Cherednik algebras and Zhelobenko operators

We study canonical intertwining operators between modules of the trigonometric Cherednik algebra, induced from the standard modules of the degenerate affine Hecke algebra. We show that these operators correspond to the Zhelobenko operators for the affine Lie algebra $\widehat{\mathfrak{sl}}_m$. To establish the correspondence, we use the functor of Arakawa, Suzuki and Tsuchiya which maps certain $\widehat{\mathfrak{sl}}_m$-modules to modules of the Cherednik algebra.

Introduction 0.1. In the present article we study the trigonometric Cherednik algebra C N corresponding to the general linear Lie algebra gl N . The complex associative algebra C N is generated by the symmetric group ring CS N , by the ring P N of Laurent polynomials in N variables x 1 , . . . , x N and by another family of pairwise commuting elements denoted by u 1 , . . . , u N . The subalgebra of C N generated by the first two rings is the crossed product S N ⋉ P N where the symmetric group S N permutes the variables x 1 , . . . , x N . The subalgebra generated by S N and u 1 , . . . , u N is the degenerate affine Hecke algebra H N introduced by Drinfeld [5] and Lusztig [11]. The other defining relations in C N are given in Subsection 2.1 of our article. In particular, the algebra C N depends on a parameter κ ∈ C.
The degenerate affine Hecke algebra H N has a distinguished family of modules which are called standard . These modules are determined by pairs of sequences λ = (λ 1 , . . . , λ m ) and µ = (µ 1 , . . . , µ m ) of length m of complex numbers such that for every a = 1, . . . , m the difference λ a − µ a is a positive integer, while λ 1 − µ 1 + · · · + λ m − µ m = N . We denote the corresponding standard module of H N by S λ µ . It is induced from a one-dimensional module of the subalgebra of H N generated by u 1 , . . . , u N and by the subgroup of S N preserving the partition of the sequence 1, . . . , N to segments of lengths λ 1 − µ 1 , . . . , λ m − µ m . This subgroup of S N acts on the one-dimensional module trivially, while u p acts as µ a − a + h where a is the number of the segment of the sequence 1, . . . , N which the index p belongs to, and h is the number of the place of the index p within that segment. Now consider the symmetric group S m which acts on sequences of length m of complex numbers by permutations. We denote by the symbol • the corresponding shifted action of S m . To define the latter action, one takes a sequence of length m, subtracts the sequence (1, . . . , m) from it, permutes the resulting sequence and adds the sequence (1, . . . , m) back. If λ a − λ b / ∈ Z or equivalently µ a − µ b / ∈ Z for all a ̸ = b, then the standard H N -module S λ µ is irreducible. Moreover, then for every permutation σ ∈ S m the standard module S σ•λ σ•µ is isomorphic to S λ µ . Hence there exists an intertwining mapping S λ µ → S σ•λ σ•µ of H N -modules, unique up to scalar multiplier. These mappings were already used by Rogawski [12].
The standard H N -module S λ µ has another realization due to Arakawa, Suzuki and Tsuchiya [1]. Take the complex Lie algebra gl m . The sequences of length m of complex numbers can be regarded as weights of gl m . There we employ the Cartan subalgebra t of gl m described in Subsection 1.3 of our article. Then the above defined action • becomes the shifted action of the Weyl group S m of gl m on weights. Take the Verma module M µ of gl m corresponding to the weight µ.
The tensor product (C m ) ⊗N ⊗ M µ of gl m -modules can be also equipped with an action of the algebra H N , which commutes with the action of gl m . Here the symmetric group S N ⊂ H N acts on (C m ) ⊗N ⊗ M µ by permutations of the N tensor factors C m , while the elements (1.2) of H N act as the operators (1.7) respectively. Let n be the nilpotent subalgebra of gl m defined in Subsection 1.3. The space ((C m ) ⊗N ⊗ M µ )) λ n of n-coinvariants of weight λ inherits an action of the algebra H N . As a H N -module it is isomorphic to S λ µ ; see our Proposition 1.3. Following Zhelobenko [16], for any λ and µ obeying the above non-integrality conditions, and for any permutation σ ∈ S m , one can define a canonical linear map See also the work of Khoroshkin and Ogievetsky [10]. In particular, the linear map (0.1) is H N -intertwining. Using Proposition 1.3, the map (0.1) determines an H N -intertwining map S λ µ → S σ•λ σ•µ . By the irreducibility of the source and of target standard H N -modules here, the latter map coincides with the intertwining map from [12] up to a scalar multiplier.
We will work with the special linear Lie algebra sl m alongside of gl m . Our n is a subalgebra of sl m . Further, the Cartan subalgebra h of sl m described in Subsection 1.3 is contained in t ⊂ gl m . Let us denote by α and β the weights of sl m corresponding to the weights λ and µ of gl m by restriction. Hence α and β are elements of the space dual to h.
The Verma module M β is isomorphic to the restriction of M µ to the subalgebra sl m ⊂ gl m . However, another action of H N on (C m ) ⊗N ⊗ M β can be defined by using only the structure of M β as a module of sl m . Namely, he symmetric group S N ⊂ H N acts on (C m ) ⊗N ⊗ M β again by permutations of the N tensor factors C m , but the elements (1.2) of H N act as the operators (1.9) respectively. The space ((C m ) ⊗N ⊗ M β )) α n of n-coinvariants of weight α inherits an action of H N . As a H Nmodule it is isomorphic to the pullback of S λ µ relative to the automorphism (1.5) of H N where f = − (µ 1 + · · · + µ m )/m; see Corollary 1.4. This automorphism acts trivially on the elements of the subalgebra S m ⊂ H N . Note that the latter space of n-coinvariants can be naturally identified with the space ((C m ) ⊗N ⊗ M µ )) λ n . The shifted action of S m on the weights of gl m factors to an action on the weights of sl m . By again following [10] and [16], one defines a canonical linear map If we identify the source vector spaces of the maps (0.1) and (0.2) as above, and also identify the target vector spaces, then the two maps become the same. Note that the shifted action • of S m on µ preserves the sum µ 1 + · · · + µ m . Hence the map (0.2) is also H N -intertwining. Via the Drinfeld duality between H N -modules and modules of the Yangians of general linear Lie algebras [5] this interpretation of intertwining maps for the standard modules of H N goes back to the work of Tarasov and Varchenko [15]; see also our work [7]. In the present work we extend this interpretation to intertwining maps for certain C N -modules. Instead of gl m and sl m above, we will use the corresponding affine Lie algebras gl m and sl m . 0.2. We will regard sl m as a one-dimensional central extension of the current Lie algebra sl m [ t, t −1 ]. We choose a basis element C in the extending one-dimensional vector space. For any ℓ ∈ C, a module of sl m is said to be of level ℓ if C acts as the scalar ℓ on that module. Let us extend the Cartan subalgebra h of sl m by the one-dimensional space spanned by C, and denote by h the Abelian subalgebra of sl m so obtained. For ℓ = κ − m we will denote by α and β the extensions of the weights α and β from h to h, determined by setting α(C) = β(C) = ℓ. We will use the Verma module M β of sl m as defined in Subsection 2.3.
