1.1 Double affine Weyl group actions on quasi-polynomials

To set the stage, it is instructive to first introduce the quasi-polynomial representations of the double affine Weyl group.

Let $\Phi _0$ be a reduced irreducible root system, realised within the linear dual of a Euclidean space E, and normalised so that long roots have squared length $2/m^2$ for some natural number $m\in \mathbb {Z}_{>0}$ . Let $\Phi =\Phi _0\times \mathbb {Z}$ be the associated irreducible affine root system of untwisted type. The coroot lattice $\widehat {Q}^{\vee }$ of $\Phi $ is isomorphic to $Q^{\vee }+m^2{\mathbb Z} K$ , where $Q^{\vee }$ is the coroot lattice of $\Phi _0$ and K is the central element in the associated untwisted affine Lie algebra. The affine Weyl group W of $\Phi $ is a semidirect product $W=W_0\ltimes Q^{\vee }$ , where $W_0$ is the Weyl group of $\Phi _0$ . The double affine Weyl group is the semidirect product

(1.1) $$ \begin{align} \mathbb{W}=W\ltimes \widehat{Q}^{\vee}=(W_0\ltimes Q^{\vee})\ltimes \widehat{Q}^{\vee}, \\[-24pt]\nonumber\end{align} $$

where the W-action $(w,\widehat {\mu })\mapsto w\cdot \widehat {\mu }$ on $\widehat {Q}^{\vee }$ is determined by the pairing of $Q^{\vee }$ into the center $m^2{\mathbb Z} K$ of $\mathbb {W}$ induced by the inner product $\langle \cdot ,\cdot \rangle $ on E.

We fix simple roots $\alpha _0,\alpha _1,\ldots , \alpha _r$ for $\Phi $ such that $\alpha _1, \ldots , \alpha _r$ are simple roots for $\Phi _0$ . Let $\alpha _j^{\vee }$ be the corresponding simple coroots. Then $\alpha _0=(-\varphi ,1)$ , where $\varphi $ is the highest root for $\Phi _0$ , and $\alpha _0^{\vee }=m^2K-\varphi ^{\vee }$ . The simple reflection associated to $\alpha _j$ is denoted by $s_j\in W$ . The affine root hyperplane configuration in E provides a decomposition of E in closed alcoves, each closed alcove being a fundamental domain for the action of W on E by reflections and translations. The choice of simple roots singles out the fundamental open alcove $C_+$ . We write $\mathcal {O}_c$ for the W-orbit in E containing $c\in \overline {C}_+$ .

The face decomposition of $\overline {C}_+$ is the disjoint union

$$\begin{align*}\overline{C}_+=\bigsqcup_{J\subsetneq \{0,\ldots,r\}}C^J \end{align*}$$

with $C^J$ the vectors c in $\overline {C}_+$ satisfying $\alpha _j(c)=0$ if and only if $j\in J$ .

Let $\mathbf {F}$ be a field of characteristic zero and fix $q\in \mathbf {F}^\times $ not a root of unity. Let $T:=\text {Hom}(Q^{\vee },\mathbf {F}^\times )$ be the $\mathbf {F}$ -torus consisting of multiplicative characters $\mathfrak {t}: Q^{\vee }\rightarrow \mathbf {F}^\times $ . We will denote the value of $\mathfrak {t}$ at $\mu \in Q^{\vee }$ by $\mathfrak {t}^\mu $ . Consider the associated affine subtorus

$$\begin{align*}T_J:=\{\mathfrak{t}\in T \,\, | \,\, \mathfrak{t}^{\alpha_j^{\vee}}=1 \,\,\, \forall\, j\in J\} \end{align*}$$

of dimension $r-\#J$ , where $\mathfrak {t}^{\mu +\ell K}:=q^\ell \mathfrak {t}^\mu $ . The field $\mathbf {F}$ should contain a sufficiently large root of q to ensure that $T_J\not =\emptyset $ for subsets J containing $0$ .

Denote by

$$\begin{align*}\mathbf{F}[E]=\bigoplus_{y\in E}\mathbf{F}x^y \end{align*}$$

the group algebra of E, viewed as additive group. It splits up as the direct sum of the $\mathcal {P}$ -submodules $\mathcal {P}^{(c)}:=\bigoplus _{y\in \mathcal {O}_c}\mathbf {F}x^y$ ( $c\in \overline {C}_+$ ).

For $c\in C^J$ , the space $\mathcal {P}^{(c)}$ of quasi-polynomials admits a natural family of $\mathbb {W}$ -actions $\cdot _{\mathfrak {t}}$ depending on q and $\mathfrak {t}\in T_J$ . The action of $W_0\ltimes \widehat {Q}^{\vee }$ is by reflections and translations of the exponents of the quasi-monomials $x^y$ , with $m^2K$ acting as multiplication by $q^{m^2}$ . In particular, this part of the action is independent of $\mathfrak {t}$ . The action of the standard abelian subgroup $\{\tau (\mu )\}_{\mu \in Q^{\vee }}$ in $W=W_0\ltimes Q^{\vee }$ is defined by

(1.2) $$ \begin{align} \tau(\mu)_{\mathfrak{t}}x^y:=\mathfrak{t}_y^{-\mu}x^y\qquad\quad (\mu\in Q^{\vee},\, y\in\mathcal{O}_c), \end{align} $$

where $\mathfrak {t}_y\in T$ is the multiplicative character $\mu \mapsto \mathfrak {t}^{w_y^{-1}\cdot \mu }$ with $w_y\in W$ the element of shortest length such that $w_yc=y$ (actually, one may take here any $w\in W$ satisfying $wc=y$ ). Note that $s_{j,\mathfrak {t}}x^y:=(s_j)_{\mathfrak {t}}x^y$ for $y\in \mathcal {O}_c$ is explicitly given by

$$\begin{align*}s_{i,\mathfrak{t}}x^y=x^{s_iy}\quad (1\leq i\leq r),\qquad\quad s_{0,\mathfrak{t}}x^y=\mathfrak{t}_y^{\varphi^{\vee}}x^{s_\varphi y} \end{align*}$$

with $s_\varphi \in W_0$ the reflection associated to the highest root $\varphi \in \Phi _0$ .

In case $J=\{1,\ldots ,r\}$ , $c=0$ and $\mathfrak {t}=1_T$ , we reobtain the standard action of $\mathbb {W}$ on $\mathcal {P}$ by polynomial q-difference reflection operators, since

$$\begin{align*}\tau(\mu)_{1_T}(x^\nu)=q^{-\langle\mu,\nu\rangle}x^\nu\qquad\quad (\mu,\nu\in Q^{\vee}). \end{align*}$$

In this case, we write $w(x^\mu )$ for $w_{1_T}x^\mu $ ( $w\in W$ ).

In this paper, we deform the $\mathbb {W}$ -action $\cdot _{\mathfrak {t}}$ on $\mathcal {P}^{(c)}$ to an action of the double affine Hecke algebra $\mathbb {H}$ . The resulting quasi-polynomial representation of $\mathbb {H}$ plays a central role in this paper.

