This paper contains two main results concerning a somewhat mysterious action of the Weyl group of a reductive Lie group on the algebra of rational functions on its torus. This action was first introduced in type A by Kazhdan and Patterson [24], and in full generality by Chinta and Gunnells [15,16], who used it to obtain formulas for the local parts ( p-parts) of Weyl group multiple Dirichlet series. The action involves an integer n and parameters g 1 , . . . , g n-1 , which in the application are specialized to certain Gauss sums; however it remains a group action even without this specialization. Chinta and Gunnells verified this fact through a computer check and they asked for a conceptual proof. Our first main result provides such a proof in complete generality. The key role in the proof is played by a certain representation of the affine Hecke algebra that we construct in Theorem 3.7 below, and which we refer to as the metaplectic representation.
There is a striking analogy between the Chinta-Gunnells setting and the theory of Macdonald polynomials [12,29,31]. The latter are a family of orthogonal polynomials on the torus that depend on two or three "root-length" parameters, and which generalize many important polynomials in representation theory and algebraic combinatorics, including spherical functions for real and p-adic groups. We show that there is much more to this analogy. Our second main result is the construction of a family of polynomials that we refer to as metaplectic polynomials. These depend on the root-length parameters as well as the g 1 , . . . , g n-1 , and are a common generalization of nonsymmetric Macdonald polynomials [11,28] and of the p-parts of Weyl group multiple Dirichlet series. A key point in our construction is extending the metaplectic representation from the affine Hecke algebra to the double affine Hecke algebra.
In the present paper we introduce, without proofs, the metaplectic polynomials in type A, where many of the essential ideas already appear. This is the setting of [24] and of Macdonald's book on symmetric functions [30], which is of considerable independent interest in algebraic combinatorics. The consideration of the metaplectic polynomials for arbitrary type requires some additional ideas. This will be presented in a forthcoming paper [36], which will also include the detailed proofs.
We recall now briefly the Chinta-Gunnells Weyl group action, referring the reader to [4,5,9,[14][15][16] and especially the survey [8] for the connection to Weyl group multiple Dirichlet series. Let W be the Weyl group of an irreducible root system , with Coxeter generators {s i } r i=1 corresponding to a choice of simple roots {α i } r i=1 . Let P be the weight lattice of . The Weyl group canonically acts on the fraction field C(P) of the group algebra C[P] by field automorphisms. Chinta and Gunnells have constructed a deformation of this action, which depends on the choice of a W -invariant quadratic form Q : P → Q taking integer values on the root lattice Q of , a natural number n, and on parameters v, g 0 , . . . , g n-1 satisfying g 0 = -1, g j g n-j = v -1 , j = 1, . . . , n -1.
Let 0 ≤ r m ( j) ≤ m -1 denote the remainder on dividing j by the natural number m, and define g j for arbitrary j ∈ Z by setting g j = g r n ( j) , let B (λ, μ) = Q (λ + μ)-Q (λ)-Q (μ) be the bilinear form associated to Q, and put m (α) = n/ gcd (n, Q (α)). It defines a new root system m := {m(α)α} α∈ , which is either isomorphic to or to ∨ . The weight lattice P m ⊆ P of m is P m = {λ ∈ P | B(λ, α ≡ 0 mod n ∀ α ∈ } (see Lemma 2.2). Then the Chinta-Gunnells action σ i = σ (s i ) of the simple reflection s i ∈ W on C(P) is given by the formula
for f ∈ C(P m ) and λ ∈ P.
It is non-trivial to show that the formula (1.1) defines a representation of W . The main issue is to verify that the braid relations are satisfied. Although this reduces to a rank 2 computation, the calculations become rather formidable, and in [16] the details are only presented for A 2 . Trying to find a natural interpretation of this representation was one of the main motivations for our work.
Chinta and Gunnells [16] employed the action (1.1) to give an explicit construction of the "local" parts of certain Weyl group multiple Dirichlet series, and to establish thus the analytic continuation and functional equations for these series. In this situation, the g i are n-th order Gauss sums for the local field, and v = p -1 with p the cardinality of the residue field. Subsequently, Chinta-Offen [18] for type A, and McNamara [32] in general, showed that these local parts are essentially Whittaker functions for principal series of certain n-fold "metaplectic" covers of quasi-split reductive groups. The resulting explicit expression for the Whittaker function in terms of the action (1.1) is the metaplectic generalization of the Casselman-Shalika formula. This result is in line with the fact that multiple Dirichlet series should themselves be Whittaker coefficients attached to metaplectic Eisenstein series [6,9].
Still more recently, Chinta-Gunnells-Puskas [17] have shown that the W -action (1.1) gives rise to a Cherednik [12] type Demazure-Lusztig action of the Hecke algebra of W . It leads to an expression of the metaplectic Whittaker functions in terms of metaplectic Demazure-Lusztig operators. Their work was partly motivated by Brubaker-Bump-Licata [7], who gave formulas for (nonmetaplectic) Iwahori-Whittaker functions in terms of Hecke operators and nonsymmetric Macdonald polynomials. The recent work of Patnaik-Puskas [33] uses the Chinta-Gunnells-Puskas Hecke algebra action to study metaplectic Iwahori-Whittaker functions. It leads to a conceptual proof [33,App. B] that (1.1) defines a representation of W when the g i are n-th order Gauss sums for a local field, and v = p -1 with p the cardinality of the corresponding residue field.
In Sects. 3 and 4, we give a uniform construction of a Weyl group representation (Theorem 3.21) and an associated Hecke algebra representation (Theorem 4.2) that generalize the Chinta-Gunnells [16] and Chinta-Gunnells-Puskas [17] representations, respectively. Our construction does not involve case-by-case considerations, and it yields a representation for the generic Hecke algebra H (k), which has independent Hecke parameters for each root length in . Our method also allows us to incorporate extra freedom in the definition of g i by allowing them to depend on the root length (see Definition 3.5 of the representation parameters). The Chinta-Gunnells and Chinta-Gunnells-Puskas representations are recovered in the equal Hecke and representation parameter case of our constructions.