Let us now regard H N as a subalgebra of C N . Denote by S λ µ the module of C N induced from the standard module S λ µ of H N . The induced module also has another realization [1]. Take the vector space P N ⊗ (C m ) ⊗N . It can be naturally identified with the tensor product of N copies of the vector space C m [t, t −1 ]. By regarding the latter space as a module of sl m of level zero, P N ⊗ (C m ) ⊗N becomes a zero level module of sl m . Further, the vector space can be equipped with an action of the algebra C N . The symmetric group S N ⊂ C N acts on (0.3) by simultaneous permutations of the variables x 1 , . . . , x N and of the N tensor factors C m while the subalgebra P N ⊂ C N acts on (0.3) via multiplication in the first tensor factor. The elements (1.2) of H N ⊂ C N act on (0.3) as the operators (2.7) respectively. In general, the action of C N on the vector space (0.3) does not commute with the action of sl m . However, let n be the nilpotent subalgebra of sl m defined in Subsection 2.1. For ℓ = κ − m the action of the algebra C N on (0.3) preserves the image of the action of n; see Corollary 2.2. Therefore the space of n-coinvariants of (0.3) of weight α inherits an action of the algebra C N . The automorphism (1.5) of H N extends to C N so that it acts on the elements of the subalgebra P N ⊂ C N trivially. As a C N -module, (0.4) is isomorphic to the pullback of S λ µ relative to the extended automorphism (1.5) where f = − (µ 1 +· · ·+µ m )/m; see our Corollary 2.2 and Proposition 2.3. Now consider the semidirect group product S m ⋉ Z m . Extend the permutation action of the group S m on sequences of length m of complex numbers to an action of S m ⋉ Z m so that the elements of Z m act by addition of the respective elements of ℓ Z m ⊂ C m . Here we set ℓ = κ − m as above. The action • of S m on the sequences also extends to an action of the group S m ⋉ Z m , where the elements of Z m however act by addition of the respective elements of κ Z m ⊂ C m . Let us denote by the symbol • the latter action of S m ⋉ Z m on the sequences. If Hence there is an intertwining mapping S λ µ → S ω•λ ω•µ of C N -modules, unique up to a scalar multiplier. These mappings were used by Suzuki [13]. Recently they were further used by Balagović [2].
The group S m ⋉ Z m is isomorphic to the extended affine Weyl group of gl m , for details see Subsection 2.4. This group is Z-graded so that the degree of any element of S m is zero, while the degree of any element of Z m is the sum of its m components. All the elements of degree zero make a subgroup of S m ⋉ Z m isomorphic to the proper affine Weyl group of gl m . Note that this subgroup is also isomorphic to the Weyl group of the affine Lie algebra sl m .
Again regard λ and µ as weights of gl m . Restrict them to the weights α and β of sl m . Extend the latter two to the weights α and β of sl m as above. The action • of S m ⋉ Z m on λ and µ determines its action on α and β. We will still denote by • the action of S m ⋉ Z m so determined. It can also be described as a shifted action of the group S m ⋉ Z m on those weights of sl m which take the value ℓ = κ − m at C ∈ h; see Subsection 3.1 for details.
By following [10] and [16], for every element ω ∈ S m ⋉ Z m we can define a canonical linear map from the vector space (0.4) to the vector space (0.5) Details of this definition are given in Subsection 4.1. Denote by g the Z-degree of ω. Our linear map is C N -intertwining only if κ = 0 or g = 0. In general, it becomes C N -intertwining if we pull the action of C N on the target space (0.5) back through the automorphism (1.5) where f = κ g/m. Here we use Proposition 2.4 and its Corollary 2.5 which seem to be new. By using Proposition 2.3 we can replace the source and the target modules of this C N -intertwining linear map by their isomorphic modules. The source module can be replaced by the pullback of S λ µ relative to the automorphism (1.5) where f = − (µ 1 + · · · + µ m )/m. Note that the sum of the terms of the sequence ω•µ is equal to µ 1 + · · · + µ m + κ g by the definition of the action • of S m ⋉ Z m on the sequences. Therefore the target module here can be replaced by the pullback of S ω•λ ω•µ relative to the automorphism (1.5) where f = − (µ 1 + · · · + µ m + κ g)/m + κ g/m = − (µ 1 + · · · + µ m )/m.
Since the values of f for the source and the target modules are the same, our canonical linear map from (0.4) to (0.5) also determines a C N -intertwining linear map S λ µ → S ω•λ ω•µ . By the irreducibility of the source and target modules here, the latter map coincides with the intertwining map from [13] up to a scalar multiplier. 0.3. Let us now briefly survey our article. In Section 1 we collect basic facts about the degenerate affine Hecke algebra H N , including the realisation of its standard modules [1]. In Section 2 we recall the definition of the trigonometric Cherednik algebra C N , and describe the action of C N on the spaces of n-coinvariants. By using this action, we give the realisation of induced modules of C N mentioned above. Towards the end of Section 2 we introduce the extended affine Weyl group of gl m , and describe its action on the spaces of n-coinvariants. Our Proposition 2.4 relates this action to the action of C N on the same spaces. This relation is the key technical result of our article. In Section 3 we define the Zhelobenko operators for the affine Lie algebra sl m . Theorem 3.6 relates these operators to the algebra C N . Further details of this relation are worked out in Section 4.
