf{k}}$

by interchanging the values $k_{0}$ and $u_{r}$ of $\mathbf{k}$ on the
$W$-orbits $W\alpha_{0}$ and $W\frac{\alpha_{r}}{2}$. We call it the duality
involution. Note that it is the identity unless $\Phi_{0}$ is of type
$\textup{C}_{r}$ ($r\geq 1$). Furthermore, its restriction to
$\mathcal{K}^{\textup{res}}$ is the identity for all types.

The standard Cherednik-Macdonald theory for parameters
$\mathbf{k}\in\mathcal{K}^{\textup{res}}$ admits an extension to parameters
$\mathbf{k}\in\mathcal{K}$ [11, 12]. It only gives new results when $\Phi_{0}$
is of type $\textup{C}_{r}$ ($r\geq 1$), since otherwise
$\mathcal{K}=\mathcal{K}^{\textup{res}}$. The resulting theory is sometimes
referred to as the Koornwinder case (since the associated analogs of the
symmetric Macdonald polynomials are the Koornwinder polynomials [6]), or as
the $\textup{C}^{\vee}\textup{C}_{r}$ case (since $\mathcal{K}$ is the natural
set of multiplicity functions on the nonreduced root system of type
$\textup{C}^{\vee}\textup{C}_{r}$). It is common in the literature on
Koornwinder ($\textup{C}^{\vee}\textup{C}_{r}$) extensions of the Cherednik-
Macdonald theory to develop the theory directly using the following explicit
realisation of the root system $\Phi_{0}$ of type $\textup{C}_{r}$ ($r\geq
1$), see, e.g., [6, 11, 12, 15]:

  1. $\bullet$

$E=\mathbb{R}^{r}$ with orthonormal basis $\\{\epsilon_{i}\\}_{i=1}^{r}$,

  2. $\bullet$

$\Phi_{0}^{s}=\\{\pm(\epsilon_{i}\pm\epsilon_{j})\\}_{1\leq i<j\leq r}$
($=\emptyset$ when $r=1$) and $\Phi_{0}^{\ell}=\\{\pm
2\epsilon_{i}\\}_{i=1}^{r}$, where we identify $E\simeq E^{*}$ via the scalar
product (in particular, $Q^{\vee}=\bigoplus_{i=1}^{r}\mathbb{Z}\epsilon_{i}$),

  3. $\bullet$

$\alpha_{i}=\epsilon_{i}-\epsilon_{i+1}$ ($1\leq i<r$) and
$\alpha_{r}=2\epsilon_{r}$.

Note that $\varphi=2\epsilon_{1}$ is the highest root, hence
$\alpha_{0}=(-2\epsilon_{1},1)$. With these choices, the type $\textup{C}_{r}$
results presented in Subsection 1.2–1.4 follow immediately from the general
results as discussed below. In the remainder of the paper $\mathbf{k}$ will be
a multiplicity function in $\mathcal{K}$ unless stated explicitly otherwise.

###  2.3. Quasi-polynomials

The space of quasi-polynomials [14] is the group algebra

$\mathbf{F}[E]=\bigoplus_{y\in E}\mathbf{F}x^{y}$

of $E$, viewed as abelian additive group. Here we denote the canonical basis
elements $x^{y}$ ($y\in E$) of $\mathbf{F}[E]$ multiplicatively, so
$x^{y}x^{y^{\prime}}=x^{y+y^{\prime}}$ and $x^{0}$ is the unit element. We
call $x^{y}$ the quasi-monomial with quasi-exponent $y\in E$. The Weyl group
$W_{0}$ acts on $\mathbf{F}[E]$ by $\mathbf{F}$-algebra automorphisms by

$w(x^{y}):=x^{wy}$

for $w\in W_{0}$ and $y\in E$.

For any subset $Z\subseteq E$ we write

$\mathbf{F}[Z]:=\bigoplus_{y\in Z}\mathbf{F}x^{y}$

for the subspace of $\mathbf{F}[E]$ spanned by the quasi-monomials $x^{y}$
($y\in Z$).

The subspace $\mathbf{F}[P^{\vee}]$ is a $W_{0}$-stable subalgebra of
$\mathbf{F}[E]$, which we call the subalgebra of Laurent polynomials in
$\mathbf{F}[E]$. For $\mu\in P^{\vee}$ we say that $x^{\mu}$ is a monomial
with exponent $\mu\in P^{\vee}$. For generic multiplicity parameter
$\mathbf{k}\in\mathcal{K}^{\textup{res}}$ the nonsymmetric Macdonald
polynomials form a basis of $\mathbf{F}[P^{\vee}]$. In this paper we are
focussing on the theory for the extended set $\mathcal{K}$ of multiplicity
parameters, in which case the corresponding nonsymmetric Macdonald-Koornwinder
polynomials form a basis of the $W_{0}$-stable subalgebra
$\mathbf{F}[Q^{\vee}]$ of $\mathbf{F}[P^{\vee}]$.

Let $E/W$ be the set of $W$-orbits in $E$. For a $W$-orbit $\mathcal{O}\in
E/W$ we denote by $c^{\mathcal{O}}$ the unique point in the intersection of
$\mathcal{O}$ and the closure $\overline{C_{+}}$ of the fundamental alcove.
Note that the $W$-orbit in $E$ containing the origin is $Q^{\vee}$. A
$W$-orbit $\mathcal{O}$ in $E$ has a finite number of $\tau(Q^{\vee})$-orbits
$\tau(Q^{\vee})y_{i}$ ($1\leq i\leq N$), hence the corresponding space
$\mathbf{F}[\mathcal{O}]$ of quasi-polynomials with quasi-exponents in
$\mathcal{O}$ is a free $\mathbf{F}[Q^{\vee}]$-module of finite rank,

(2.10)
$\mathbf{F}[\mathcal{O}]=\bigoplus_{i=1}^{N}\mathbf{F}[Q^{\vee}]x^{y_{i}}.$

Furthermore, the space $\mathbf{F}[E]$ of quasi-polynomials decomposes as

(2.11) $\mathbf{F}[E]=\bigoplus_{\mathcal{O}\in E/W}\mathbf{F}[\mathcal{O}].$

For generic multiplicity functions $\mathbf{k}\in\mathcal{K}^{\textup{res}}$
and an arbitrary $W$-orbit $\mathcal{O}$, quasi-polynomial extensions of the
nonsymmetric Macdonald polynomials were introduced in [14]. They depend on a
generic dilation parameter $q_{\varphi}\in\mathbf{F}^{\times}$ and additional
$\mathcal{O}$-dependent representation parameters, and they form a basis of
$\mathbf{F}[\mathcal{O}]$. For $\mathcal{O}=Q^{\vee}$ they are the
nonsymmetric Macdonald polynomials. The goal of this paper is to extend this
result to multiplicity functions $\mathbf{k}$ in $\mathcal{K}$.

This boils down to introducing the Koornwinder
($\textup{C}^{\vee}\textup{C}_{r}$) analogs of the quasi-polynomial extensions
of the type $\textup{C}_{r}$ nonsymmetric Macdonald polynomials from [14].
They will now depend on five (four in case of $r=1$) multiplicity parameters
instead of two (one in case of $r=1$). To stay close to the notations and
results from [14] we will give a uniform treatment of the theory for
multiplicity functions $\mathbf{k}\in\mathcal{K}$ when the root system
$\Phi_{0}$ is of arbitrary type. For type $\textup{C}_{r}$ the results are
made more concrete in Subsections 1.2-1.4.

###  2.4. The affine Hecke algebra

The affine Hecke algebra $H=H(\mathbf{k})$ is the unital associative
$\mathbf{F}$-algebra with generators $T_{j}$ ($0\leq j\leq r$) and relations

  1. (a)

The $(W,\\{s_{0},\ldots,s_{r}\\})$-braid relations for $T_{0},\ldots,T_{r}$,

  2. (b)

$(T_{j}-k_{j})(T_{j}+k_{j}^{-1})=0$ for $j=0,\ldots,r$.

Here (a) means

$T_{i}T_{j}T_{i}\cdots=T_{j}T_{i}T_{j}\cdots\qquad(0\leq i\not=j\leq r)$

with on each side $m_{ij}$ terms, where $m_{ij}$ is the order of $s_{i}s_{j}$
in $W$.

A reduced expression of $g\in W$ is an expression $g=s_{j_{1}}\cdots
s_{j_{\ell(g)}}$ of $g$ as product of simple reflections with $\ell(g)$
minimal. The length function $\ell:W\rightarrow\mathbb{Z}_{\geq 0}$ satisfies
$\ell(g)=\\#\Pi(g)$, with

$\Pi(g):=\Phi^{+}\cap g^{-1}\Phi^{-}.$

The braid relations ensure that the element

$T_{g}:=T_{j_{1}}\cdots T_{j_{\ell(g)}}$

in $H$ does not depend on the choice of reduced expression $g=s_{j_{1}}\cdots
s_{j_{\ell(g)}}$, and $\\{T_{g}\\}_{g\in W}$ is a basis of $H$.

The finite Hecke algebra $H_{0}=H_{0}(\mathbf{k})$ is the subalgebra of $H$
generated by $T_{1},\ldots,T_{r}$. The defining relations of $H_{0}$ in terms
of $T_{1},\ldots,T_{r}$ are the $(W_{0},\\{s_{1},\ldots,s_{r}\\})$-braid
relations and the quadratic relations $(T_{i}-k_{i})(T_{i}+k_{i}^{-1})=0$ for
$i=1,\ldots,r$. For $w\in W_{0}\subset W$ a reduced expression
$w=s_{i_{1}}\ldots s_{i_{\ell}}$ in $W$ can be chosen with simple reflections
from $W_{0}$, i.e., with $1\leq i_{j}\leq r$. Furthermore,
$\Pi(w)=\Phi_{0}^{+}\cap w^{-1}\Phi_{0}^{-}$ for $w\in W_{0}$, and
$\\{T_{w}\\}_{w\in W_{0}}$ is a basis of $H_{0}$.

Define $\chi:\Phi_{0}\rightarrow\\{\pm 1\\}$ by

(2.12) $\chi(\alpha):=\begin{cases}1\qquad&\hbox{ if
}\,\alpha\in\Phi_{0}^{+},\\\ -1\qquad&\hbox{ if
}\,\alpha\in\Phi_{0}^{-}.\end{cases}$

In other words, $\chi=\chi_{+}-\chi_{-}$ with

(2.13) $\chi_{\pm}:=\chi_{\Phi_{0}^{\pm}}:\Phi_{0}\rightarrow\\{0,1\\}$

the indicator function of $\Phi_{0}^{\pm}$ in $\Phi_{0}$. For $j=0,\dots,r$
and $g\in W$ we have

(2.14) $\ell(s_{j}g)=\ell(g)+\chi(g^{-1}\alpha_{j}),$

and hence

(2.15)
$T_{j}T_{g}=\chi_{-}(g^{-1}\alpha_{j})(k_{j}-k_{j}^{-1})T_{g}+T_{s_{j}g}$

in the affine Hecke algebra $H$.

We now describe the Bernstein decomposition of $H$ (see [7] for details). Let
$H^{\times}$ be the group of units in $H$. There exists a unique group
homomorphism

$Q^{\vee}\rightarrow H^{\times},\qquad\lambda\mapsto Y^{\lambda}$

such that $Y^{\lambda}=T_{\tau(\lambda)}$ for $\lambda\in\overline{E_{+}}\cap
Q^{\vee}$. The resulting algebra map

(2.16) $\mathbf{F}[Q^{\vee}]\hookrightarrow H,\qquad p\mapsto p(Y),$

mapping $x^{\lambda}$ to $Y^{\lambda}$ for all $\lambda\in Q^{\vee}$, is
injective. The image of the embedding is denoted by
$\mathbf{F}_{Y}[Q^{\vee}]$. The multiplication map of $H$ restricts to a
linear isomorphism

$H_{0}\otimes\mathbf{F}_{Y}[Q^{\vee}]\overset{\sim}{\longrightarrow}H.$

The commutation relations between elements in $H_{0}$ and
$\mathbf{F}_{Y}[Q^{\vee}]$ are described (and determined) by the Bernstein-
Lusztig cross relations

(2.17)
$Y^{\lambda}T_{i}-T_{i}Y^{s_{i}\lambda}=\left(\frac{\widetilde{\mathbf{k}}_{\alpha_{i}}-\widetilde{\mathbf{k}}_{\alpha_{i}}^{-1}+(\widetilde{\mathbf{k}}_{\frac{\alpha_{i}}{2}}-\widetilde{\mathbf{k}}_{\frac{\alpha_{i}}{2}}^{-1})Y^{-\alpha_{i}^{\vee}}}{1-Y^{-2\alpha_{i}^{\vee}}}\right)(Y^{\lambda}-Y^{s_{i}\lambda})$

for $i=1,\ldots,r$ and $\lambda\in Q^{\vee}$.

The right hand side of (2.17) appears to be in the quotient field
$\mathbf{F}_{Y}(Q^{\vee})$ of $\mathbf{F}_{Y}[Q^{\vee}]$, but it lies in
$\mathbf{F}_{Y}[Q^{\vee}]$. Indeed, if $\alpha_{i}(Q^{\vee})=\mathbb{Z}$ then
$\widetilde{\mathbf{k}}_{\frac{\alpha_{i}}{2}}=\widetilde{\mathbf{k}}_{\alpha_{i}}=k_{i}$,
hence the right hand side of (2.17) reduces to

$(k_{i}-k_{i}^{-1})\left(\frac{Y^{\lambda}-Y^{s_{i}\lambda}}{1-Y^{-\alpha_{i}^{\vee}}}\right)=(k_{i}-k_{i}^{-1})Y^{\lambda}\left(\frac{1-Y^{-\alpha_{i}(\lambda)\alpha^{\vee}}}{1-Y^{-\alpha_{i}^{\vee}}}\right),$

which lies in $\mathbf{F}_{Y}[Q^{\vee}]$ since
$\alpha_{i}(\lambda)\in\mathbb{Z}$. If $\alpha_{i}(Q^{\vee})=\mathbb{Z}_{e}$
then $\Phi_{0}$ is of type $\textup{C}_{r}$ ($r\geq 1$) and $i=r$, in view of
our convention that $\alpha_{r}$ is a long root. The right hand side of (2.17)
then reads

$\Bigl{(}k_{r}-k_{r}^{-1}+(k_{0}-k_{0}^{-1})Y^{-\alpha_{r}^{\vee}}\Bigr{)}Y^{\lambda}\left(\frac{1-Y^{-\alpha_{r}(\lambda)\alpha_{r}^{\vee}}}{1-Y^{-2\alpha_{r}^{\vee}}}\right),$

which lies in $\mathbf{F}_{Y}[Q^{\vee}]$ since
$\alpha_{r}(\lambda)\in\mathbb{Z}_{e}$.

Denote by

$\mathbf{F}[Q^{\vee}]^{W_{0}}\subset\mathbf{F}[Q^{\vee}]$

the subalgebra of $W_{0}$-invariant elements in $\mathbf{F}[Q^{\vee}]$, and
$\mathbf{F}_{Y}[Q^{\vee}]^{W_{0}}$ for its image in $H$ under the embedding
(2.16). Then

$Z(H)=\mathbf{F}_{Y}[Q^{\vee}]^{W_{0}},$

where $Z(H)$ denotes the center of $H$.

###  2.5. The double affine Hecke algebra

Fix a parameter $q_{\varphi}\in\mathbf{F}^{\times}$ and set

$q_{\alpha}:=q_{\varphi}^{\|\varphi\|^{2}/\|\alpha\|^{2}}\quad\,\hbox{ for
}\,\,\alpha\in\Phi_{0}.$

It equals either $q_{\varphi}$, $q_{\varphi}^{2}$ or $q_{\varphi}^{3}$ (see
[5, §9.4, Table 1]), and $q_{w\alpha}=q_{\alpha}$ for all $\alpha\in\Phi_{0}$.
We like to think of $q_{\varphi}$ as being equal to
$q^{\frac{2}{\|\varphi\|^{2}}}$ for $q$ in some field extension of
$\mathbf{F}$ (this is done in Subsection 1.2-1.4, where we described the
results explicitly for $\Phi_{0}$ of type $\textup{C}_{r}$). To circumvent
field extensions we will introduce instead a group homomorphism
$q:\frac{2}{\|\varphi\|^{2}}\mathbb{Z}\rightarrow\mathbf{F}^{\times}$,
$m\mapsto q^{m}$, with

(2.18) $q^{m}:=q_{\varphi}^{m\|\varphi\|^{2}/2}\quad\,\,\hbox{ for
}\,m\in\tfrac{2}{\|\varphi\|^{2}}\mathbb{Z}.$

Note that $q^{m}\in\mathbf{F}$ is meaningful for $m\in\langle
Q^{\vee},Q^{\vee}\rangle$ since $\langle
Q^{\vee},Q^{\vee}\rangle\subseteq\frac{2}{\|\varphi\|^{2}}\mathbb{Z}$.

Consider the $\mathbf{F}$-torus $\mathbf{T}$ of rank $r$ consisting of the
group homomorphisms

$Q^{\vee}\rightarrow\mathbf{F}^{\times}.$

We write $t^{\mu}\in\mathbf{F}^{\times}$ for the value of $t\in\mathbf{T}$ at
$\mu\in Q^{\vee}$. Clearly $\mathbf{T}$ is an abelian group by pointwise
multiplication: $(st)^{\mu}:=s^{\mu}t^{\mu}$ for $s,t\in\mathbf{T}$ and
$\mu\in Q^{\vee}$.

For $\lambda\in Q^{\vee}$ we define the torus element

$q^{\lambda}\in\mathbf{T}$

by $\mu\mapsto q^{\langle\lambda,\mu\rangle}$ ($\mu\in Q^{\vee}$). Then
$\mathbf{T}$ admits a left $W$-action by

(2.19) $\begin{split}(wt)^{\mu}:=&t^{w^{-1}\mu},\\\
(\tau(\lambda)t)^{\mu}:=&(q^{\lambda}t)^{\mu}=q^{\langle\lambda,\mu\rangle}t^{\mu}\end{split}$

for $t\in\mathbf{T}$, $w\in W_{0}$ and $\lambda,\mu\in Q^{\vee}$.