Our starting point was the observation that (1.1) has many features in common with formulas obtained by the process of "Baxterization" [12]. The key idea behind this process is that the group algebra of the affine Weyl group and the affine Hecke algebra become isomorphic after a suitable localization, which allows one to relate certain representations of the two algebras. This inspired our search for a natural representation of the affine Hecke algebra whose associated localized affine Weyl group representation produces (1.1) for its W -action. Its first form can be recovered from the Chinta-Gunnells-Puskas Hecke algebra action as follows.
Note that the Chinta-Gunnells W -action
with ∇ i the following metaplectic version of the divided-difference operator
But now we want to have an a priori proof that (1.2) defines a H (k)-action on C[P] and conclude from it that (1.1) defines a W -action on C(P) via the localization technique.
Although the formulas (1.2) are much simpler than (1.1), a direct case-by-case check that it defines a H (k)-representation will be close to being as cumbersome as for the Chinta-Gunnells action. Our first result is to circumvent the case-by-case check by proving that π is isomorphic to a quotient of the induced module H m (k) ⊗ H (k) V C for an appropriate H (k)-representation V C . This isomorphism in addition allows us to generalize π and the Chinta-Gunnells Weyl group action to the context of generic affine Hecke algebras.
The H (k)-representation V C is defined as follows. Let V = λ∈P Cv λ be the complex vector space with basis the weight lattice P. It has a natural left H (k)-module structure reducing to the canonical C[W ]-module structure when k = 1 (see Lemma 3.1). We call V the reflection representation of H (k). For each W -invariant subset D ⊆ P, the subspace
The appropriate choice of W -invariant subset C of P in the above realization of π now turns out to be
Note that C contains a complete set of coset representatives of P/P m .
The following trivial example is instructive to get a feeling for what is going on. Suppose that m(α) = 1 for all α ∈ . Then P m = P and ∇ i is the standard divideddifference operator on C[P]. In this case it is well known that (1.2) is equivalent to the induced module H m (k) ⊗ H (k) V {0} by the Bernstein-Zelevinsky [27] presentation of H m (k). The W -subset C in this case is oversized, with C \ {0} being the set of nonzero minuscule weights in P.
In Sect. 5 we construct the metaplectic polynomials in type A. The extension to arbitrary types will be treated in the forthcoming paper [36]. The GL r double affine Hecke algebra H (m) has generators T 0 , . . . , T r -1 , ω ±1 , x ±m 1 , . . . , x ±m r , with T 0 , . . . , T r -1 , ω ±1 . Coxeter type generators of a copy of the GL r affine Hecke algebra in H (m) (ω is the generator of the abelian group of group elements of length zero), and T 1 , . . . , T r -1 , x ±m 1 , . . . , x ±m r Bernstein-Zelevinsky type generators of the second copy of the GL r affine Hecke algebra in H (m) (the x ±m j ( j = 1, . . . , r ) are generating its commutative subalgebra). The metaplectic representation of the second copy of the GL r affine Hecke algebra is acting on Laurent polynomials in x ±1 1 , . . . , x ±1 r , where T i for 1 ≤ i < r act by (1.2) and x ν (ν ∈ mZ r ) act by multiplication. It extends to a representation π of H (m) , with ω acting as a twisted-cyclic permutation of the variables and T 0 by an appropriate affine version of the metaplectic Demazure-Lusztig operator (see Theorem 5.4). The representation π is a metaplectic generalization of Cherednik's basic representation [12,31], which we call the metaplectic basic representation.
The GL r affine Hecke algebra generated by T 0 , . . . , T r -1 , ω ±1 in its Bernstein-Zelevinsky presentation contains an abelian subalgebra generated by elements Y ±m i (i = 1, . . . , r ). We define the metaplectic polynomials E (m) μ (μ ∈ Z r ) in Theorem 5.7 as the simultaneous eigenfunctions of π(Y m i ) (i = 1, . . . , r ). It depends, besides the standard Macdonald parameters, on the additional representation parameters g j . The subfamily indexed by mZ r recovers the nonsymmetric Macdonald polynomials in the variables x m 1 , . . . , x m r (see Remark 5.10). At the end of Sect. 5 we provide examples of G L 3 -metaplectic polynomials, highlighting some important phenomena. In a followup paper [36] other important properties, such as triangularity and orthogonality will be established in the context of arbitrary root systems.
We now briefly discuss the content of the paper. We introduce in Sect. 2 the appropriate metaplectic structures on the root systems and affine Weyl and Hecke algebras. Section 3 is devoted to the metaplectic representation theory of the affine Weyl groups and generic affine Hecke algebras. We introduce the reflection representation in Sect. In Sect. 4 we form the associated metaplectic Demazure-Lusztig operators and generalize some of the results from [17] to the setting of unequal Hecke and representation parameters. We also simplify some of the proofs from that paper by using the standard symmetrizer and antisymmetrizer elements in the Hecke algebra. This allows us to define a natural class of "Whittaker functions" for generic Hecke algebras. It is natural to ask whether these more general functions arise as actual matrix coefficients for some class of representations of p-adic groups. This question is of particular interest since generic Hecke algebras have begun to play an increasing role in the study of the Bernstein components within the categories of smooth representations of p-adic groups, see, e.g., [10,21] and references therein.
In Sect. 5, we construct the metaplectic polynomials in type A. We begin by setting up the notation and modifications specific to the G L r case. The double affine Hecke algebra H m is presented in Sect. 5.2, and the metaplectic basic representation in Sect. 5.3. The characterization of the metaplectic polynomials as eigenfunctions of the metaplectic operators π(Y ν ) (ν ∈ mZ r ) may be found in Sect. 5.4. We also discuss the dependence on parameters, showing that we do not lose any generality by taking the quadratic form Q to satisfy Q(α) = 1 for α a root. Finally, in the "Appendix", we provide a list of metaplectic polynomials for r = 3 and 1 ≤ m ≤ 5.
