Generalized Macdonald-Ruijsenaars systems

We consider the polynomial representation of Double Affine Hecke Algebras (DAHAs) and construct its submodules as ideals of functions vanishing on the special collections of affine planes. This generalizes certain results of Kasatani in types A_n, (C_n^\vee,C_n). We obtain commutative algebras of difference operators given by the action of invariant combinations of Cherednik-Dunkl operators in the corresponding quotient modules of the polynomial representation. This gives known and new generalized Macdonald-Ruijsenaars systems. Thus in the cases of DAHAs of types A_n and (C_n^\vee,C_n) we derive Chalykh-Sergeev-Veselov operators and a generalization of the Koornwinder operator respectively, together with complete sets of quantum integrals in the explicit form.

The main goal of this paper is to obtain commutative algebras of difference operators containing generalizations of Macdonald-Ruijsenaars operators using special representations of double affine Hecke algebras (DAHAs). In the work [1] the Macdonald-Ruijsenaars operator related to the root system R = A N −1 appeared as the relativistic version of quantum Calogero-Moser system, this operator has the form where T x i = e ∂x i is the shift operator and η, are parameters. Macdonald derived and considered operator (1.1) and its generalizations for any root system and (quasi-)minuscule coweight in the context of multi-variable orthogonal polynomials [2]. Commutative algebras of difference operators containing the operators constructed by Macdonald were realized by Cherednik with the help of his DAHA and its polynomial representation [3,4]. These algebras are centralizers of the Macdonald-Ruijsenaars operators in the algebras of Weylinvariant difference operators [5]. The whole area of DAHA is a very rich mathematical subject with deep connections and interactions with combinatorics, theory of (quantum) symmetric spaces, representation theory, noncommutative geometry [6]. The first generalization of the Macdonald-Ruijsenaars operator which is not directly associated to a root system was found by Chalykh [7]. The operator is a difference version of the Hamiltonian which describes interaction of the system of Calogero-Moser particles of two possible masses [8]. The approach of [7] (see also [9]) is based on the multidimensional Baker-Akhiezer functions, in this case all but one particles have equal masses. The generalization of Chalykh's operator to the system of N 1 + N 2 particles of two types was introduced by Sergeev and Veselov [10], the corresponding operator has the form m =ℓ e y ℓ − e ym+ e y ℓ − e ym T η y ℓ . (1.2) In the work [10] the operator (1.2) was obtained from the action of the Macdonald-Ruijsenaars operator (1.1) extended to the space of symmetric functions in the infinite number of variables with subsequent restriction of the operator (1.1) to the special discriminant subvariety. Further rational generalized Macdonald-Ruijsenaars operators were derived in [11], [12] starting from the generalized Calogero-Moser systems and using the bispectrality.
In the present work we develop a uniform approach to the generalized Macdonald-Ruijsenaars systems based on the special representations of DAHAs. DAHA can be associated to any affine root system [2]. The main cases are DAHA of type R and of type R ∨ where R is a reduced irreducible root system, R is its affinization and R ∨ denotes the dual affine root system (see Cherednik's papers [13] and [3] respectively), and DAHA of type (C ∨ n , C n ) (see [14], [15]). The original Cherednik's construction [3,13] allowed to obtain Macdonald-Ruijsenaars systems for any root system R from the polynomial representation of the corresponding DAHA. The difference operators from the commutative algebra are obtained by the averaging of the Cherednik-Dunkl operators over the Weyl group with subsequent restriction of the result to the space of invariant polynomials. In [14] Noumi generalized the construction for the (C ∨ n , C n ) case which allowed to obtain commutative algebra containing the Koornwinder operator [16]. In order to obtain generalized Macdonald-Ruijsenaars systems we first specialize DAHA-invariant ideals in the polynomial representation, these ideals consist of functions vanishing on certain collections of planes. Then the action of the invariant combinations of Cherednik-Dunkl operators is defined on the functions on these planes. Upon restriction to Weyl group invariants we obtain commutative algebras of difference operators. In this way we recover operator (1.2) (under assumption that /η ∈ Z) for the case of DAHA of type A and derive further known and new generalized Macdonald-Ruijsenaars operators starting from other root systems. Similar approach to the generalized Calogero-Moser systems was developed by one of the authors in [17]. It is based on finding special ideals invariant under the rational Cherednik algebras with subsequent restriction of invariant combinations of Dunkl operators thus generalizing Heckman's construction of the usual Calogero-Moser systems [18].
The idea to describe submodules of the polynomial representations of DAHAs by vanishing conditions was very successfully used by Kasatani in [19], [20] (see also discussion in the introduction of [21] where the polynomial representation is studied from another perspective). Thus in the work [19] a filtration of the polynomial representation in type A by the ideals given by vanishing ("wheel") conditions was introduced so that it provides composition factors of the polynomial module (see also [22]). The paper [20] deals with the polynomial representation of DAHA associated to the affine root system (C ∨ n , C n ). Ideals in the ring of symmetric polynomials invariant under the Macdonald-Ruijsenaars operator (1.1) were earlier defined by wheel conditions in [23]. The structure of the paper is as follows. In Section 2 we recall Cherednik's construction of the commutative algebra of difference operators using DAHA of type R. In Section 3 we introduce arrangements of planes D corresponding to subsystems S 0 in the root system R and we define ideals of functions I D vanishing on these planes. Our first main result (Theorem 3.3) states that for the special geometry of the subsystem S 0 and for the related choice of parameters for DAHA the ideals I D are DAHA-invariant. The arising arrangements D are shifted versions of the linear arrangements appearing in the study of invariant ideals for the rational Cherednik algebras [17]. In Subsection 3.4 we construct further invariant ideals given by vanishing conditions where we allow parallel planes. This generalizes construction of such ideals for DAHA for classical root systems from [19,20].
In Section 4 we present our next main result (Theorem 4.1) which explains how commutative algebras of difference operators arise in the quotients of the polynomial representations by the above ideals I D . We show in Subsection 4.2 that these algebras contain complete sets of algebraically independent operators. We verify in Subsection 4.3 that in type A we obtain operators (1.2). We also construct explicitly a complete set of commuting operators for (1.2) (Theorem 4.5). We consider further examples illustrating the approach in Subsections 4.4, 4.5. Thus we demonstrate that starting from the Macdonald operator of type B we obtain trigonometric version of the operators known from [12], [11]. Subsystems in the exceptional root systems provide new families of commutative algebras, we demonstrate this by getting a restricted operator starting from the Macdonald operator for the root system R = F 4 .
In Section 5 we extend our constructions for the (C ∨ n , C n ) DAHA. We specify invariant ideals in the polynomial representation by imposing vanishing condition on suitable arrangements of planes (Theorem 5.2). These submodules correspond to the intersections of the submodules investigated in [20]. We introduce generalized Koornwinder operator and derive it as a restriction of the Koornwinder operator in Propositions 5.4, 5.5. This operator depends on six parameters like the original Koornwinder operator, and also it reflects that there are interacting particles of two types. Although the restriction construction imposes some integrality relation(s) on the parameters we show that the operator can be embedded into a large commutative algebra of difference operators for any values of the parameters. In Theorem 5.7 we find a complete set of quantum integrals in the explicit form. This generalizes van Diejen's quantum integrals for the Koornwinder operator [30].
In Section 6 we present our constructions for DAHA of type R ∨ [3]. The invariant ideals are specified in Theorem 6.1.