We will view a polynomial $p=\sum_{\mu}d_{\mu}x^{\mu}\in\mathbf{F}[Q^{\vee}]$
as regular function on $\mathbf{T}$ by

$p(t):=\sum_{\mu}d_{\mu}t^{\mu}\quad\,\,\hbox{ for }\,\,t\in\mathbf{T}.$

The formula

$(gp)(t):=p(g^{-1}t)$

for $p\in\mathbf{F}[Q^{\vee}]$, $g\in W$ and $t\in\mathbf{T}$ then turns
$\mathbf{F}[Q^{\vee}]$ into a $W$-module, with $W$ acting by algebra
automorphisms. Concretely, the action on the basis of monomials is given by

(2.20)
$w(x^{\mu})=x^{w\mu},\qquad\quad\tau(\lambda)(x^{\mu})=q^{-\langle\mu,\lambda\rangle}x^{\mu}$

for $w\in W_{0}$ and $\lambda,\mu\in Q^{\vee}$.

Note that by (2.3),

(2.21)
$s_{a}(x^{\mu})=q_{\overline{a}}^{-a(0)\overline{a}(\mu)}x^{s_{\overline{a}}\mu}$

for $a\in\Phi$ and $\mu\in Q^{\vee}$. In particular,

(2.22) $s_{0}(x^{\mu})=q_{\varphi}^{\varphi(\mu)}x^{s_{\varphi}\mu}.$

In various computations it is convenient to use co-roots of affine roots and
incorporate $q$-powers in the exponents of the monomials. The co-root
$b^{\vee}$ of $b\in\Phi$ is defined by

$b^{\vee}:=\Bigl{(}\overline{b}^{\vee},\frac{2b(0)}{\|\overline{b}\|^{2}}\Bigr{)}\in
E\times\mathbb{R}.$

The resulting set $\Phi^{\vee}:=\\{b^{\vee}\\}_{b\in\Phi}$ of co-roots is an
affine root system in $E\times\mathbb{R}$ [8]. Note that

(2.23)
$(s_{a}b)^{\vee}=b^{\vee}-\overline{a}(\overline{b}^{\vee})a^{\vee}\quad\,\,\hbox{
for }\,a,b\in\Phi$

in view of (2.4) and the fact that
$\frac{2}{\|\beta\|^{2}}\beta(\alpha^{\vee})=\frac{2}{\|\alpha\|^{2}}\alpha(\beta^{\vee})$
for $\alpha,\beta\in\Phi_{0}$ (indeed, both sides are equal to
$\langle\alpha^{\vee},\beta^{\vee}\rangle$ by (2.1)).

We set

$x^{\widehat{\mu}}:=q^{m}x^{\mu}\in\mathbf{F}[Q^{\vee}]\qquad\textup{ for
}\quad\widehat{\mu}=(\mu,m)\in
Q^{\vee}\times\tfrac{2}{\|\varphi\|^{2}}\mathbb{Z}$

and we write $t^{\widehat{\mu}}$ for the evaluation of
$x^{\widehat{\mu}}\in\mathbf{F}[Q^{\vee}]$ at $t\in\mathbf{T}$ (so
$t^{\widehat{\mu}}=q^{m}t^{\mu}$). Note that $\Phi^{\vee}\subset
Q^{\vee}\times\frac{2}{\|\varphi\|^{2}}\mathbb{Z}$, hence $x^{b^{\vee}}$ makes
sense for all $b\in\Phi$. Concretely,

(2.24)
$x^{b^{\vee}}:=q_{\overline{b}}^{b(0)}x^{\overline{b}^{\vee}}\in\mathbf{F}[Q^{\vee}],$

which reduces to the monomial $x^{\beta^{\vee}}$ in case $b=(\beta,0)$.

Formula (2.21) can then be rewritten as

(2.25) $s_{a}(x^{\mu})=x^{\mu-\overline{a}(\mu)a^{\vee}}.$

We furthermore have

(2.26) $g(x^{b^{\vee}})=x^{(gb)^{\vee}}\quad\,\,\textup{ for }\,g\in W\,\hbox{
and }\,b\in\Phi.$

Indeed, it suffices to check (2.26) for $w=s_{a}$ ($a\in\Phi$). By (2.24) and
(2.25) we have

$s_{a}(x^{b^{\vee}})=x^{b^{\vee}-\overline{a}(\overline{b}^{\vee})a^{\vee}}$

which equals $x^{(s_{a}b)^{\vee}}$ by (2.23).

######  Definition 2.1 ([2, 12]).

The double affine Hecke algebra
$\mathbb{H}=\mathbb{H}(\mathbf{k};q_{\varphi})$ is the unital associative
$\mathbf{F}$-algebra generated by $T_{j}$ ($0\leq j\leq r$) and $x^{\mu}$
($\mu\in Q^{\vee}$) subject to the following relations:

  1. (a)

The $(W,\\{s_{0},\ldots,s_{r}\\})$-braid relations for $T_{0},\ldots,T_{r}$,

  2. (b)

The quadratic relations $(T_{j}-k_{j})(T_{j}+k_{j}^{-1})=0$ for
$j=0,\ldots,r$,

  3. (c)

$x^{\mu}x^{\nu}=x^{\mu+\nu}$ ($\mu,\nu\in Q^{\vee}$) and $x^{0}$ is the unit
element of $\mathbb{H}$,

  4. (d)

The cross relations

(2.27)
$T_{j}x^{\mu}-s_{j}(x^{\mu})T_{j}=\left(\frac{k_{j}-k_{j}^{-1}+(u_{j}-u_{j}^{-1})x^{\alpha_{j}^{\vee}}}{1-x^{2\alpha_{j}^{\vee}}}\right)\bigl{(}x^{\mu}-s_{j}(x^{\mu})\bigr{)}$

for $j=0,\ldots,r$ and $\mu\in Q^{\vee}$.

The Poincaré-Birkhoff-Witt (PBW) theorem for $\mathbb{H}$ states that the
canonical algebra maps $H\rightarrow\mathbb{H}$ and
$\mathbf{F}[Q^{\vee}]\rightarrow\mathbb{H}$ are embeddings, and that the
multiplication map of $\mathbb{H}$ restricts to a linear isomorphism

$\mathbf{F}[Q^{\vee}]\otimes H\overset{\sim}{\longrightarrow}\mathbb{H}.$

By the Bernstein presentation of the affine Hecke algebra (see Subsection
2.4), the subalgebra

$H^{X}\subset\mathbb{H}$

generated by $\mathbf{F}[Q^{\vee}]$ and $H_{0}$ is isomorphic to the affine
Hecke algebra $\widetilde{H}:=H(\widetilde{\mathbf{k}})$.

The double affine Hecke algebra with dual multiplicity parameters will be
denoted by

$\widetilde{\mathbb{H}}:=\mathbb{H}(\widetilde{\mathbf{k}},q_{\varphi}).$

To keep the notations manageable, we will use the same notations
$T_{j},T_{g},Y^{\mu},x^{\mu}$ in both $\mathbb{H}$ and
$\widetilde{\mathbb{H}}$.

The duality anti-isomorphism [3, 12] is the unique anti-algebra isomorphism

$\delta=\delta_{\mathbf{k}}:\mathbb{H}\overset{\sim}{\longrightarrow}\widetilde{\mathbb{H}}$

satisfying

$\delta(T_{i})=T_{i},\qquad\delta(Y^{\lambda})=x^{-\lambda},\qquad\delta(x^{\mu})=Y^{-\mu}$

for $i=1,\ldots,r$ and $\lambda,\mu\in Q^{\vee}$. Its inverse is
$\widetilde{\delta}:=\delta_{\widetilde{\mathbf{k}}}$.

##  3\. Quasi-polynomial representations

###  3.1. The quasi-polynomial representation of $H^{X}$

In this subsection we introduce the quasi-polynomial representation of the
dual affine Hecke algebra $H^{X}$. In case
$\mathbf{k}\in\mathcal{K}^{\textup{res}}$, the representation we obtain was
derived before in [13, 14].

Let

$\lfloor\cdot\rfloor:\mathbb{R}\rightarrow\mathbb{Z}$

be the floor function, so $\lfloor s\rfloor$ is the largest integer $\leq s$.
We denote by

$\lfloor\cdot\rfloor_{e}:\mathbb{R}\rightarrow\mathbb{Z}_{e},\qquad\quad\lfloor\cdot\rfloor_{o}:\mathbb{R}\rightarrow\mathbb{Z}_{o}$

the functions which map $s\in\mathbb{R}$ to the smallest even and odd integer
$\leq s$, respectively. Note that

$\lfloor s\rfloor_{e}:=2\lfloor s/2\rfloor,\qquad\quad\lfloor
s\rfloor_{o}=\lfloor s+1\rfloor_{e}-1$

for $s\in\mathbb{R}$.

######  Definition 3.1.

For $a\in\Phi$ define the even and odd truncated divided difference operator
$\nabla_{a}=\nabla_{a}(\mathbf{k})\in\textup{End}(\mathbf{F}[E])$ by

(3.1)
$\begin{split}\nabla_{a}^{e}(x^{y})&:=\left(\frac{1-x^{-\lfloor\overline{a}(y)\rfloor_{e}a^{\vee}}}{1-x^{2a^{\vee}}}\right)x^{y},\\\
\nabla_{a}^{o}(x^{y})&:=\left(\frac{x^{a^{\vee}}-x^{-\lfloor\overline{a}(y)\rfloor_{o}a^{\vee}}}{1-x^{2a^{\vee}}}\right)x^{y}\end{split}$

for $y\in E$. We furthermore write $\nabla_{j}^{e}:=\nabla_{\alpha_{j}}^{e}$
and $\nabla_{j}^{o}:=\nabla_{\alpha_{j}}^{o}$ for $j=0,\ldots,r$.

Truncation in Definition 3.1 refers to the fact that the real numbers
$\overline{a}(y)$ in formula (3.1) are truncated using the even and odd floor
operations. These truncations are necessary to turn the two quotients in (3.1)
into well defined elements in $\mathbf{F}[Q^{\vee}]$. Note that
$\nabla_{a}^{e}$ and $\nabla_{a}^{o}$ depend on $q_{\varphi}$ when
$a\in\Phi\setminus\Phi_{0}$, in view of (2.24). In this subsection we only
need the truncated divided difference operators for $a\in\Phi_{0}$, and there
will be no dependence on $q_{\varphi}$.

We write

(3.2) $\nabla_{a}:=\nabla_{a}^{e}+\nabla_{a}^{o}$

for the sum of the even and odd truncated divided difference operator, and
$\nabla_{j}:=\nabla_{\alpha_{j}}$ for $j=0,\ldots,r$. Then

(3.3) $\nabla_{a}(x^{y})=\left(\frac{1-x^{-\lfloor\overline{a}(y)\rfloor
a^{\vee}}}{1-x^{a^{\vee}}}\right)x^{y}$

for $y\in E$ since

(3.4) $\\{\lfloor s\rfloor_{e},\lfloor s\rfloor_{o}\\}=\\{\lfloor
s\rfloor,\lfloor s\rfloor-1\\}$

as unordered $2$-sets for any $s\in\mathbb{R}$. The truncated difference
operator $\nabla_{a}\in\textup{End}(\mathbf{F}[E])$ was introduced before in
[14, §4.2].

The link of the various truncated divided difference operators to the usual
divided difference operator is as follows (the first part of the lemma was
observed before in [14, Lem. 4.4]).

######  Lemma 3.2.

  1. (1)

If $a\in\Phi$ then

$\nabla_{a}(x^{\mu})=\frac{x^{\mu}-s_{a}(x^{\mu})}{1-x^{a^{\vee}}}$

for $\mu\in Q^{\vee}$.

  2. (2)

If $\Phi_{0}$ is of type $\textup{C}_{r}$ ($r\geq 1$) and $a\in\Phi$ is an
affine root such that $\overline{a}\in\Phi_{0}^{\ell}$ (i.e., $a\in
W\alpha_{0}\sqcup W\alpha_{r}$), then

$\nabla_{a}^{e}(x^{\mu})=\frac{x^{\mu}-s_{a}(x^{\mu})}{1-x^{2a^{\vee}}}=x^{-a^{\vee}}\nabla_{a}^{o}(x^{\mu})$

for $\mu\in Q^{\vee}$.

###### Proof.

(1) This is immediate from (2.25) and the fact that
$\overline{a}(Q^{\vee})\subseteq\mathbb{Z}$.
(2) Under these assumptions we have $\overline{a}(Q^{\vee})=\mathbb{Z}_{e}$,
hence

$\lfloor\overline{a}(\mu)\rfloor_{e}=\overline{a}(\mu)=\lfloor\overline{a}(\mu)\rfloor_{o}+1$

and the result follows again from (2.25). ∎

We will often make use of the indicator functions
$\chi_{\mathbb{Z}_{e}},\chi_{\mathbb{Z}_{o}}:\mathbb{R}\rightarrow\\{0,1\\}$
of the subsets of even and odd integers, respectively. We will denote them by
$\chi_{e}$ and $\chi_{o}$, respectively.

######  Theorem 3.3.

The formulas

(3.5) $\begin{split}\pi(x^{\mu})x^{y}&:=x^{y+\mu},\\\
\pi(T_{i})x^{y}&:=k_{i}^{\chi_{e}(\alpha_{i}(y))}u_{i}^{\chi_{o}(\alpha_{i}(y))}x^{s_{i}y}+(k_{i}-k_{i}^{-1})\nabla_{i}^{e}(x^{y})+(u_{i}-u_{i}^{-1})\nabla_{i}^{o}(x^{y})\end{split}$

for $\mu\in Q^{\vee}$, $i\in\\{1,\ldots,r\\}$ and $y\in E$ define a
representation $\pi:H^{X}\rightarrow\textup{End}(\mathbf{F}[E])$.

###### Proof.

Formula (3.5) uniquely defines linear operators $\pi(x^{\mu})$ and
$\pi(T_{i})$ on $\mathbf{F}[E]$, which in turn restrict to linear operators

(3.6)
$\pi^{\mathcal{O}}(x^{\mu}):=\pi(x^{\mu})|_{\mathbf{F}[\mathcal{O}]},\qquad\pi^{\mathcal{O}}(T_{i}):=\pi(T_{i})|_{\mathbf{F}[\mathcal{O}]}$

on $\mathbf{F}[\mathcal{O}]$ for every $W$-orbit $\mathcal{O}$ in $E$. In view
of (2.11) it thus suffices to show that the linear operators (3.6) define a
representation
$\pi^{\mathcal{O}}=\pi(\cdot)|_{\mathbf{F}[\mathcal{O}]}:H^{X}\rightarrow\textup{End}(\mathbf{F}[\mathcal{O}])$.

Consider $H^{X}$ as left regular $H^{X}$-module. By (2.15) and (2.27) the
$H^{X}$-action can be written down explicitly relative to the basis

$\\{x^{\lambda}T_{w}\,\,|\,\,\lambda\in Q^{\vee},w\in W_{0}\\}$

of $H^{X}$. The resulting formulas are

(3.7) $\begin{split}x^{\mu}x^{\lambda}T_{w}&=x^{\lambda+\mu}T_{w},\\\
T_{i}x^{\lambda}T_{w}&=x^{s_{i}\lambda}T_{s_{i}w}+(k_{i}-k_{i}^{-1})\left(\frac{1-x^{(2\chi_{-}(w^{-1}\alpha_{i})-\alpha_{i}(\lambda))\alpha_{i}^{\vee}}}{1-x^{2\alpha_{i}^{\vee}}}\right)x^{\lambda}T_{w}\\\
&\qquad\qquad\quad+(u_{i}-u_{i}^{-1})\left(\frac{x^{\alpha_{i}^{\vee}}-x^{(1-\alpha_{i}(\lambda))\alpha_{i}^{\vee}}}{1-x^{2\alpha_{i}^{\vee}}}\right)x^{\lambda}T_{w}\end{split}$

for $\lambda,\mu\in Q^{\vee}$, $w\in W_{0}$ and $i=1,\ldots,r$.

Let $\kappa_{w}^{\mathcal{O}}\in\mathbf{F}^{\times}$ ($w\in W_{0}$) be a
collection of nonzero scalars. Consider the surjective linear map

(3.8)
$\psi^{\mathcal{O}}:H^{X}\twoheadrightarrow\mathbf{F}[\mathcal{O}],\qquad\quad\psi^{\mathcal{O}}(x^{\lambda}T_{w}):=\kappa_{w}^{\mathcal{O}}\,x^{\lambda+wc^{\mathcal{O}}}\quad(\lambda\in
Q^{\vee},\,w\in W_{0}).$

We will fine tune the scalars $\kappa_{w}^{\mathcal{O}}\in\mathbf{F}^{\times}$
($w\in W_{0}$) in such a way that the kernel of $\psi^{\mathcal{O}}$ is a left
ideal in $H^{X}$. This will allow us to push the left regular $H^{X}$-action
through $\psi^{\mathcal{O}}$, giving rise to a $H^{X}$-action on
$\mathbf{F}[\mathcal{O}]$. We then show that the resulting action of $x^{\mu}$
and $T_{i}$ on $\mathbf{F}[\mathcal{O}]$ is by the linear operators
$\pi^{\mathcal{O}}(x^{\mu})$ and $\pi^{\mathcal{O}}(T_{i})$, which completes
the proof of the theorem.