Let us conclude with remarking that the localization procedure we use in this paper is instrumental in Cherednik's construction of quantum affine Knizhnik-Zamolodchikov equations attached to affine Hecke algebra modules. Closely related to it is the role of the localization procedure for type A in the context of integrable vertex models with U q ( sl n )-symmetry, in the special cases that the associated braid group action descends to an affine Hecke algebra action, in which case the localization procedure is often referred to as Baxterization (see, e.g., [12,37] and references therein). This is exactly the context in which the metaplectic Whittaker function can be realized as a partition function, the corresponding integrable model being "metaplectic ice", see [1][2][3]. It is an intriguing open question whether there is a conceptual connection with the current interpretation of the Chinta-Gunnells action through localization.
Let E be an Euclidean space with scalar product (•, •) and norm • . Let ⊂ E be an irreducible reduced root system, and W ⊂ O(E) its Weyl group. The reflection in α ∈ is denoted by s α ∈ W , and its co-root is α ∨ := 2α/ α 2 .
Fix a base {α 1 , . . . , α r } of . Let + be the corresponding set of positive roots and write s i := s α i for i = 1, . . . , r . Let
be the weight lattice of with i ∈ E the fundamental weights, defined by
Zα i be the root lattice of .
In the theory of metaplectic Whittaker functions, a new root system m is attached to the metaplectic covering data of the reductive group over the non-archimedean local field, cf. [16,17] and references therein. We recall in this subsection this additional metaplectic data on the root system. Fix a W -invariant quadratic form Q : P → Q which takes integral values on Q and write B : P × P → Q for the associated symmetric bilinear pairing
Then Q(•) = κ 2 • 2 for some κ ∈ R × , and hence B(λ, μ) = κ(λ, μ) for all λ, μ ∈ P. In particular, for all λ ∈ P and α ∈ ,
Let n ∈ Z >0 and define
Set m := {α m := m(α)α} α∈ ⊂ E. Then m is a root system. In fact, if m is constant then m is isomorphic to , while if m is nonconstant then m is isomorphic to the co-root system ∨ = {α ∨ } α∈ (this follows from the definition of m(α) and the fact that Q(•) = κ 2 • 2 ). In particular, {α m 1 , . . . , α m r } is a base of m and W is the Weyl group of m .
Write Q m for the root lattice of m and P m for the weight lattice of m . Since (α m ) ∨ = m(α) -1 α ∨ for α ∈ , we have
Lemma 2.1 (a) For α ∈ and λ ∈ P we have
(b) For α ∈ and λ ∈ P we have
Proof. (a) For λ ∈ P and α ∈ we have
(b) For λ ∈ P and α ∈ we have
Proof The first equality follows from the fact that (α m ) ∨ = m(α) -1 α ∨ for α ∈ .
The second equality follows immediately from part (b) of Lemma 2.1.
We start with the definition of the finite Hecke algebra. Let k : → C × be a W -invariant function and write k α for the value of k at α ∈ . Set k i := k α i for i = 1, . . . , r .
The Hecke algebra H (k) associated to the root system is the unital associative algebra over C generated by T 1 , . . . , T r with defining relations
each side, with m i j the order of s i s j in W ).
Define the length of w ∈ W by
The T w (w ∈ W ) are well defined and form a linear basis of H (k).
We now introduce the extended affine Hecke algebra H m (k) associated to the finite root system m through its Bernstein-Zelevinsky presentation (see [27]). It contains as subalgebras the finite Hecke algebra H (k) and the group algebra C[P m ] of the weight lattice P m of m . We write the canonical basis elements of C[P m ] in exponential form x μ (μ ∈ P m ), so that x μ x ν = x μ+ν and x 0 = 1. The Weyl group W acts naturally on C[P m ] by algebra automorphisms.
For 1 ≤ i ≤ r there exists a well defined linear operator
It is called the divided difference operator associated to the simple root α m i .
The extended affine Hecke algebra H m (k) is the unital associative algebra over C generated by the algebras H (k) and C[P m ], with additional defining relations
It is well known that the multiplication map defines a linear isomorphism
3 Metaplectic representations
Set
It inherits a left W -action by the linear extension of the canonical action of W on P.
For a W -invariant subset D ⊂ P we write
the W -orbit of λ in P and P + ⊂ P the cone of dominant weights of with respect to the base {α 1 , . . . , α r }. In this subsection we deform the W -action on V D and V to a H (k)-action. Fix λ ∈ P + . The stabilizer subgroup
It is generated by the simple reflections s i (i ∈ I λ ), with I λ the index subset
C as W -modules, with C regarded as the trivial W λmodule. This description leads to a natural Hecke deformation of the W -action on V O λ as follows.
Let W λ be the minimal coset representatives of W /W λ , which can be characterized by
satisfies the braid relations, and hence defines a one-dimensional H λ (k)-module, which we denote C λ . Consider now the linear isomorphism
can be explicitly described as follows.
Proof Write μ = wλ with λ ∈ P + and w ∈ W λ . We claim that
In this case we have s i w ∈ W λ . Since each w ∈ W λ satisfies exactly one of these three conditions, it suffices to prove the ⇐'s.
If s i wλ = wλ then s i w would be a representative of wW λ of smaller length than w, which is absurd. Hence s i wλ = wλ, and consequently (μ, α ∨ i ) < 0. If s i w / ∈ W λ then the minimal length representative w ∈ W λ of the coset s i wW λ has length strictly smaller than (s i w) = (w) -1. But then wW λ contains an element of length strictly smaller than (w), which is absurd. Hence s i w ∈ W λ .