Acknowledgements. The first named author has been supported by the RSF grant 16-11-10316. The second named author has been supported by the EPSRC grant N023919 and by a Santander International Connections Award. We dedicate this article to Professor Alexandre Kirillov to celebrate his eightieth birthday.
1. Hecke algebras 1.1. We begin with the definition of the degenerate affine Hecke algebra H N corresponding to the general linear group GL N over a local non-Archimedean field. This algebra has been introduced by Drinfeld [D2]; see also the work of Lusztig [L]. The complex associative algebra H N is generated by the symmetric group algebra CS N and by pairwise commuting elements u 1 , . . . , u N with the cross relations for p = 1, . . . , N − 1 and q = 1, . . . , N σ p u q = u q σ p for q ̸ = p, p + 1; σ p u p = u p+1 σ p − 1.
Here and in what follows σ p ∈ S N denotes the transposition of numbers p and p + 1. More generally, σ pq ∈ S N will denote the transposition of the numbers p and q. The group algebra CS N can be then regarded as a subalgebra in H N . Furthermore, it follows from the defining relations of H N that a homomorphism H N → CS N , identical on the subalgebra CS N ⊂ H N , can be defined by mapping (1.1) We will also use the elements of the algebra H N z p = u p − σ 1p − · · · − σ p−1,p where p = 1, . . . , N .
The elements z 1 , . . . , z N do not commute, but satisfy the commutation relations It follows from the definition of H N that for any f ∈ C an automorphism of this algebra, identical on the subalgebra CS N ⊂ H N , can be defined by mapping (1.5) Then by (1.2) z p → z p + f for p = 1, . . . , N .
By pulling the trivial one-dimensional module of the algebra CS N back through the homomorphism (1.1), and further back through the automorphism (1.5), we get a one-dimensional module of H N . On the latter module each of the elements z 1 , . . . , z N ∈ H N acts as multiplication by f . Let us denote this module by S f +N f ; this peculiar choice of notation will be justified next. Now fix a postive integer m. Take any two sequences λ = (λ 1 , . . . , λ m ) and µ = (µ 1 , . . . , µ m ) of length m of complex numbers. For each a = 1, . . . , m denote ν a = λ a − µ a and suppose that ν a is a non-negative integer. Note that unlike in the Introduction, here we allow the equality ν a = 0. We still suppose that ν 1 +· · ·+ν m = N . Denote ν = (ν 1 , . . . , ν m ). Let S ν be the corresponding subgroup of the symmetric group S N . This subgroup is naturally isomorphic to the direct product S ν1 × · · · × S νm . The tensor product H ν1 ⊗ · · · ⊗ H νm can be naturally identified with the subalgebra of H N generated by the subgroup S ν ⊂ S N and by all the pairwise commuting elements u 1 , . . . , u N . Denote by H ν this subalgebra. The induced module of H N is called standard and denoted by S λ µ . If m = 1 and µ 1 = f then λ 1 = f + N and . The reason to use, in the definition of S λ µ , the numbers λ a − a + 1 and µ a − a + 1 rather than λ a and µ a will become clear in Subsection 1.3.

1.2.
Let us now recall a construction due to Cherednik [4, Example 2.1]. It has been further developed by Arakawa, Suzuki and Tsuchiya [1,Subsect. 5.3]. Let U be any module over the complex general linear Lie algebra gl m . Let E ab ∈ gl m with a, b = 1, . . . , m be the standard matrix units. We will also regard the matrix units E ab as elements of the algebra End(C m ); this should not cause any confusion. Let us consider the tensor product (C m ) ⊗N ⊗ U of gl m -modules. Here each of the N tensor factors C m is a copy of the natural gl m -module. We shall use the indices 1, . . . , N to label these N tensor factors. For any index p = 1, . . . , N we will denote by E (p) ab the operator on the vector space (C m ) ⊗N acting as (1.6) Proposition 1.1. (i) By using the gl m -module structure of U , an action of the algebra H N on the vector space (C m ) ⊗N ⊗ U is defined as follows: the symmetric group S N ⊂ H N acts by permutations of the N tensor factors C m , and the element For a proof of this proposition see [7, Sect. 1]. By using Proposition 1.1 we obtain a functor E N : U → (C m ) ⊗N ⊗ U from the category of all gl m -modules to the category of bimodules over gl m and H N . We will also use a version of this proposition for the special linear Lie algebra sl m instead of gl m . Let us denote Hence an action of can be defined on the vector space (C m ) ⊗N ⊗ U by using only the sl m -module structure of U . Because the element I ∈ gl m is central, the operators (1.9) with p = 1, . . . , N satisfy the same commutation relations (1.4) as the operators (1.7) respectively instead of z 1 , . . . , z N . Using Corollary 1.2 we get a functor F N : U → (C m ) ⊗N ⊗ U from the category of all sl m -modules to the category of bimodules over sl m and H N . Our principal tool will be an analogue of this functor for the affine Lie algebra sl m instead of sl m . The role of the degenerate affine algebra H N will be then played by the trigonometric Cherednik algebra C N .

1.3.
Consider the triangular decomposition of the Lie algebra gl m , Here t is the Cartan subalgebra of gl m with the basis vectors E 11 , . . . , E mm . Every element of the vector space t * dual to t is called a weight of gl m . We will regard any sequence µ = (µ 1 , . . . , µ m ) of length m of complex numbers as such a weight, by setting µ(E aa ) = µ a for a = 1, . . . , m. For any gl m -module U , its subspace consisting of all vectors of weight µ is denoted by U µ . In the above display n is the nilpotent subalgebra of gl m spanned by all the elements E ab with a > b, while n ′ is spanned by all E ab with a < b. We will denote by U n the vector space U/n U of the coinvariants of the action of the subalgebra n on U . Note that the Cartan subalgebra h ⊂ gl m acts on the vector space U n .