For any choice of scalars $\kappa_{w}^{\mathcal{O}}\in\mathbf{F}^{\times}$
($w\in W_{0}$) the kernel of $\psi^{\mathcal{O}}$ is invariant under left
multiplication by $\mathbf{F}[Q^{\vee}]$. Note that the kernel
$\psi^{\mathcal{O}}$ is invariant under left multiplication by $T_{i}$ if
there exists a linear operator
$\mathcal{D}_{i}^{\mathcal{O}}\in\textup{End}(\mathbf{F}[\mathcal{O}])$ such
that

(3.9)
$\mathcal{D}_{i}^{\mathcal{O}}(\psi^{\mathcal{O}}(h))=\psi^{\mathcal{O}}(T_{i}h)$

for all $h\in H^{X}$. By (3.7) we have

(3.10)
$\begin{split}\psi^{\mathcal{O}}(T_{i}x^{\lambda}T_{w})&=\kappa_{s_{i}w}^{\mathcal{O}}s_{i}(x^{\lambda+wc})\\\
&+\kappa_{w}^{\mathcal{O}}(k_{i}-k_{i}^{-1})\left(\frac{1-x^{(2\chi_{-}(w^{-1}\alpha_{i})-\alpha_{i}(\lambda))\alpha_{i}^{\vee}}}{1-x^{2\alpha_{i}^{\vee}}}\right)x^{\lambda+wc}\\\
&+\kappa_{w}^{\mathcal{O}}(u_{i}-u_{i}^{-1})\left(\frac{x^{\alpha_{i}^{\vee}}-x^{(1-\alpha_{i}(\lambda))\alpha_{i}^{\vee}}}{1-x^{2\alpha_{i}^{\vee}}}\right)x^{\lambda+wc},\end{split}$

so we look for conditions on the scalars $\kappa_{w}^{\mathcal{O}}$ such that
(3.10) can be expressed as a linear operator
$\mathcal{D}_{i}^{\mathcal{O}}\in\textup{End}(\mathbf{F}[\mathcal{O}])$ acting
on
$\psi^{\mathcal{O}}(x^{\lambda}T_{w})=\kappa_{w}^{\mathcal{O}}x^{\lambda+wc^{\mathcal{O}}}$
for all $\lambda\in Q^{\vee}$ and $w\in W_{0}$.

To simplify notations we write

$y=\lambda+wc^{\mathcal{O}}$

with $w\in W_{0}$ and $\lambda\in Q^{\vee}$. We first prove that

(3.11)
$\begin{split}\frac{1}{\kappa_{w}^{\mathcal{O}}}\,\psi^{\mathcal{O}}(T_{i}x^{\lambda}T_{w})&=\Bigl{(}\frac{\kappa_{s_{i}w}^{\mathcal{O}}}{\kappa_{w}^{\mathcal{O}}}+(k_{i}-k_{i}^{-1})\chi_{-}(w^{-1}\alpha_{i})\chi_{e}(\alpha_{i}(wc^{\mathcal{O}}))\\\
&\qquad\qquad\qquad\qquad+(u_{i}-u_{i}^{-1})\chi_{+}(w^{-1}\alpha_{i})\chi_{o}(\alpha_{i}(wc^{\mathcal{O}}))\Bigr{)}x^{s_{i}y}\\\
&+(k_{i}-k_{i}^{-1})\nabla_{i}^{e}(x^{y})+(u_{i}-u_{i}^{-1})\nabla_{i}^{o}(x^{y})\end{split}$

by rewriting the two quotients in (3.10) in terms of the odd and even
truncated difference operators.

We first consider the proof of (3.11) when
$\alpha_{i}(Q^{\vee})=\mathbb{Z}_{e}$, i.e., when $\Phi_{0}$ is of type
$\textup{C}_{r}$ ($r\geq 1$) and $i=r$. Then (2.7) and
$\alpha_{r}(Q^{\vee})=\mathbb{Z}_{e}$ imply that

$2\chi_{-}(w^{-1}\alpha_{r})-\alpha_{r}(\lambda)=\begin{cases}2-\lfloor\alpha_{r}(y)\rfloor_{e}\quad&\hbox{if
}\,w^{-1}\alpha_{r}\in\Phi_{0}^{-}\,\hbox{ and
}\,\alpha_{r}(wc^{\mathcal{O}})=0,\\\
-\lfloor\alpha_{r}(y)\rfloor_{e}\quad&\textup{otherwise}.\end{cases}$

By (2.7) this can be reformulated as

(3.12)
$2\chi_{-}(w^{-1}\alpha_{r})-\alpha_{r}(\lambda)=\begin{cases}2-\lfloor\alpha_{r}(y)\rfloor_{e}\quad&\hbox{
if
}\,\,\,\chi_{-}(w^{-1}\alpha_{r})\chi_{e}(\alpha_{r}(wc^{\mathcal{O}}))=1,\\\
-\lfloor\alpha_{r}(y)\rfloor_{e}\quad&\hbox{ if
}\,\,\,\chi_{-}(w^{-1}\alpha_{r})\chi_{e}(\alpha_{r}(wc^{\mathcal{O}}))=0.\end{cases}$

This allows us to rewrite the second line of (3.10) for $\Phi_{0}$ of type
$\textup{C}_{r}$ and $i=r$ in terms of the even truncated divided difference
operator,

(3.13)
$\left(\frac{1-x^{(2\chi_{-}(w^{-1}\alpha_{r})-\alpha_{r}(\lambda))\alpha_{r}^{\vee}}}{1-x^{2\alpha_{r}^{\vee}}}\right)x^{y}=\nabla_{r}^{e}(x^{y})+\chi_{-}(w^{-1}\alpha_{r})\chi_{e}(\alpha_{r}(wc^{\mathcal{O}}))x^{s_{r}y}$

(in case $\chi_{-}(w^{-1}\alpha_{r})\chi_{e}(\alpha_{r}(wc^{\mathcal{O}}))=1$
use the fact that $\lfloor\alpha_{r}(y)\rfloor_{e}=\alpha_{r}(y)$). To rewrite
the third line of (3.10) for $\Phi_{0}$ of type $\textup{C}_{r}$ and $i=r$,
note that

$1-\alpha_{r}(\lambda)=\begin{cases}2-\lfloor\alpha_{r}(y)\rfloor_{o}\quad&\hbox{
if }\alpha_{r}(wc^{\mathcal{O}})=1,\\\
-\lfloor\alpha_{r}(y)\rfloor_{o}\quad&\hbox{ otherwise},\end{cases}$

which can be reformulated as

(3.14)
$1-\alpha_{r}(\lambda)=\begin{cases}2-\lfloor\alpha_{r}(y)\rfloor_{o}\quad&\hbox{
if }\,\,\chi_{+}(w^{-1}\alpha_{r})\chi_{o}(\alpha_{r}(wc^{\mathcal{O}}))=1,\\\
-\lfloor\alpha_{r}(y)\rfloor_{o}\quad&\hbox{ if
}\,\,\chi_{+}(w^{-1}\alpha_{r})\chi_{o}(\alpha_{r}(wc^{\mathcal{O}}))=0\end{cases}$

in view of (2.7). It follows that

(3.15)
$\left(\frac{x^{\alpha_{r}^{\vee}}-x^{(1-\alpha_{r}(\lambda))\alpha_{r}^{\vee}}}{1-x^{2\alpha_{r}^{\vee}}}\right)x^{y}=\nabla_{r}^{o}(x^{y})+\chi_{+}(w^{-1}\alpha_{r})\chi_{o}(\alpha_{r}(wc^{\mathcal{O}}))x^{s_{r}y}$

(in case $\chi_{+}(w^{-1}\alpha_{r})\chi_{o}(\alpha_{r}(wc^{\mathcal{O}}))=1$
use the fact that $\lfloor\alpha_{r}(y)\rfloor_{o}=\alpha_{r}(y)$).
Substituting (3.13) and (3.15) into formula (3.10) for $i=r$, we obtain (3.11)
for $\Phi_{0}$ of type $\textup{C}_{r}$ and for $i=r$.

We now prove (3.11) when $\alpha_{i}(Q^{\vee})=\mathbb{Z}$. Then $k_{i}=u_{i}$
and using that

$\\{2\chi_{-}(w^{-1}\alpha_{i})-\alpha_{i}(\lambda),1-\alpha_{i}(\lambda)\\}=\\{\chi_{-}(w^{-1}\alpha_{i})-\alpha_{i}(\lambda),1+\chi_{-}(w^{-1}\alpha_{i})-\alpha_{i}(\lambda)\\}$

as unordered $2$-sets, formula (3.10) simplifies to

(3.16)
$\begin{split}\psi^{\mathcal{O}}(T_{i}x^{\lambda}T_{w})&=\kappa_{s_{i}w}^{\mathcal{O}}x^{s_{i}y}\\\
&+\kappa_{w}^{\mathcal{O}}(k_{i}-k_{i}^{-1})\left(\frac{1-x^{(\chi_{-}(w^{-1}\alpha_{i})-\alpha_{i}(\lambda))\alpha_{i}^{\vee}}}{1-x^{\alpha_{i}^{\vee}}}\right)x^{y}.\end{split}$

Using (2.7), we have for $w^{-1}\alpha_{i}\in\Phi_{0}^{-}$ that

$\chi_{-}(w^{-1}\alpha_{i})-\alpha_{i}(\lambda)=\begin{cases}-\lfloor\alpha_{i}(y)\rfloor\quad\,\,&\hbox{
if }\,\,\chi_{e}(\alpha_{i}(wc^{\mathcal{O}}))=0,\\\
1-\lfloor\alpha_{i}(y)\rfloor\quad\,\,&\hbox{ if
}\,\,\chi_{e}(\alpha_{i}(wc^{\mathcal{O}}))=1\end{cases}$

and for $w^{-1}\alpha_{i}\in\Phi_{0}^{+}$ that

$\chi_{-}(w^{-1}\alpha_{i})-\alpha_{i}(\lambda)=\begin{cases}-\lfloor\alpha_{i}(y)\rfloor\quad\,\,&\hbox{
if }\,\,\chi_{o}(\alpha_{i}(wc^{\mathcal{O}}))=0,\\\
1-\lfloor\alpha_{i}(y)\rfloor\quad\,\,&\hbox{ if
}\,\,\chi_{0}(\alpha_{i}(wc^{\mathcal{O}}))=1.\end{cases}$

This leads to the formula

$\begin{split}\left(\frac{1-x^{(\chi_{-}(w^{-1}\alpha_{i})-\alpha_{i}(\lambda))\alpha_{i}^{\vee}}}{1-x^{\alpha_{i}^{\vee}}}\right)x^{y}&=\nabla_{i}(x^{y})\\\
+&\bigl{(}\chi_{-}(w^{-1}\alpha_{i})\chi_{e}(\alpha_{i}(wc^{\mathcal{O}}))+\chi_{+}(w^{-1}\alpha_{i})\chi_{o}(\alpha_{i}(wc^{\mathcal{O}}))\bigr{)}x^{s_{i}y}.\end{split}$

Substituting into (3.16) and using the formulas (3.2) and (3.3), we now also
obtain (3.11) in case $\alpha_{i}(Q^{\vee})=\mathbb{Z}$.

The next step is choosing the normalization factors $\kappa_{w}^{\mathcal{O}}$
($w\in W_{0}$) in such a way that the coefficient of $x^{s_{i}\lambda}$ in
(3.11) only depends on $y=\lambda+wc^{\mathcal{O}}$. We will show that this is
the case for the scalars

(3.17)
$\kappa_{w}^{\mathcal{O}}:=\prod_{\alpha\in\Pi(w)}\mathbf{k}_{\alpha}^{\chi_{e}(\alpha(c^{\mathcal{O}}))}\mathbf{k}_{\alpha/2}^{-\chi_{o}(\alpha(c^{\mathcal{O}}))}$

(a small computation using (2.7) shows that $\kappa_{w}^{\mathcal{O}}$ reduces
for $\mathbf{k}\in\mathcal{K}^{\textup{res}}$ to the normalization factor [14,
(4.11)]). Since $\Pi(w)=\Phi_{0}^{+}\cap w^{-1}\Phi_{0}^{-}$ for $w\in W_{0}$,
we have

$\Pi(s_{i}w)=\begin{cases}\Pi(w)\cup\\{w^{-1}\alpha_{i}\\}\quad&\hbox{ if
}\,\,\,\,\chi_{+}(w^{-1}\alpha_{i})=1,\\\
\Pi(w)\setminus\\{-w^{-1}\alpha_{i}\\}\quad&\hbox{ if
}\,\,\,\,\chi_{-}(w^{-1}\alpha_{i})=1.\end{cases}$

Combined with the fact that $\chi_{e},\chi_{o}:\mathbb{R}\rightarrow\\{0,1\\}$
are even functions, we obtain

$\frac{\kappa_{s_{i}w}^{\mathcal{O}}}{\kappa_{w}^{\mathcal{O}}}=\begin{cases}k_{i}^{\chi_{e}(\alpha_{i}(wc^{\mathcal{O}}))}u_{i}^{-\chi_{o}(\alpha_{i}(wc^{\mathcal{O}}))}\qquad&\hbox{
if }\,\,\,\chi_{+}(w^{-1}\alpha_{i})=1,\\\
k_{i}^{-\chi_{e}(\alpha_{i}(wc^{\mathcal{O}}))}u_{i}^{\chi_{o}(\alpha_{i}(wc^{\mathcal{O}}))}\qquad&\hbox{
if }\,\,\,\chi_{-}(w^{-1}\alpha_{i})=1\end{cases}$

and hence

(3.18)
$\begin{split}\frac{\kappa_{s_{i}w}^{\mathcal{O}}}{\kappa_{w}^{\mathcal{O}}}&+(k_{i}-k_{i}^{-1})\chi_{-}(w^{-1}\alpha_{i})\chi_{e}(\alpha_{i}(wc^{\mathcal{O}}))\\\
&\quad\,\,\,\,+(u_{i}-u_{i}^{-1})\chi_{+}(w^{-1}\alpha_{i})\chi_{o}(\alpha_{i}(wc^{\mathcal{O}}))=k_{i}^{\chi_{e}(\alpha_{i}(wc^{\mathcal{O}}))}u_{i}^{\chi_{o}(\alpha_{i}(wc^{\mathcal{O}}))}\end{split}$

for $w\in W_{0}$ and $i=1,\ldots,r$. So formula (3.11) reduces for the
specific choice (3.17) of $\kappa_{w}^{\mathcal{O}}$ to

(3.19)
$\begin{split}\frac{1}{\kappa_{w}^{\mathcal{O}}}\psi^{\mathcal{O}}(T_{i}x^{\lambda}T_{w})&=k_{i}^{\chi_{e}(\alpha_{i}(wc^{\mathcal{O}}))}u_{i}^{\chi_{o}(\alpha_{i}(wc^{\mathcal{O}}))}x^{s_{i}y}\\\
&+(k_{i}-k_{i}^{-1})\nabla_{i}^{e}(x^{y})+(u_{i}-u_{i}^{-1})\nabla_{i}^{o}(x^{y}).\end{split}$

Now note that the map

(3.20) $E\rightarrow\mathbf{F}^{\times},\qquad y\mapsto
k_{i}^{\chi_{e}(\alpha_{i}(y))}u_{i}^{\chi_{o}(\alpha_{i}(y))}$

is $\tau(Q^{\vee})$-invariant. This is trivial when $\Phi_{0}$ is of type
$\textup{C}_{r}$ ($r\geq 1$) and $i=r$, since then
$\alpha_{r}(Q^{\vee})=\mathbb{Z}_{e}$. In all other cases
$\alpha_{i}(Q^{\vee})=\mathbb{Z}$ hence $k_{i}=u_{i}$, in which case it
follows from the fact that the function (3.20) reduces to $y\mapsto
k_{i}^{\chi_{\mathbb{Z}}(\alpha_{i}(y))}$. In (3.19) we may thus replace the
coefficient
$k_{i}^{\chi_{e}(\alpha_{i}(wc^{\mathcal{O}}))}u_{i}^{\chi_{o}(\alpha_{i}(wc^{\mathcal{O}}))}$
of $x^{s_{i}y}$ by
$k_{i}^{\chi_{e}(\alpha_{i}(y))}u_{i}^{\chi_{o}(\alpha_{i}(y))}$.

In conclusion, for $\kappa_{w}^{\mathcal{O}}$ given by (3.17) the associated
linear map $\psi^{\mathcal{O}}$ (3.8) satisfies

$\psi^{\mathcal{O}}(T_{i}h)=\mathcal{D}_{i}^{\mathcal{O}}(\psi^{\mathcal{O}}(h))\qquad\forall\,h\in
H^{X}$

with $\mathcal{D}_{i}^{\mathcal{O}}\in\textup{End}(\mathbf{F}[\mathcal{O}])$
defined by

$\mathcal{D}_{i}^{\mathcal{O}}(x^{y}):=k_{i}^{\chi_{e}(\alpha_{i}(y))}u_{i}^{\chi_{o}(\alpha_{i}(y))}x^{s_{i}y}+(k_{i}-k_{i}^{-1})\nabla_{i}^{e}(x^{y})+(u_{i}-u_{i}^{-1})\nabla_{i}^{o}(x^{y})$

for $y\in\mathcal{O}$. The kernel of
$\psi^{\mathcal{O}}:H^{X}\twoheadrightarrow\mathbf{F}[\mathcal{O}]$ thus is a
left-ideal, and $\mathbf{F}[\mathcal{O}]$ inherits a $H^{X}$-action from
$H^{X}/\textup{ker}(\psi^{\mathcal{O}})$ with $T_{i}$ acting by
$\mathcal{D}_{i}^{\mathcal{O}}=\pi^{\mathcal{O}}(T_{i})$ and $x^{\mu}$ acting
by $\pi^{\mathcal{O}}(x^{\mu})$. This completes the proof of the theorem. ∎

######  Corollary 3.4.

  1. (1)

$\mathbf{F}[E]=\bigoplus_{\mathcal{O}\in E/W}\mathbf{F}[\mathcal{O}]$ is a
decomposition of $\mathbf{F}[E]$ in $H^{X}$-submodules.

  2. (2)

Let $\mathcal{O}\in E/W$. Then

(3.21)
$\pi(x^{\lambda}T_{w})x^{c^{\mathcal{O}}}=\kappa_{w}^{\mathcal{O}}x^{\lambda+wc^{\mathcal{O}}}$

for $\lambda\in Q^{\vee}$ and $w\in W_{0}$, with $\kappa_{w}^{\mathcal{O}}$
defined by (3.17).