It is now easy to conclude the proof of the lemma: Case (1) (s i w) = (w) + 1 and s i w ∈ W λ . Then
in H (k), and hence
For s ∈ Z >0 and t ∈ Z let r s (t) ∈ {0, . . . , s -1} be the remainder of t modulo s. Define q, r : P → P by
Lemma 3.2 q(P) ⊆ P m .
Proof. For i = 1, . . . , r and λ ∈ P we have
Let C[P] = span{x λ } λ∈P be the group algebra of the weight lattice P. The Weyl group W acts naturally on C[P] by algebra automorphisms.
Note that the divided difference operator ∇ m i featuring in the Bernstein-Zelevinsky cross relations (2.2) of the extended affine Hecke algebra H (k) satisfies
for i = 1, . . . , r .
For i = 1, . . . , r there exists a unique linear map
for λ ∈ P. Furthermore,
is a well defined linear operator by the previous lemma. In fact,
The second statement follows from the observation that
Remark 3.4 Note that the action of ∇ i can alternatively be described by
Write m = m sh ∪ m lg for the division of m into short and long roots, with the convention m = m lg if all roots have the same length. Write size : m → {sh, lg} for the function on m satisfying size(α) = sh iff α ∈ m sh . Write k sh and k lg for the value of k on m sh and m lg respectively. Definition 3.5 (Representation parameters) Let g j (y) ∈ C × for j ∈ Z and y ∈ {sh, lg} be parameters satisfying the following conditions: 6 The special case where g j (y) = g j , i.e., the parameters do not depend on root length, was considered in [15,16] and motivated the generalization above. In the applications considered in those papers, the g i are taken to be certain Gauss sums.
Write λ for the class of λ ∈ P in the finite abelian quotient group P/P m . By Lemma 2.2,
is a well defined function p i :
The following theorem is the main result of this subsection.
In particular for m ≡ 1 (which happens for instance when n = 1), the representation π itself is isomorphic to
Note that is automatically W -invariant. The lattice 0 := ∩ P m then satisfies Q m ⊆ 0 ⊆ P m , and
The remainder of this subsection is devoted to the proof of Theorem 3.7. The strategy is to realize the H m (k)-module (π, C[P]) as a quotient of the induced H m (k)-module
The elements
form a linear basis of N C and, by the Bernstein-Zelevinsky commutation relations (2.2), the H m (k)-action on N C is explicitly given by
Note that the group algebra C[P] := span{x λ } λ∈P is a free left C[P m ]-module via the action
for the morphism of C[P m ]-modules satisfying
We fix from now on the W -invariant subset C ⊆ P to be
Proof. We need to show that ψ c C is surjective. Consider the action of W m = W P m on P and E by reflections and translations. Since C is W -invariant it suffices to show that each W m -orbit in P intersects C. We prove the stronger statement that each W Q m -orbit in P intersects C ∩ P + in exactly one point.
Write
for the closure of the fundamental Weyl chamber of E with respect to + . Let θ m ∈ m+ be the highest short root with respect to the base {α m 1 , . . . , α m r } of m . Then θ m∨ ∈ m∨+ is the highest root of m∨ .
By [22, §4.3] each W Q m -orbit in E intersects the fundamental alcove
in exactly one point. Hence each W Q m -orbit in P intersects A o ∩ P in exactly one point. Now note that
The map ψ c C gives rise to an isomorphism
of C[P m ]-modules by Lemma 3.10. We now show how to fine-tune the normalizing factor c so that the kernel ker(
We start with deriving some elementary properties of the metaplectic divided difference operators ∇ i (i = 1, . . . , r ). Lemma 3.11 Let i ∈ {1, . . . , r }. (i) For λ ∈ P and ν ∈ P m we have
(ii) For λ ∈ P and ν ∈ P m we have
Proof (i) This follows by a direct computation.
(ii) Note that
The following lemma will play an important role in finding the proper choice of normalizing factor c. Lemma 3.12 For ν ∈ P m , λ ∈ C and i = 1, . . . , r we have
with d i : C → C × given by
Proof By a direct computation using (3.5), we have
for λ ∈ C and ν ∈ P m . We analyze the right hand side using Lemma 3.1. We now consider four cases. Case 1: -m(α i ) ≤ (λ, α ∨ i ) < 0. Then
Substituting into (3.9) and using Lemma 3.11 we get the desired formula
Case 2: (λ, α ∨ i ) = 0. Now we have
Substituting into (3.9) and using Lemma 3.11 we now get the desired formula λ+ν) .
Substitution into (3.9) and using Lemma 3.11 gives the desired formula
hence substitution into (3.9) and using Lemma 3.11 gives
as desired.
We now continue with the proof of Theorem 3.7. Define parameters h j (y) ∈ C × for j ∈ Z and y ∈ {sh, lg} by
Equations (2.1), (3.8) and Lemma 2.1b one verifies that for i = 1, . . . , r and λ ∈ C,
Rewriting in terms of the representation parameters g j (y) and using Lemma 2.1(b) we get
for i = 1, . . . , r and λ ∈ C, with p i (λ) given by (3.3). Now let S i : C[P] → C[P] be the linear map defined by
then Lemma 3.12 and (3.12) show that for i = 1, . . . , r and λ ∈ C, ν ∈ P m , , is explicitly given by (3.4). This completes the proof of Theorem 3.7.
In subsequent sections, we will work with some conjugations of π , so the following lemma will be useful.