1.2 The quasi-polynomial representation

Cherednik’s [Reference Cherednik8] double affine Hecke algebra (DAHA) is a certain flat deformation $\mathbb {H}=\mathbb {H}(\mathbf {k},q)$ of $\mathbf {F}[\mathbb {W}]/(m^2K-q^{m^2})\simeq W\ltimes \mathcal {P}$ depending on a multiplicity function $\mathbf {k}=(k_a)_{a\in \Phi }$ . We consider in this paper the case that $\mathbf {k}$ is invariant for the extended affine Weyl group. This means that $k_a\in \mathbf {F}^\times $ only depends on the length $\|\alpha \|$ of the gradient $\alpha \in \Phi _0$ of the affine root $a=(\alpha ,\ell )\in \Phi $ . In particular, there are at most two distinct values of the $k_a$ . We write $k_j=k_{\alpha _j}$ .

The double affine Hecke algebra $\mathbb {H}$ has an explicit definition in terms of generators and relations (see Definition 2.12). The generators are $T_j$ ( $0\leq j\leq r$ ) and $x^\mu $ ( $\mu \in Q^{\vee }$ ). The subalgebra generated by $T_j$ ( $0\leq j\leq r$ ) is the affine Hecke algebra $H=H(\mathbf {k})$ of W. Its defining relations are the $(W,\{s_0,\ldots ,s_r\})$ -braid relations and the Hecke relations $(T_j-k_j)(T_j+k_j^{-1})=0$ . The subalgebra generated by $x^\mu $ ( $\mu \in Q^{\vee }$ ) is $\mathcal {P}$ . The commutation relations between the $T_j$ and $x^\mu $ are $\mathbf {k}$ -deformations of $s_jx^\mu =s_{j}(x^\mu )s_j$ in $W\ltimes \mathcal {P}$ , given explicitly by

(1.3) $$ \begin{align} T_jx^\mu-s_{j}(x^\mu)T_j=(k_j-k_j^{-1})\left(\frac{x^\mu-s_{j}(x^\mu)}{1-x^{\alpha_j^{\vee}}}\right). \end{align} $$

Note here that $x^{\alpha _0^{\vee }}=x^{m^2K^2-\varphi ^{\vee }}=q^{m^2}x^{-\varphi ^{\vee }}$ . The right-hand side can be alternatively written as

$$\begin{align*}(k_j-k_j^{-1})\left(\frac{1-x^{-D\alpha_j(\mu)\alpha_j^{\vee}}}{1-x^{\alpha_j^{\vee}}}\right)x^\mu \end{align*}$$

with $D\alpha _j\in \Phi _0$ the gradient of $\alpha _j$ . In particular, (1.3) makes sense in $\mathbb {H}$ , since $D\alpha _j(\mu )\in \mathbb {Z}$ . Together with the well-known fact that $H\otimes \mathcal {P}\simeq \mathbb {H}$ as vector spaces by multiplication (Cherednik’s [Reference Cherednik8] PBW theorem for $\mathbb {H}$ ), this gives an explicit generalisation of the first isomorphism in (1.1).

Also, the decomposition $W\simeq W_0\ltimes Q^{\vee }$ has a natural analog in H, known as the Bernstein-Zelevinsky decomposition of H (see [Reference Lusztig25]). It provides a subalgebra $\mathcal {P}_Y=\bigoplus _{\mu \in Q^{\vee }}\mathbf {F}Y^\mu $ of H, isomorphic to $\mathcal {P}$ , such that $H_0\otimes \mathcal {P}_Y\simeq H$ as vector spaces, with $H_0$ the subalgebra of $\mathbb {H}$ generated by $T_i$ ( $1\leq i\leq r$ ). The duality anti-involution $\delta $ of $\mathbb {H}$ interchanges the two copies of $\mathcal {P}$ inside $\mathbb {H}$ . Concretely, it satisfies $\delta (Y^\mu )=x^{-\mu }$ and $\delta (T_i)=T_i$ for $\mu \in Q^{\vee }$ and $1\leq i\leq r$ .

Let $\chi _{B}: \mathbb {R}\rightarrow \{0,1\}$ for a subset $B\subset \mathbb {R}$ be the characteristic function of B, and let $\lfloor \cdot \rfloor : \mathbb {R}\rightarrow \mathbb {Z}$ be the floor-function (i.e., $\lfloor d\rfloor $ for $d\in \mathbb {R}$ is the largest integer $\leq d$ ). For $0\leq j\leq r$ , the truncated divided-difference operator $\nabla _j$ is the linear operator on $\mathbf {F}[E]$ defined by

$$\begin{align*}\nabla_j(x^y):=\left(\frac{1-x^{-\lfloor D\alpha_j(y)\rfloor\alpha_j^{\vee}}}{1-x^{\alpha_j^{\vee}}}\right)x^y\qquad\quad (y\in E). \end{align*}$$

Its restriction to $\mathcal {P}$ is the usual divided-difference operator.

The first main result of this paper is the following DAHA analog of the $\mathbb {W}$ -module $(\mathcal {P}^{(c)},\cdot _{\mathfrak {t}})$ ( $c\in C^J$ , $\mathfrak {t}\in T_J$ ), in which $T_j$ acts by a Demazure-Lusztig type operator involving the truncated divided-difference operator $\nabla _j$ . We call $\pi _{c,\mathfrak {t}}$ the quasi-polynomial representation of $\mathbb {H}$ .

Theorem 1.1. Let $J\subsetneq \{0,\ldots ,r\}$ , $c\in C^J$ and $\mathfrak {t}\in T_J$ . The formulas

$$ \begin{align*} \begin{aligned} \pi_{c,\mathfrak{t}}(T_j)x^y&:=k_j^{\chi_{\mathbb{Z}}(\alpha_j(y))}s_{j,\mathfrak{t}}x^y+(k_j-k_j^{-1}) \nabla_j(x^y),\\ \pi_{c,\mathfrak{t}}(x^\mu)x^y&:=x^{y+\mu} \end{aligned} \end{align*} $$

for $0\leq j\leq r$ , $\mu \in Q^{\vee }$ and $y\in \mathcal {O}_c$ define a representation $\pi _{c,\mathfrak {t}}: \mathbb {H}\rightarrow \text {End}(\mathcal {P}^{(c)})$ .

The representation $\pi :=\pi _{0,1_T}: \mathbb {H}\rightarrow \text {End}(\mathcal {P})$ is Cherednik’s [Reference Cherednik8] polynomial representation of $\mathbb {H}$ . In this case, $J=\{1,\ldots ,r\}$ , and the commuting operators $\pi (Y^\mu )$ ( $\mu \in Q^{\vee }$ ) are the Cherednik operators. The polynomial representation $\pi $ can alternatively be obtained by inducing the trivial one-dimensional H-representation to $\mathbb {H}$ . As a second main result of this paper we extend this result to the quasi-polynomial representation $\pi _{c,\mathfrak {t}}$ ( $c\in C^J$ ) by inducing a one-dimensional representation of the Y-parabolic subalgebra $\mathbb {H}_J^Y$ of $\mathbb {H}$ generated by $\delta (T_j)$ ( $j\in J$ ) and $\mathcal {P}_Y$ .

The relevant one-dimensional $\mathbb {H}_J^Y$ -representations are parametrised by the affine subtorus

$$\begin{align*}L_J:=\{\mathfrak{t}\in T \,\, | \,\, \mathfrak{t}^{\alpha_j^{\vee}}=k_j^{-2}\quad \forall\, j\in J\} \end{align*}$$

of T. The one-dimensional $\mathbb {H}_J^Y$ -representation associated to $t\in L_J$ is then defined by

$$\begin{align*}\chi_{J,t}(Y^\mu):=t^{-\mu}\quad (\mu\in Q^{\vee}),\qquad \chi_{J,t}(\delta(T_j)):=k_j\quad (j\in J). \end{align*}$$

We write $\mathbb {M}_{J,t}$ for the resulting induced $\mathbb {H}$ -module, and $m_{J,t}$ for its canonical cyclic vector.