###### Proof.

(1) This was remarked in the first paragraph of the proof of Theorem 3.3.
(2) It follows from the last paragraph of the proof of Theorem 3.3 that the
epimorphism
$\psi^{\mathcal{O}}:H^{X}\twoheadrightarrow\mathbf{F}[\mathcal{O}]$, mapping
$x^{\lambda}T_{w}$ to $\kappa_{w}^{\mathcal{O}}x^{\lambda+wc^{\mathcal{O}}}$
for $\lambda\in Q^{\vee}$ and $w\in W_{0}$, is $H^{X}$-linear for the special
choice (3.17) of the scalars $\kappa_{w}^{\mathcal{O}}$. Hence

$\pi(x^{\lambda}T_{w})x^{c^{\mathcal{O}}}=\pi(x^{\lambda}T_{w})\psi^{\mathcal{O}}(1)=\psi^{\mathcal{O}}(x^{\lambda}T_{w})=\kappa_{w}^{\mathcal{O}}x^{\lambda+wc^{\mathcal{O}}}$

for $\lambda\in Q^{\vee}$ and $w\in W_{0}$. ∎

For the upgrade of Theorem 3.3 to the double affine Hecke algebra $\mathbb{H}$
(see Subsection 3.3) it is useful to introduce the notation

(3.22) $\pi^{\mathcal{O}}(\cdot):=\pi(\cdot)|_{\mathbf{F}[\mathcal{O}]}$

for the representation map of the $H^{X}$-submodule $\mathbf{F}[\mathcal{O}]$
(as we in fact already have done in the proof of Theorem 3.3). The reason for
this is that the extension of
$\pi^{\mathcal{O}}:H^{X}\rightarrow\textup{End}(\mathbf{F}[\mathcal{O}])$ to a
$\mathbb{H}$-representation on $\mathbf{F}[\mathcal{O}]$ involves additional
$\mathcal{O}$-dependent representation parameters.

######  Remark 3.5.

  1. (1)

By (3.2) we have

(3.23)
$\pi(T_{i})x^{y}=k_{i}^{\chi_{\mathbb{Z}}(\alpha_{i}(y))}x^{s_{i}y}+(k_{i}-k_{i}^{-1})\nabla_{i}(x^{y})\qquad\hbox{
if }\,\,k_{i}=u_{i},$

from which it follows that the $H^{X}$-representation $\pi^{\mathcal{O}}$ for
$\mathbf{k}\in\mathcal{K}^{\textup{res}}$ is the restriction to $H^{X}$ of the
quasi-polynomial $\mathbb{H}$-representation defined in [14, Thm 1.1].

  2. (2)

Let $\Lambda\subset E$ be a $W_{0}$-invariant lattice containing $Q^{\vee}$.
Then $\mathbf{F}[\Lambda]$ is a $\pi(H^{X})$-submodule, and the action of
$\pi(T_{i})|_{\mathbf{F}[\Lambda]}$ can be written in terms of metaplectic
Demazure-Lusztig type operators when $k_{i}=u_{i}$, see [14]. The resulting
$H^{X}$-representation $\pi(\cdot)|_{\mathbf{F}[\Lambda]}$ for
$\mathbf{k}\in\mathcal{K}^{\textup{res}}$ is essentially the one introduced in
[13, Thm. 3.7]. The proof of Theorem 3.3 basically follows the same strategy
as the proof of [13, Thm. 3.7].

###  3.2. Parabolic data and parabolically induced modules

We show in this subsection that the representation $\pi^{\mathcal{O}}$ is a
parabolically induced $H^{X}$-module when $\alpha_{0}(c^{\mathcal{O}})\not=0$.

For a subset $I\subseteq\\{1,\ldots,r\\}$ write

  1. –

$W_{0,I}$ for the parabolic subgroup of $W_{0}$ generated by $s_{i}$ ($i\in
I$),

  2. –

$W_{0}^{I}$ for the minimal coset representatives of $W_{0}/W_{0,I}$,

  3. –

$H_{0,I}$ for the subalgebra of $H_{0}$ generated by $T_{i}$ ($i\in I$).

The finite Hecke algebra $H_{0}$ is a free right $H_{0,I}$-module with basis
$\\{T_{v}\\}_{v\in W_{0}^{I}}$, since

(3.24) $\ell(vw)=\ell(v)+\ell(w)\quad\hbox{ for }v\in W_{0}^{I}\hbox{ and
}w\in W_{0,I}.$

Here is an immediate lift to the affine Hecke algebra $H^{X}$.

######  Lemma 3.6.

  1. (1)

$\\{\tau(\lambda)v\\}_{(\lambda,v)\in Q^{\vee}\times W_{0}^{I}}$ is a complete
set of representatives of $W/W_{0,I}$.

  2. (2)

$H^{X}$ is a free right $H_{0,I}$-module with basis
$\\{x^{\lambda}T_{v}\\}_{(\lambda,v)\in Q^{\vee}\times W_{0}^{I}}$.

###### Proof.

The first part follows from the fact that $(\lambda,w)\mapsto\tau(\lambda)w$
defines a bijection $Q^{\vee}\rtimes W_{0}\overset{\sim}{\longrightarrow}W$.
For the second part, note that the multiplication map of $H^{X}$ restricts to
a linear isomorphism $\mathbf{F}[Q^{\vee}]\otimes
H_{0}\overset{\sim}{\longrightarrow}H^{X}$. ∎

The more familiar parabolic structures on $W$ and on the associated affine
Hecke algebra $H$ arise from their Coxeter type presentations. In this case it
depends on a subset $J$ of $\\{0,\ldots,r\\}$. We write

  1. –

$W_{J}$ for the parabolic subgroup of $W$ generated by $s_{j}$ ($j\in J$),

  2. –

$W^{J}$ for the minimal coset representatives of $W/W_{J}$,

  3. –

$H_{J}$ for the subalgebra of $H$ generated by $T_{j}$ ($j\in J$).

The length identity (3.24) now also holds true for $v\in W^{J}$ and $w\in
W_{J}$. As a consequence, $H$ is a free right $H_{J}$-module with basis
$\\{T_{g}\\}_{g\in W^{J}}$.

The closure $\overline{C_{+}}$ of the fundamental alcove $C_{+}$ splits in a
disjoint union of facets

(3.25) $\overline{C_{+}}=\bigsqcup_{J\subsetneq\\{0,\ldots,r\\}}C_{+}^{J},$

with $C_{+}^{J}$ the set of vectors $y\in\overline{C_{+}}$ for which
$\alpha_{j}(y)=0$ iff $j\in J$. For $y\in E$ denote by $W_{y}\subset W$ the
subgroup of $W$ fixing $y$. It is well known that

$W_{c}=W_{J}\quad\,\hbox{ for }\,\,c\in C_{+}^{J}.$

######  Definition 3.7.

For a $W$-orbit $\mathcal{O}$ in $E$ we write $J(\mathcal{O})$ for the subset
of $\\{0,\ldots,r\\}$ such that $c^{\mathcal{O}}\in C_{+}^{J(\mathcal{O})}$.
We furthermore write

(3.26) $I(\mathcal{O}):=J(\mathcal{O})\cap\\{1,\ldots,r\\}.$

Note that $I(\mathcal{O})=J(\mathcal{O})$ iff
$\alpha_{0}(c^{\mathcal{O}})\not=0$. We will use the shorthand notations

$C_{+}^{\mathcal{O}},W_{\mathcal{O}},W^{\mathcal{O}},H_{\mathcal{O}},...$

for
$C_{+}^{J(\mathcal{O})},W_{J(\mathcal{O})},W^{J(\mathcal{O})},H_{J(\mathcal{O})},...$
and

$W_{0,\mathcal{O}},W^{\mathcal{O}}_{0},H_{0,\mathcal{O}},...$

for $W_{0,I(\mathcal{O})},W_{0}^{I(\mathcal{O})},H_{0,I(\mathcal{O})},...$.
The following lemma refines the decomposition (2.10) of the space of quasi-
polynomials $\mathbf{F}[\mathcal{O}]$ as $\mathbf{F}[Q^{\vee}]$-module when
$\alpha_{0}(c^{\mathcal{O}})\not=0$.

######  Lemma 3.8.

If $\mathcal{O}$ is a $W$-orbit such that $\alpha_{0}(c^{\mathcal{O}})\not=0$
then $\mathbf{F}[\mathcal{O}]$ decomposes as

(3.27) $\mathbf{F}[\mathcal{O}]=\bigoplus_{v\in
W_{0}^{\mathcal{O}}}\mathbf{F}[Q^{\vee}]x^{vc^{\mathcal{O}}}.$

###### Proof.

The assignment $gW_{0,\mathcal{O}}\mapsto gc^{\mathcal{O}}$ gives rise to a
bijection

$W/W_{0,\mathcal{O}}\overset{\sim}{\longrightarrow}\mathcal{O}$

since $W_{c^{\mathcal{O}}}=W_{\mathcal{O}}=W_{0,\mathcal{O}}$ by the
assumption on $\mathcal{O}$. Furthermore, $W_{0}^{\mathcal{O}}$ is a complete
set of representatives of the double coset space $\tau(Q^{\vee})\backslash
W/W_{0,\mathcal{O}}$ by Lemma 3.6(1). Hence $\\{vc^{\mathcal{O}}\\}_{v\in
W_{0}^{\mathcal{O}}}=W_{0}c^{\mathcal{O}}$ is a complete set of
representatives of the $\tau(Q^{\vee})$-orbits in $\mathcal{O}$, and the lemma
follows. ∎

For a $W$-orbit $\mathcal{O}$ in $E$ let $\mathbf{F}1^{\mathcal{O}}$ the
trivial $H_{\mathcal{O}}$-module, defined by

$T_{j}1^{\mathcal{O}}=k_{j}1^{\mathcal{O}}\quad\,\,\hbox{ for }\,\,j\in
J(\mathcal{O}).$

We will also view $\mathbf{F}1^{\mathcal{O}}$ as $H_{0,\mathcal{O}}$-module by
restricting the action to $H_{0,\mathcal{O}}$. Consider the induced
$H^{X}$-module

$\textup{Ind}_{H_{0,\mathcal{O}}}^{H^{X}}(\mathbf{F}1^{\mathcal{O}})=H^{X}\otimes_{H_{0,\mathcal{O}}}\mathbf{F}1^{\mathcal{O}}$

and write

$\mathbf{1}^{\mathcal{O}}=1\otimes_{H_{0,\mathcal{O}}}1^{\mathcal{O}}$

for its canonical cyclic vector.

######  Proposition 3.9.

There exists a unique $H^{X}$-linear epimorphism

(3.28)
$\textup{Ind}_{H_{0,\mathcal{O}}}^{H^{X}}(\mathbf{F}1^{\mathcal{O}})\twoheadrightarrow(\mathbf{F}[\mathcal{O}],\pi^{\mathcal{O}})$

mapping $\mathbf{1}^{\mathcal{O}}$ to $x^{c^{\mathcal{O}}}$. It is an
isomorphism when $\alpha_{0}(c^{\mathcal{O}})\not=0$.

###### Proof.

For the first statement we need to show that the assignment
$h\mathbf{1}^{\mathcal{O}}\mapsto\pi(h)x^{c^{\mathcal{O}}}$ for $h\in H^{X}$
is well defined. It suffices to note that

$\pi(T_{i})x^{c^{\mathcal{O}}}=k_{i}x^{c^{\mathcal{O}}}\quad\,\,\hbox{ for
}\,i\in I(\mathcal{O}).$

But for $i\in I(\mathcal{O})$ we have $s_{i}c^{\mathcal{O}}=c^{\mathcal{O}}$,
and hence

$\pi(T_{i})x^{c^{\mathcal{O}}}=\kappa_{s_{i}}^{\mathcal{O}}x^{s_{i}c^{\mathcal{O}}}=k_{i}x^{c^{\mathcal{O}}}$

by Corollary 3.4(2) and (3.17).

For the second statement, note first that

(3.29) $\\{x^{\lambda}T_{v}\mathbf{1}^{\mathcal{O}}\,\,\,|\,\,\,(\lambda,v)\in
Q^{\vee}\times W_{0}^{\mathcal{O}}\\}$

is a basis of
$\textup{Ind}_{H_{0,\mathcal{O}}}^{H^{X}}(\mathbf{F}1^{\mathcal{O}})$. By
Corollary 3.4(2) the basis element $x^{\lambda}T_{v}\mathbf{1}^{\mathcal{O}}$
is mapped by the epimorphism (3.28) to

$\pi(x^{\lambda}T_{v})x^{c^{\mathcal{O}}}=\kappa_{v}^{\mathcal{O}}x^{\lambda+vc^{\mathcal{O}}}$

By Lemma 3.8 we conclude that the epimorphism (3.28) maps the basis (3.29) of
$\textup{Ind}_{H_{0,\mathcal{O}}}^{H^{X}}(\mathbf{F}1^{\mathcal{O}})$ to a
basis of $\mathbf{F}[\mathcal{O}]$, which concludes the proof of the second
statement. ∎

The natural analog of Proposition 3.9 for $W$-orbits $\mathcal{O}$ with
$\alpha_{0}(c^{\mathcal{O}})=0$ (i.e., with $0\in J(\mathcal{O})$) requires
the extension of the $H^{X}$-action $\pi^{\mathcal{O}}$ on
$\mathbf{F}[\mathcal{O}]$ to an action of the double affine Hecke algebra
$\mathbb{H}$. This will be the subject of the next subsection.

###  3.3. The quasi-polynomial representation of $\mathbb{H}$

We now promote the quasi-polynomial representation $\pi^{\mathcal{O}}$ of the
affine Hecke algebra $H^{X}$ to a family of representations of the double
affine Hecke algebra $\mathbb{H}$. The number of additional parameters depends
on the facet $C^{J(\mathcal{O)}}_{+}$ containing $c^{\mathcal{O}}$. For
$\mathbf{k}\in\mathcal{K}^{\textup{res}}$ the extended representations are the
quasi-polynomial $\mathbb{H}$-representations $\pi_{c^{\mathcal{O}},t}$
introduced in [14, Thm. 1.1]. For $\mathbf{k}\in\mathcal{K}$ and $\Phi_{0}$ of
type $\textup{C}_{r}$ ($r\geq 1$) they will give quasi-polynomial extensions
of the polynomial representation of the type $\textup{C}^{\vee}\textup{C}_{r}$
double affine Hecke algebra $\mathbb{H}$ [11, 12].

For $J\subsetneq\\{0,\ldots,r\\}$ consider the affine subtori

$\mathbf{T}_{J}:=\\{t\in\mathbf{T}\,\,\,|\,\,\,t^{\alpha_{j}^{\vee}}=1\qquad\,\forall\,j\in
J\\}$

of $\mathbf{T}$, and for a $W$-orbit $\mathcal{O}$ in $E$ we write

$\mathbf{T}_{\mathcal{O}}:=\mathbf{T}_{J(\mathcal{O})}.$

Note that

(3.30) $s_{a}t=t(t^{-a^{\vee}})^{\overline{a}}\quad\hbox{ for
}\,a\in\Phi\,\hbox{ and }\,t\in\mathbf{T},$

where $t(t^{-a^{\vee}})^{\overline{a}}\in\mathbf{T}$ is viewed as character of
$Q^{\vee}$ by $\lambda\mapsto
t^{\lambda}(t^{-a^{\vee}})^{\overline{a}(\lambda)}$ for $\lambda\in Q^{\vee}$.
By (3.30) we have $s_{a}t=t$ if $t^{a^{\vee}}=1$, and so

(3.31)
$\mathbf{T}_{J}\subseteq\mathbf{T}^{W_{J}}:=\\{t\in\mathbf{T}\,\,\,|\,\,\,gt=t\quad\,\,\forall\,g\in
W_{J}\\}.$

Note that $\mathbf{T}_{J}$ and $\mathbf{T}^{W_{J}}$ are sub-tori of
$\mathbf{T}$ when $0\not\in J$.

In [14, Lemma 4.2] the $W_{0}$-action on $\mathbf{F}[\mathcal{O}]$ was
extended to a family of $W$-actions on $\mathbf{F}[\mathcal{O}]$, parametrised
by $t\in\mathbf{T}_{\mathcal{O}}$. These actions are compatible with the
$W$-action (2.20) on $\mathbf{F}[Q^{\vee}]$ by $q$-dilations and reflections.
The definition of this action requires the following definition.

######  Definition 3.10.

For $y\in E$ write $\mathbf{g}_{y}\in W$ for the unique element of shortest
length in $W$ such that $\mathbf{g}_{y}^{-1}y\in\overline{C_{+}}$.

Note that if $y$ lies in the $W$-orbit $\mathcal{O}$ of $E$, then
$\mathbf{g}_{y}$ is the unique element in $W^{\mathcal{O}}$ such that
$y=\mathbf{g}_{y}c^{\mathcal{O}}$.

######  Lemma 3.11.

Let $\mathcal{O}$ be a $W$-orbit in $E$. For $t\in\mathbf{T}_{\mathcal{O}}$
the formulas

(3.32) $\begin{split}w_{t}x^{y}&:=w(x^{y})=x^{wy}\qquad\qquad\,\,\,\,(w\in
W_{0}),\\\
\tau(\lambda)_{t}x^{y}&:=(\mathbf{g}_{y}t)^{-\lambda}x^{y}\qquad\qquad\qquad(\lambda\in
Q^{\vee})\end{split}$

for $y\in\mathcal{O}$ define a linear left $W$-action on
$\mathbf{F}[\mathcal{O}]$ satisfying

(3.33) $g_{t}(pf)=(gp)(g_{t}f)$

for $g\in W$, $p\in\mathbf{F}[Q^{\vee}]$ and $f\in\mathbf{F}[\mathcal{O}]$.