Macdonald-Ruijsenaars systems from DAHA
In this section we recall the construction of the Macdonald-Ruijsenaars systems from the Cherednik-Dunkl operators. We consider here DAHAs of type R (other cases will be considered in Sections 5, 6 and Appendix A). Following [13,4,6,24] we define DAHA for the R case, then we define the Cherednik-Dunkl operators as elements of DAHA acting in the polynomial representation and describe the Macdonald-Ruijsenaars systems in these terms.

Double Affine Hecke Algebra
Let R ⊂ R n ⊂ V = C n be an irreducible reduced root system of rank n. Let (·, ·) be the standard bilinear symmetric non-degenerated form in V and let β ∨ = 2β/(β, β). We have (α, β ∨ ) ∈ Z for all α, β ∈ R. Consider the orthogonal reflections s α corresponding to the roots α ∈ R, they act on x ∈ V as s α (x) = x − (α, x)α ∨ . The group W generated by the orthogonal reflections s α , α ∈ R, is a Weyl group.
Let R + be a set of positive roots and ∆ = {α 1 , . . . , α n } ⊂ R + be the set of simple roots. The reflections s α 1 , . . . , s αn generate the Weyl group W . Let us introduce lattices of coroots, of coweights and of weights: where b i and ω i are fundamental coweights and fundamental weights: Introduce also the subsets of dominant and anti-dominant coweights: Let V be the space of affine functionals on V . Each functional α ∈ V acts on x ∈ V as α(x) = (α (1) , x) + α (0) , where α (1) ∈ V and α (0) ∈ C, we write α = [α (1) , α (0) ] in this case. The roots γ ∈ R can be considered as elements of V acting as γ(x) = (γ, x). Define the bilinear form on the space V by the formula (α, β) = (α (1) Let ∈ C be a non-zero complex parameter. Introduce the functional α 0 = [−θ, ], where θ ∈ R is the maximal root for R with respect to the basis ∆. For λ ∈ V denote by τ (λ) the shift operator τ (λ)x = x + λ, ∀x ∈ V .
Extended affine Weyl group is defined as W = W ⋉τ (P ∨ ) with the commutation relations wτ (λ) = τ (wλ)w, where w ∈ W , λ ∈ P ∨ . The action of the group W on V induces the following linear action on the space V : Let Π = {π ∈ W | π ∆ = ∆} ⊂ W be a subgroup of elements preserving the set of simple affine roots. Then W = Π ⋉ W . The group Π is finite: Π = {π r | r ∈ O}, where O ⊂ {0, 1, . . . , n} and each element π r is uniquely determined by the condition π r (α 0 ) = α r . They have the form π 0 = 1, π r = τ (b r )u −1 r for r ∈ O ′ , where O ′ = O\{0} and u r ∈ W are transformations preserving the set {−θ, α 1 , . . . , α n } and satisfying u r (α r ) = −θ. The corresponding fundamental coweights b r form the entire set of minuscule coweights, that is We consider the weight lattice P as a sublattice of P . Now we define DAHA corresponding to the affine system R. Let t : R → C\{0} be a W -invariant function on R, that is t β = tŵ (β) , for all β ∈ R,ŵ ∈ W , where we denote t β = t(β). We also denote t i = t α i for i = 0, 1, . . . , n, these values uniquely determine the function t. Let q = e /2 . We will assume that / ∈ πiQ, so q is not a root of unity.
Definition 2.1. ( [13], see also [24]). Double Affine Hecke Algebra (for R case) is defined as the algebra H H t = H H R ,t generated by the elements T 0 , T 1 , . . . , T n , by the set of pairwise commuting elements {X µ } µ∈P , and by the group Π = {π r | r ∈ O} with the relations where i, j = 0, 1, . . . , n.
Note that the generators T 0 , T 1 , . . . , T n , X µ , π r of the algebra H H t are invertible elements. We define an element Tŵ ∈ H H t for each elementŵ ∈ W according to its reduced expression: In particular, T πr = π r and T s i = T i . Forŵ,v ∈ W one has the relation Since for the fundamental coweights we have l τ . . , n. Let us define the commutative family of elements Y λ for λ ∈ P ∨ : (2.14) They span a subalgebra C[Y ] of H H t isomorphic to C[P ∨ ] and they satisfy the relations (see [24]) where i = 1, . . . , n. These formulae imply that the generators T 1 , . . . , T n commute with the elements of the subalgebra consisting of the polynomials invariant under the usual action w : Y λ → Y wλ of the Weyl group W . Let F = C ∞ (V ) 2πiQ ∨ be the space of smooth complex-valued periodic functions f on V with the lattice of periods 2πiQ ∨ , that is f (x + 2πiα ∨ j ) = f (x) for j = 1, . . . , n. The group W acts on F asŵ (2.17) The elements of the algebra H H t can be represented as operators acting in F : where µ ∈ P , r ∈ O, i = 0, 1, . . . , n. Rearranging the last formula into the form we see that T i f ∈ F for all f ∈ F . Indeed, for any i = 0, 1, . . . , n the periodic function is periodic and smooth:g ∈ F . Therefore, the function also belongs to F . The formulae (2.18) define a faithful representation of DAHA H H t on the space F called polynomial representation (extended to smooth functions). The elements Y λ in this representation are called Cherednik-Dunkl operators.