Proof Since π(x ν ) for ν ∈ 0 commutes with multiplication by x μ , we need only check that x -μ π(T i )x μ preserves C[ ] for 1 ≤ i ≤ r . Let λ ∈ . By Theorem 3.7, we have
We have
since
For the other term, by (3.1) and Lemma 3.2, we have
The defining relations of H m loc (k) with respect to the decomposition (3.14) are captured by the extended cross relations If the multiplicity function k is identically equal to one then H m loc (k) is isomorphic to the semi-direct product algebra
with algebra structure given by (v ⊗ f )(w ⊗ g) := vw ⊗ (w -1 f )g for v, w ∈ W and f , g ∈ C(P m ). We write gw for the element
We write c i := c α i (i = 1, . . . , r ) for the c-functions at the simple roots. Note that w(c α ) = c wα for w ∈ W and α ∈ . By [23] we have the following result.
Theorem 3.14 There exists a unique algebra isomorphism
given by ϕ( f ) = f for f ∈ C(P m ) and
The ϕ(s i ) are the so-called normalized intertwiners of the extended affine Hecke algebra H m (k) (see [23] and, e.g., [12, §3.3.3]). They play an instrumental role in the representation theory of H m (k).
Note that for i = 1, . . . , r we have
in W C(P m ), which are the Demazure-Lusztig operators [27].
with representation map ρ loc : W C(P m ) → End(M loc ) defined by
and m ∈ M (the map is well defined by the Bernstein-Zelevinsky presentation of H m loc (k)). Remark 3.17 Identifying M as subspace of M loc by the linear embedding
Remark 3.18 A Bethe integrable system with extended affine Hecke algebra symmetry is a H m (k)-module V endowed with the integrable structure obtained from the action of the associated dual intertwiners on C(P m ) ⊗ V . The integrable structure is thus encoded by solutions of (braid versions of) generalized quantum Yang-Baxter equations with spectral parameter. In the literature on integrable systems one sometimes says that the integrable structure arises from Baxterizing the affine Hecke algebra module structure on the quantum state space. See e.g. [37] for an example involving the Heisenberg XXZ spin-1 2 chain. The intertwiners are also instrumental in the construction of the quantum affine KZ equations, see, e.g., [12, §1.3.2].
Recall the metaplectic affine Hecke algebra representation (π, C[P]) from Theorem 3.7. In the following proposition we explicitly describe (π loc , C[P] loc ).
(iii) The π loc -action of W C(P m ) on C(P) (identifying C[P] loc with C(P) using the linear isomorphism from (ii)) is explicitly given by
for f , g ∈ C(P m ), λ ∈ P and i = 1, . . . , r (recall that p i (λ) is given by (3.3)).
Proof (i) Let G be the group of characters of the finite abelian group P/P m . It acts by field automorphisms on C(P) by
Decomposing C(P) in G-isotypical components yields
(ii) Using (3.14) we get
with the last isomorphism mapping f ⊗ C[P m ] g to f g for f ∈ C(P m ) and g ∈ C[P]. This is well defined and an isomorphism due to the second formula of (3.4) and due to part (i) of the proposition. The result now immediately follows. (iii) For f , g ∈ C(P m ) and λ ∈ P we have
this establishes the second formula. For the first formula it then suffices to prove that
for i = 1, . . . , r and λ ∈ P. By the first formula of (3.4) we have
Substituting the definition of the c-function c i (see (3.15)) gives
Simplifying the expression gives (3.18).
Since q(0) = 0 and p i (0) = k i we have π loc (s i )1 = 1. Hence C(P m ) is a π loc -submodule of C(P) with the W C(P m )-action reducing to the standard one,
Recall the definition of the representation parameters g j (y) ( j ∈ Z, y ∈ {sh, lg}), see Definition 3.5. We conjugate the π loc -action by a certain factor, in order to line it up with the Weyl group action of Chinta-Gunnells [15,16].
The following formulas turn
for f , g ∈ C(P m ), λ ∈ P and i = 1, . . . , r.
Proof Write ρ := 1 2 α∈ + α and ρ m := 1 2 α∈ + α m for the half sum of positive roots of and m respectively. Then s i (ρ) = ρ -α i and s i (ρ m ) = ρ m -α m i , in particular ρ = r i=1 i ∈ P and
Consider now the action of W C(P m ) on C(P) defined by
Then σ (g) f = g f for g ∈ C(P m ) and f ∈ C(P), and
Furthermore,
Substituting these two formulas in (3.21) gives the desired result.
As in Remark 3.8(ii), fix a lattice ⊆ E satisfying Q ⊆ ⊆ P and set 0 := ∩ P m . Then Q m ⊆ 0 ⊆ P m and recall that 0 can alternatively be described as
which places us directly in the context of [16]. Note that and 0 are automatically W -stable. In particular the subalgebra of W C(P m ) generated by W and C( 0 ) is isomorphic to the semi-direct product algebra W C( 0 ). Let C( 0 ) and C( ) be the subfields of C(P) generated by x ν (ν ∈ 0 ) and x λ (λ ∈ ) respectively. Similarly to Proposition 3.19(i) we have the decomposition
Then C( ) ⊆ C(P) is a W C( 0 )-submodule with respect to the action σ . Writing
for the resulting representation map, we get Corollary 3. 22 In the setup as above, the representation map σ is explicitly given by
for f , g ∈ C( 0 ), λ ∈ and i = 1, . . . , r.
Consider the special case that k : m → C × is constant and the representation parameters g j (y) satisfy g j (sh) = g j (lg) for all j ∈ Z. We call this the equal Hecke and representation parameter case. Then σ is exactly the Chinta-Gunnells [15,16] Weyl group action. This is immediately apparent by comparing (3.22) with [17, (7)] (the parameter v in [17] corresponds to k 2 ). Note that our technique gives an independent and uniform proof that the formulas of Chinta-Gunnells do indeed give an action of the Weyl group.
Note that σ reduces at n = 1 to the standard W -action. However, it is in fact not the standard action on C(P m ), due to the fact that we have conjugated π loc by x ρ-ρ m (compare with Remark 3.20).