It is instructive to recall the proof of (3.33) for $g=\tau(\lambda)$
($\lambda\in Q^{\vee}$). First note that by (3.31), we may replace
$\mathbf{g}_{y}$ in (3.32) by any other representative of the coset
$\mathbf{g}_{y}W_{\mathcal{O}}$. We then have for $\mu\in Q^{\vee}$,

(3.34)
$\begin{split}\tau(\lambda)_{t}x^{\mu+y}&=(\mathbf{g}_{\mu+y}t)^{-\lambda}x^{\mu+y}\\\
&=(\tau(\mu)\mathbf{g}_{y}t)^{-\lambda}x^{\mu+y}\\\
&=q^{-\langle\lambda,\mu\rangle}(\mathbf{g}_{y}t)^{-\lambda}x^{\mu+y}\\\
&=(q^{-\langle\lambda,\mu\rangle}x^{\mu})\bigl{(}(\mathbf{g}_{y}t)^{-\lambda}x^{y}\bigr{)},\end{split}$

where we used
$\mathbf{g}_{y+\mu}W_{\mathcal{O}}=\tau(\mu)\mathbf{g}_{y}W_{\mathcal{O}}$ for
the second equality, and (2.19) for the third equality. The last line in
(3.34) clearly equals $(\tau(\lambda)x^{\mu})(\tau(\lambda)_{t}x^{y})$. Note
in particular that the family of $W$-actions (3.32) depends on $q$ via the
$W$-action (2.19) on $\mathbf{T}$.

Note that by (3.34) we have the formula

$\tau(\lambda)_{t}|_{\mathbf{F}[Q^{\vee}]x^{y}}=(\mathbf{g}_{y}t)^{-\lambda}\bigl{(}x^{y}\circ\tau(\lambda)\circ
x^{-y}\bigr{)}|_{\mathbf{F}[Q^{\vee}]x^{y}},$

where $x^{\pm y}$ are regarded as multiplication operators on $\mathbf{F}[E]$.

The following result generalizes [14, Thm. 1.1] to multiplicity functions
$\mathbf{k}\in\mathcal{K}$.

######  Theorem 3.12.

Let $\mathcal{O}$ be a $W$-orbit in $E$ and $t\in\mathbf{T}_{\mathcal{O}}$.
The formulas

(3.35)
$\begin{split}\pi_{t}^{\mathcal{O}}(x^{\lambda})x^{y}&:=x^{y+\lambda},\\\
\pi_{t}^{\mathcal{O}}(T_{j})x^{y}&:=k_{j}^{\chi_{e}(\overline{\alpha}_{j}(y))}u_{j}^{\chi_{o}(\overline{\alpha}_{j}(y))}s_{j,t}x^{y}+(k_{j}-k_{j}^{-1})\nabla_{j}^{e}(x^{y})+(u_{j}-u_{j}^{-1})\nabla_{j}^{o}(x^{y})\end{split}$

for $j=0,\ldots,r$, $\lambda\in Q^{\vee}$ and $y\in\mathcal{O}$ define a
representation
$\pi_{t}^{\mathcal{O}}:\mathbb{H}\rightarrow\textup{End}(\mathbf{F}[\mathcal{O}])$.

######  Remark 3.13.

Note that $\pi_{t}^{\mathcal{O}}|_{H^{X}}$ is the restriction
$\pi^{\mathcal{O}}$ of the quasi-polynomial $H^{X}$-representation $\pi$ from
Theorem 3.3 to $\mathbf{F}[\mathcal{O}]$. Furthermore, for restricted
multiplicity parameters $\mathbf{k}\in\mathcal{K}^{\textup{res}}$ we have

$\pi_{t}^{\mathcal{O}}(T_{j})=k_{j}^{\chi_{\mathbb{Z}}(\overline{\alpha}_{j}(y))}s_{j,t}x^{y}+(k_{j}-k_{j}^{-1})\nabla_{j}(x^{y})$

since $k_{j}=u_{j}$, hence $\pi_{t}^{\mathcal{O}}$ then coincides with the
quasi-polynomial representation $\pi_{c^{\mathcal{O}},t}$ defined in [14, Thm.
1.1].

###### Proof.

Consider $\mathbb{H}$ as left regular $\mathbb{H}$-module. The $H$-action
relative to the $\mathbf{F}$-basis
$\\{x^{\lambda}T_{w}Y^{\mu}\,\,|\,\,\lambda,\mu\in Q^{\vee},w\in W_{0}\\}$ of
$\mathbb{H}$ is explicitly given by

(3.36)
$\begin{split}T_{j}x^{\lambda}T_{w}Y^{\mu}&=s_{j}(x^{\lambda})T_{s_{\overline{\alpha}_{j}}w}Y^{\mu-w^{-1}s_{j}(0)}\\\
&+(k_{j}-k_{j}^{-1})\left(\frac{1-x^{(2\chi_{-}(w^{-1}\overline{\alpha}_{j})-\overline{\alpha}_{j}(\lambda))\alpha_{j}^{\vee}}}{1-x^{2\alpha_{j}^{\vee}}}\right)x^{\lambda}T_{w}Y^{\mu}\\\
&+(u_{j}-u_{j}^{-1})\left(\frac{x^{\alpha_{j}^{\vee}}-x^{(1-\overline{\alpha}_{j}(\lambda))\alpha_{j}^{\vee}}}{1-x^{2\alpha_{j}^{\vee}}}\right)x^{\lambda}T_{w}Y^{\mu}\end{split}$

for $w\in W_{0}$, $\lambda,\mu\in Q^{\vee}$ and $j=0,\ldots,r$. For
$j\in\\{1,\ldots,r\\}$ formula (3.36) follows immediately from (3.7). For
$j=0$ formula (3.36) follows by commuting $T_{0}$ and $x^{\lambda}$ using the
cross relation (2.27) and then applying the identity

(3.37)
$T_{0}T_{w}=T_{s_{\overline{\alpha}_{0}}w}Y^{-w^{-1}s_{0}(0)}+\chi_{-}(w^{-1}\overline{\alpha}_{0})(k_{0}-k_{0}^{-1})T_{w}$

in $H$. For the proof of (3.37), first note that it is equivalent to the
identity

(3.38)
$T_{0}^{\chi(w^{-1}\varphi)}T_{s_{\varphi}w}=T_{w}Y^{w^{-1}\varphi^{\vee}}$

in $H$ since $s_{0}(0)=\varphi^{\vee}$, $\overline{\alpha}_{0}=-\varphi$ and
$T_{0}^{-1}=T_{0}-k_{0}+k_{0}^{-1}$. For a proof of (3.38) see, e.g., [9,
(3.3.6)].

Let $\mathcal{O}$ be a $W$-orbit in $E$. Recall the linear epimorphism
$\psi^{\mathcal{O}}:H^{X}\twoheadrightarrow\mathbf{F}[\mathcal{O}]$, defined
by (3.8), which is an epimorphism of $H^{X}$-modules by Corollary 3.4(2). The
goal is to find extensions of $\psi^{\mathcal{O}}$ to linear epimorphisms
$\mathbb{H}\twoheadrightarrow\mathbf{F}[\mathcal{O}]$ such that

  1. (1)

their kernels are left ideals in $\mathbb{H}$,

  2. (2)

right multiplication by $\mathbf{F}_{Y}[Q^{\vee}]$ is turned into
multiplication by a linear character of $\mathbf{F}_{Y}[Q^{\vee}]$.

This will upgrade the $H^{X}$-action on $\mathbf{F}[\mathcal{O}]$ to a family
of $\mathbb{H}$-actions satisfying the additional property that the quasi-
monomial $x^{c^{\mathcal{O}}}$ will be a simultaneous eigenvector for the
action of $\mathbf{F}_{Y}[Q^{\vee}]$. The family of extended
$\mathbb{H}$-actions on $\mathbf{F}[\mathcal{O}]$ will be natural parametrised
by the associated linear characters of $\mathbf{F}_{Y}[Q^{\vee}]$, which in
turn can be described by an affine subtorus of the form
$\mathfrak{s}\mathbf{T}_{\mathcal{O}}$ for a specific basepoint
$\mathfrak{s}\in\mathbf{T}$, so be determined in due course.

So our starting point will be the desired property (2). Fix
$t\in\mathbf{T}_{\mathcal{O}}$ and consider the extension of
$\psi^{\mathcal{O}}$ to a ($t$-dependent) linear map
$\psi_{t}^{\mathcal{O}}:\mathbb{H}\twoheadrightarrow\mathbf{F}[\mathcal{O}]$
by

(3.39)
$\psi_{t}^{\mathcal{O}}(x^{\lambda}T_{w}Y^{\mu}):=(\mathfrak{s}t)^{-\mu}\psi^{\mathcal{O}}(x^{\lambda}T_{w})=\kappa_{w}^{\mathcal{O}}(\mathfrak{s}t)^{-\mu}x^{\lambda+wc^{\mathcal{O}}}$

for $w\in W_{0}$ and $\lambda,\mu\in Q^{\vee}$ (recall here that
$\kappa_{w}^{\mathcal{O}}\in\mathbf{F}^{\times}$ is the explicit scalar
defined by (3.17)). By the first formula of (3.39) and Corollary 3.4(2) it
follows that

$\psi_{t}^{\mathcal{O}}(hh^{\prime})=\pi_{t}^{\mathcal{O}}(h)\psi_{t}^{\mathcal{O}}(h^{\prime})\quad\hbox{
for }\,h\in H^{X}\,\hbox{ and }\,h^{\prime}\in\mathbb{H}.$

In particular, the kernel of $\psi_{t}^{\mathcal{O}}$ is a left
$H^{X}$-submodule in $\mathbb{H}$. To meet the first property (1), we now
fine-tune the choice of $\mathfrak{s}\in\mathbf{T}$ such that

$\psi_{t}^{\mathcal{O}}(T_{0}h^{\prime})=\mathcal{D}_{0}(\psi_{t}^{\mathcal{O}}(h^{\prime}))\quad\,\textup{
for all }\,h^{\prime}\in\mathbb{H}$

for some $\mathcal{D}_{0}\in\textup{End}(\mathbf{F}[\mathcal{O}])$.

Note that for $\lambda\in Q^{\vee}$ and $w\in W_{0}$,

$s_{0,t}x^{\lambda+wc^{\mathcal{O}}}=s_{0}(x^{\lambda})(s_{0,t}x^{wc^{\mathcal{O}}})=t^{w^{-1}\varphi^{\vee}}s_{0}(x^{\lambda})x^{s_{\varphi}wc^{\mathcal{O}}}$

and hence, by (3.36),

(3.40)
$\begin{split}\frac{(\mathfrak{s}t)^{\mu}}{\kappa_{w}^{\mathcal{O}}}\psi_{t}^{\mathcal{O}}(T_{0}x^{\lambda}T_{w}Y^{\mu})&=\frac{\kappa_{s_{\varphi}w}^{\mathcal{O}}\,\mathfrak{s}^{w^{-1}\varphi^{\vee}}}{\kappa_{w}^{\mathcal{O}}}s_{0,t}x^{\lambda+wc^{\mathcal{O}}}\\\
&+(k_{0}-k_{0}^{-1})\left(\frac{1-x^{(2\chi_{-}(w^{-1}\overline{\alpha}_{0})-\overline{\alpha}_{0}(\lambda))\alpha_{0}^{\vee}}}{1-x^{2\alpha_{0}^{\vee}}}\right)x^{\lambda+wc^{\mathcal{O}}}\\\
&+(u_{0}-u_{0}^{-1})\left(\frac{x^{\alpha_{0}^{\vee}}-x^{(1-\overline{\alpha}_{0}(\lambda))\alpha_{0}^{\vee}}}{1-x^{2\alpha_{0}^{\vee}}}\right)x^{\lambda+wc^{\mathcal{O}}}.\end{split}$

So we need to fine-tune $\mathfrak{s}\in\mathbf{T}$ such that the right hand
side of (3.40) can be written $\mathcal{D}_{0}(x^{\lambda+wc})$ for some
$\mathcal{D}_{0}\in\textup{End}(\mathbf{F}[\mathcal{O}])$. We follow the proof
of Theorem 3.3, but it requires the necessary additional computations due to
the presence of the additional parameters $\mathfrak{s}$ and $t$ (compare also
with [14, §5], which deals with the case that
$\mathbf{k}\in\mathcal{K}^{\textup{res}}$).

We first consider the case that $\Phi_{0}$ is of type $\textup{C}_{r}$ ($r\geq
1$), so that $\alpha_{0}(Q^{\vee})=\mathbb{Z}_{o}$. Fix $\lambda\in Q^{\vee}$
and $w\in W_{0}$ and set

$y:=\lambda+wc^{\mathcal{O}}.$

The second line of (3.40) can then be rewritten using the formula

(3.41)
$\left(\frac{1-x^{(2\chi_{-}(w^{-1}\overline{\alpha}_{0})-\overline{\alpha}_{0}(\lambda))\alpha_{0}^{\vee}}}{1-x^{2\alpha_{0}^{\vee}}}\right)x^{y}=\nabla_{0}^{e}(x^{y})+\chi_{-}(w^{-1}\overline{\alpha}_{0})\chi_{e}(\overline{\alpha}_{0}(wc^{\mathcal{O}}))s_{0,t}x^{y}.$

To prove (3.41), note that by (2.7) we have

(3.42)
$2\chi_{-}(w^{-1}\overline{\alpha}_{0})-\overline{\alpha}_{0}(\lambda)=\begin{cases}2-\lfloor\overline{\alpha}_{0}(y)\rfloor_{e}\,\,&\hbox{
if }\,w^{-1}\overline{\alpha}_{0}\in\Phi_{0}^{-}\,\hbox{ and
}\,\overline{\alpha}_{0}(wc^{\mathcal{O}})=0,\\\
-\lfloor\overline{\alpha}_{0}(y)\rfloor_{e}\,\,&\hbox{ otherwise},\end{cases}$

and that $\overline{\alpha}_{0}(wc^{\mathcal{O}})=0$ iff
$\chi_{e}(\overline{\alpha}_{0}(wc^{\mathcal{O}}))=1$. This immediately
implies (3.41) unless $w^{-1}\overline{\alpha}_{0}\in\Phi_{0}^{-}$ and
$\overline{\alpha}_{0}(wc^{\mathcal{O}})=0$. So suppose now that
$w^{-1}\overline{\alpha}_{0}\in\Phi_{0}^{-}$ and
$\overline{\alpha}_{0}(wc^{\mathcal{O}})=0$. Then
$s_{\varphi}wc^{\mathcal{O}}=wc^{\mathcal{O}}$, and hence

$s_{0,t}x^{y}=s_{0}(x^{\lambda})(s_{0,t}x^{wc^{\mathcal{O}}})=(x^{\lambda-\overline{\alpha}_{0}(\lambda)\alpha_{0}^{\vee}})(t^{w^{-1}\varphi^{\vee}}x^{wc^{\mathcal{O}}})=t^{w^{-1}\varphi^{\vee}}x^{\lambda-\overline{\alpha}_{0}(\lambda)\alpha_{0}^{\vee}+wc^{\mathcal{O}}}.$

But $(w^{-1}\varphi)(c^{\mathcal{O}})=0$, hence
$w^{-1}\varphi^{\vee}\in\Phi_{0}^{\vee}\cap\bigoplus_{i\in
I(\mathcal{O})}\mathbb{Z}\alpha_{i}^{\vee}$. Since
$t\in\mathbf{T}_{\mathcal{O}}$ we conclude that $t^{w^{-1}\varphi^{\vee}}=1$,
so

(3.43)
$s_{0,t}x^{y}=x^{\lambda-\overline{\alpha}_{0}(\lambda)\alpha_{0}^{\vee}+wc^{\mathcal{O}}}.$

Then (3.41) follows by combining the first case of (3.42) and (3.43).

Similarly we rewrite the third line of (3.40) using the formula

(3.44)
$\left(\frac{x^{\alpha_{0}^{\vee}}-x^{(1-\overline{\alpha}_{0}(\lambda))\alpha_{0}^{\vee}}}{1-x^{2\alpha_{0}^{\vee}}}\right)x^{y}=\nabla_{0}^{o}(x^{y})+\chi_{+}(w^{-1}\overline{\alpha}_{0})\chi_{o}(\overline{\alpha}_{0}(wc^{\mathcal{O}}))s_{0,t}x^{y}.$

For the proof of (3.44) we now use that

(3.45)
$1-\overline{\alpha}_{0}(\lambda)=\begin{cases}2-\lfloor\overline{\alpha}_{0}(y)\rfloor_{o}\quad\,&\hbox{
if }\overline{\alpha}_{0}(wc^{\mathcal{O}})=1,\\\
-\lfloor\overline{\alpha}_{0}(y)\rfloor_{o}\quad\,&\hbox{
otherwise},\end{cases}$

and the observation that $\overline{\alpha}_{0}(wc^{\mathcal{O}})=1$ is
equivalent to
$\chi_{+}(w^{-1}\overline{\alpha}_{0})\chi_{o}(\overline{\alpha}_{0}(wc^{\mathcal{O}}))=1$.
Then (3.44) is immediate if $\overline{\alpha}_{0}(wc^{\mathcal{O}})\not=1$.
So suppose now that $\overline{\alpha}_{0}(wc^{\mathcal{O}})=1$. Then

(3.46)
$\begin{split}s_{0,t}x^{y}&=t^{w^{-1}\varphi^{\vee}}x^{\lambda-\overline{\alpha}_{0}(\lambda)\alpha_{0}^{\vee}+s_{\varphi}wc^{\mathcal{O}}}\\\
&=q_{\varphi}t^{w^{-1}\varphi^{\vee}}x^{\lambda-(1+\overline{\alpha}_{0}(\lambda))\alpha_{0}^{\vee}+wc^{\mathcal{O}}}=x^{\lambda-(1+\overline{\alpha}_{0}(\lambda))\alpha_{0}^{\vee}+wc^{\mathcal{O}}}.\end{split}$

The last equality follows from the fact that the affine root
$a:=(w^{-1}\varphi,1)$ satisfies $a(c^{\mathcal{O}})=0$, so
$a^{\vee}\in\Phi^{+}\cap\bigoplus_{j\in
J(\mathcal{O})}\mathbb{Z}\alpha_{j}^{\vee}$ and hence
$q_{\varphi}t^{w^{-1}\varphi^{\vee}}=t^{a^{\vee}}=1$ (cf. [14, Lemma 3.4]).
Then (3.44) follows from the first case of (3.45) and (3.46).