Macdonald-Ruijsenaars systems
It is spanned as a vector space by the elements where W λ = {wλ | w ∈ W } is the orbit of the coweight λ ∈ P ∨ and W λ = {w ∈ W | wλ = λ} is the stabilizer of λ. We will normally be choosing λ ∈ P ∨ − for the operators m λ (Y ). In the representation (2.18) the elements m λ (Y ) have the form where g λ ν,w (x) are some meromorphic functions and τ (ν) are the shift operators acting on The subspace F W ⊂ F can be characterized as a space annihilated by all the operators T 1 −t 1 , . . . , T n −t n . Since the elements of C[Y ] W commute with T 1 , . . . , T n they preserve this subspace: m λ (Y ) F W ⊂ F W . Thus we can restrict the elements of C[Y ] W to F W [3,27]. The restriction of (2.22) is Let Mer(V ) be the space of meromorphic functions ϕ(x) on V such that ϕ(x + 2πiα ∨ j ) = ϕ(x) for all j = 1, . . . , n. Define the subspace M ⊂ Mer(V ) consisting of functions ϕ ∈ Mer(V ) holomorphic out of the union of hyperplanes β∈ R {x ∈ V | β(x) ∈ 2πiZ}. Consider the smash-product algebra D = M # C[τ (P ∨ )] defined by the commutation relation τ (λ)ϕ(x) = ϕ(x − λ)τ (λ). This is the algebra of difference operators acting on the space M. Any element a ∈ D has the form a = λ∈P ∨ w∈W g λ,w (x)τ (λ)w, where g λ,w ∈ M and the sum over λ is finite. Note that M λ ∈ D for any λ ∈ P ∨ .
Remark 2.1. One can show that the operator (2.31) has the form c λ λ M λ + ν∈P ∨ − ,ν>λ c λ ν M ν such that c λ λ = 0. Then [29, VI, §3, 4, Lemma 4] implies that the operators (2.31) form a basis of the space R. It follows the commutative algebra R is isomorphic to the algebra of polynomials C[z 1 , . . . , z n ], and the isomorphism can be given by M −b j → z j . In particular the elements M −b 1 , . . . , M −bn generate the algebra R.
The subalgebra R ⊂ D is called the Macdonald-Ruijsenaars system. The Macdonald-Ruijsenaars operator (2.24) or the operator (2.26) can be considered as a Hamiltonian of this system.

Invariant shifted parabolic ideals
The goal of this section is to construct some H H t -submodules of F , which will be used in the next section. Note that any ideal of the algebra F is invariant under the action of the generators X µ ∈ H H t . We define some ideals as the sets of functions vanishing on certain planes. We show that these ideals are invariant under the actions of other generators of DAHA if parameters satisfy special conditions. Let η : R → C be a W -invariant function on the irreducible reduced root system R, that is η w(α) = η α = η(α), ∀α ∈ R, ∀w ∈ W . We will also denote η i = η α i for the simple roots α i . To each subset S ⊂ R we associate the plane (3.1) Note that H η (wS) = wH η (S) for any w ∈ W . Let S 0 be a subset of the set of simple roots ∆ and let W 0 ⊂ W be the subgroup generated by s α , α ∈ S 0 . We associate to S 0 the following subset Let V 0 = S 0 be the linear subspace of V spanned by the set S 0 . Then R 0 = V 0 ∩ R is the root system with the set of simple roots S 0 . Note that the transformations w ∈ W + restricted to R 0 respect the standard order. More exactly, if β γ, β, γ ∈ R 0 then wβ wγ. (Here Remark 3.1. The elements of W + are the shortest coset representatives of the left cosets of W 0 in W (see [26]).
Let us consider the plane and the union of planes The arrangement D is obtained by shifting the linear arrangement W H 0 (S 0 ), where H 0 (S 0 ) is defined by the formula (3.3) with the function η ≡ 0. More exactly, the following two statements take place.
Definition 3.1. Let θ = n i=1 n i α i be the maximal root of R with respect to the set of simple roots ∆. The number is called generalized Coxeter number for the root system R and the invariant function η.
We note that the generalised Coxeter number for the root system R does not depend on the choice of R + (or, equivalently, on the choice of ∆). For a constant function η α ≡ η the generalized Coxeter number h η = η · h, where h = 1 + n i=1 n i is the usual Coxeter number (see [29]).
Remark 3.2. The generalized Coxeter number can be equivalently defined as the coefficient of proportionality of the following W -invariant inner products (c.f. [17]): Note that the restriction of the function η : R → C to the subsystems R ℓ ⊂ R is a W ℓ -invariant function on R ℓ . Since R ℓ is irreducible one can define the generalized Coxeter numbers for this subsystem: where η ℓ j = η α ℓ j . Define the ideal of functions vanishing on the union of planes (3.4): (3.12) The main result of this section is the following theorem. We will prove this theorem in the next tree subsections. We establish the invariance under T i , 1 i n in Subsection 3.1, the invariance under π r is established in Subsection 3.2 and the invariance under T 0 is established in Subsection 3.3.
Remark 3.3. The degenerations of the ideals I D at η → 0 give invariant parabolic ideals for the rational Cherednik algebra H c (W ) with c = η/ . Any radical ideal invariant under H c (W ) can be obtained this way (see [17]).
Remark 3.4. The invariant ideals I D for the A series are the ideals obtained by Kasatani in [19], namely these are the ideals I (k,r) m with r = 2 and the same m as we use. (The ideals in [19] are considered in the space of polynomials of multiplicative variables).