Consider

$$\begin{align*}\mathfrak{s}_y:=\prod_{\alpha\in\Phi_0^+}k_\alpha^{\eta(\alpha(y))\alpha}\in T\qquad (y\in E), \end{align*}$$

with $\eta :=\chi _{\mathbb {Z}_{>0}}-\chi _{\mathbb {Z}_{\leq 0}}: \mathbb {R}\rightarrow \{-1,0,1\}$ (a discrete analog of the Heaviside function). The map $y\mapsto \mathfrak {s}_y$ is constant on faces, and we write $\mathfrak {s}_J\in T$ for its value on $C^J$ . We then have $L_J=\mathfrak {s}_JT_J$ and

In particular, $\pi _{c,\mathfrak {t}}\simeq \pi _{c^\prime ,\mathfrak {t}}$ if c and $c^\prime $ lie in the same face of $\overline {C}_+$ .

Quite remarkably, for all $\mu \in Q^{\vee }$ and $v\in W_0$ , the vector $x^\mu T_vm_{J,\mathfrak {s}_J\mathfrak {t}}\in \mathbb {M}_{J,\mathfrak {s}_J\mathfrak {t}}$ is mapped to an explicit constant multiple of the quasi-monomial $x^{\mu +vc}$ (see Theorem 4.5(3)). Actually, it is this formula which forms the starting point of the proof of both Theorem 1.1 and Theorem 1.2.

1.3 Quasi-polynomial eigenfunctions

The quasi-polynomial representations $\pi _{c,\mathfrak {t}}$ give rise to a family of quasi-polynomials, which generalize the nonsymmetric Macdonald polynomials. These quasi-polynomials enjoy many properties paralleling the classical case; for example, they are joint eigenfunctions of $\pi _{c,\mathfrak {t}}(Y^{\mu })$ for $\mu \in Q^{\vee }$ , satisfy a triangularity condition, and can be constructed explicitly via intertwiners. Their definition is as follows.

Let $c\in C^J$ . Define a partial order on $\mathcal {O}_c$ by

$$\begin{align*}y\leq y^\prime\quad \Leftrightarrow \quad w_y\leq_B w_{y^\prime}, \end{align*}$$

with $\leq _B$ the Bruhat order on $(W,\{s_0,\ldots ,s_r\})$ . Denote by $T_J^\prime \subseteq T_J$ the set of characters $\mathfrak {t}\in T_J$ for which the $\mathfrak {s}_y\mathfrak {t}_y$ ( $y\in \mathcal {O}_c$ ) are pairwise different in T (it does not depend on the choice of $c\in C^J$ ).

Theorem 1.3. Let $c\in C^J$ and $\mathfrak {t}\in T_J^\prime $ . For each $y\in \mathcal {O}_c$ , there exists a unique joint eigenfunction $E_y^J(x;\mathfrak {t})\in \mathcal {P}^{(c)}$ of the commuting operators $\pi _{c,\mathfrak {t}}(Y^\mu )$ ( $\mu \in Q^{\vee }$ ) satisfying

(1.4) $$ \begin{align} E_y^J(x;\mathfrak{t})=x^y+\sum_{y^\prime<y}e_{y,y^\prime;\mathfrak{t}}^Jx^{y^\prime}\qquad\quad (e_{y,y^\prime;\mathfrak{t}}^J\in\mathbf{F}). \end{align} $$

Furthermore,

(1.5) $$ \begin{align} \pi_{c,\mathfrak{t}}(Y^\mu)E_y^J(x;\mathfrak{t})=(\mathfrak{s}_y\mathfrak{t}_y)^{-\mu}E_y^J(x;\mathfrak{t})\qquad\quad \forall\, \mu\in Q^{\vee}. \end{align} $$

In particular, $\{E_y^J(x;\mathfrak {t})\}_{y\in \mathcal {O}_c}$ is a basis of $\mathcal {P}^{(c)}$ consisting of joint eigenfunctions of the commuting operators $\pi _{c,\mathfrak {t}}(Y^\mu )$ ( $\mu \in Q^{\vee }$ ). Furthermore, the quasi-polynomial eigenfunction of smallest degree is trivial,

$$\begin{align*}E_c^J(x;\mathfrak{t})=x^c. \end{align*}$$

The choice of the vector $c\in C^J$ only influences the quasi-exponents in the quasi-monomial expansion of the $E_y^J(x;\mathfrak {t})$ . In other words, the coefficients

$$\begin{align*}e_{w,w^\prime;\mathfrak{t}}^J:=e_{wc,w^\prime c;\mathfrak{t}}^J\qquad\quad (w,w^\prime\in W) \end{align*}$$

are independent of $c\in C^J$ (see Theorem 6.3). Furthermore, if $J^\prime \supseteq J$ , then $C^{J^\prime }$ lies in the closure of $C^J$ , and the quasi-polynomial $E^{J^\prime }_{y^\prime }(x;\mathfrak {t})$ for $\mathfrak {t}\in T_{J^\prime }^\prime $ can be recovered from $E^J_y(x;\mathfrak {s}_J^{-1}\mathfrak {s}_{J^\prime }\mathfrak {t})$ (see Proposition 6.16).

Example 1.4. Consider $\mathbf {F}=\mathbb {Q}(q,\mathbf {k},z_1,\ldots ,z_r)$ with indeterminates $q,k_a,z_1,\ldots ,z_r$ , where $k_a=k_b$ if $\|Da\|=\|Db\|$ . For $I\subseteq \{1,\ldots ,r\}$ , let $I^{\text {co}}$ be its complement. Then

$$\begin{align*}\prod_{i\in I^{\text{co}}}z_i^{\varpi_i}\in T_I^\prime, \end{align*}$$

where $\varpi _i$ denotes the $i^{\text {th}}$ fundamental weight of $\Phi _0$ . Then

$$\begin{align*}E_y^I(x;z_1,\ldots,z_r):=E_y^I\left(x;\prod_{i \in I^{\text{co}}}z_i^{\varpi_i}\right)\qquad\quad (y\in\mathcal{O}_c) \end{align*}$$

depend on the indeterminates $q, \mathbf {k}$ and $z_i$ ( $i\in I^{\text {co}}$ ). Quasi-polynomial eigenfunctions over $\mathbb {Q}(q,\mathbf {k})$ can be obtained by specialisation of the $z_i$ ’s.

The situation is a bit more intricate when $0\in J$ since the $z_i$ ’s will no longer be independent; see Subsection 6.1.