Set (w) := + ∩ w -1 -(w ∈ W ) and let w 0 ∈ W be the longest Weyl group element. Definition 3.25 For λ ∈ P + define W λ ∈ C(P) by
In the equal Hecke and representation parameter case, McNamara's [32,Thm. 15.2] metaplectic Casselman-Shalika formula relates W λ to the spherical Whittaker function of metaplectic covers of unramified reductive groups over local fields, see also [17,Thm. 16]. It is a natural open problem what the corresponding representation theoretic interpretation is of W λ in the unequal Hecke and/or representation parameter case.
In the following section we will obtain in Theorem 4.9 an expression of W λ in terms of metaplectic analogues of Demazure-Lusztig operators, generalizing [17,Thm. 16].
In the previous section we used the localization isomorphism ϕ : W C(P m ) ∼ -→ H m loc (k) to obtain the metaplectic Weyl group representation σ from the metaplectic affine Hecke algebra representation π . In this section we use the localization isomorphism to turn the metaplectic Weyl group representation σ into a localized affine Hecke algebra representation involving metaplectic Demazure-Lusztig type operators. This leads to a generalization of some of the results in [17, §3] to unequal Hecke and representation parameters, and simplifies some of the proofs in [17, §3].
Define the algebra map
In particular, the restriction of τ to H m (k) preserves C[P], and the restriction of
Proof The formula follows from (3.20), Proposition 3.19(ii) and Remark 3.17, and then the statements about restrictions follow from Theorem 3.7 and Lemma 3.13.
for f , g ∈ C(P m ), λ ∈ P and i = 1, . . . , r.
Proof This is immediate from the fact that ϕ -1 (T i ) = k i +k -1 i c i (s i -1) and ϕ -1 (g) = g for i = 1, . . . , r and g ∈ C(P m ). By a direct computation,
They restrict to well-defined linear operators on C( ) for any lattice in V satisfying Q ⊆ ⊆ P. In case of equal Hecke and representation parameters they reduce to the Demazure-Lusztig operators [17, (11)]. The realization (4.1) of the T i 's through the H m loc (k)-representation τ directly imply that the metaplectic Demazure-Lusztig operators T i (i = 1, . . . , r ) satisfy the braid relations of W and the quadratic Hecke relations
(this in particular provides an alternative and uniform proof of the braid relations and quadratic Hecke relations of the metaplectic Demazure-Lusztig operators in [17], see [17,Prop. 5(ii)] and formula (13) in [17,Prop. 7]). For
for f ∈ C(P). Hence
Remark 4. 6 Let be a lattice in E satisfying Q ⊆ ⊆ P. The localization isomorphism ϕ restricts to an isomorphism of algebras
for f , g ∈ C( 0 ), λ ∈ and i = 1, . . . , r , where 0 := ∩ P m . Note that
The metaplectic Demazure-Lusztig operators T i then restrict to the following linear operators on C( ),
where Ad
We now use these results to generalize results from [17, §3] to the case of unequal Hecke and representation parameters. We first analyze certain symmetrizer and antisymmetrizer elements in H m loc (k). We then use the metaplectic Weyl group representation σ to obtain generalizations of the formula [17,Thm. 16] for the metaplectic Whittaker function.
Recall from Sect. 2.3 that k : → C × is a W -invariant function and
Note that, in the special case that k is a constant function (the equal Hecke algebra parameters case), we have k w = k (w) . Also let
Define the symmetrizer 1 + ∈ H (k) and antisymmetrizer 1 -∈ H (k) by
It is well known (see e.g., [23, 1.19.1] and [12]) that the symmetrizer 1 + and antisymmetrizer 1 -satisfy the following properties.
Proposition 4. 7 We have the following identities in H (k):
The equations T i 1 ± = ±k ±1 i 1 ± for i = 1, . . . , r characterize 1 ± as an element in H (k) up to a multiplicative constant. It follows from this observation that
The multiplicative constant is determined by comparing the coefficient of T w 0 in the linear expansion in terms of the basis {T w } w∈W of H (k).
Recall the definition (3.15) of the c-functions c α (α ∈ ).
We have the following identities in W C(P m ):
Proof See [31, (5.5.14)].
We now obtain the following main result of this section.
Theorem 4.9 We have the following identity of operators in End(C(P)):
w∈W
In particular, for λ ∈ P + we have
Proof By (4.1) and (4.10) we have
The first formula now follows directly using τ = σ •ϕ -1 and the previous proposition.
The second formula follows from the observation that
for w ∈ W .
In this section we present metaplectic variants of GL r Macdonald polynomials. Full proofs and additional results will be provided in the forthcoming paper [36], in which we will also introduce the metaplectic polynomials for arbitrary root systems.
Let r ≥ 2. Fix the standard orthonormal basis { i } r i=1 of R r . The associated scalar product is denoted by •, • and the corresponding norm by • . Then
is the root system of type A r -1 , with basis of simple roots and associated set + of positive roots given by := {α 1 , . . . , α r -1 } ⊂ + = { ij } 1≤i< j≤r with α i := i -i+1 . The associated highest root is θ = 1r . The root lattice is Q := Z , which is contained in the GL r weight lattice r i=1 Z i Z r . The Weyl group is the symmetric group S r in r letters.
Let Q : Z r → Q be a non-zero S r -invariant quadratic form which is integral-valued on Q. Then
for some nonzero integer κ = κ Q (we suppress the dependence of κ on Q if it is clear from the context). In particular, Q(α) = κ for all α ∈ . Write
for the associated symmetric S r -invariant bilinear form B : Z r × Z r → Q. By the S r -invariance of B we then have
(in the present context α ∨ = α for α ∈ , but we distinguish them in anticipation of the results for arbitrary root systems in our followup paper [36]). In particular, B(λ, α) ∈ Z for λ ∈ Z r and α ∈ .