Let us record the introduced planes in terms of affine functionals. For a root
(3.14) We will write H β 1 , β 2 , . . . instead of H {β 1 , β 2 , . . .} . In these notations the definitions (3.1) and (3.4) take the form 3.1 Invariance under T 1 ,. . . , T n Lemma 3.4. Let S = wS 0 where S 0 ⊂ ∆, and w ∈ W + . Let β ∈ ∆. If β ∈ S then β cannot be expressed as a linear combination of the elements of S.
Taking into account that wα ∈ R + for any α ∈ S 0 we see that the relation β = α∈S 0 γ α wα holds only if β ∈ S, but this contradicts the condition of the lemma. Proof. Let f ∈ I D . Consider the subset S = wS 0 ⊂ R + , where w ∈ W + . We need to prove that the function T i f (x) vanishes on the plane H(S ′ ).
Consider first the case α i ∈ S. Consider the expression (2.20) for the function T i f (x). Let x ∈ H(S ′ ), then the first term in this expression vanishes since the function f (x) vanishes on the plane H(S ′ ) and the second term vanishes since t −1 vanish: T i f (x) = 0. Note that for any p ∈ Z the intersection of the planes H(S ′ ) and and this implies, in particular, that α i can be obtained as a linear combination of the elements of S, but this is impossible by Lemma 3.4. Thus the subset Remark 3.5. If the W -invariant function η : R → C is fixed then one can choose a Winvariant function t : R → C satisfying the conditions of Proposition 3.5 by the formula t [β, k] = e η β /2 for all β ∈ R and k ∈ Z.

Invariance under the group Π
Lemma 3.6. (c.f. [29]). Let θ = n i=1 n i α i be the maximal root of R with respect to the set of simple roots ∆. Then n r = 1 for some r if and only if there exists u r ∈ W such that Proof. In the notations of Subsection 2.1 the statement means O ′ = {r | n r = 1} and it is proved in [29]. Now let r ∈ O ′ . Define the number r * ∈ O ′ such that π r * = π −1 r , then u r * = u −1 r , which leads to u r (−θ) = α r * . Taking into account that u r (α r ) = −θ, u r (α i ) = α j , n r = 1 and u r (∆\{α r }) = ∆\{α r * }, we rearrange u r (− n i=1 n i α i ) = α r * as whereñ k ∈ Z. We obtain n i = n j since j = r * by extracting the coefficient at α j .
Proposition 3.7. Suppose that the subsystem S 0 ⊂ ∆ and the function η are such that h ℓ η = for all ℓ = 1, . . . , m. Then the arrangement (3.4) and the ideal (3.12) satisfy π r D = D and π r I D = I D for any r ∈ O.
Proof. It is sufficient to establish that π −1 r D ⊂ D for any r ∈ O ′ . Consider the subset wS 0 ⊂ R + , where w ∈ W + . We will show that there existsw ∈ W + such that Indeed since the transformation π −1 r preserves the set ∆ it also preserves the set R + . Therefore u r α = π −1 r α ∈ R + ∩ R = R + , where we used u r = π −1 r τ (b r ) and u r α ∈ R. Lemma 3.6 and r ∈ O ′ imply n r = 1. Since the roots wα ℓ j and wθ ℓ = k ℓ j=1 n ℓ j wα ℓ j are positive and coweight b r is minuscule the products (wθ ℓ , b r ) and (wα ℓ j , b r ) are equal to 0 or 1. So for each ℓ one of the following two possibilities holds: We can fix p ℓ = 1 by renumbering the roots α ℓ 1 , . . . , α ℓ k ℓ if necessary. Consider first the case (i): γ ℓ rj = δ j1 and n ℓ 1 = 1. Let us first rearrange the plane H(wS ′ ℓ ) as follows. We can replace in ; so taking into account wθ ℓ = k ℓ j=1 n ℓ j wα ℓ j and the formula (3.11) we obtain This gives (1) , α (0) ] ∈ V , It follows from (3.17) that u r wα ℓ j ∈ R + for j = 2, . . . , k ℓ . We note that −u r wθ ℓ ∈ R + since (u r wθ ℓ , u r b r ) = 1 and u r b r ∈ P ∨ − . The latter property is proved as follows. Since Π is a group there exists r * ∈ O ′ such that π −1 r = π r * . Rearranging . Since n ℓ 1 = 1 it follows by Lemma 3.6 applied to the root system R ℓ that there exists an In the case (ii) the formulae (3.20) are valid for v ℓ = 1. Indeed π −1 r H(wS ′ ℓ ) = H(u r wS ′ ℓ ) and it follows from (3.17) that u r wα ℓ j ∈ R + for all j = 1, . . . , k ℓ .
hencew ∈ W + , and Thus the invariance of D and, as consequence, the invariance of I D are proven.

Invariance under T 0
If O = {0} then there exists r ∈ O ′ and the invariance of I D under T 0 follows from the formula we need to prove the invariance under T 0 separately. We will prove it in the general case. We start with two preliminary lemmas.
Lemma 3.8. Let w ∈ W + . Suppose the maximal roots for the system R and the subsystem R ℓ for some ℓ satisfy wθ ℓ = θ. Then θ ∈ wR ℓ .
Now the invariance of the ideal I D under the action of the operator T 0 can be established. Proof. We need to prove that for any f ∈ I D and w ∈ W + the function . Therefore the right hand side in (2.20) with i = 0 vanishes.
Otherwise wθ ℓ = θ for all ℓ and by virtue of Lemma 3.9 we have s 0 H(S ′ ) ⊂ D. Hence the function But this is impossible due to Lemma 3.8. Thus the continuous function T 0 f (x) vanishes on all the plane H(S ′ ). Theorem 3.3 follows from Propositions 3.5, 3.7, 3.10.