Returning now to (3.40), we conclude from (3.41) and (3.44) that

(3.47)
$\frac{(\mathfrak{s}t)^{\mu}}{\kappa_{w}^{\mathcal{O}}}\psi_{t}^{\mathcal{O}}(T_{0}x^{\lambda}T_{w}Y^{\mu})=\textup{coeff}_{w}^{\mathcal{O}}s_{0,t}x^{y}+(k_{0}-k_{0}^{-1})\nabla_{0}^{e}(x^{y})+(u_{0}-u_{0}^{-1})\nabla_{0}^{o}(x^{y})$

with

(3.48)
$\begin{split}\textup{coeff}_{w}^{\mathcal{O}}&=\frac{\kappa_{s_{\varphi}w}^{\mathcal{O}}\mathfrak{s}^{w^{-1}\varphi^{\vee}}}{\kappa_{w}^{\mathcal{O}}}+(k_{0}-k_{0}^{-1})\chi_{-}(w^{-1}\overline{\alpha}_{0})\chi_{e}(\overline{\alpha}_{0}(wc^{\mathcal{O}}))\\\
&+(u_{0}-u_{0}^{-1})\chi_{+}(w^{-1}\overline{\alpha}_{0})\chi_{o}(\overline{\alpha}_{0}(wc^{\mathcal{O}})).\end{split}$

Formula (3.47) also holds true when $\Phi_{0}$ is not of type
$\textup{C}_{r}$. In fact, if $\Phi_{0}$ is not of type $\textup{C}_{r}$ then
$\mathcal{K}=\mathcal{K}^{\textup{res}}$ and $k_{0}=u_{0}$, and the formula
can be recovered from [14, §5]. It can also be derived directly, similarly as
the proof of (3.11) when $\alpha_{i}(Q^{\vee})=\mathbb{Z}$. So we will now
continue the proof of the theorem for $\Phi_{0}$ of any type, taking (3.47) as
the starting point.

Properties of the normalization factors $\kappa_{w}^{\mathcal{O}}$ (3.17) were
derived in [14] for restricted parameters
$\mathbf{k}\in\mathcal{K}^{\textup{res}}$, which led to an explicit expression
of the quotient $\kappa_{s_{\varphi}w}^{\mathcal{O}}/\kappa_{w}^{\mathcal{O}}$
(see [14, Lemma 5.6(1)], as well as case 2 of the proof of [14, Lemma 5.9]).
This can be easily extended to $\kappa_{w}^{\mathcal{O}}$ for arbitrary
parameters $\mathbf{k}\in\mathcal{K}$. It leads to the formula

$\frac{\kappa_{s_{\varphi}w}^{\mathcal{O}}}{\kappa_{w}^{\mathcal{O}}}=\mathbf{k}_{\varphi}^{-\chi(w^{-1}\varphi)\chi_{e}(\varphi(wc^{\mathcal{O}}))}\mathbf{k}_{\frac{\varphi}{2}}^{\chi(w^{-1}\varphi)\chi_{o}(\varphi(wc^{\mathcal{O}}))}\prod_{\alpha\in\Phi_{0}^{+}}\mathbf{k}_{\alpha}^{\chi_{e}(\alpha(c^{\mathcal{O}}))\alpha(w^{-1}\varphi^{\vee})}\mathbf{k}_{\frac{\alpha}{2}}^{-\chi_{o}(\alpha(c^{\mathcal{O}}))\alpha(w^{-1}\varphi^{\vee})},$

where $\chi$ is given by (2.12). Hence

$\begin{split}\textup{coeff}_{w}^{\mathcal{O}}&=k_{r}^{-\chi(w^{-1}\varphi)\chi_{e}(\varphi(wc^{\mathcal{O}}))}u_{r}^{\chi(w^{-1}\varphi)\chi_{o}(\varphi(wc^{\mathcal{O}}))}\Bigl{(}\mathfrak{s}\prod_{\alpha\in\Phi_{0}^{+}}\mathbf{k}_{\alpha}^{\chi_{e}(\alpha(c^{\mathcal{O}}))\alpha}\,\mathbf{k}_{\frac{\alpha}{2}}^{-\chi_{o}(\alpha(c^{\mathcal{O}}))\alpha}\Bigr{)}^{w^{-1}\varphi^{\vee}}\\\
&+(k_{0}-k_{0}^{-1})\chi_{+}(w^{-1}\varphi)\chi_{e}(\varphi(wc^{\mathcal{O}}))+(u_{0}-u_{0}^{-1})\chi_{-}(w^{-1}\varphi)\chi_{o}(\varphi(wc^{\mathcal{O}})),\end{split}$

where the factor in big brackets in the first line is considered as element in
$\mathbf{T}$ with value at $\lambda\in Q^{\vee}$ given by

$\mathfrak{s}^{\lambda}\prod_{\alpha\in\Phi_{0}^{+}}\mathbf{k}_{\alpha}^{\chi_{e}(\alpha(c^{\mathcal{O}}))\alpha(\lambda)}\,\mathbf{k}_{\frac{\alpha}{2}}^{-\chi_{o}(\alpha(c^{\mathcal{O}}))\alpha(\lambda)}.$

Now we pick $\mathfrak{s}\in\mathbf{T}$ to be

(3.49)
$\mathfrak{s}_{\mathcal{O}}:=\prod_{\alpha\in\Phi_{0}^{+}}\bigl{(}\mathbf{k}_{\alpha}\mathbf{k}_{(\alpha,1)}\bigr{)}^{-\frac{\chi_{e}(\alpha(c^{\mathcal{O}}))}{2}\alpha}\bigl{(}\mathbf{k}_{\frac{\alpha}{2}}\mathbf{k}_{(\frac{\alpha}{2},\frac{1}{2})}\bigr{)}^{\frac{\chi_{o}(\alpha(c^{\mathcal{O}}))}{2}\alpha},$

whose value at $\lambda\in Q^{\vee}$ is

$\mathfrak{s}^{\lambda}_{\mathcal{O}}:=\prod_{\alpha\in\Phi_{0}^{+}}\bigl{(}\mathbf{k}_{\alpha}\mathbf{k}_{(\alpha,1)}\bigr{)}^{-\frac{\chi_{e}(\alpha(c^{\mathcal{O}}))\alpha(\lambda)}{2}}\bigl{(}\mathbf{k}_{\frac{\alpha}{2}}\mathbf{k}_{(\frac{\alpha}{2},\frac{1}{2})}\bigr{)}^{\frac{\chi_{o}(\alpha(c^{\mathcal{O}}))\alpha(\lambda)}{2}}.$

When $\alpha(Q^{\vee})=\mathbb{Z}$ the factor in this product should be read
as

$\mathbf{k}_{\alpha}^{-\chi_{e}(\alpha(c^{\mathcal{O}}))\alpha(\lambda)}\,\mathbf{k}_{\frac{\alpha}{2}}^{\chi_{o}(\alpha(c^{\mathcal{O}}))\alpha(\lambda)}=\mathbf{k}_{\alpha}^{(\chi_{o}(\alpha(c^{\mathcal{O}}))-\chi_{e}(\alpha(c^{\mathcal{O}})))\alpha(\lambda)}$

(recall that
$\mathbf{k}_{\alpha}=\mathbf{k}_{(\alpha,1)}=\mathbf{k}_{\frac{\alpha}{2}}=\mathbf{k}_{(\frac{\alpha}{2},\frac{1}{2})}$
when $\alpha(Q^{\vee})=\mathbb{Z}$).

For the remainder of the proof we set
$\mathfrak{s}=\mathfrak{s}_{\mathcal{O}}$, hence the linear map
$\psi_{t}^{\mathcal{O}}:\mathbb{H}\twoheadrightarrow\mathbf{F}[\mathcal{O}]$
is now given by

(3.50)
$\psi_{t}^{\mathcal{O}}\bigl{(}x^{\lambda}T_{w}Y^{\mu})=\kappa_{w}^{\mathcal{O}}(\mathfrak{s}_{\mathcal{O}}t)^{-\mu}x^{\lambda+wc^{\mathcal{O}}}$

for $\lambda,\mu\in Q^{\vee}$ and $w\in W_{0}$. We get

(3.51)
$\begin{split}\textup{coeff}_{w}^{\mathcal{O}}&=k_{r}^{-\chi(w^{-1}\varphi)\chi_{e}(\varphi(wc^{\mathcal{O}}))}u_{r}^{\chi(w^{-1}\varphi)\chi_{o}(\varphi(wc^{\mathcal{O}}))}\\\
&\,\,\,\,\times\prod_{\alpha\in\Phi_{0}^{+}:\alpha(Q^{\vee})=\mathbb{Z}_{e}}\bigl{(}\mathbf{k}_{\alpha}\mathbf{k}_{(\alpha,1)}^{-1}\bigr{)}^{\frac{\chi_{e}(\alpha(c^{\mathcal{O}}))\alpha(w^{-1}\varphi^{\vee})}{2}}\bigl{(}\mathbf{k}_{\frac{\alpha}{2}}^{-1}\mathbf{k}_{(\frac{\alpha}{2},\frac{1}{2})}\bigr{)}^{\frac{\chi_{o}(\alpha(c^{\mathcal{O}}))\alpha(w^{-1}\varphi^{\vee})}{2}}\\\
+&(k_{0}-k_{0}^{-1})\chi_{+}(w^{-1}\varphi)\chi_{e}(\varphi(wc^{\mathcal{O}}))+(u_{0}-u_{0}^{-1})\chi_{-}(w^{-1}\varphi)\chi_{o}(\varphi(wc^{\mathcal{O}})).\end{split}$

Recall that there only exist roots $\alpha\in\Phi_{0}$ with
$\alpha(Q^{\vee})=\mathbb{Z}_{e}$ when $\Phi_{0}$ is of type $\textup{C}_{r}$.
In this case $\\{\alpha\in\Phi_{0}\,\,|\,\,\alpha(Q^{\vee})=\mathbb{Z}_{e}\\}$
is the set $\Phi_{0,\ell}$ of long roots in $\Phi_{0}$, and

$\\{\alpha\in\Phi_{0,\ell}\,\,|\,\,\alpha(w^{-1}\varphi^{\vee})\not=0\\}=\\{w^{-1}\varphi,-w^{-1}\varphi\\}.$

We thus have

(3.52)
$\begin{split}\prod_{\alpha\in\Phi_{0}^{+}:\alpha(Q^{\vee})=\mathbb{Z}_{e}}&\bigl{(}\mathbf{k}_{\alpha}\mathbf{k}_{(\alpha,1)}^{-1}\bigr{)}^{\frac{\chi_{e}(\alpha(c^{\mathcal{O}}))\alpha(w^{-1}\varphi^{\vee})}{2}}\bigl{(}\mathbf{k}_{\frac{\alpha}{2}}^{-1}\mathbf{k}_{(\frac{\alpha}{2},\frac{1}{2})}\bigr{)}^{\frac{\chi_{o}(\alpha(c^{\mathcal{O}}))\alpha(w^{-1}\varphi^{\vee})}{2}}=\\\
&\qquad\qquad=\bigl{(}k_{0}^{-1}k_{r}\bigr{)}^{\chi(w^{-1}\varphi)\chi_{e}(\varphi(wc^{\mathcal{O}}))}\bigl{(}u_{0}u_{r}^{-1}\bigr{)}^{\chi(w^{-1}\varphi)\chi_{o}(\varphi(wc^{\mathcal{O}}))}.\end{split}$

Note that this formula is correct for $\Phi_{0}$ of arbitrary type. Indeed, if
$\Phi_{0}$ is not of type $C_{r}$ then the product on the left hand side is an
empty product, and the right hand side also reduces to $1$. Substituting
(3.52) into (3.51), we see that the dependence on $k_{r}$ and $u_{r}$ drops
out, and we end up with the formula

$\begin{split}\textup{coeff}_{w}^{\mathcal{O}}&=k_{0}^{-\chi(w^{-1}\varphi)\chi_{e}(\varphi(wc^{\mathcal{O}}))}u_{0}^{\chi(w^{-1}\varphi)\chi_{o}(\varphi(wc^{\mathcal{O}}))}\\\
&+(k_{0}-k_{0}^{-1})\chi_{+}(w^{-1}\varphi)\chi_{e}(\varphi(wc^{\mathcal{O}}))+(u_{0}-u_{0}^{-1})\chi_{-}(w^{-1}\varphi)\chi_{o}(\varphi(wc^{\mathcal{O}})).\end{split}$

Now note that

$\begin{split}k_{0}^{-\chi(w^{-1}\varphi)\chi_{e}(\varphi(wc^{\mathcal{O}}))}u_{0}^{\chi(w^{-1}\varphi)\chi_{o}(\varphi(wc^{\mathcal{O}}))}&=\bigl{(}k_{0}\chi_{-}(w^{-1}\varphi)+k_{0}^{-1}\chi_{+}(w^{-1}\varphi)\bigr{)}\chi_{e}(\varphi(wc^{\mathcal{O}}))\\\
&+\bigl{(}u_{0}\chi_{+}(w^{-1}\varphi)+u_{0}^{-1}\chi_{-}(w^{-1}\varphi)\bigr{)}\chi_{o}(\varphi(wc^{\mathcal{O}})),\end{split}$

and hence

(3.53)
$\textup{coeff}_{w}^{\mathcal{O}}=k_{0}\chi_{e}(\varphi(wc^{\mathcal{O}}))+u_{0}\chi_{o}(\varphi(wc^{\mathcal{O}}))=k_{0}^{\chi_{e}(\varphi(wc^{\mathcal{O}}))}u_{0}^{\chi_{o}(\varphi(wc^{\mathcal{O}}))}.$

Note that $y\mapsto k_{0}^{\chi_{e}(\varphi(y))}u_{0}^{\chi_{o}(\varphi(y))}$
is $\tau(Q^{\vee})$-invariant (compare with the proof of Theorem 3.3), hence
(3.47) and (3.53) lead to the formula

$\frac{(\mathfrak{s}_{\mathcal{O}}t)^{\mu}}{\kappa_{w}^{\mathcal{O}}}\psi_{t}^{\mathcal{O}}(T_{0}x^{\lambda}T_{w}Y^{\mu})=\mathcal{D}_{0}(x^{y})$

for $y=\lambda+wc^{\mathcal{O}}\in\mathcal{O}$ and $\mu\in Q^{\vee}$, where
$\mathcal{D}_{0}$ is the linear operator on $\mathbf{F}[\mathcal{O}]$ defined
by

$\mathcal{D}_{0}(x^{y}):=k_{0}^{\chi_{e}(\overline{\alpha}_{0}(y))}u_{0}^{\chi_{o}(\overline{\alpha}_{0}(y))}s_{0,t}x^{y}+(k_{0}-k_{0}^{-1})\nabla_{0}^{e}(x^{y})+(u_{0}-u_{0}^{-1})\nabla_{0}^{o}(x^{y})$

for $y\in\mathcal{O}$.

In conclusion, the kernel of
$\psi_{t}^{\mathcal{O}}:\mathbb{H}\twoheadrightarrow\mathbf{F}[\mathcal{O}]$
(see (3.50)) is a left $\mathbb{H}$-module, and the resulting isomorphism
$\mathbb{H}/\textup{ker}(\psi_{t}^{\mathcal{O}})\overset{\sim}{\longrightarrow}\mathbf{F}[\mathcal{O}]$
extends the quasi-polynomial $H^{X}$-action $\pi$ on $\mathbf{F}[\mathcal{O}]$
to a $\mathbb{H}$-action with $T_{0}\in H\subset\mathbb{H}$ acting by
$\mathcal{D}_{0}$, hence the corresponding representation map is
$\pi_{t}^{\mathcal{O}}$. This concludes the proof. ∎

######  Remark 3.14.

For the $W$-orbit $\mathcal{O}=Q^{\vee}$ containing the origin we have
$\mathbf{T}_{Q^{\vee}}=\\{1_{\mathbf{T}}\\}$ since
$J(Q^{\vee})=\\{1,\ldots,r\\}$. By Lemma 3.2 and by the fact that
$g_{1_{\mathbf{T}}}f=g(f)$ for $g\in W$ and $f\in\mathbf{F}[Q^{\vee}]$ (see
Lemma 3.11), we conclude that

$\pi_{1_{\mathbf{T}}}^{Q^{\vee}}(T_{j})x^{\mu}=k_{j}s_{j}(x^{\mu})+\Bigl{(}k_{j}-k_{j}^{-1}+(u_{j}-u_{j}^{-1})x^{\alpha_{j}^{\vee}}\Bigr{)}\left(\frac{x^{\mu}-s_{j}(x^{\mu})}{1-x^{2\alpha_{j}^{\vee}}}\right)$

for $j=0,\ldots,r$ and $\mu\in Q^{\vee}$. Hence
$\pi_{1_{\mathbf{T}}}^{Q^{\vee}}:\mathbb{H}\rightarrow\textup{End}(\mathbf{F}[Q^{\vee}])$
is the basic representation of $\mathbb{H}$, due to Cherednik [2] for
$\mathbf{k}\in\mathcal{K}^{\textup{res}}$ and due to Noumi [11] and Sahi [12]
when $\mathbf{k}\in\mathcal{K}$ and $\Phi_{0}$ is of type $\textup{C}_{r}$.

######  Corollary 3.15.

We have

$\pi_{t}^{\mathcal{O}}\bigl{(}x^{\lambda}T_{w}Y^{\mu})x^{c^{\mathcal{O}}}=\kappa_{w}^{\mathcal{O}}(\mathfrak{s}_{\mathcal{O}}t)^{-\mu}x^{\lambda+wc^{\mathcal{O}}}$

for $\lambda,\mu\in Q^{\vee}$ and $w\in W_{0}$, with
$\mathfrak{s}_{\mathcal{O}}\in\mathbf{T}$ given by (3.49) and
$\kappa_{w}^{\mathcal{O}}$ given by (3.17).