Consider the Verma module M µ of the Lie algebra gl m . It can be described as the quotient of the universal enveloping algebra U(gl m ) by the left ideal generated by all the elements E ab with a < b and by the elements E aa − µ a . The elements of the Lie algebra gl m act on this quotient via left multiplication. Let us apply the functor E N to the gl m -module U = M µ . By using Proposition 1.2 we obtain a bimodule E N (M µ ) of gl m and H N . For any λ = (λ 1 , . . . , λ m ) consider the space E N (M µ ) λ n of those coinvariants of this bimodule relative to n which are of the weight λ. This space comes with an action of the algebra H N .
n of the algebra H N is isomorphic to the standard module S λ µ . Proof. By repeatedly using [8, Thm. 1.3] the proof reduces to its particular case when m = 1. In the latter case the proposition is immediate.
Let us now give a counterpart [1, Prop. 5.3.1] of Proposition 1.3 for the Lie algebra sl m instead of gl m . Let h be the Cartan subalgebra of sl m with the basis vectors where n and n ′ are the same as above. We will denote respectively by α and β the restrictions of the weights λ and µ of gl m to the subspace h ⊂ t. Thus α and β will be weights of sl m . Note that the restriction of the gl m -module M µ to the subalgebra sl m ⊂ gl m is isomorphic to the Verma module M β , while the central element I ∈ gl m acts on M µ as multiplication by µ 1 + · · · + µ m . Therefore by using the definition of F N we get a corollary to Proposition 1.3.
Note that by pulling the standard module S λ µ back through the automorphism (1.5) with any f we get another standard module, corresponding to the sequences (λ 1 + f , . . . , λ m + f ) and (µ 1 + f , . . . , µ m + f ) instead of λ and µ. However, we will use Corollary 1.4 as stated.

Cherednik algebras
be the ring of of Laurent polynomials in N variables x 1 , . . . , x N with complex coefficients. We will denote by ∂ 1 , . . . , ∂ N the derivation operators in P N relative to these variables. The trigonometric Cherednik algebra C N depending on a parameter κ ∈ C is the complex associative algebra generated by H N and P N , subject to the relations σ x p σ −1 = x σ(p) for all σ ∈ S N and to the commutation relations We can also employ the pairwise commuting generators u 1 , . . . , u N ∈ H N instead of z 1 , . . . , z N ; see (1.2). Then instead of the above displayed relations in C N we get The latter set of defining relations shows that (1.5) extends to an automorphism of the algebra C N identical in the subalgebras CS N and P N . By [6, Thm. 1.3] multiplication in the algebra C N yields a bijective linear map Next we will state the generalizations of Proposition 1.1 and Corollary 1.2 to C N . They go back to the work of Cherednik [3].

2.2.
First consider the affine Lie algebra gl m over the field C. We will define it as a central extension of the current Lie algebra gl m [ t, t −1 ] by a one-dimensional complex vector space with a fixed basis element which will be denoted by C. Here t is a formal variable. Choose the basis of gl m [ t, t −1 ] consisting of the elements E cd t j where c, d = 1, . . . , m whereas j ranges over Z. The commutators in the Lie algebra gl m [ t, t −1 ] are taken pointwise so that for the basis elements. In the extended Lie algebra gl m we have the relations We will also work with the affine Lie algebra sl m . This is a subalgebra of gl m spanned by the subspace sl m [ t, t −1 ] ⊂ gl m [ t, t −1 ] and by the central element C. Let h be the Abelian subalgebra of sl m spanned by C and by the Cartan subalgebra h ⊂ sl m . The vector spaces are also Lie subalgebras of sl m by the relations (2.1). As a vector space, Let V be any module of gl m such that for any given vector in V , there exists a degree i such that the subspace t i gl m [t] ⊂ gl m annihilates the vector. Consider the vector space Due to our condition on V for any p = 1, . . . , N there is a well-defined linear ab is the operator (1.6) acting on (C m ) ⊗N . Further, the symmetric group S N acts on the tensor factor P N of W by permutations of the variables x 1 , . . . , x N . There is another copy of the group S N acting on the N tensor factors C m of W by permutation. Using these two actions of S N for any p = 1, . . . , N introduce the Cherednik operator on W The vector space P N ⊗ (C m ) ⊗N can be naturally identified with the tensor product of N copies of the space C m [t, t −1 ]. The latter space can be regarded as a gl m -module where the central element C acts as zero. By taking the tensor product of N copies of this module with V we turn the vector space W to a gl m -module. The element E cd t j ∈ gl m acts on W as For any complex number ℓ, a module of the Lie algebra gl m or sl m is said to be of level ℓ if the element C acts on this module as that complex number. In particular, the gl m -module C m [t, t −1 ] used above is of level zero. We can now state the main properties of Cherednik operators on W from [1], [14]. These properties immediately follow from [ Proposition 2.1. (i) By using the gl m -module structure on V , an action of the algebra C N on the vector space W is defined as follows: the elements , the group S N ⊂ H N acts by simultaneous permutations of the variables x 1 , . . . , x N and of the N tensor factors C m , and the element z p ∈ H N acts as (2.4).
(ii) This action of C N on W commutes with that of the Lie subalgebra gl m ⊂ gl m .
Below is a version of this proposition in the case when V is a module not of gl m but only of sl m , also due to [1]. There for any vector in V we assume the existence of i such that the subspace t i sl m [t] ⊂ sl m annihilates the vector. Let I = E 11 + · · · + E mm as before. By (1.8) an action of can be defined on (2.2) by using only the sl m -module structure of V . Then for every p = 1, . . . , N we have a modification of the Cherednik operator (2.4) on W , (2.7) Here we use the sum (2.6) instead of (2. Using Corollary 2.2(i) and the definition (2.2), we get a functor A N : V → W from the category of all sl m -modules satisfying the annihilation condition stated just before (2.6). Note that the resulting actions of sl m and C N on W do not commute in general. However, this will be our analogue for sl m of the functor F N introduced in the end of Subsection 1.2.

2.3.