If $1_T\in T_{\{1,\ldots ,r\}}^\prime $ (which imposes generic conditions on q and $\mathbf {k}$ ), then

$$\begin{align*}E_\mu(x):=E_\mu^{\{1,\ldots,r\}}(x;1_T)\in\mathcal{P} \end{align*}$$

is the monic nonsymmetric Macdonald polynomial of degree $\mu \in Q^{\vee }$ . The Macdonald polynomials and their normalised and (anti)symmetric variants have remarkable properties. They satisfy orthogonality relations with explicit quadratic norm formulas [Reference Macdonald28, Reference Cherednik6, Reference Cherednik8, Reference Macdonald29], have explicit evaluation formulas, and are self-dual (see, for example, [Reference Macdonald29, Reference Cherednik8] and references therein). They admit combinatorial formulas [Reference Ram and Yip36, Reference Haglund, Haiman and Loehr18], arise as spherical functions on quantum groups (see, for example, [Reference Noumi33, Reference Letzter24, Reference Etingof and Kirilliv17]), admit interpretations as partition functions on integrable lattice models [Reference Borodin and Wheeler2], and degenerate to Heckman-Opdam polynomials [Reference Macdonald27], spherical functions on p-adic reductive groups [Reference Macdonald27], and Iwahori-Whittaker functions [Reference Ion22, Reference Cherednik and Ma11]. In this paper, we discuss generalisations of a number of these properties to the quasi-polynomial setup:

1. 1. We define a normalized version $P_y^J(x;\mathfrak {t})$ of $E_y^J(x;\mathfrak {t})$ using the action of normalised Y-intertwiners on the lowest-degree quasi-polynomial eigenfunction $x^c$ . It leads to an explicit expression of $E_y^J(x;\mathfrak {t})$ in terms of Y-intertwiners (Theorem 6.13). This entails a quasi-polynomial analog of the evaluation formula for Macdonald polynomials.

2. 2. We obtain a weak form of duality for $P_y^J(x;\mathfrak {t})$ , called pseudo-duality, by showing that the H-action on $P_y^J(x;\mathfrak {t})$ can be alternatively described in terms of an H-action on $y\in \mathcal {O}_c$ by discrete Demazure-Lusztig operators (Theorem 6.26).

3. 3. We establish orthogonality relations and quadratic norm formulas for the quasi-polynomial eigenfunctions $E_y^J(x;\mathfrak {t})$ and $P_y^J(x;\mathfrak {t})$ (Theorem 6.43).

4. 4. We construct (anti)symmetric variants $E_y^{J,\pm }(x;\mathfrak {t})$ of the quasi-polynomial eigenfunctions $E_y^J(x;\mathfrak {t})$ by (anti)symmetrisation with respect to the trivial (resp., sign) idempotent of $H_0$ . They generalize the (anti)symmetric Macdonald polynomials. The symmetric variants $E_{y}^{J, +}(x; \mathfrak {t})$ are invariant under the $W_0$ -action obtained by localizing $\pi _{c, \mathfrak {t}}$ (see Theorem 4.9 and Lemma 6.32). We obtain explicit expansion formulas of $E_y^{J,\pm }(x;\mathfrak {t})$ in terms of the quasi-polynomials $\{E_{y^\prime }^J(x;\mathfrak {t})\}_{y^\prime \in W_0y}$ (Corollary 6.38 and Proposition 6.41).

5. 5. We establish the existence of the Whittaker limit $q\rightarrow \infty $ of $E_y^J(x;\mathfrak {t})$ and $E_y^{J,-}(x;\mathfrak {t})$ (Proposition 6.47 and Corollary 6.52). For y in the closure of the negative Weyl chamber and $y^\prime \in W_0y$ , we obtain an explicit formula for the limit $q \rightarrow \infty $ of $E_{y^\prime }^J(x;\mathfrak {t})$ in terms of the $\pi _{c,\mathfrak {t}}\vert _{H_0}$ -action on $x^y$ (Theorem 6.51). It generalises a formula due to Patnaik-Puskas [Reference Patnaik and Puskas34, Cor. 5.4], which expresses the metaplectic Iwahori-Whittaker function in terms of metaplectic Demazure-Lusztig operators. We will say more about the link with metaplectic representation theory in Subsection 1.5.

We also consider the theory for reductive root data, when both $\mathcal {P}$ and $\mathcal {P}_Y$ in $\mathbb {H}$ are extended to group algebras of a finitely generated abelian subgroup $\Lambda \subset E$ satisfying

(1.6) $$ \begin{align} Q^{\vee}\subseteq\Lambda\,\, \mbox{ and }\,\, \alpha(\Lambda)\subseteq\mathbb{Z}\,\,\,\, \forall\alpha\in\Phi_0. \end{align} $$

When only $\mathcal {P}_Y$ is extended to the group algebra of such a lattice $\Lambda ^\prime $ , then the associated Y-extended double affine Hecke algebra $\mathbb {H}_{Q^{\vee },\Lambda ^\prime }$ is a smashed product algebra $\Omega _{\Lambda ^\prime }\ltimes \mathbb {H}$ , with $\Omega _{\Lambda ^\prime }\simeq \Lambda ^\prime /Q^{\vee }$ the subgroup of the extended affine Weyl group $W_{\Lambda ^\prime }:=W_0\ltimes \Lambda ^\prime $ consisting of elements of length zero. The associated extended quasi-polynomial representation $\pi _{c,\mathfrak {t}}^{Q^{\vee },\Lambda ^\prime }:\mathbb {H}_{Q^{\vee },\Lambda ^\prime }\rightarrow \text {End}(\mathcal {P}^{(c)})$ now depends on parameters $\mathfrak {t}\in T_{\Lambda ^\prime ,J}$ , where $T_{\Lambda ^\prime ,J}$ is the affine subtorus of $T_{\Lambda ^\prime }=\text {Hom}(\Lambda ^\prime ,\mathbf {F}^\times )$ consisting of the elements $\mathfrak {t}\in T_{\Lambda ^\prime }$ satisfying $\mathfrak {t}^{\alpha _j^{\vee }}=1$ for all $j\in J$ .

The extra representation parameters in $\pi _{c,\mathfrak {t}}^{Q^{\vee },\Lambda ^\prime }$ compared to $\pi _{c,\mathfrak {t}\vert _{Q^{\vee }}}$ are captured by the kernel $T_{\Lambda ^\prime ,\{1,\ldots ,r\}}$ of the restriction map $T_{\Lambda ^\prime }\rightarrow T$ . It identifies with the character group of $\Omega _{\Lambda ^\prime }$ and naturally embeds into the automorphism group of $\mathbb {H}_{Q^{\vee },\Lambda ^\prime }\simeq \Omega _{\Lambda ^\prime }\ltimes \mathbb {H}$ fixing $\mathbb {H}$ point-wise. We call these extra parameters twist parameters because $\pi _{c,\mathfrak {t}\mathfrak {t}^\prime }^{Q^{\vee },\Lambda ^\prime }$ with $\mathfrak {t}^\prime \in T_{\Lambda ^\prime ,\{1,\ldots ,r\}}$ coincides with $\pi _{c,\mathfrak {t}}^{Q^{\vee },\Lambda ^\prime }$ twisted by the automorphism of $\mathbb {H}_{Q^{\vee },\Lambda ^\prime }$ associated to $\mathfrak {t}^\prime $ . Since these automorphisms rescale the $Y^\mu $ ( $\mu \in Q^{\vee }$ ), the quasi-polynomials $E_y^J(x;\mathfrak {t}\vert _{Q^{\vee }})$ ( $y\in \mathcal {O}_c$ ) are eigenfunctions of all the commuting operators $\pi _{c,\mathfrak {t}}^{Q^{\vee },\Lambda ^\prime }(Y^\mu )$ ( $\mu \in \Lambda ^\prime $ ) under suitable generic conditions on the parameters (Theorem 7.26).