###### Proof.

In the proof of Theorem 3.12 we showed that the epimorphism
$\psi_{t}^{\mathcal{O}}:\mathbb{H}\twoheadrightarrow\mathbf{F}[\mathcal{O}]$,
defined by (3.50), is $\mathbb{H}$-linear. Since
$\psi_{t}^{\mathcal{O}}(1)=x^{c^{\mathcal{O}}}$, the result now follows
immediately from formula (3.50), cf. the proof of Corollary 3.4(2). ∎

Corollary 3.15 is the first step towards realizing $\pi_{t}^{\mathcal{O}}$ as
a $Y$-parabolically induced $\mathbb{H}$-module. We will discuss it in detail
in Section 5.

##  4\. Quasi-polynomial analogs of the nonsymmetric Macdonald-Koornwinder
polynomials

We fix a $W$-orbit $\mathcal{O}$ in $E$ and $t\in\mathbf{T}_{\mathcal{O}}$
throughout this section.

In the first part of this section we show that the commuting operators
$\pi_{t}^{\mathcal{O}}(Y^{\mu})$ ($\mu\in Q^{\vee}$) on
$\mathbf{F}[\mathcal{O}]$ are triangular with respect to an appropriate
partial order on the basis $\\{x^{y}\\}_{y\in\mathcal{O}}$ of quasi-monomials.
This will lead to the definition of the quasi-polynomial analogs of the
nonsymmetric Macdonald-Koornwinder polynomials as the simultaneous
eigenfunctions of the operators $\pi_{t}^{\mathcal{O}}(Y^{\mu})$ ($\mu\in
Q^{\vee}$). The techniques in this section again closely follow the paper
[14], in which these results are derived for
$\mathbf{k}\in\mathcal{K}^{\textup{res}}$.

We first establish triangularity for a family $G_{t}^{\mathcal{O}}(a)$
($a\in\Phi$) of operators closely related to the
$\pi_{t}^{\mathcal{O}}(T_{j})$. The linear operator $G_{t}^{\mathcal{O}}(a)$
on $\mathbf{F}[\mathcal{O}]$ is defined by the formula

(4.1)
$\begin{split}G_{t}^{\mathcal{O}}(a)x^{y}:=\mathbf{k}_{a}^{\chi_{e}(\overline{a}(y))}\mathbf{k}_{\frac{a}{2}}^{\chi_{o}(\overline{a}(y))}x^{y}+&(\mathbf{k}_{a}-\mathbf{k}_{a}^{-1})\left(\frac{1-x^{\lfloor\overline{a}(y)\rfloor_{e}a^{\vee}}}{1-x^{-2a^{\vee}}}\right)s_{a,t}x^{y}\\\
+&(\mathbf{k}_{\frac{a}{2}}-\mathbf{k}_{\frac{a}{2}}^{-1})\left(\frac{x^{-a^{\vee}}-x^{\lfloor\overline{a}(y)\rfloor_{o}a^{\vee}}}{1-x^{-2a^{\vee}}}\right)s_{a,t}x^{y}\end{split}$

for $y\in\mathcal{O}$.

######  Lemma 4.1.

We have

  1. (1)

$G_{t}^{\mathcal{O}}(\alpha_{j})=s_{j,t}\pi_{t}^{\mathcal{O}}(T_{j})$ for
$j=0,\ldots,r$.

  2. (2)

$G_{t}^{\mathcal{O}}(ga)=g_{t}G_{t}^{\mathcal{O}}(a)g_{t}^{-1}$ for $g\in W$.

###### Proof.

This follows from a direct computation using (3.33) and (2.26), cf. [14,
§5.5]. ∎

Recall the definition of $\mathbf{g}_{y}\in W^{\mathcal{O}}$ from Definition
3.10. Denote by $\leq$ the partial order on $E$ defined by

$y\leq z\qquad\Leftrightarrow\qquad y\in Wz\,\,\hbox{ and
}\,\,\mathbf{g}_{y}\leq_{B}\mathbf{g}_{z}$

with $\leq_{B}$ the Bruhat order of $(W,\\{s_{0},\ldots,s_{r}\\})$. Note that
for each $z\in E$,

$\\{y\in E\,\,|\,\,y\leq z\\}$

is a finite set contained in the $W$-orbit $Wz$ of $z$. Various other
properties of this partial order are obtained in [14, §5.4].

######  Definition 4.2.

For $f\in\mathbf{F}[E]$ we write

$f=dx^{y}+\textup{ l.o.t.}$

if $f\in dx^{y}+\bigoplus_{z<y}\mathbf{F}x^{z}$ with
$d\in\mathbf{F}^{\times}$. We then say that $f$ is of degree $y$ with leading
coefficient $d$.

Define $\eta_{e},\eta_{o}:\mathbb{R}\rightarrow\\{-1,0,1\\}$ by

(4.2) $\eta_{e}=\chi_{2\mathbb{Z}_{\geq 1}}-\chi_{2\mathbb{Z}_{\leq
0}},\qquad\eta_{o}=\chi_{1+2\mathbb{Z}_{\geq 0}}-\chi_{1+2\mathbb{Z}_{<0}}.$

We also set

(4.3) $\eta:=\eta_{e}+\eta_{o},$

which is equal to $\chi_{\mathbb{Z}_{>0}}-\chi_{\mathbb{Z}_{\leq 0}}$. We have
the following extension of [14, Lem. 5.27].

######  Lemma 4.3.

For $a\in\Phi_{0}^{+}\times\mathbb{Z}$ we have

$G_{t}^{\mathcal{O}}(a)x^{y}=\mathbf{k}_{a}^{-\eta_{e}(\overline{a}(y))}\mathbf{k}_{\frac{a}{2}}^{-\eta_{o}(\overline{a}(y))}x^{y}+\textup{
l.o.t.}$

for all $y\in\mathcal{O}$.

###### Proof.

This is covered by [14, Lem. 5.27] unless $\Phi_{0}$ is of type
$\textup{C}_{r}$. For $\Phi_{0}$ of type $\textup{C}_{r}$, one checks using
[14, Lem. 4.3(1)], [14, Prop. 5.20] and [14, Lem. 5.24] that for
$a\in\Phi_{0}^{+}\times\mathbb{Z}$,

$\begin{split}\left(\frac{1-x^{\lfloor\overline{a}(y)\rfloor_{e}a^{\vee}}}{1-x^{-2a^{\vee}}}\right)s_{a,t}x^{y}&=-\chi_{2\mathbb{Z}_{\geq
1}}(\overline{a}(y))x^{y}+\textup{ l.o.t.},\\\
\left(\frac{x^{-a^{\vee}}-x^{\lfloor\overline{a}(y)\rfloor_{o}a^{\vee}}}{1-x^{-2a^{\vee}}}\right)s_{a,t}x^{y}&=-\chi_{1+2\mathbb{Z}_{\geq
0}}(\overline{a}(y))x^{y}+\textup{ l.o.t.},\end{split}$

and hence

$\begin{split}&G_{t}^{\mathcal{O}}(a)x^{y}=\\\
&=\Bigl{(}\mathbf{k}_{a}^{\chi_{e}(\overline{a}(y))}\mathbf{k}_{\frac{a}{2}}^{\chi_{o}(\overline{a}(y))}+(\mathbf{k}_{a}^{-1}-\mathbf{k}_{a})\chi_{2\mathbb{Z}_{\geq
1}}(\overline{a}(y))+(\mathbf{k}_{\frac{a}{2}}^{-1}-\mathbf{k}_{\frac{a}{2}})\chi_{1+2\mathbb{Z}_{\geq
0}}(\overline{a}(y))\Bigr{)}x^{y}+\textup{ l.o.t.}\\\
&=\mathbf{k}_{a}^{-\eta_{e}(\overline{a}(y))}\mathbf{k}_{\frac{a}{2}}^{-\eta_{o}(\overline{a}(y))}x^{y}+\textup{
l.o.t.},\end{split}$

as desired. ∎

######  Definition 4.4.

For $y\in E$ define $\mathfrak{s}_{y}\in\mathbf{T}$ by

$\mathfrak{s}_{y}:=\prod_{\alpha\in\Phi_{0}^{+}}\bigl{(}\mathbf{k}_{\alpha}\mathbf{k}_{(1,\alpha)}\bigr{)}^{\frac{\eta_{e}(\alpha(y))}{2}\alpha}\bigl{(}\mathbf{k}_{\frac{\alpha}{2}}\mathbf{k}_{(\frac{1}{2},\frac{\alpha}{2})}\bigr{)}^{\frac{\eta_{o}(\alpha(y))}{2}\alpha}.$

In other words, the value of $\mathfrak{s}_{y}$ at $\lambda\in Q^{\vee}$ is

(4.4)
$\mathfrak{s}_{y}^{\lambda}:=\prod_{\alpha\in\Phi_{0}^{+}}\bigl{(}\mathbf{k}_{\alpha}\mathbf{k}_{(\alpha,1)}\bigr{)}^{\frac{\eta_{e}(\alpha(y))\alpha(\lambda)}{2}}\bigl{(}\mathbf{k}_{\frac{\alpha}{2}}\mathbf{k}_{(\frac{\alpha}{2},\frac{1}{2})}\bigr{)}^{\frac{\eta_{o}(\alpha(y))\alpha(\lambda)}{2}}.$

If $\alpha(Q^{\vee})=\mathbb{Z}_{e}$ then the factors in this product are
clearly well defined. If $\alpha(Q^{\vee})=\mathbb{Z}$ then
$\mathbf{k}_{\alpha}=\mathbf{k}_{(\alpha,1)}=\mathbf{k}_{\frac{\alpha}{2}}=\mathbf{k}_{(\frac{\alpha}{2},\frac{1}{2})}$,
and the corresponding factor in the product should be read as
$\mathbf{k}_{\alpha}^{\eta(\alpha(y))\alpha(\lambda)}$. In particular,

$\mathfrak{s}_{y}=\prod_{\alpha\in\Phi_{0}^{+}}\mathbf{k}_{\alpha}^{\eta(\alpha(y))\alpha}\quad\,\,\textup{
for }\,\,\,\mathbf{k}\in\mathcal{K}^{\textup{res}},$

which is the base-point considered in [14] (see [14, Def. 5.1]). By [14, Lem.
2.5] the function $E\rightarrow\mathbf{T}$, $y\mapsto\mathfrak{s}_{y}$ is
constant on the faces of the affine root hyperplane arrangement.

######  Remark 4.5.

If $c\in\overline{C_{+}}$ then (2.7) implies that

$\mathfrak{s}_{c}=\prod_{\alpha\in\Phi_{0}^{+}}\bigl{(}\mathbf{k}_{\alpha}\mathbf{k}_{(\alpha,1)}\bigr{)}^{-\frac{\chi_{e}(\alpha(c))}{2}\alpha}\bigl{(}\mathbf{k}_{\frac{\alpha}{2}}\mathbf{k}_{(\frac{\alpha}{2},\frac{1}{2})}\bigr{)}^{\frac{\chi_{o}(\alpha(c))}{2}\alpha}.$

In particular

$\mathfrak{s}_{c^{\mathcal{O}}}=\mathfrak{s}_{\mathcal{O}},$

with $\mathfrak{s}_{\mathcal{O}}\in\mathbf{T}$ the torus element appearing
before in Corollary 3.15 (see (3.49)).

######  Corollary 4.6.

For all $\mu\in Q^{\vee}$ and $y\in\mathcal{O}$ we have

$\pi_{t}^{\mathcal{O}}(Y^{\mu})x^{y}=\bigl{(}\mathfrak{s}_{y}\mathbf{g}_{y}t\bigr{)}^{-\mu}x^{y}+\textup{
l.o.t.}$

###### Proof.

This is [14, Prop. 5.28] when $\mathbf{k}\in\mathcal{K}^{\textup{res}}$. Using
Lemma 4.3 as replacement of [14, Lem. 5.27], the proof of [14, Prop. 5.28]
extends to the case $\mathbf{k}\in\mathcal{K}$. ∎

We now first derive some further properties of the map
$E\rightarrow\mathbf{T}$, $y\mapsto\mathfrak{s}_{y}$. The following lemma
extends [14, Lem. 5.3].

######  Lemma 4.7.

Let $W_{y}:=\\{w\in W\,\,|\,\,wy=y\\}$ be the subgroup of $W$ fixing $y\in E$
and $j\in\\{0,\ldots,r\\}$.

  1. (1)

If $s_{j}\in W_{y}$ then

$\mathfrak{s}_{y}^{\overline{\alpha}_{j}^{\vee}}=\widetilde{\mathbf{k}}_{\alpha_{j}}^{-1}\widetilde{\mathbf{k}}_{\frac{\alpha_{j}}{2}}^{-1}\quad\hbox{
and }\quad
s_{\overline{\alpha}_{j}}\mathfrak{s}_{y}=\bigl{(}\widetilde{\mathbf{k}}_{\alpha_{j}}\widetilde{\mathbf{k}}_{\frac{\alpha_{j}}{2}}\bigr{)}^{\overline{\alpha}_{j}}\mathfrak{s}_{y}.$

  2. (2)

If $s_{j}\not\in W_{y}$ then
$s_{\overline{\alpha}_{j}}\mathfrak{s}_{y}=\mathfrak{s}_{s_{j}y}$.

###### Proof.

We give here the required adjustments to the proof of [14, Lem. 5.3].

For $1\leq i\leq r$ we have $\overline{\alpha_{i}}=\alpha_{i}$,
$s_{\overline{\alpha}_{i}}=s_{i}$ and $\Pi(s_{i})=\\{\alpha_{i}\\}$, hence

$s_{i}\mathfrak{s}_{y}=\bigl{(}\mathbf{k}_{\alpha_{i}},\mathbf{k}_{(\alpha_{i},1)}\bigr{)}^{-(\eta_{e}(\alpha_{i}(y))+\eta_{e}(-\alpha_{i}(y)))\frac{\alpha_{i}}{2}}\bigl{(}\mathbf{k}_{\frac{\alpha_{i}}{2}}\mathbf{k}_{(\frac{\alpha_{i}}{2},\frac{1}{2})}\bigr{)}^{-(\eta_{o}(\alpha_{i}(y))+\eta_{o}(-\alpha_{i}(y)))\frac{\alpha_{i}}{2}}\mathfrak{s}_{y},$

with the obvious interpretation of the right hand side when
$\alpha_{i}(Q^{\vee})=\mathbb{Z}$. Then (1) and (2) follow from the fact that

$\eta_{e}(z)+\eta_{e}(-z)=-2\chi_{\\{0\\}}(z),\qquad\quad\eta_{o}(z)+\eta_{o}(-z)=0$

for $z\in\mathbb{R}$.

We now prove the lemma for $s_{0}$ when $\Phi_{0}$ is of type $\textup{C}_{r}$
(the other types are covered by [14, Lem. 5.3]). Write $\Phi_{0,\ell}^{\pm}$
(resp. $\Phi_{0,s}^{\pm}$) for the positive and negative long (resp. short)
roots in $\Phi_{0}$. Clearly

$\Pi(s_{\varphi})=\Pi_{\ell}(s_{\varphi})\sqcup\Pi_{s}(s_{\varphi})$

with $\Pi_{\ell}(w):=\Phi_{0,\ell}^{+}\cap w^{-1}\Phi_{0,\ell}^{-}$ and
$\Pi_{s}(w):=\Phi_{0,s}^{+}\cap w^{-1}\Phi_{0,s}^{-}$ for $w\in W_{0}$.
Furthermore,

$\Pi_{\ell}(s_{\varphi})=\\{\varphi\\},\qquad\quad\Pi_{s}(s_{\varphi})=\\{\alpha\in\Phi_{0,s}^{+}\,\,\,|\,\,\,\alpha(\varphi^{\vee})=1\\},$

where we used for the first equality that there are no long positive roots
$\alpha$ with $\alpha(\varphi^{\vee})=1$, because $\Phi_{0}$ is of type
$\textup{C}_{r}$. Following the proof of [14, Lem. 5.3] and using that
$\mathbf{k}_{\beta}=\mathbf{k}_{(\beta,1)}=\mathbf{k}_{\frac{\beta}{2}}=\mathbf{k}_{(\frac{\beta}{2},\frac{1}{2})}$
for $\beta\in\Phi_{0,s}^{+}$, we get

(4.5)
$\begin{split}s_{\varphi}\mathfrak{s}_{y}&=\mathfrak{s}_{s_{0}y}\bigl{(}k_{0}k_{r}\bigr{)}^{-(\eta_{e}(\varphi(y))+\eta_{e}(2-\varphi(y)))\frac{\varphi}{2}}\bigl{(}u_{0}u_{r}\bigr{)}^{-(\eta_{o}(\varphi(y))+\eta_{o}(2-\varphi(y)))\frac{\varphi}{2}}\\\
&\times\prod_{\beta\in\Phi_{0,s}^{+}:\,\beta(\varphi^{\vee})=1}\mathbf{k}_{\beta}^{-(\eta(-\beta(s_{\varphi}y))+\eta(1+\beta(s_{\varphi}y)))\beta}.\end{split}$

Now the product in the second line of (4.5) is $1_{\mathbf{T}}$ since
$\eta(z)+\eta(1-z)=0$ for $z\in\mathbb{R}$. Applying the elementary formulas

(4.6)
$\eta_{e}(z)+\eta_{e}(2-z)=0,\qquad\quad\eta_{o}(z)+\eta_{o}(2-z)=2\chi_{\\{1\\}}(z)$

for $z\in\mathbb{R}$ to the first line of (4.5), the identity (4.5) reduces to

$s_{\varphi}\mathfrak{s}_{y}=\mathfrak{s}_{s_{0}y}\bigl{(}u_{0}u_{r}\bigr{)}^{-\chi_{\\{1\\}}(\varphi(y))\varphi},$

from which the lemma for $j=0$ follows immediately. ∎

We denote by $\overline{w}\in W_{0}$ the image of $w\in W$ under the group
homomorphism $W\twoheadrightarrow W_{0}$, $v\tau(\lambda)\mapsto v$ ($v\in
W_{0}$, $\lambda\in Q^{\vee}$). Note that $\overline{s_{a}}=s_{\overline{a}}$
for $a\in\Phi$.