Further invariant ideals
In this subsection we generalize previously constructed ideals I D by requiring functions to vanish on the families of parallel planes containing the planes wD 0 with w ∈ W + . For DAHA associated with the classical root systems such submodules were considered by Kasatani in [19,20]. Consider a subset wS 0 ⊂ R, where w ∈ W , and its decomposition wS 0 = m ℓ=1 wS ℓ . For a function ξ : For an element w ∈ W we define the following set of functions Let η : R → C be a fixed W -invariant function. To a subset S ⊂ R and a function ξ : S → Z 0 we associate the plane (3.27) Starting from the subset S 0 and r we define the union of planes We will assume that r is such that where GCD stands for the greatest common divisor. Note that in the case r = 0 the union D r coincides with D = W + D 0 defined by the formula (3.4). We also generalize the conditions h ℓ η = as h ℓ η = (r ℓ + 1) .
where D r is given by (3.28), is invariant under the action of DAHA H H t .
We will need the following technical lemma for the rest of the proof of Theorem 3.11.
Proof. It is sufficient to establish π −1 r D r ⊂ D r for arbitrary r ∈ O ′ . Consider an element w ∈ W and a function ξ ∈ Ξ r (w). We will show that there exists an elementw ∈ W and a function ξ ∈ Ξ r (w) such that π −1 r H η (wS 0 ; ξ) = H η (wS 0 ;ξ). Note that we have where ξ ℓ = ξ wS ℓ . For each ℓ we have two possible cases: For u = u r and λ = b r we can apply Lemma 3.13. Indeed, it easy to see that u r b r ∈ P ∨ − (see the proof of Proposition 3.7). The transformation u = u r τ (−b r ) = π −1 r preserves the set ∆ and henceûR + ⊂ R + . Since b r is a minuscule coweight m ℓ i ∈ {−1, 0, 1} for i = 0, 1, . . . , k ℓ (we use the notations of Lemma 3.13) and hence |ξ| w ℓ + m ℓ 0 r ℓ + 1. Further we have ξ ℓ j + m ℓ j 0 because otherwise ξ ℓ j = 0 and m ℓ j = −1; since ξ −1 (0) ⊂ R + the first equality means that wα ℓ j ∈ R + , but this contradicts the second equality.

Generalized Macdonald-Ruijsenaars systems
In this section we use the invariant ideals I D found in Section 3 to obtain commutative algebras of difference operators acting in the space of functions defined on the affine planes. We work in the assumptions of Theorem 3.3, so that the ideal I D is invariant and the quotient module is defined. The symmetrized Cherednik-Dunkl operators m λ (Y ) considered in the quotient representation define a commutative algebra of difference operators. We consider explicit examples of operators arising this way.

The operation of restriction
In Section 2 we reviewed how the Macdonald-Ruijsenaars systems are obtained in terms of the polynomial representation of DAHAs. In this section we show how the generalized Macdonald-Ruijsenaars systems are obtained in terms of the representation This operator as an operator on D 0 is given more explicitly by the next theorem. Before its formulation we introduce some notations.
Let V = {x ∈ V | (α, x) = 0, ∀α ∈ S 0 }. Denote byλ the orthogonal projection of a vector λ ∈ V onto the subspace V and by P ∨ the set of projections of the coweights: P ∨ = {λ | λ ∈ P ∨ } ⊂ V . The shift operators τ (λ) preserve the affine plane D 0 and therefore they act on functions ψ on this plane: τ (λ)ψ(x) = ψ(x − λ ), where x ∈ D 0 . Let M be the space of meromorphic functions on D 0 which are restriction ψ| D 0 , ψ ∈ M. Define the algebra D of difference operators on D 0 by D = M # τ (P ∨ ). Note that any element a ∈ H H t in the representation (2.18) has the form where g λ,w ∈ M Formula (4.3) defines the algebra homomorphism res : Let f x 0 λ (x) ∈ F W be a function which equals 1 in the points x ≈ x 0 − λ and vanishes outside the small neighbourhoods of these points. Note that for any f ∈ F W the function (af )(x) is smooth and that f D 0 = 0 implies (af ) D 0 = 0 since aI D ⊂ I D . Substitute f = f x 0 λ 0 for λ 0 ∈ P ∨ and x = x 0 into the formula We get that for any Decomposing the element w ∈ W as w = w ′ w 0 with w ′ ∈ W + , w 0 ∈ W 0 (see Remark 3.1 and [26]) we obtain w ′ V 0 = V 0 . Then by applying Proposition 3.2 we derive w ′ D 0 = D 0 and hence w 0 (x 0 − λ) ∈ D 0 + 2πiQ ∨ .
Let the W -invariant function η satisfying the conditions h ℓ η = be generic. Then one can show that w 0 (x 0 − λ) ∈ D 0 + 2πiQ ∨ implies x 0 − λ ≈ x 0 − λ . Note that the right hand side of (4.4) at x = x 0 has the form (af )( where we used the W -invariance and 2πiQ ∨ -periodicity of the function f (x). One can prove that if x 0 − λ ≈ x 0 − ν for some λ, ν ∈ P ∨ thenλ =ν. Thus we getν =λ if G x 0 λ (x 0 ) = 0 and ν ∈ P x 0 λ . Hence the formula (4.5) can be rearranged to We established the equality (4.6) for generic x 0 ∈ D 0 and η. Since the left hand side is defined for all x 0 ∈ D 0 and η the equality is valid for all x 0 ∈ D 0 and η. Since res ρ D (a)f D 0 (x 0 ) = (af )(x 0 ) for f ∈ F W the equality (4.3) follows. Note that by putting and generic x 0 ∈ D 0 . Notice that if A ∈ D is such that Aφ = 0 for any φ ∈ F 0 then A = 0. Indeed consider the Then one yields g λ ′ 0 (x 0 ) = 0 for generic x 0 ∈ D 0 and all λ ′ 0 ∈ P ∨ . It follows that the linear map res : ρ D C[Y ] W → D defined by the formula (4.3) is an algebra homomorphism.
We are going to derive generalized Macdonald-Ruijsenaars operators using the operation resρ D from Theorem 4.1. First it is useful to note that restrictions to the invariant functions and restrictions to the quotient module can be interchanged. More specifically, let D ∈ D be an operator of the form D = λ∈P ∨ g λ τ (λ). Define the operator in the case when for any λ ′ ∈ P ∨ the restricted function is well defined meromorphic function on D 0 . The algebra R coincides with the image of the map ρ D : R → D, thus it is spanned by the elements M λ , λ ∈ P ∨ . The following diagram of surjective linear maps is commutative:

Integrability of generalized Macdonald-Ruijsenaars systems
Letn = dim D 0 = n − |S 0 |, where |S 0 | is a number of elements in the subset S 0 ⊂ ∆. The set of indexes J = {j ∈ Z >0 | α j / ∈ S 0 } has sizen, let J = {j 1 , . . . , jn}. If j ∈ J then the corresponding fundamental coweight belongs to the subspace V , that is b j = b j , b j ∈ P ∨ and τ (±b j ) preserve the plane D 0 . Proof. By the formula (2.30) with λ = −b j we have Indeed, if a weight ν ∈ P ∨ satisfies ν ≺ −b j then it follows the corresponding inequality between the lengths: |ν| | − b j |. If ν = ν thenν = ν = −b j . Ifν = ν then one has |ν| < |ν| hence |ν| < | − b j | and thereforē ν = −b j . Taking into account that b j = b j one yields Let us prove that the function g −b j D 0 does not vanish identically. Indeed, otherwise there exists an affine root α = [α (1) α for all x ∈ D 0 . This implies that α (1) ∈ R is a linear combination of the elements of S 0 and, consequently, (α (1) , b j ) = 0. But on the other hand

A n type examples
The root system R = A n is defined as {e i − e j | i, j = 1, . . . n + 1, i = j} considered in the where {e i } is the standard orthonormal basis in C n+1 . The simple roots are α 1 = e 1 − e 2 , α 2 = e 2 − e 3 , . . . , α n = e n − e n+1 and the maximal root is θ = e 1 − e n+1 All fundamental coweights b r , r = 1, . . . , n are minuscule. Any W -invariant function t : R → C\{0} is constant.
We will consider functions f on V as functions of n + 1 variables satisfying the condi- Taking the coweight b 1 =ê 1 in (2.24) we obtain the operator Any root subsystem R 0 has irreducible components R ℓ of type A only in its decomposition R 0 = m ℓ=1 R ℓ . The invariant function η = const. The generalised Coxeter number is h ℓ η = kη for R ℓ ∼ = A k−1 . Thus the conditions (3.13) lead to R 1 ∼ = R 2 ∼ = . . . ∼ = R m ∼ = A k−1 and = kη for some k ∈ Z 2 . Let us choose where N 1 = n + 1 − mk and N 2 = m. Introduce new variables Proposition 4.4. Let t = e η/2 . The restriction of the operator (4.14) onto D 0 takes the form g(x N 1 +kℓ ′ −s − a; t, 1) = g(y ℓ ′ − a; t k , 1), where a ∈ C and g(z; b, c) s=0 e N 1 +kℓ−s and y ℓ = (e i , x) − (k − 1)η/2 (see the end of Subsection 4.2). Taking into account the condition = kη and its exponential form q = t k we can write the restriction of (4.14) in the form (4.17).
Note that the operator (4.17) does not effectively depend on k, it depends on N 1 , N 2 , , η only, where it is assumed that /η ∈ Z >0 . The operator (4.17) was previously obtained and investigated in [10] for general parameters and η and earlier in [7] in the case N 2 = 1.
The formula (2.24) gives the explicit form of n Macdonald operators where |I| is the number of elements in the set I ⊂ {1, . . . , n + 1}. Similar to the proof of Proposition 4.4 we calculate the restrictions of the operators (4.18) onto D 0 . We obtain where q = e /2 , t = e η/2 . In the case r = 1 the operator (4.19) takes the form (4.17). Proof. For any r, r ′ ∈ Z >0 the commutator has the form where the coefficients C k 1 ,...,k N 1 m 1 ,...,m N 2 (x, y; q, t) are rational functions of q = e /2 and t = e η/2 . It follows from Proposition 4.2 that these functions equal zero if , η ∈ C\πiQ such that /η ∈ Z >1 and N 1 + η N 2 > max(r, r ′ ). Thus the coefficients vanish for any parameters and η, and the operators commute.
The second part follows from the fact that the operators M −br at = η are specialisations of the Macdonald operators M −br with n = N 1 +N 2 −1, which are algebraically independent.
If α n ∈ S 0 then we suppose that the numeration of the components S ℓ is such that α n ∈ S m . Then  The variables x 1 , . . . , x N 1 together with (4.23) are coordinates on D 0 .