The full double affine Hecke algebra $\mathbb {H}_{\Lambda ,\Lambda ^\prime }$ depends on two lattices $\Lambda ,\Lambda ^\prime $ . The extension of the quasi-polynomial representation $\pi _{c,\mathfrak {t}}^{Q^{\vee },\Lambda ^\prime }$ to $\mathbb {H}_{\Lambda ,\Lambda ^\prime }$ requires adding $(c,\Lambda )$ -dependent constraints on $\mathfrak {t}\in T_{\Lambda ^\prime ,J}$ , as well as enlarging the representation space. It will be discussed in detail in Section 7. In case of the $\text {GL}_{r+1}$ root datum, the quasi-polynomial eigenfunctions give rise to the metaplectic polynomials introduced in our previous work [Reference Sahi, Stokman and Venkateswaran38, §5.4]. We will say more about the root system generalisations of these metaplectic polynomials in Subsection 1.5.

1.4 The uniform quasi-polynomial representation

The quasi-polynomial representations defined in Theorem 1.1 can be naturally combined into a family of $\mathbb {H}$ -representations on $\mathbf {F}[E]$ . The parametrisation of this family of uniform quasi-polynomial representations requires that $q\in \mathbf {F}^\times $ has a $(2h)^{th}$ root $q^{\frac {1}{2h}}$ in $\mathbf {F}$ , where h is the Coxeter number of $W_0$ .

To motivate the parametrisation, it is instructive to consider an important special case first, which requires the stronger assumption that $q\in \mathbf {F}^\times $ is part of an injective group homomorphism

(1.7) $$ \begin{align} \mathbb{R}\rightarrow\mathbf{F}^\times,\qquad d\mapsto q^d. \end{align} $$

Then

(1.8) $$ \begin{align} \bigoplus_{c\in\overline{C}_+}\pi_{c,q^c}: \mathbb{H}\rightarrow\text{End}(\mathbf{F}[E]) \end{align} $$

is the prototypical example of a uniform quasi-polynomial representation, where $q^y\in T$ ( $y\in E$ ) is the character $\mu \mapsto q^{\langle y,\mu \rangle }$ of $Q^{\vee }$ . Note that for the underlying $\mathbb {W}$ -action on $\mathbf {F}[E]$ (given by (1.8) for $\mathbf {k}\equiv 1$ ), the translations $\tau (\mu )\in W$ are simply acting by q-dilations $x^y\mapsto q^{-\langle \mu ,y\rangle }x^y$ ( $y\in E$ ). Furthermore, for $c^\prime \in C^J$ and $\mathfrak {t}\in T_J$ , we have $\pi _{c^\prime ,\mathfrak {t}}\simeq \pi _{c,q^c}$ for some $c\in C^J$ if $\mathfrak {t}$ takes values in $\{q^d\}_{d\in \mathbb {R}}$ . The new set of parameters that we will assign to the uniform quasi-polynomial representation (1.8) is the tuple $\{\widehat {p}_\alpha \}_{\alpha \in \Phi _0}$ with $\widehat {p}_\alpha : \mathbb {R}\rightarrow \mathbf {F}^\times $ given by $\widehat {p}_\alpha (d):=q_\alpha ^{d/h}$ , where $q_\alpha :=q^{2/\|\alpha \|^2}$ . The link with the original representation parameters of the quasi-polynomial representation is through the formula

$$\begin{align*}q^c=\prod_{\alpha\in\Phi_0^+} \widehat{p}_{\alpha}(\alpha(c))^{\alpha} \end{align*}$$

for the character $q^c\in T$ .

In general, we will consider the group $\mathcal {G}^{\text {amb}}$ of tuples $\mathbf {f}=(f_\alpha )_{\alpha \in \Phi _0}$ consisting of $\mathbf {F}^\times $ -valued functions $f_\alpha $ on $\mathbb {R}$ . Let $\mathcal {G}$ be the subgroup of $\mathcal {G}^{\text {amb}}$ consisting of the tuples $\mathbf {g}=(g_\alpha )_{\alpha \in \Phi _0}$ satisfying

(1.9) $$ \begin{align} g_{v\alpha}(d+\ell)=g_\alpha(d)\qquad (v\in W_0,\,\ell\in\mathbb{Z}), \end{align} $$

$g_\alpha (-d)=g_\alpha (d)^{-1}$ and $g_\alpha (0)=1$ . If we replace (1.9) by the quasi-invariance condition

$$\begin{align*}\widehat{g}_{v\alpha}(d+\ell)=q_\alpha^{\ell/h}\widehat{g}_\alpha(d)\qquad (v\in W_0,\,\ell\in\mathbb{Z}), \end{align*}$$

we obtain a $\mathcal {G}$ -coset in $\mathcal {G}^{\text {amb}}$ , which we will denote by $\widehat {\mathcal {G}}$ .

For a tuple $\mathbf {f}=(f_\alpha )_{\alpha \in \Phi _0}\in \mathcal {G}^{\text {amb}}$ and $y\in E$ , we set

(1.10) $$ \begin{align} \mathfrak{t}_y(\mathbf{f}):=\prod_{\alpha\in\Phi_0^+}f_\alpha(\alpha(y))^\alpha\in T. \end{align} $$

Then $\mathfrak {t}_c(\widehat {\mathbf {g}})\in T_J$ for $c\in C^J$ when $\widehat {\mathbf {g}}\in \widehat {\mathcal {G}}$ .

Theorem 1.5. Let $\widehat {\mathbf {p}}\in \widehat {\mathcal {G}}$ and $\mathbf {g}\in \mathcal {G}$ . The formulas

$$ \begin{align*} \begin{aligned} \pi_{\mathbf{g},\widehat{\mathbf{p}}}(T_i)x^y&:=k_i^{\chi_{\mathbb{Z}}(\alpha_i(y))}g_{\alpha_i}(\alpha_i(y))^{-1}x^{s_iy}+(k_i-k_i^{-1})\nabla_i(x^y) \qquad (1\leq i\leq r),\\ \pi_{\mathbf{g},\widehat{\mathbf{p}}}(T_0)x^y&:=k_0^{\chi_{\mathbb{Z}}(\alpha_0(y))}g_{\varphi}(\varphi(y))\mathfrak{t}_y(\widehat{\mathbf{p}})^{\varphi^{\vee}}x^{s_\varphi y} +(k_0-k_0^{-1})\nabla_0(x^y) \end{aligned} \end{align*} $$

and $\pi _{\mathbf {g},\widehat {\mathbf {p}}}(x^\lambda )x^y:=x^{y+\lambda }$ for $\lambda \in Q^{\vee }$ define a $\mathbb {H}$ -representation on $\mathbf {F}[E]$ , which is isomorphic to $\bigoplus _{c\in \overline {C}_+}\pi _{c,\mathfrak {t}_c(\widehat {\mathbf {p}}\cdot \mathbf {g})}$ .