Let λ and µ be same sequences of length m of complex numbers as in Subsection 1.1. Take the standard module S λ µ over the algebra H N . By regarding H N as a subalgebra of C N , consider the induced module Denote the latter module by S λ µ . Its underlying vector space can be identified with that of P N ⊗ S λ µ whereon the subalgebra P N ⊂ C N acts via multiplication in the first tensor factor. Notice that by transitivity of induction and by the definition of S λ µ , the C N -module S λ µ is isomorphic to Now suppose that ℓ = κ − m, so that our Corollary 2.2(iii) applies. Regard λ and µ as weights of gl m . Their restrictions to the subspace h ⊂ t are denoted by α and β respectively. Define the weight β of sl m as the element of the space dual to h such that β(C) = ℓ and β(X) = β(X) for all X ∈ h. (2.8) Consider the Verma module M β of sl m . By definition, this is the quotient of the universal enveloping algebra U( sl m ) by the left ideal generated by n ′ and by all the elements X − β(X) where X ranges over h. Since the element C ∈ sl m is central, the first equality in (2.8) implies that the sl m -module M β is of level ℓ. Moreover, V = M β satisfies the annihilation condition stated just before (2.6). Therefore we can apply the functor A N to this V . Further, let us define the weight α of sl m similarly to β and consider the space A N (M β ) α n of those coinvariants of A N (M β ) relative to n which have the weight α. This space comes with an action of the algebra C N due to Corollary 2.2(iii). The latter action is described by the next proposition [1,Prop. 5.2.3]. The proof given in [1] was different, however.

2.4.
Denote by T m the affine Weyl group of the Lie algebra gl m . This group is generated by the elements τ c where c = 0, 1, . . . , m − 1. However, we will let the indices of the generators τ c run through Z, assuming that τ c+m = τ c for c ∈ Z.
Then the defining relations of T m are The corresponding extended affine Weyl group is generated by T m and an element π such that π τ c = τ c+1 π.
Let us denote the extended group by R m . The group R m acts on the set Z by permutations of period m. Namely, each generator τ c of T m exchanges c + j m with c+1+j m for each j ∈ Z, leaving all other integers fixed. The extra generator π maps any integer d to d + 1.
The group R m is Z-graded so that the element π has degree one, while all elements of T m have degree zero. Further, the group R m is isomorphic to the semidirect product S m ⋉ Z m . We will use the isomorphism R m → S m ⋉ Z m defined by mapping The group R m acts by automorphisms of the Lie algebra gl m so that the central element C is invariant, Here we let a, b = 1, . . . , m. If any of the indices of the matrix units appearing in the last three displayed formulas is 0 or m + 1, it should be then replaced respectively by m or 1.
Take the level zero module C m [t, t −1 ] of gl m . Let e 1 , . . . , e m be the standard basis vectors of C m . The group R m acts on the vector space C m [t, t −1 ] so that τ c : e a t i → e τc(a) t i for c = 1, . . . , m − 1 while τ 0 : e a t i → e τ0(a) t i+δa1−δam and π : e a t i → e a+1 t i−δam .
Then π −1 : e a t i → e a−1 t i+δa1 .
Here we use the same interpretation of the indices of the standard basis vectors of C m as of the indices of the matrix units above. One can easily verify that the actions of gl m and of R m on C m [t, t −1 ] extend to an action of the crossed product algebra R m ⋉ U( gl m ). This algebra is defined by the above described action of the group R m on gl m . In the crossed product algebra, π Xπ −1 = π(X) for X ∈ gl m .

2.5.
Suppose that the gl m -module V is also equipped with an action of the extended affine Weyl group R m . Moreover, suppose that the actions of both gl m and R m on V extend to an action of the crossed product algebra R m ⋉ U( gl m ). By identifying the tensor product of N copies of C m [t, t −1 ] with P N ⊗ (C m ) ⊗N we define an action of the group R m on the latter vector space, and hence on its tensor product (2.2) with V .
By the definition given in Subsection 2.4, the action of the element π ∈ R m on the Lie algebra gl m preserves the subalgebra n. Therefore the element π acts on the space W n of n-coinvariants of the gl m -module W . On the other hand, under the assumption ℓ = κ − m, the Cherednik operator (2.4) also acts on W n due to Proposition 2.1. Let us denote by ζ p the operator on W n corresponding to (2.4). The next property of ζ p will be crucial for us. Proposition 2.4. If the gl m -module V has level κ − m then for p = 1, . . . , N we have an equality of operators on W n π ζ p π −1 = ζ p + id.
Proof. Extend the vector space W in (2.2) by replacing its first tensor factor P N by the space of all complex valued rational functions in the variables x 1 , . . . , x N with the permutation action of the symmetric group S N . Extend the action of the element π on W accordingly. To this end, identify the tensor product P N ⊗(C m ) ⊗N in (2.2) with the tensor product of N copies of C m [t, t −1 ] as above. Then restate the definition of the action of π on the vector space C m [t, t −1 ] by regarding the latter as the tensor product C m ⊗ C[t, t −1 ]. Here we also use the given action of the group element π ∈ R m on the tensor factor V of (2.2).
For p = 1, . . . , N consider the following operators on the extended vector space, Then (2.4) is the restriction of the operator D p − R p + T p to the space (2.2). By identifying P N ⊗ (C m ) ⊗N with the tensor product of N copies of C m [t, t −1 ] and using the action of the element π on the pth of these N copies as defined in The action of π on the tensor product P N ⊗(C m ) ⊗N commutes with the multiplication by any element of P N in the first tensor factor. It also commutes with the operator σ pr ⊗ σ pr for any r ̸ = p. Therefore Consider the operator T p . In its definition, the summand corresponding to any r ̸ = p can be rewritten as Hence π T p π −1 equals the sum over the indices i = 0, 1, . . . and a, b = 1, . . . , m of ∑ r̸ =p (2.10) Here we use the action of π ∈ R m on U( gl m ) as defined in Subsection 2.4. By the definition of T p , the sum over the indices i and a, b of the expressions displayed in the two lines (2.10) equals By adding to this result the right-hand side of (2.9) and by subtracting R p , we get back the Cherednik operator (2.4) plus the sum ∑ r̸ =p ∑ a̸ =1 (2.11) Here we used the equality κ − ℓ = m and replaced the indices a + 1, b + 1 by a, b. The sum displayed in the two lines (2.11) can be rewritten as Here the sum over b = 1, . . . , m is the identity operator on W . For any a ̸ = 1 the element E 1a t −1 ∈ n acts on W as the sum in the brackets in the first of the last two displayed lines; see (2.5). Hence the whole expression displayed in the first line vanishes on the quotient W n . Further, for any b ̸ = 1 the element E b1 ∈ n acts on W as the sum in the brackets in the second of the last two displayed lines.