Fix n ∈ Z >0 once and for all. Given a quadratic form Q as in the previous paragraph with associated normalisation scalar κ = κ Q , we define positive integers κ = κ Q and m = m Q by κ := gcd(n, κ),
Note that m = n/gcd(n, Q(α)) for all α ∈ , in particular m = m . Furthermore,
Set F := C(q, k), and let K (n) be the field extension of F obtained by adjoining
For n ≥ 1 we now define representation parameters g n) for all integers j ∈ Z as follows (it depends on a choice of a sign ∈ {±1} when n is even, which we fix once and for all). We set g (n)
For n even, we set g (n) n 2
Finally, the representation parameters g
r n ( j) , with r n ( j) ∈ {0, . . . , n -1} the remainder modulo n. Note that g
Lemma 5.1 There exists a unique F-homomorphism ι κ : K (m) → K (n) mapping g
j+1 for 0 ≤ j < r (with the indices taken modulo r ). We write x v (v ∈ R r ) for the canonical basis of the group algebra F[R r ] of R r over F, so that x u x v = x u+v and x 0 = 1. We write for c ∈ Z and v ∈ R r ,
Let F[x ±1 ] be the F-algebra of Laurent polynomials in x 1 , . . . , x r , viewed as the Fsubalgebra of F[R r ] generated by Z r ⊂ R r via x i := x i (1 ≤ i ≤ r ). The extended affine Weyl group W (m) acts by F-algebra automorphisms on F[x ±1 ] by w x (λ,c) := x w(λ,c) (5.3) for w ∈ W (m) and (λ, c) ∈ Z r ⊕ Z. In particular, for λ ∈ Z r , σ ∈ S r and ν ∈ Z r , (σ τ (ν))x λ = q -(ν,λ) x σ λ .
(5.4)
For λ ∈ Z r we thus have
x ω (m) λ = q -mλ r x s 1 ...s r -1 λ , x s (m) 0 λ = q m(λ,θ ∨ ) x s θ λ and x b (m) 0 = q m 2 x -mθ .
The GL r double affine Hecke algebra H (m) is the unital associative F-algebra generated by T 0 , . . . , T r -1 , ω ±1 and F[x ±m ] := F[x ±m 1 , . . . , x ±m r ] with defining relations:
(1) The type A r -1 braid relations for T 0 , . . . , T r -1 .
(2) The Hecke relations (T jk)(T j + k -1 ) = 0.
(3) ωω -1 = 1 = ω -1 ω and ωT j = T j+1 ω (indices modulo r ). ( 4) The cross relations
for λ ∈ mZ r and 0 ≤ j < r .
Consider the subalgebras H (m) Y := F T 0 , . . . , T r -1 , ω ±1 and H (m) = F T 1 , . . . , T r -1 of H (m) . The subalgebra H (m) is the finite Hecke algebra (of type A r -1 ). A key structure theoretic fact (see e.g. §3.2.1 in [12]) is that the elements and that δ(ω -1 )Y λ = q mλ r Y s 1 •••s r -1 λ δ(ω -1 ) for λ ∈ mZ r in H (m) .
Set t m (s) := sr m (s) ∈ mZ. The metaplectic divided difference operators ∇ (m) j (0 ≤ j < r ) are the F-linear operators on F[x ±1 ] defined by
for 0 ≤ j < r . For a field extension F ⊆ K, the K-linear extension of ∇ (m) j to a linear operator on K[x ±1 ] will also be denoted by
Recall that the metaplectic data (n, Q) provide us with the nonzero integer κ := Q(α) (α ∈ ), from which κ and m are determined by (5.2). In addition, we have fixed a sign ∈ {±1} in case n is even, through the definition of the representation parameter g
Theorem 5. 4 The formulas
(5.7)
We write π (m) := π (m,1) (we suppress here the dependence on ).
Proof Set
Note that contains the weight lattice P of . The quadratic form Q has a unique extension to a Q-valued S r -invariant quadratic form → Q, which we also denote by Q. We write B : × → Q for the associated symmetric S r -invariant bilinear form. Adjoin a r th root q 1 r of q to K (n,κ) (by abuse of notation, we denote it again by K (n,κ) ). Let K (n,κ) [ ] be the K (n,κ) -submodule of K (n,κ) [R r ] := K (n,κ) ⊗ F F[R r ] generated by x λ (λ ∈ ). The extended affine Weyl group W (m) acts on K (n,κ) [ ] by K (n,κ) -algebra automorphisms by the formula (5.3). The K (n,κ) -subalgebra K (n,κ) [P] generated by x λ (λ ∈ P) is a W (m) -submodule.
By Theorem 3.7 (which holds true with formal parameters), the first two lines of (5.7),
Using the decomposition
it extends to a representation π :
X , s ∈ Z and λ ∈ P. The formulas (5.8) are then valid for all λ ∈ . A direct check shows that the operators π(T j ) and π(x μ ) on K (n,κ) [ ] satisfy the cross relation (5.5) for 1 ≤ j < r and μ ∈ mZ r . Furthermore, direct computations show that
for λ ∈ Z r and 0 ≤ j < r , hence π(ω) π(T j ) = π(T j+1 ) π(ω) as operators on K (n,κ) [x ±1 ], with the indices modulo r . From this the defining double affine Hecke algebra relations involving T 0 are easily verified. The result now follows directly.
We call π (n,κ) the metaplectic basic representation of the double affine Hecke algebra H (m) . Remark 5.5 (i) Since the double affine Hecke algebra H (m) is defined over F, the representation parameters should be thought of as representation parameters of the representation π (n,κ) . Since the representation is defined over the subfield K (n,κ) of K (n) , the representation π (n,κ) only depends on the representation parameters g
2 ) and, if m is even, on . (ii) It follows from
defines an isomorphism
of H (m) -modules. In particular, (K (n,κ) [x ±1 ], π (n,κ) ) only depends on if m is even.
(iii) π (1) :
] is Cherednik's basic representation for GL r , see, e.g., [12, §3.7] and [20].