In the main text, we prove a version of this theorem for the extended double affine Hecke algebra $\mathbb {H}_{\Lambda ,\Lambda ^\prime }$ ; see Theorem 8.19. In that case, the extra twisting parameters are added by replacing $\mathfrak {t}_y(\widehat {\mathbf {p}})$ , viewed as a multiplicative character of $\Lambda ^\prime $ , by

$$\begin{align*}\mathfrak{t}_y(\widehat{\mathbf{p}},\mathfrak{c}):=\mathfrak{t}_y(\widehat{\mathbf{p}})\mathfrak{c}(\text{pr}_{E_{\text{co}}}(y)). \end{align*}$$

Here, $\text {pr}_{E_{\text {co}}}$ is the orthogonal projection of E onto the orthocomplement $E_{\text {co}}$ of $\mathbb {R}\Phi _0^{\vee }$ in E, and $\mathfrak {c}: E_{\text {co}}\rightarrow T_{\Lambda ^\prime ,[1,r]}$ satisfies $\mathfrak {c}(y+\mu )=q^\mu \mathfrak {c}(y)$ for $\mu \in \text {pr}_{E_{\text {co}}}(\Lambda ^\prime )$ .

The isomorphism $\pi _{\mathbf {g},\widehat {\mathbf {p}}}\simeq \bigoplus _{c\in \overline {C}_+}\pi _{c,\mathfrak {t}_c(\widehat {\mathbf {p}}\cdot \mathbf {g})}$ in (the extended version of) Theorem 1.5 will be realised by an explicit $\mathbf {g}$ -dependent linear automorphism of $\mathbf {F}[E]$ , which is diagonalised by the quasi-monomial basis $\{x^y\}_{y\in E}$ of $\mathbf {F}[E]$ . As a consequence, we obtain simultaneous quasi-polynomial eigenfunctions $\mathcal {E}_y(x;\mathbf {g},\widehat {\mathbf {p}})$ of $\pi _{\mathbf {g},\widehat {\mathbf {p}}}(Y^\mu )$ ( $\mu \in Q^{\vee }$ ) by appropriately rescaling the quasi-monomials in the quasi-monomial expansion (1.4) of $E_y^J(x;\mathfrak {t}_c(\widehat {\mathbf {p}}\cdot \mathbf {g}))$ (see Corollary 8.24).

If $L\subset E$ is a $W_0$ -invariant, finitely generated abelian subgroup containing $\Lambda $ , then $\mathcal {P}_L:=\bigoplus _{\lambda \in L}\mathbf {F}x^\lambda $ is a $\mathbb {H}$ -subrepresentation of $\pi _{\mathbf {g},\widehat {\mathbf {p}}}$ containing the quasi-polynomial eigenfunctions $\mathcal {E}_\lambda (x;\mathbf {g},\widehat {\mathbf {p}})$ of degree $\lambda \in L$ . Since $\mathcal {P}_L$ may be viewed as the space of polynomials on the torus $T_L:=\text {Hom}(L,\mathbf {F}^\times )$ , we thus obtain a polynomial representation of $\mathbb {H}$ with $\pi _{\mathbf {g},\widehat {\mathbf {p}}}(T_j)\vert _{\mathcal {P}_L}$ acting by truncated Demazure-Lusztig operators (these operators are still truncated since the roots $\alpha \in \Phi _0$ are allowed to take non-integral values on L). The decomposition of L into W-orbits provides a decomposition of $\pi _{\mathbf {g},\widehat {\mathbf {p}}}(\cdot )\vert _{\mathcal {P}_L}$ as a direct sum of quasi-polynomial representations, with the number of free parameters in $\pi _{\mathbf {g},\widehat {\mathbf {p}}}(\cdot )\vert _{\mathcal {P}_L}$ depending on L. This representation plays an important role in establishing the connection to metaplectic representation theory, which we discuss in the next subsection.

1.5 The metaplectic theory

The metaplectic root system $\Phi _0^m$ and the associated metaplectic double affine Hecke algebra $\mathbb {H}^m$ are defined in terms of a metaplectic datum. A metaplectic datum is a pair $(n,\mathbf {Q})$ consisting of a positive integer n and a rational $W_0$ -invariant quadratic form $\mathbf {Q}$ on $\Lambda $ taking integral values on $Q^{\vee }$ . The associated symmetric bilinear form on $\Lambda $ will be denoted by $\mathbf {B}$ . The metaplectic root system $\Phi _0^m:=\{\alpha ^m\}_{\alpha \in \Phi _0}$ consists of the rescaled roots $\alpha ^m=m(\alpha )^{-1}\alpha $ where

$$\begin{align*}m(\alpha):=\frac{n}{\text{gcd}(n,\mathbf{Q}(\alpha^{\vee}))}. \end{align*}$$

This is a root system with basis $\{\alpha _1^m,\ldots ,\alpha _r^m\}$ , which is isomorphic to either $\Phi _0$ or $\Phi _0^{\vee }$ . We denote by $\vartheta \in \Phi _0^+$ the root such that $\vartheta ^m$ is the long root of $\Phi _0^m$ . The simple affine root for the associated metaplectic affine root system $\Phi ^m$ is $\alpha _0^m=(-\vartheta ^m,1)$ . The co-root lattice $Q^{m\vee }$ of $\Phi _0^m$ is contained in $\Lambda $ , and $\alpha ^m(\Lambda )\subseteq m(\alpha )^{-1}\mathbb {Z}$ for $\alpha \in \Phi _0$ . The metaplectic double affine Hecke algebra $\mathbb {H}^m$ is the double affine Hecke algebra relative to the metaplectic root system $\Phi _0^m$ . It depends on a choice $\mathbf {k}=(k_{a^m})_{a^m\in \Phi ^m}$ of a multiplicity function of $\Phi ^m$ .

To connect the uniform quasi-polynomial representation and the associated quasi-polynomial eigenfunctions with the metaplectic representation theory and the associated metaplectic polynomials from [Reference Sahi, Stokman and Venkateswaran38], we consider the polynomial representation $\pi _{\mathbf {g},\widehat {\mathbf {p}}}(\cdot )\vert _{\mathcal {P}_L}$ with $\Phi _0$ and $\mathbb {H}$ replaced by their metaplectic variants $\Phi _0^m$ and $\mathbb {H}^m$ , with L a lattice $\Lambda $ satisfying (1.6) relative to the original, nonmetaplectic root system $\Phi _0$ , and with base-point $\widehat {\mathbf {p}}$ satisfying $\mathfrak {t}_\lambda (\widehat {\mathbf {p}})=q^\lambda $ for all $\lambda \in L$ . The remaining parameter dependencies are captured by the set $\mathcal {M}_{(n,\mathbf {Q})}$ of tuples $\underline {h}=(h_s(\alpha ))_{s\in \mathbf {Q}(\alpha ^{\vee })\mathbb {Z},\alpha \in \Phi _0}$ consisting of $W_0$ -invariant functions $h_s: \Phi _0\rightarrow \mathbf {F}^\times $ ( $s\in \mathbf {Q}(\alpha ^{\vee })\mathbb {Z}$ ) satisfying the additional requirements that $s\mapsto h_s(\alpha )$ is a $\text {lcm}(n,\mathbf {Q}(\alpha ^{\vee }))$ -periodic function, $h_0(\alpha )=-1$ , and

$$\begin{align*}h_s(\alpha)h_{-s}(\alpha)=k_{\alpha^m}^{-2}\qquad\qquad \forall\, s\in\mathbf{Q}(\alpha^{\vee})\mathbb{Z}\setminus\text{lcm}(n,\mathbf{Q}(\alpha^{\vee}))\mathbb{Z}. \end{align*}$$

The explicit relation with the parameter set $\mathcal {G}$ for the uniform quasi-polynomial representation is described in Lemma 9.2. This specialisation leads to the following metaplectic basic representation of $\mathbb {H}^m$ . Let $r_\ell (s)\in \{0,\ldots ,\ell -1\}$ be the remainder of $s\in \mathbb {Z}$ modulo $\ell \in \mathbb {Z}_{>0}$ , and write $k_j=k_{\alpha _j^m}$ for $0\leq j\leq r$ .