Hence the whole expression displayed in the second line acts on W n as the identity operator.
Below is a version of Proposition 2.4 in the case when V is a module not of gl m but only of sl m . Here we regard W as sl m -module, and use the action of the element π ∈ R m on the corresponding space W n of n-coinvariants. Under the assumption ℓ = κ − m, the modified Cherednik operator (2.7) acts on W n due to Corollary 2.2. Let us denote by θ p the operator on W n corresponding to (2.7).
Corollary 2.5. If the sl m -module V has level κ − m then for p = 1, . . . , N we have an equality of operators on W n π θ p π −1 = θ p + κ m id.
Proof. The modified Cherednik operator (2.7) is obtained by subtracting from (2.4) the sum 1 m But the action of π ∈ R m on the latter sum amounts to subtracting from it the operator see Subsection 2.4. Since the module V has level κ−m, Proposition 2.4 implies that

Zhelobenko operators
3.1. Lett be the subalgebra of gl m with the basis vectors C and E 11 , . . . , E mm . Note thatt contains the subalgebra h of sl m . Consider the action of the extended affine Weyl group R m on gl m defined in Subsection 2.4. This action preserves the subalgebrat ⊂ gl m . By definition, we have π(C) = C and τ (C) = C for all τ ∈ T m . Further, we have while the generators τ 1 , . . . , τ m−1 act on the basis vectors E 11 , . . . , E mm naturally, that is by transpositions of the indices 1, . . . , m. Note that here we also have We will also use the action of the group R m on the vector spacet * , dual to the above action ont. To describe the dual action explicitly, let C * and E * 11 , . . . , E * mm be the basis vectors oft * dual to our chosen basis vectors oft. Then while the generators τ 1 , . . . , τ m−1 leave C * invariant and act on the vectors E * 11 , . . . . . . , E * mm by transpositions of the indices 1, . . . , m. Now for any given ℓ ∈ C and for any weight µ ∈ t * define an element µ ∈t * by setting µ(C) = ℓ and µ(X) = µ(X) for all X ∈ t.
Equivalently, µ = ℓ C * + µ 1 E * 11 + · · · + µ m E * mm . Then In particular, for any given ℓ ∈ C the action of the group R m ont * preserves the set of weights of the form µ. We will also use the shifted action of R m ont * . It is defined by adding to the elements oft * , then applying the above described action of R m , and then subtracting (3.1). We will employ the symbol • to denote the shifted action. Put For any given ℓ the shifted action of R m ont * preserves the set of weights of the form µ. Hence we get a shifted action of R m on the set of sequences of length m of complex numbers. We will use the symbol • to denote it as well. Then for any sequence µ = (µ 1 , . . . , µ m ) Note that via our isomorphism R m → S m ⋉ Z m , the same shifted action of the group R m on the sequences can be obtained by using the last displayed formula and by letting the elements of the subgroup Z m ⊂ S m ⋉ Z m act by addition of the respective elements of κ Z m where κ = ℓ + m. Indeed, because the group R m is generated by τ 1 , . . . , τ m−1 and π, it suffices to check the coincidence of two actions of the element π only. Its image under the isomorphism R m → S m ⋉ Z m is the product (1, 0, . . . , 0) σ 1 . . . σ m−1 ; see Subsection 2.4. But by our definition of the shifted action of the group S m ⋉ Z m on the sequences we have σ 1 · · · σ m−1 • µ = (µ m − m + 1, µ 1 + 1, . . . , µ m−1 + 1), (1, 0, . . . , 0) σ 1 · · · σ m−1 • µ = (µ m − m + κ + 1, µ 1 + 1, . . . , µ m−1 + 1) = (µ m + ℓ + 1, µ 1 + 1, . . . , µ m−1 + 1).

3.2.
Consider the tensor product of N copies of the sl m -module C m [t, t −1 ]. In Subsection 2.2 we identified the vector space of this tensor product with Following [9] regard B as bimodule over the associative algebra U( sl m ) by setting for X ∈ sl m while P ∈ P N ⊗(C m ) ⊗N and A ∈ U( sl m ). So the left module structure on B is defined by regarding U( sl m ) as a module over itself via left multiplication, and then taking its tensor product with the module P N ⊗ (C m ) ⊗N by using the standard comultiplication on U( sl m ). The right module structure on B is defined by using only the right multiplication in the tensor factor U( sl m ) of B. We will also use the adjoint action of U( sl m ) on B. Here The action of the group R m on the Lie algebra gl m preserves the subalgebra sl m . By again identifying the tensor product of N copies of C m [t, t −1 ] with P N ⊗ (C m ) ⊗N , we get an action of the group R m on the former vector space, and hence on the vector space of B.
Take the universal enveloping algebra U( h ) of the Abelian Lie algebra h ⊂ sl m . Let U( h ) be the ring of fractions of the commutative algebra U( h ) with the set of denominators generated by  .3) there exists k ∈ Z also depending on Y , such that ad Eaa−E bb +i C (Y ) = k Y .
Hence the vector space B becomes a bimodule over the algebra U( sl m ). The action of the group R m on the Lie algebra gl m preserves the subalgebra h ⊂ sl m . Moreover, the resulting action of R m on U( h ) preserves the set of denominators generated by (3.3). So the action of R m extends from B to B. We will use the extended action later.

The Lie algebra sl m is generated by the elements
For each c = 0, 1, . . . , m − 1 the elements E c , F c , H c span a subalgebra of sl m isomorphic to sl 2 . We will also use the element ε c ∈t * defined in Subsection 3.1.