By the second part of the remark, the dependence of the metaplectic basic representation on the metaplectic data is essentially only a dependence on m. The metaplectic basic representation π (m) can be recovered from π (n) as follows.
By a direct check one verifies that the assignments
Let j κ : K (m) → K (n) be the C(q)-homomorphism mapping q to q κ 2 and g (m) j to g (n)
κ j for all j ∈ Z. Note the difference with ι κ : K (m) → K (n) (Lemma 5.1), which fixes q. The image K (n,κ ) j of the homomorphism j κ : K (m) → K (n) is the subfield of K (n) obtained by adjoining q κ 2 and g (n)
κ j (1 ≤ j < m 2 ) to C(k). We now have the following proposition.
defines an isomorphism
of H (m) -modules. In particular, (K (n,n) j [x ±n ], π (n) • φ n ) realizes Cherednik's basic representation π (1) with the role of q replaced by q n 2 .
We keep the notation from the previous subsections. In particular, (n, Q) is the fixed metaplectic data and κ := Q(α) (α ∈ ), leading to the positive integers κ and m by (5.2); moreover the sign ∈ {±1} is fixed through the definition of the representation parameter g (n) n 2 := -1 k -1 if n is even.
The commuting linear operators π (n,κ)
are metaplectic analogues of Cherednik's Y -operators. The following theorem establishes the existence of a family of Laurent polynomials which are simultaneous eigenfunctions of the metaplectic Y -operators.
For
In other words, the value γ
takes values in K (n,κ) . The proof of the following theorem, including its extension to arbitrary root systems, will be given in the forthcoming paper [36].
(ii) The coefficient of x μ in the expansion of E (n,κ) μ (x) in the monomial basis {x ν } ν∈Z r , is one. 8 Many key properties of Macdonald polynomials can be proved via the technique of intertwiners, introduced in [25,26,34] for type A, in [13] for arbitrary root systems and in [35] for the Koornwinder setting. In our forthcoming paper [36] we develop the metaplectic analog of the theory of intertwiners, which allows us to prove the above result and also establish a key triangularity property of the polynomials E
The following proposition is a consequence of Remark 5.5(ii) and Proposition 5.6. Proposition 5.9 For all μ ∈ Z r ,
(5.9)
By the first line of (5.9), the metaplectic polynomial E (n,κ) (x) essentially only depends on m, q, k, the representation parameters g
2 ) and, if m is even, on .
Remark 5.10 By Remark 5.5(iii), E
(1) μ (x) is the monic nonsymmetric Macdonald polynomial of degree μ (compared to the standard conventions on nonsymmetric Macdonald polynomials as in e.g. [20], k 2 corresponds to t). Furthermore, as a special case of the second line of (5.9), E (n) nμ (x) realizes the monic nonsymmetric Macdonald polynomial of degree μ ∈ Z r in the variables x n 1 , . . . , x n r , with the role of q replaced by q n 2 .
We give formulas for E (m) λ (x), where 1 ≤ m ≤ 5 and λ ∈ Z 3 has weight at most 2. For convenience of notation, we write g j instead of g (m) j . The technique used to compute these polynomials will be provided in the forthcoming paper [36].
(5) (0,1,0) (x) = (k -1) (k + 1) g 1 k 4 g 3 1 q 5 + 1
(0,0,1) (x) = (k -1) (k + 1)
(0,0,1) (x) = -(k -1) (k + 1) g 2 1 k 2 g 3 1 q 4 + 1
x 1 + (k -1) (k + 1) g 1 k 2 g 3 1 q 4 + 1
x 2 + x 3
(5) (0,0,1) (x) = -(k -1) (k + 1) g 2 1 k 2 g 3 1 q 5 + 1
x 1 + (k -1) (k + 1) g 1 k 2 g 3 1 q 5 + 1
(k -1) (k + 1)
(1,0,1) (x) = (k -1) (k + 1)
(1,0,1) (x) = (k -1) (k + 1) k kq 2 + x 1 x 2 + x 3 x 1 E (3) (1,0,1) (x) = (k -1) (k + 1) g 1 k 4 g 3 1 q 3 + 1
(1,0,1) (x) = (k -1) (k + 1) g 1 k 4 g 3 1 q 4 + 1
(1,0,1) (x) = (k -1) (k + 1) g 1 k 4 g 3 1 q 5 + 1
(5) (0,2,0) (x) = (k -1) (k + 1) g 2 k 4 g 3 2 q 10 + 1
(1) (0,0,2) (x) = (k -1) (k + 1) (kq -1) (kq + 1)
x 2 1 + (q + 1) (k -1) 2 (k + 1) 2 (kq -1) (kq + 1) qk 2 -1
x 1 x 2 + (q + 1) (k -1) (k + 1) (kq -1) (kq + 1)
x 2 2 + (q + 1) (k -1) (k + 1) (kq -1) (kq + 1) (2) (0,0,2) (x) (see Remark 5.10). In particular, the formulas given above for these polynomials match the ones provided in the appendix of [20] (with k 2 replaced by t).
(2) More generally, for any a ∈ Z ≥1 , the metaplectic polynomial E (am) aλ (x) may be obtained from E (m) λ (x) via the substitutions x i → x a i , q → q a 2 and g aj . This follows directly from Proposition 5.9 with κ = a. We list the pairs E (see (3.6) for the definition of C m ). If λ ∈ C m + , the metaplectic polynomial E (m) λ (x) is equal to the monomial x λ . This will be proved in the followup paper [36] in the context of arbitrary root systems. Note that this result applies to the following polynomials listed above: E (2,0,0) (x) (any m ∈ Z ≥2 ). Note that for λ ∈ mZ 3 ∩ C m + , this recovers the wellknown fact that the nonsymmetric Macdonald polynomial corresponding to the minuscule weight λ is a monomial.