Theorem 1.6. Let $\underline {h}\in \mathcal {M}_{(n,\mathbf {Q})}$ . The formulas

$$ \begin{align*} \pi_\Lambda^m(T_0)x^\lambda&=-k_0h_{\mathbf{B}(\lambda,\vartheta^{\vee})}(\vartheta)q_\vartheta^{m(\vartheta)\vartheta(\lambda)}x^{s_{\vartheta}\lambda}\\ &\quad+ (k_0-k_0^{-1})\left(\frac{1-(q_\vartheta^{m(\vartheta)}x^{-\vartheta^{\vee}})^{(r_{m(\vartheta)}(-\vartheta(\lambda))+\vartheta(\lambda))}}{1-(q_\vartheta^{m(\vartheta)}x^{-\vartheta^{\vee}})^{m(\vartheta)}}\right)x^\lambda,\\ \pi_\Lambda^m(T_i)x^\lambda&=-k_ih_{-\mathbf{B}(\lambda,\alpha_i^{\vee})}(\alpha_i)x^{s_i\lambda}+(k_i-k_i^{-1})\left(\frac{1-x^{(r_{m(\alpha_i)}(\alpha_i(\lambda))-\alpha_i(\lambda))\alpha_i^{\vee}}}{1-x^{m(\alpha_i)\alpha_i^{\vee}}}\right)x^\lambda,\\ \pi_\Lambda^m(x^\mu)x^\lambda&=x^{\lambda+\mu} \end{align*} $$

for $1\leq i\leq r$ , $\mu \in Q^{m\vee }$ and $\lambda \in \Lambda $ define a representation $\pi _\Lambda ^m: \mathbb {H}^m\rightarrow \text {End}(\mathcal {P}_\Lambda )$ .

The metaplectic basic representation $\pi _\Lambda ^m$ naturally extends to the metaplectic extended double affine Hecke algebra; see Theorem 9.5.

At this stage, we can make the link with the (announced) results in [Reference Sahi, Stokman and Venkateswaran38]. The representation $\pi _\Lambda ^m$ , restricted to the metaplectic affine Hecke algebra generated by $\mathcal {P}$ and $T_1,\ldots ,T_r$ is the metaplectic affine Hecke algebra representation from [Reference Sahi, Stokman and Venkateswaran38, Thm. 3.7]. When the multiplicity functions $\mathbf {k}$ and $h_s$ are constant, it recovers, after localisation, Chinta’s and Gunnells’ [Reference Chinta and Gunnells13, Reference Chinta and Gunnells14] $W_0$ -action on a suitable subspace of rational functions on $T_\Lambda $ , which plays an important role in the construction of the local parts of Weyl group multiple Dirichlet series. By X-localisation of the double affine Hecke algebra, Theorem 1.6 gives rise to families of W-actions which generalise the Chinta-Gunnells action; see Corollary 9.16 (compare also with [Reference Patnaik and Puskas35, §3.3] and [Reference Lee and Zhang23] where the Chinta-Gunnells action is extended to an arbitrary Coxeter group). These W-actions actually extend to actions of the double affine Weyl group $\mathbb {W}$ , with the second coroot lattice $Q^{\vee }$ acting by translations of the exponents of the monomials. The localisation procedure can also be applied to the (uniform) quasi-polynomial representations $\pi _{c,\mathfrak {t}}$ and $\pi _{\mathbf {g},\widehat {\mathbf {p}}}$ ; see Theorem 4.9 and Theorem 8.27 for the resulting $\mathbb {W}$ -actions.

Rescaling the quasi-monomials in the expansion (1.4) gives rise to simultaneous eigenfunctions $E_\lambda ^m(x)\in \mathcal {P}_\Lambda $ of the commuting operators $\pi _{\Lambda }^m(Y^\mu )$ ( $\mu \in Q^{m\vee }$ ) satisfying

$$\begin{align*}E_\lambda^m(x)=x^\lambda+\sum_{\lambda^\prime<\lambda}e_{\lambda,\lambda^\prime}^m x^{\lambda^\prime}\qquad (e_{\lambda,\lambda^\prime}^m\in\mathbf{F}) \end{align*}$$

under suitable generic conditions on the metaplectic parameters $\underline {h}\in \mathcal {M}_{(n,\mathbf {Q})}$ ; see Theorem 9.18. An antisymmetric version of the metaplectic polynomial $E_\lambda ^m(x)$ , denoted $E_{\lambda }^{m, -}(x)$ , is obtained by acting by the sign-idempotent of the finite Hecke algebra in $\mathbb {H}^m$ on $E_\lambda ^m(x)$ . For the root datum of type $\text {GL}_{r+1}$ , the representation $\pi _\Lambda ^m$ of the extended metaplectic double affine Hecke algebra recovers the representation introduced in [Reference Sahi, Stokman and Venkateswaran38, Thm. 5.4], and the $E_\lambda ^m(x)$ are the metaplectic polynomials introduced in [Reference Sahi, Stokman and Venkateswaran38, Thm. 5.7].

We consider the $q\rightarrow \infty $ limit of the metaplectic polynomials, for which we need to assume that $\mathbf {F}=\mathbf {K}(q^{\frac {1}{2h}})$ and that the remaining parameters $\mathbf {k}$ and $\underline {h}$ take values in the multiplicative group $\mathbf {K}^\times $ of the field $\mathbf {K}$ of characteristic zero. The Whittaker limit

$$\begin{align*}\overline{E}_\lambda^m(x):=E_\lambda^m(x)\vert_{q^{-\frac{1}{2h}}=0}\qquad (\lambda\in\Lambda) \end{align*}$$

of $E_\lambda ^m(x)$ then defines a Laurent polynomial in $\mathcal {P}_\Lambda $ with coefficients in $\mathbf {K}$ . They are related to metaplectic Iwahori-Whittaker functions [Reference Patnaik and Puskas34] as follows.

Suppose that $\mathbf {K}$ is a non-archimedean local field containing the $n^{\text {th}}$ roots of unity, with the cardinality $\upsilon $ of its residue field congruent $1$ module $2n$ . In [Reference Patnaik and Puskas34], Patnaik and Puskas introduced Iwahori-Whittaker functions for metaplectic covers of reductive groups over $\mathbf {K}$ . They expressed them in terms of certain metaplectic Demazure-Lusztig operators which depend on certain Gauss sums $\mathbf {g}_s$ , as well as on $\upsilon $ ; see [Reference Patnaik and Puskas34, Cor. 5.4]. Comparing this with the analogous formula for $\overline {E}_{\lambda }^{m}(x)$ in terms of truncated Demazure-Lusztig operators (Theorem 6.51), we establish in Subsection 10 the following link between $\overline {E}_\lambda ^m(x)$ and metaplectic Iwahori-Whittaker functions. Let I the linear automorphism of $\mathcal {P}_\Lambda $ satisfying $I(x^\mu )=x^{-\mu }$ for $\mu \in \Lambda $ , let $\overline {E}_\lambda ^{m,-}(x)$ be the Whittaker limit of $E_\lambda ^{m,-}(x)$ , and denote by $\rho ^{\vee }$ the half-sum of positive co-roots.