Consider the vector spaces B and B introduced in Subsection 3.2. For every c = 0, 1, . . . , m − 1 define a linear map ξ c : B → B by setting for any Y ∈ B and we take the nth power of the adjoint operator corresponding to the element F c ∈ sl m . For any given Y ∈ B only finitely many terms of the sum (3.4) differ from zero, so the map ξ c is well defined. The definition (3.4) and the next proposition go back to [16,Sect. 2]. By using the left action of the Lie subalgebra n ⊂ sl m , introduce the vector subspaces J = n B ⊂ B and J = n B ⊂ B.
Proposition 3.1. For any X ∈ h and Y ∈ B we have Let us use the symbol ≡ to indicate equalities in B modulo the subspace J. By (3.7), for any element Y ∈ B we get Here the relation ≡ is obtained as in the proof of [7, Proposition 3.1]. By following another calculation, as given in the end of the proof of [7, Proposition 3.1], 3.4. The property (3.5) allows us to define a linear mapξ c : B → B/J by settinḡ where the element Z ∈ U( h ) is obtained from A by regarding it as a rational function on the dual vector space h * , and then adding ε c to the argument of that rational function. Recall that in the end of Subsection 3. Proof. Note that τ c (F c ) = E c . If c > 0 then let n c be the subspace of sl m spanned by all the elements E ab t i where i < 0, and by those elements E ab where a > b but (a, b) ̸ = (c + 1, c). Further, let n 0 be the subspace of sl m spanned by the elements E ab where a > b, and by those elements E ab t i where i < 0 but (a, b, i) ̸ = (1, m, −1). Then for any c = 0, 1, . . . , m − 1 the image τ c (J ) ⊂ B is spanned by the subspaces n c B and E c B.
By using the relations (2.1) one can check that the subspace n c ⊂ sl m is preserved by the adjoint action of the elements E c , F c , H c . So we have ξ c (XY ) ∈ J for any X ∈ n c and any Y ∈ B; see (3.4). To prove Proposition 3.2 it remains to show that ξ c (E c Y ) ∈ J for any Y ∈ B. By using the relations (3.7), this can be shown by the same calculation as in the proof of [7,Prop. 3.2]. as the compositionξ c τ c applied to the elements of B which are taken modulo the subspace J. This definition also goes back to [16], and we will call η 0 , η 1 , . . . , η m−1 the Zhelobenko operators on B/J. The next proposition states their key property; for its proof see [10,Sect. 6]. As in the beginning of Subsection 2.4, here we will let the indices c of the operators η c run through Z, assuming that η c+m = η c .  By the definition given in Subsection 2.4, the action of the element π ∈ R m on the Lie algebra sl m maps E c , F c , H c respectively to E c+1 , F c+1 , H c+1 . If the index c + 1 here is m, it should be then replaced by 0. Furthermore, the action of the element π on sl m preserves the subalgebra n. Hence the action of π on B determines its action on the quotient B/J. It now follows from the definition (3.4) that on B/J we have π η c = η c+1 π. By using the relations (2.1) one can check that the subspace n ′ c ⊂ sl m is preserved by the adjoint action of the element F c . Hence we have ξ c (XY ) ∈ J ′ for any X ∈ n ′ c and any Y ∈ B; see the definition (3.4). Further, note that Observe that the vector space B/(J + J ′ +Ī ) coincides with the space of ncoinvariants of the sl m -module (2.2) where the tensor factor V is the universal Verma module of level ℓ. Namely, here V is the quotient of the universal enveloping algebra U( sl m ) by the left ideal generated by n ′ and by the element C − ℓ. This V satisfies the annihilation condition stated before (2.6). By applying Corollary 2.2 to this sl m -module V , we define an action of the Cherednik algebra C N on the quotient vector space B/(J + J ′ +Ī ).
Note that the action of the element π on B also determines a linear map (3.9). For κ ̸ = 0 this map does not commute with the action of C N ; see Corollary 2.5. However, we still have the following theorem. Consider the linear map (3.9) determined by the Zhelobenko operator η 0 . We can write η 0 = π −1 η 1 π; see the end of Subsection 3.4. The action of the element π ∈ R m on B commutes with multiplication by the variables x 1 , . . . , x N in the tensor factor P N of B, and also commutes with simultaneous permutations of these variables and of the corresponding N tensor factors C m of B. By using the argument from the previous paragraph when c = 1, and by applying Corollary 2.5 when V is the universal Verma module of level ℓ, we can now complete the proof of our theorem. If we denote simply by υ the linear map (3.9) determined by the Zhelobenko operator η 1 , then for any p = 1, . . . , N we have The group T m generated by τ 0 , τ 1 , . . . , τ m−1 can be regarded as the Weyl group of the affine Lie algebra sl m . The quotient of R m by the relation π m = 1 can be then regarded as the extended Weyl group of sl m . These two facts underline our definition of the operators η 0 , η 1 , . . . , η m−1 . In the next section we will apply Theorem 3.6 when the universal Verma module V of level ℓ appearing above is replaced by the usual Verma module M β of sl m .

Intertwining operators
4.1. Using the right action of the Lie subalgebra h ⊂ sl m , take the vector subspace This subspace depends on the weight µ ∈ t * via its restriction β to h ⊂ t . Let I β be the sum of all these subspaces and of the subspace I ⊂ B introduced just before stating Theorem 3.6. As an sl m -module, the quotient B/( J ′ + I β ) can be identified with Here sl m acts on the quotient via the left action of the algebra U( sl m ) on its bimodule B. Now suppose that the sequence (µ 1 , . . . , µ m ) of complex numbers obeys the conditions Note that then the sequence λ = (λ 1 , . . . , λ m ) obeys the same conditions, since λ a − µ a ∈ Z for a = 1, . . . , m by our assumption. Similarly to I β , denote byĪ β the sum of all subspaces and of the subspaceĪ. As a module of sl m , the quotient B/( J ′ +Ī β ) can be also identified with (4.1). The quotient B/( J + J ′ +Ī β ) is then identified with the space A N (M β ) n of n-coinvariants of (4.1). The shifted action of the affine Weyl group R m ont * determines an action of R m on h * . We use the same symbol • to denote the latter action. Then by (3.2) Here the summand ε c defined in Subsection 3.1 is regarded as a linear function on the vector space h by restriction fromt. Due to (3.6) and to the last displayed equality, we haveξ c ( τ c (Ī β )) ⊂ J +Ī τc• β . So the Zhelobenko operator η c defined in Subsection 3.4 determines a linear map see also Subsection 3.5. Via the identifications described above, (4.3) becomes a linear map Then by (3.5) the restriction of (4.3) to the subspace of vectors of weight α becomes a linear map Now consider the action of the element π ∈ R m on B. This action preserves the subspaces J and J ′ of B. Further, by the definition of the subspaceĪ β of B we have π(Ī β ) =Ī π( β ) .