######  Corollary 4.8.

For $y\in\mathcal{O}$ we have

$\mathfrak{s}_{y}=\overline{\mathbf{g}_{y}}\mathfrak{s}_{\mathcal{O}}\quad\hbox{
and
}\quad\mathfrak{s}_{y}\mathbf{g}_{y}t=\mathbf{g}_{y}(\mathfrak{s}_{\mathcal{O}}t).$

Furthermore,
$\mathfrak{s}_{\mathcal{O}}^{\overline{\alpha}_{j}^{\vee}}=\widetilde{\mathbf{k}}_{\alpha_{j}}^{-1}\widetilde{\mathbf{k}}_{\frac{\alpha_{j}}{2}}^{-1}$
for all $j\in J(\mathcal{O})$.

###### Proof.

Similar to the proof of [14, Prop. 5.4] and [14, Cor. 5.5]. ∎

By Corollary 4.8 we have
$\mathfrak{s}_{\mathcal{O}}\mathbf{T}_{\mathcal{O}}=\mathbf{L}_{\mathcal{O}}$
with

(4.7)
$\mathbf{L}_{\mathcal{O}}:=\bigl{\\{}\gamma\in\mathbf{T}\,\,\,|\,\,\,\gamma^{\alpha_{j}^{\vee}}=\widetilde{\mathbf{k}}_{\alpha_{j}}^{-1}\widetilde{\mathbf{k}}_{\frac{\alpha_{j}}{2}}^{-1}\quad\,\,\,\forall\,j\in
J(\mathcal{O})\bigr{\\}}.$

Note here that
$\widetilde{\mathbf{k}}_{\alpha_{0}}\widetilde{\mathbf{k}}_{\frac{\alpha_{0}}{2}}=u_{0}u_{r}$,
$\widetilde{\mathbf{k}}_{\alpha_{i}}\widetilde{\mathbf{k}}_{\frac{\alpha_{i}}{2}}=k_{i}^{2}=k^{2}$
for $1\leq i<r$, and
$\widetilde{\mathbf{k}}_{\alpha_{r}}\widetilde{\mathbf{k}}_{\frac{\alpha_{r}}{2}}=k_{0}k_{r}$.

Write

$\mathbf{T}_{\mathcal{O}}^{\prime}:=\\{t\in\mathbf{T}_{\mathcal{O}}\,\,|\,\,\textup{
the map }\,\,W^{\mathcal{O}}\rightarrow\mathbf{T},\,g\mapsto
g(\mathfrak{s}_{\mathcal{O}}t)\,\,\hbox{ is injective}\\}.$

Then $\mathbf{T}_{\mathcal{O}}^{\prime}\not=\emptyset$ for generic
$q\in\mathbf{F}^{\times}$ and $\mathbf{k}\in\mathcal{K}$. We are now in the
position to extend the definition of the quasi-polynomial analogs of the monic
nonsymmetric Macdonald polynomials, introduced in [14, Thm. 6.2] for
multiplicity functions $\mathbf{k}\in\mathcal{K}^{\textup{res}}$, to
multiplicity functions $\mathbf{k}\in\mathcal{K}$.

######  Theorem 4.9.

For $t\in\mathbf{T}_{\mathcal{O}}^{\prime}$ and $y\in\mathcal{O}$ there exists
a unique quasi-polynomial

$E^{\mathcal{O}}_{y}(x)=E_{y}^{\mathcal{O}}(x;\mathbf{k},t;q)\in\mathbf{F}[\mathcal{O}]$

satisfying the following two properties:

  1. (a)

$E_{y}^{\mathcal{O}}(x)=x^{y}+\textup{ l.o.t.}$,

  2. (b)

$\pi_{t}^{\mathcal{O}}(Y^{\mu})E_{y}^{\mathcal{O}}(x)=(\mathbf{g}_{y}(\mathfrak{s}_{\mathcal{O}}t))^{-\mu}E_{y}^{\mathcal{O}}(x)$
for all $\mu\in Q^{\vee}$.

###### Proof.

This is a direct consequence of Corollary 4.6 and Corollary 4.8. ∎

Only the Koornwinder/$\textup{C}^{\vee}\textup{C}_{r}$-case of Theorem 4.9 is
new compared to [14, Thm. 6.2]. In this case $\Phi_{0}$ is of type
$\textup{C}_{r}$ ($r\geq 1$) and $E_{y}^{\mathcal{O}}(x)$ depends on five
multiplicity parameters $k_{0},u_{0},k_{r},u_{r},k\in\mathbf{F}^{\times}$
(four in case of $r=1$), on the dilation parameter $q$, and on the
representation parameter $t\in\mathbf{T}_{\mathcal{O}}$.

To see how Sahi’s [12] monic nonsymmetric Koornwinder polynomials fit into
this picture, consider the special case that $\mathcal{O}=Q^{\vee}$. Then
$J(Q^{\vee})=\\{1,\ldots,r\\}$ and
$\mathbf{T}_{Q^{\vee}}=\\{1_{\mathbf{T}}\\}$. Then Theorem 4.9 requires that
$1_{\mathbf{T}}\in\mathbf{T}_{\\{1,\ldots,r\\}}^{\prime}$, which amounts to
generic conditions on $q\in\mathbf{F}^{\times}$ and $\mathbf{k}\in\mathcal{K}$
(including, typically, the condition that $q$ is not a root of unity). By
Remark 3.14 the resulting Laurent polynomial

$E_{\mu}^{Q^{\vee}}(x;\mathbf{k},1_{\mathbf{T}};q)\in\mathbf{F}[Q^{\vee}]$

is Sahi’s [12, Thm. 6.2] monic nonsymmetric Koornwinder polynomial $E_{\mu}$
of degree $\mu\in Q^{\vee}$, with $n$ and the multiplicity parameters
$t_{0},u_{0},t_{n},u_{n},t_{i}$ ($i\not=0,n$) in [12] corresponding to $r$ and
$k_{0},u_{0},k_{r},u_{r},k$.

Various properties of the quasi-polynomial generalisations of the Macdonald
polynomials obtained in [14] have direct analogs in the Koornwinder case, such
as the face limit transitions [14, Prop. 6.15], the creation formulas [14,
Thm. 6.12] in terms of double affine Hecke algebra $Y$-intertwiners, the
orthogonality relations [14, Thm. 6.42], and (anti)symmetrisation [14, §6.6].
We do not give the details here. The quasi-polynomial generalisations of the
symmetric Macdonald-Koornwinder polynomials will be the topic of an upcoming
paper.

##  5\. The quasi-polynomial representation as $Y$-parabolically induced
module

In this section $\mathcal{O}$ is a $W$-orbit in $E$ and
$t\in\mathbf{T}_{\mathcal{O}}$. Then
$\mathfrak{s}_{\mathcal{O}}t\in\mathfrak{s}_{\mathcal{O}}\mathbf{T}_{\mathcal{O}}=\mathbf{L}_{\mathcal{O}}$
with $\mathbf{L}_{\mathcal{O}}$ the affine subtorus of $\mathbf{T}$ defined by
(4.7) and $\mathfrak{s}_{\mathcal{O}}$ given by (3.49). Recall the definition
of the subset $I(\mathcal{O})\subseteq\\{1,\ldots,r\\}$ from Definition 3.7.

######  Lemma 5.1.

The $\mathbb{H}$-module $(\mathbf{F}[\mathcal{O}],\pi_{t}^{\mathcal{O}})$ is
cyclic with cyclic vector $x^{c^{\mathcal{O}}}$. Furthermore,

(5.1)
$\begin{split}\pi_{t}^{\mathcal{O}}(T_{i})x^{c^{\mathcal{O}}}&=k_{i}x^{c^{\mathcal{O}}}\qquad\qquad\,\,(i\in
I(\mathcal{O})),\\\
\pi_{t}^{\mathcal{O}}(Y^{\mu})x^{c^{\mathcal{O}}}&=(\mathfrak{s}_{\mathcal{O}}t)^{-\mu}x^{c^{\mathcal{O}}}\qquad(\mu\in
Q^{\vee}).\end{split}$

###### Proof.

This follows immediately from Corollary 3.15 and the fact that
$\kappa_{s_{i}}^{\mathcal{O}}=k_{i}$ for $i\in I(\mathcal{O})$. ∎

Lemma 5.1 prompts the following definition.

######  Definition 5.2.

We write $H_{0,\mathcal{O}}[Y]$ for the subalgebra of the affine Hecke algebra
$H=H(\mathbf{k})$ generated by $H_{0,\mathcal{O}}$ and
$\mathbf{F}_{Y}[Q^{\vee}]$.

By Lemma 5.1, $\mathbf{F}x^{c^{\mathcal{O}}}$ is a one-dimensional
$H_{0,\mathcal{O}}[Y]$-submodule of
$\bigl{(}\mathbf{F}[\mathcal{O}],\pi_{t}^{\mathcal{O}}|_{H_{0,\mathcal{O}}[Y]}\bigr{)}$,
with the action defined by the unique linear character
$\zeta_{t}^{\mathcal{O}}:H_{0,\mathcal{O}}[Y]\rightarrow\mathbf{F}$ satisfying

(5.2) $\begin{split}\zeta_{t}^{\mathcal{O}}(T_{i})&=k_{i}\qquad\qquad\textup{
for }\,\,i\in I(\mathcal{O}),\\\
\zeta_{t}^{\mathcal{O}}(Y^{\mu})&=(\mathfrak{s}_{\mathcal{O}}t)^{-\mu}\qquad\textup{
for }\,\,\mu\in Q^{\vee}.\end{split}$

The existence of the linear character
$\zeta_{t}^{\mathcal{O}}:H_{0,\mathcal{O}}[Y]\rightarrow\mathbf{F}$ can also
be established without reference to Lemma 5.1 using the Bernstein presentation
of $H_{0,\mathcal{O}}[Y]\subseteq H$ in terms of the algebraic generators
$T_{i}$ ($i\in I(\mathcal{O})$) and $Y^{\mu}$ ($\mu\in Q^{\vee}$). From now on
we write $\mathbf{F}1_{t}^{\mathcal{O}}$ for the one-dimensional
$H_{0,\mathcal{O}}[Y]$-representation with representation map
$\zeta_{t}^{\mathcal{O}}$.

######  Proposition 5.3.

We have a unique surjection of $\mathbb{H}$-modules

(5.3)
$\textup{Ind}_{H_{0,\mathcal{O}}[Y]}^{\mathbb{H}}\bigl{(}\mathbf{F}1_{t}^{\mathcal{O}}\bigr{)}\twoheadrightarrow(\mathbf{F}[\mathcal{O}],\pi_{t}^{\mathcal{O}})$

mapping $1\otimes_{H_{0,\mathcal{O}}[Y]}1_{t}^{\mathcal{O}}$ to
$x^{c^{\mathcal{O}}}$. It is an isomorphism when
$\alpha_{0}(c^{\mathcal{O}})\not=0$ (i.e., when
$I(\mathcal{O})=J(\mathcal{O})$).

###### Proof.

The first statement is immediate from Lemma 5.1.

In view of the PBW theorem for $\mathbb{H}$,

$\\{x^{\lambda}T_{v}\otimes_{H_{0,\mathcal{O}}[Y]}1_{t}^{\mathcal{O}}\,\,\,\,|\,\,\,\,\lambda\in
Q^{\vee},\,v\in W_{0}^{\mathcal{O}}\\}$

is a $\mathbf{F}$-basis of
$\textup{Ind}_{H_{0,\mathcal{O}}[Y]}^{\mathbb{H}}\bigl{(}\mathbf{F}1_{t}^{\mathcal{O}}\bigr{)}$.
This is mapped to

$\\{\kappa_{v}^{\mathcal{O}}x^{\lambda+vc^{\mathcal{O}}}\,\,|\,\,\lambda\in
Q^{\vee},\,v\in W_{0}^{\mathcal{O}}\\}$

by Corollary 3.15, which is a basis of $\mathbf{F}[\mathcal{O}]$ when
$\alpha_{0}(c^{\mathcal{O}})\not=0$ by Corollary 3.8. ∎

######  Remark 5.4.

For an associative $\mathbf{F}$-algebra $A$ denote by $\textup{mod}_{A}$ the
category of left $A$-modules. The image of
$\textup{Ind}_{H_{0,\mathcal{O}}[Y]}^{\mathbb{H}}\bigl{(}\mathbf{F}1_{t}^{\mathcal{O}}\bigr{)}$
under the restriction functor

$\textup{Res}_{H^{X}}^{\mathbb{H}}:\textup{mod}_{\mathbb{H}}\rightarrow\textup{mod}_{H^{X}}$

is isomorphic to
$\textup{Ind}_{H_{0,\mathcal{O}}}^{H^{X}}(\mathbf{F}1^{\mathcal{O}})$ because
$\textup{Ind}_{H_{0,\mathcal{O}}[Y]}^{\mathbb{H}}\bigl{(}\mathbf{F}1_{t}^{\mathcal{O}}\bigr{)}$
is already generated by $\mathbf{1}_{t}^{\mathcal{O}}$ as a $H^{X}$-module and
$\zeta_{t}^{\mathcal{O}}|_{H_{0,\mathcal{O}}}$ is the trivial linear character
of $H_{0,\mathcal{O}}$. Hence Proposition 3.9 follows from Proposition 5.3 by
applying the restriction functor $\textup{Res}_{H^{X}}^{\mathbb{H}}$.

We finish this section by realising
$(\mathbf{F}[\mathcal{O}],\pi_{t}^{\mathcal{O}})$ as a $Y$-parabolically
induced $\mathbb{H}$-module when $\alpha_{0}(c^{\mathcal{O}})=0$. For $y\in E$
and $w\in W$ set

$k_{w}(y):=\prod_{\alpha\in\Phi_{0}^{+}}\mathbf{k}_{\alpha}^{\frac{\eta_{e}(\alpha(wy))-\eta_{e}(\alpha(y))}{2}}\mathbf{k}_{\frac{\alpha}{2}}^{\frac{\eta_{o}(\alpha(wy))-\eta_{o}(\alpha(y))}{2}},$

which is well defined since the product involves integer powers of the
multiplicity parameters. Indeed, this follows from the observation that

(5.4) $k_{ww^{\prime}}(y)=k_{w}(w^{\prime}y)k_{w^{\prime}}(y)$

for $w,w^{\prime}\in W$ and the formulas

(5.5)
$k_{s_{i}}(y)=\begin{cases}\mathbf{k}_{\alpha_{i}}^{-\eta_{e}(\alpha_{i}(y))}\mathbf{k}_{\frac{\alpha_{i}}{2}}^{-\eta_{o}(\alpha_{i}(y))}\quad&\hbox{
if }\alpha_{i}(y)\not=0,\\\ 1\qquad&\hbox{ if }\alpha_{i}(y)=0\end{cases}$

for $i=1,\ldots,r$ and

(5.6)
$k_{s_{0}}(y)=\begin{cases}\prod_{\alpha\in\Pi(s_{\varphi})}\mathbf{k}_{\alpha}^{-\eta_{e}(\alpha(y))}\mathbf{k}_{\frac{\alpha}{2}}^{-\eta_{o}(\alpha(y))}\quad&\hbox{
if }\,\,\alpha_{0}(y)\not=0,\\\
\prod_{\alpha\in\Pi(s_{\varphi})\setminus\\{\varphi\\}}\mathbf{k}_{\alpha}^{-\eta_{e}(\alpha(y))}\mathbf{k}_{\frac{\alpha}{2}}^{-\eta_{o}(\alpha(y))}\qquad&\hbox{
if }\,\,\alpha_{0}(y)=0,\end{cases}$

which in turn follow by a computation in the spirit of Lemma 4.7.

The following result extends Lemma 5.1, [14, Lem. 5.11] and [14, Prop. 5.29].
Recall the definition of the duality anti-algebra isomorphism
$\delta=\delta_{\mathbf{k}}:\mathbb{H}\rightarrow\widetilde{\mathbb{H}}$ with
inverse $\widetilde{\delta}=\delta_{\widetilde{\mathbf{k}}}$ from Subsection
2.5.

######  Proposition 5.5.

We have

(5.7)
$\begin{split}\pi_{t}^{\mathcal{O}}(\widetilde{\delta}(T_{j}))x^{c^{\mathcal{O}}}&=\widetilde{\mathbf{k}}_{\alpha_{j}}x^{c^{\mathcal{O}}}\,\,\,\qquad\qquad\qquad\,\,\,\textup{
for }\,\,j\in J(\mathcal{O}),\\\
\pi_{t}^{\mathcal{O}}(\widetilde{\delta}(T_{w^{-1}}))x^{c^{\mathcal{O}}}&=k_{w}(c^{\mathcal{O}})x^{wc^{\mathcal{O}}}+\textup{
l.o.t.}\quad\,\,\,\textup{ for }\,\,w\in W^{\mathcal{O}}.\end{split}$

Note that for $j\in I(\mathcal{O})$ we have $\widetilde{\delta}(T_{j})=T_{j}$
and $\widetilde{\mathbf{k}}_{\alpha_{j}}=k_{j}$, so the first line of (5.7) is
consistent with the first line of (5.1). Before proving Proposition 5.5, let
me explain how it leads to the interpretation of
$(\mathbf{F}[\mathcal{O}],\pi_{t}^{\mathcal{O}})$ as a $Y$-parabolically
induced $\mathbb{H}$-module.

Consider the following algebras:

  1. –

the subalgebra $\widetilde{H}_{\mathcal{O}}$ of
$\widetilde{H}=H(\widetilde{\mathbf{k}})$, generated by $T_{j}$ ($j\in
J(\mathcal{O})$),

  2. –

the subalgebra
$H_{\mathcal{O}}^{\delta}:=\widetilde{\delta}(\widetilde{H}_{\mathcal{O}})$ of
$H^{X}$,

  3. –

the subalgebra $\widetilde{H}_{\mathcal{O}}[X]$ of $\widetilde{\mathbb{H}}$,
generated by $\widetilde{H}_{\mathcal{O}}$ and $\mathbf{F}[Q^{\vee}]$,

  4. –

the subalgebra
$H_{\mathcal{O}}^{\delta}[Y]:=\widetilde{\delta}(\widetilde{H}_{\mathcal{O}}[X])$