Theorem 1.7. Suppose that the multiplicity function $\mathbf {k}$ is constantly equal to $\upsilon ^{-2}$ and the multiplicity functions $h_s$ of the metaplectic parameters $\underline {h}$ are constantly equal to $\mathbf {g}_s$ . For $v\in W_0$ and for $\lambda \in \Lambda $ lying in the closure of the positive Weyl chamber,

$$\begin{align*}x^{-\rho^{\vee}}I\big(\overline{E}^m_{-v(\lambda+\rho^{\vee})}(x)\big)\quad \mbox{ and }\quad x^{-\rho^{\vee}}I\big(\overline{E}^{m,-}_{-\lambda-\rho^{\vee}}(x)\big) \end{align*}$$

are constant multiples of metaplectic Iwahori-Whittaker functions and metaplectic spherical Whittaker functions, respectively.

In Subsection 10, we also recover McNamara’s [Reference McNamara30] Casselman-Shalika type formula for the metaplectic spherical Whittaker function as a consequence of the explicit expansion formula of $E_y^{J,-}(x;\mathfrak {t})$ as a linear combination of the quasi-polynomial eigenfunctions $E_{y^\prime }^J(x;\mathfrak {t})$ ( $y^\prime \in W_0 y$ ), derived earlier in Subsection 6.6.

1.6 Structure of the paper

In Section 2, we collect basic properties of the affine Weyl group W and the double affine Hecke algebra $\mathbb {H}$ .

In Section 3, we introduce the class of Y-parabolically induced cyclic $\mathbb {H}$ -modules $\mathbb {M}_t^J$ .

In Section 4, we introduce the quasi-polynomial representation $\pi _{c,\mathfrak {t}}$ . We state the main theorem that the quasi-polynomial representation is well defined and isomorphic to $\mathbb {M}_{\mathfrak {s}_J\mathfrak {t}}^J$ in Subsection 4.3. We discuss the dependence of $\pi _{c,\mathfrak {t}}$ on $c\in C^J$ and J in Subsection 4.4, and localise it to obtain a nontrivial W-action on quasi-rational functions in Subsection 4.5.

Section 5 is devoted to the proof of the main theorem. As part of the proof, we establish in Subsection 5.5 triangularity properties of the commuting operators $\pi _{c,\mathfrak {t}}(Y^\mu )$ ( $\mu \in Q^{\vee }$ ) with respect to the basis of quasi-monomials, ordered by the parabolic Bruhat order on their exponents.

In Section 6, we define the quasi-polynomial generalisations $E_y^J(x;\mathfrak {t})$ of the monic Macdonald polynomials for adjoint root data. We show how they can be explicitly created from the quasi-monomial $x^c$ by acting by Y-intertwiners in Subsection 6.3. Their dependence on $J\subsetneq \{0,\ldots ,r\}$ is discussed in Subsection 6.4. Pseudo-duality for the normalised versions $P_y^J(x;\mathfrak {t})$ of the quasi-polynomials $E_y^J(x;\mathfrak {t})$ is derived in Subsection 6.5, quasi-polynomial generalisations $E_y^{J,\pm }(x;\mathfrak {t})$ of the (anti)symmetric Macdonald polynomials are discussed in Subsection 6.6, and orthogonality relations and quadratic norm formulas are obtained in Subsection 6.7. The Whittaker limit $q\rightarrow \infty $ of $E_y^J(x;\mathfrak {t})$ and $E_y^{J,-}(x;\mathfrak {t})$ is considered in Subsection 6.8.

In Section 7, we generalise the results of Sections 4-6 to reductive root data (including the $\text {GL}_{r+1}$ root datum as a special case). We reinterpret the role of the extra parameters in terms of twisting by automorphisms of the extended double affine Hecke algebra in Subsection 7.4. In Subsection 7.5, we introduce the analogs of the quasi-polynomial eigenfunctions $E_y^J(x;\mathfrak {t})$ for reductive root datum. We furthermore show that, under suitable generic conditions on the parameters, they are independent of the twist parameters and reduce to the quasi-polynomial eigenfunctions for the underlying adjoint root datum.

In Section 8, we introduce the uniform quasi-polynomial representation (Subsection 8.2), the associated uniform quasi-polynomial eigenfunctions (Subsection 8.3), and we explicitly relate the latter to the quasi-polynomial eigenfunctions $E_y^J(x;\mathfrak {t})$ , using an automorphism that rescales the quasi-monomials.

In Section 9, we attach to a metaplectic datum a polynomial representation of the metaplectic double affine Hecke algebra, called the metaplectic basic representation (Subsection 9.2), and we show that it is isomorphic to a suitable sub-representation of the uniform quasi-polynomial representation. We obtain an affine version of the Chinta-Gunnells’ $W_0$ -action on rational functions by X-localising the metaplectic basic representation in Subsection 9.3. We introduce metaplectic polynomials, which are simultaneous eigenfunctions for the action of $\mathcal {P}_Y$ under the metaplectic basic representation, and relate them to the quasi-polynomial eigenfunctions $E_y^J(x;\mathfrak {t})$ in Subsection 9.4.

In Section 10, we show the existence of the Whittaker limit of the metaplectic polynomial. In the context of representation theory of metaplectic covers of reductive groups over non-archimedean local fields, we prove that the Whittaker limit of the metaplectic polynomial is a metaplectic Iwahori-Whittaker function up to some elementary twists. We rederive McNamara’s [Reference McNamara30] Casselman-Shalika type formula for the metaplectic spherical Whittaker function from the Whittaker limit of an explicit expansion formula for the anti-symmetrised version of the quasi-polynomial eigenfunction derived in Subsection 6.6.

1.7 Conventions

We take $\mathbf {F}$ to be a field of characteristic zero, and we fix $q\in \mathbf {F}^\times $ not a root of unity (it contains the case of formal variable q by taking $\mathbf {F}=\mathbf {K}(q)$ for some field $\mathbf {K}$ of characteristic zero). For a commutative ring R and an abelian group A, we write $R[A]= \bigoplus _{y\in A}Rx^y$ for the group algebra of A over R (i.e., $x^yx^{y^\prime }=x^{y+y^\prime }$ for $y,y^\prime \in A$ and $x^0=1$ ). We will often apply this to the case that A is a Euclidean space.

For groups G and H, we write $\text {Hom}(G,H)$ for the class of group homomorphisms $G\rightarrow H$ . For a finitely generated free abelian group $\Lambda $ , we write $T_\Lambda :=\text {Hom}(\Lambda ,\mathbf {F}^\times )$ for the $\mathbf {F}$ -torus of multiplicative characters $\Lambda \rightarrow \mathbf {F}^\times $ , and $\mathcal {P}_\Lambda =\mathbf {F}[\Lambda ]$ for the group algebra of $\Lambda $ over $\mathbf {F}$ . Alternatively, it is the algebra of regular $\mathbf {F}$ -valued functions on $T_\Lambda $ , where the monomial $x^\lambda $ maps $t\in T_\Lambda $ to its value $t^\lambda $ at $\lambda \in \Lambda $ . Finally, for $\ell ,m\in \mathbb {Z}$ with $\ell \leq m$ , we write $[\ell ,m]:=\{\ell ,\ell +1,\ldots ,m\}$ .