Here we employ the usual, not shifted action of R m ont * . But we also have π( β ) = π• β; see again Subsection 3.1. Hence the action of π on B determines a linear map B/( J + J ′ +Ī β ) → B/( J + J ′ +Ī π• β ). (4.5) Via the identifications described above, (4.5) becomes a linear map Then the restriction of (4.5) to the subspace of vectors of weight α becomes a linear map We can replace the source and target C N -modules in (4.4) by their isomorphic modules, using Proposition 2.3. The value of f appearing in that proposition is the same for the sequence µ and for the sequence τ c • µ instead of µ; see the end of Subsection 3.2. Hence our replacement modules in (4.4) will be pullbacks of respectively S λ µ and S τc•λ τc•µ relative to the same automorphism (1.5). By applying the inverse of this automorphism, the Zhelobenko operator η c now determines an C N -intertwining linear map S λ µ → S τc•λ τc•µ . (4.7) Note that because ℓ = κ − m, the conditions (4.2) on the sequence µ can be restated as µ a − µ b / ∈ Z + κ Z for 1 a < b m.
Under the latter conditions both the source and target C N -modules in (4.7) are irreducible by [1,Prop. 2.4.3]. Hence any intertwining linear map between them is unique up to a factor from C. In the next subsection we will determine this scalar factor for the intertwining map determined by the Zhelobenko operator η c . Now consider the map (4.5) and the corresponding map (4.6), which are determined by the action of the element π ∈ R m on B. The map (4.6) is not C Nintertwining unless κ = 0; see Corollary 2.5. However, it will become intertwining if we replace the target C N -module in (4.6) by its pullback via the automorphism (1.5) of C N where f = κ/m.
Here we used the formula for π•µ given in Subsection 3.1.
Since the values of f for the source and the target replacement modules are the same, the action of π on B now determines a C N -intertwining operator S λ µ → S π•λ π•µ . (4.8) By the irreducibility of the source and of target induced C N -modules here, the latter operator must coincide with the intertwining operator from [13] up to a scalar multiplier. Any element of R m has a reduced decomposition of the form π g τ c · · · τ d where g is the degree of this element relative to the Z-grading defined in Subsection 2.4. By using Corollary 2.4 and the relation (3.8), the composition of linear maps π g η c · · · η d : B/J → B/J now determines a C N -intertwining operator S λ µ → S π g τc···τ d •λ π g τc···τ d •µ . It is defined as a composition of intertwining operators of the form (4.7),(4.8) corresponding to generators of R m . This is the intertwining operator mentioned in the Introduction, where ω is now the image of the element π g τ c · · · τ d under the isomorphism of groups R m → S m ⋉ Z m . Indeed, under this isomorphism the shifted actions of the two groups on λ and µ correspond to each other.
Due to the Poincaré-Birkhoff-Witt theorem for the universal enveloping algebra U( sl m ), the images of all these elements in the source quotient vector space of the maps (4.3) and (4.5) make a basis in this quotient. Now suppose that the numbers 1, . . . , m occur respectively ν 1 , . . . , ν m times in the sequence a 1 , . . . , a N . Then the image of the element Y i1...i N a1...a N in the source quotient has weight α relative to the left h-module structure on the quotient. This is because the image of the element 1 ∈ U( sl m ) in the Verma module M β has weight β.
where b 1 , . . . , b N is a sequence obtained from a 1 , . . . , a N by changing ν c − h terms c to c + 1, and also changing ν c+1 − h terms c + 1 to c.
Proof. Applying the action of τ c on B to the element Y i1...i N a1...a N amounts to replacing every c in the sequence a 1 , . . . , a N by c+1, and the other way round. Let d 1 , . . . , d N be the sequence so obtained. Apply the operator ξ c to the resulting element of B by using (3.4). Note that for any n = 0, 1, 2, . . . we have an equality in B E n c ad n Fc (Y i1...i N d1...d N ) = x i1 1 · · · x i N N ⊗ E n c (F n c (e d1 ⊗ · · · ⊗ e d N ) ⊗ 1).
Modulo the subspace J ′ ⊂ B, the element of B displayed here at the right-hand side equals x i1 1 · · · x i N N ⊗ E n c F n c (e d1 ⊗ · · · ⊗ e d N ) ⊗ 1.
In its turn, the last displayed element equals the sum of elements of the form Y i1...i N b1...b N taken with certain multiplicities. Here the multiplicity is the number of ways the sequence b 1 , . . . , b N can be obtained from d 1 , . . . , d N by consecutively replacing any n occurences of c by c + 1, and then consecutively replacing any n occurences of c + 1 by c in the result.
Denote by h the number of those terms of the sequence d 1 , . . . , d N which are equal to c, but change to c + 1 in the sequence b 1 , . . . , b N . Obviously h ν c+1 . When passing from d 1 , . . . , d N to b 1 , . . . , b N as above, the numbers of occurences of 1, . . . , m remain the same. Therefore h is also the number of those terms of the sequence d 1 , . . . , d N which are equal to c + 1, but change to c in b 1 , . . . , b N . Hence where j p = i p + δ ap1 − δ bp1 = i p − δ apm + δ bpm for p = 1, . . . , N whereas b 1 , . . . , b N is a sequence obtained from a 1 , . . . , a N by changing ν 1 −h terms 1 to m, and also changing ν m − h terms m to 1.