Remark 4.1. The operator (4.32) can be rewritten in the form
).
Remark 4.3. It is possible to obtain a commuting operator to (4.39) by restriction of the relevant Macdonald operators for the small weights [31]. Similar one can get Hamiltonians and quantum integrals starting from the Macdonald operators for the small weights in the case of other exceptional root systems. 5 The case (C ∨ n , C n ) (C ∨ n , C n ) (C ∨ n , C n ) The affine root system of type (C ∨ n , C n ) is given by (see [25,2]). Let W be the group generated by the affine reflections s α with α ∈ S. The corresponding DAHA is defined in [15] (see also [14]). We recall the associated commuting difference operators below.
Let R = C n , that is is the affine Weyl group for C n . Let Q ∨ , P ∨ , P and W be the lattices of coroots, of coweights, of weights and the Weyl group for the root system R = C n respectively. Let also P ⊂ V be the subset defined in Subsection 2.1 for R = C n . 5.1 Noumi representation of (C ∨ n , C n ) (C ∨ n , C n ) (C ∨ n , C n ) type DAHA and Koornwinder operator Let t, u : R → C\{0} be such that t α = tw α = t(α), u α = uw α = u(α) for anyw ∈ W , α ∈ R. Put u ±e i ±e j = 1, t i = t α i , u i = u α i . So we have t 1 = t 2 = . . . = t n−1 and u 1 = u 2 = . . . = u n−1 = 1.
Definition 5.1. [15] Double Affine Hecke Algebra of type (C ∨ n , C n ) is the algebra H H t,u generated by the elements T 0 , T 1 , . . . , T n , and by the set of pairwise commuting elements {X µ } µ∈ P with the relations (2.4)-(2.7), (2.10) and
The algebra H H t,u depends on six independent parameters t 1 , t n , t 0 , u n , u 0 and . Note that the elements π r are not included in this case. One can also define an 'extended' version of (C ∨ n , C n ) type DAHA where affine Weyl group W = W ⋉ τ (Q ∨ ) is replaced with the extended affine Weyl group W = W ⋉ τ (P ∨ ). However the W -invariance of the functions t, u leads to less independent parameters in this algebra and subsequently in the commuting difference operators. Note also that if u 0 = u n = 1 then the algebra H H t,u can be interpreted as a 'non-extended' version of DAHA of type C n .
Consider the action of the generators of H H t,u on the space F = C ∞ (V ) 2πiQ ∨ given by the Noumi representation [14] X µ = e µ(x) , where µ ∈ P , i = 0, 1, . . . , n. The elements Tw ∈ H H t,u withw ∈ W and Cherednik-Dunkl operators Y λ ∈ H H t,u with λ ∈ Q ∨ are defined by the formula (2.12), where π r = 1, and The operators Y λ commute with each other and satisfy the relations [14,2] where λ ∈ Q ∨ ,t i = t i for i = 1, . . . , n − 1 andt n = t 0 .
Let C[Y ] be the commutative subalgebra of H H t,u spanned by the Cherednik-Dunkl operators (5.6). The relation (5.7) implies that the elements of the subalgebra C[Y ] W ⊂ C[Y ] preserve the subspace of W -invariant functions F , that is m λ (Y )F W ⊂ F W , where λ ∈ Q ∨ and m λ (Y ) is the operator defined by the formula (2.21). By restricting these operators to the subspace F W we obtain the difference operators M λ given by formula (2.23) with (2.22).
Thus we get a commutative algebra R spanned by M λ with λ ∈ Q ∨ . It is generated by the algebraically independent elements M ν i = M −ν i , where ν i = e 1 + e 2 + . . . + e i and i = 1, . . . , n. The first of the generators is the Koornwinder operator [14,16] ).

Invariant ideals and generalized Koornwinder operator
Let η : R → C be a W -invariant function. It is defined by independent parameters η 1 = η α 1 and η n = η αn . Let S 0 ⊂ ∆ and let D 0 , W + , D and I D be defined in the same way as in Section 3 for the case R = C n . Let S 0 = m ℓ=1 S ℓ and R 0 = m ℓ=1 R ℓ be the decompositions defined in Subsection 3.2. We denote k ℓ = |S ℓ |.
There are two possible cases: α n / ∈ S 0 and α n ∈ S 0 . In the former case the components R ℓ ∼ = A k ℓ for all ℓ = 1, . . . , m. In the latter case one of the components is of type C. We suppose α n ∈ S m so that R ℓ ∼ = A k ℓ for all ℓ = 1, . . . , m − 1 and R m ∼ = C km .
Theorem 5.2. Suppose the subset S 0 ⊂ ∆ and the function η are such that the condition (5.9) is satisfied. Suppose the parameters of DAHA H H t,u are such that t 2 1 = e η 1 if n > 1 and ǫ n t n u ǫn n = e ηn/2 for some ǫ n = ±1. Suppose also that if α n ∈ S 0 then ǫ 0 ǫ n t n t 0 u ǫn n u ǫ 0 0 t 2(k−1) 1 for some ǫ 0 = ±1, wherek = |S m | is the rank of the irreducible component of type C. Then the ideal I D ⊂ F is invariant under the action of H H t,u in the Noumi representation (5.5).
Proof. Let f ∈ I D and x ∈ wD 0 for some w ∈ W + . The operators T 1 , . . . , T n−1 have the same form as the operators T 1 , . . . , T n−1 of Section 3 in the case R = C n , and the proof of invariance under their action follows by Proposition 3.5.
To prove the invariance under T n note that if α n / ∈ wS 0 then we have f (x) = f (s n x) = 0 and, hence, T n f (x) = 0 (see the proof of Proposition 3.5). If α n ∈ wS 0 then 1 −t n u n e −xn = 0 or 1 + t n u −1 n e −xn = 0 hence T n f (x) = 0. To prove the invariance under T 0 suppose first that wθ ℓ = θ for some ℓ. This is possible only if α n ∈ S 0 and ℓ = m since α n ∈ S m . In this case we obtain −α 0 ( In the case when wθ ℓ = θ for all ℓ = 1, . . . , m the plane s 0 (wD 0 ) = s 0 H η (wS 0 ) is contained in D due to Lemma 5.1 and hence f (s 0 x) = 0. As in the proof of Proposition 3.10 it follows that T 0 f (x) = 0 for all x ∈ wD 0 .
Theorem 5.2 guarantees that the arguments of Subsection 4.1 remain valid in the (C ∨ n , C n ) case. Thus we obtain the commutative algebra R spanned by the operators M λ , λ ∈ Q ∨ , defined by the formula (4.7).
Indeed, the operators M λ in this case have the form (2.30) with .
The algebra R contains the generalized Koornwinder operator. Let us introduce this operator explicitly. It acts on functions of two sets of variables x 1 , . . . , x N 1 , y 1 , . . . , y N 2 and it depends on six independent parameters: two shift parameters , ξ and the parameters a, b, c, d. It is defined as where q = e /2 , s = e ξ/2 and g(ǫy ℓ − x j ; q 1/2 s 1/2 , q 1/2 s −1/2 )g(ǫy ℓ + x j ; q 1/2 s 1/2 , q 1/2 s −1/2 )× . (5.14) For N 2 = 0 this operator coincides with the Koornwinder operator (5.8) up to a constant and a factor (where a, b, c, d, ξ are expressed through the parameters of DAHA H H t,u ). We note that this operator is invariant (up to a factor) under the involution Now we explain how this operator arises by the restriction of the Koornwinder operator M −e 1 . Consider first the case α n / ∈ S 0 . In this case we assume that all the subsets S ℓ coincide with (4.22) and t 1 = e η 1 /2 . Then N 1 = n − mk, N 2 = m, = kη 1 , q = t k 1 . We introduce orthogonal coordinates x 1 , . . . , x N 1 , y 1 , . . . , y N 2 on the corresponding plane D 0 by (4.23).
The propositions can be established by direct calculations. In both cases the obtained operators M −e 1 depend on six independent parameters. Namely, in the case α n / ∈ S 0 we have five continuous parameters ξ, a, b, c, d and one discrete parameter k (where = kξ), and in the case α n ∈ S 0 we have four continuous parameters ξ, b, c, d and two discrete ones k,k ( = kξ, a = c −1 e (k+1)ξ ).
It follows from above that the operator M N 1 ,N 2 comes together with the complete algebra of commuting difference operators. We specify these operators explicitly and establish commutativity (without extra integrality restriction for a combination of parameters). We start with the explicit formula for the higher Koornwinder operators obtained by van Diejen [30]: where r = 1, . . . , n, (5.20) where b = 1, 2, J 3 = ∅ and q = e /2 . These operators commute with each other, and for r = 1 we have H 1 = t 0 t n (M −e 1 − κ t ), where M −e 1 is the Koornwinder operator (5.8).
The operators H r are W -invariant. Since for generic parameters the centralizer of M −e 1 in the algebra of W -invariant difference operators (with suitable coefficients) coincides with the algebra R (see [5,Theorem 3.15]) we obtain H r ∈ R. Hence H r ∈ R for any values of the parameters, and the restrictions of the operators H r onto the corresponding plane D 0 can be defined and they commute with each other.
Consider restriction of the operators (5.18) in the case α n / ∈ S 0 . The structure of the functions (5.20) implies that if U b is not zero on the plane D 0 then J b have the form where I 2 ⊂ I 1 ⊂ {1, . . . , N 1 } and m ± ℓb are non-negative integers such that m ± ℓ2 m ± ℓ1 , m − ℓ1 + m + ℓ1 k and either m − ℓ2 or m + ℓ2 vanishes for each ℓ. Moreover, the signs ǫ i are given by We recall the coordinates x 1 , . . . , x N 1 , y 1 , . . . , y N 2 defined by (4.23) on the plane D 0 .
Proposition 5.6. The restriction of the operator H r takes the form with b = 1, 2, I 3 = ∅, m ± ℓ3 = 0 and q = e /2 , t 1 = e η 1 /2 . Note that we do not impose the condition m − ℓ1 + m + ℓ1 k in the sum in (5.23) since the product U 1 U 2 vanishes if m − ℓ1 +m + ℓ1 > k, so the operators H r do not have explicit dependence on k.

(5.25)
Here we used the formulae where m, M ∈ Z are such that m M + 1, a, z ∈ C (the second formula is used in the case b = 1 only). If (5.26) while if ℓ = ℓ ′ we derive the following two factors. The factor corresponding to ǫ i = −ǫ j = +1 is equal to ). (5.27) (In the case b = 2 both sides of the equality (5.27) equal 1). The factor corresponding to ǫ i = ǫ j = ǫ = ±1 is equal to Now consider the restriction of the second product in the expression (5.20) which is taken If N 1 < i < j then i = N 1 +kℓ−s and j = N 1 +kℓ ′ −s ′ , where 1 ℓ, ℓ ′ N 2 , 0 s, s ′ k−1.
In the case ℓ = ℓ ′ we obtain ), (5.30) and for ℓ = ℓ ′ we derive (the first factor in the right hand side corresponds to ǫ ′ = −ǫ and the second one corresponds to ǫ ′ = ǫ). By combining the factors (5.27), (5.28) and (5.31) we obtain the beginning and ending of the formula (5.24). Then combining (5.26) and the factors of (5.30) with ℓ < ℓ ′ we obtain the corresponding factors of (5.24). By combining (5.25) with (5.29) and taking into account all the remaining factors we obtain the formula (5.24). For r = 1 we obtain the generalized Koornwinder operator H 1 = M N 1 ,N 2 with the parameters as in Proposition 5.4, so we get a collection of operators commuting with the operator (5.12). Note that if = η 1 the operators H r coincide with the specialization of H r for n = N 1 + N 2 . Thus we obtain the following theorem.
where R is an irreducible reduced root system in R n ⊂ V ∼ = C n with set of simple roots ∆ = {α 1 , . . . , α n }. We have R ∨ ∼ = R only for R = B n , C n , F 4 and G 2 , but we will consider the general R. Let W , W , Π, O be the same as in Section 2 and let t : Then P ∨ is a sublattice of P ∨ . The associated DAHA is defined as follows.
Proof. First we define the W -invariant function η ′ : R → C by the formula η ′ α = (α,α) 2 η α ∨ . Note that one can represent (6.7) as By virtue of (6.10) the ideal (6.8) for the function η : R ∨ → C coincides with the ideal (3.12) for the function η ′ , moreover the conditionsȟ ℓ η = are equivalent to h ℓ η ′ = where h ℓ η ′ is the generalized Coxeter number for the function η ′ . Hence the invariance under the action of π r follows from Proposition 3.7.
Let f ∈ I D and x ∈ wD 0 for some w ∈ W + . Consider first the invariance under T i with i = 1, . . . , n. In the case α i ∈ wS 0 it follows from f (x) = 0 and (t −1 i − t i e −α ∨ i (x) ) = 0. In the case α i / ∈ wS 0 we have f (x) = f (s i x) = 0 and the plane wD 0 = H η ′ (wS 0 ) does not lie in any plane H([α ∨ i , 2πik]) = H([α i , πik(α i , α i )]) due to Lemma 3.4 (see the proof of Proposition 3.5 for detail).
Let M λ be the restriction of the difference operator M λ to the plane D 0 defined as in Subsection 4.1. Repeating the reasoning of Subsections 4.1 and 4.2 we conclude that the operators M λ span the commutative algebra R containingn = dim D 0 algebraically independent elements. This is the generalized Macdonald-Ruijsenaars system for the R ∨ case.

Concluding remarks
We considered submodules in the polynomial representation of DAHA given by vanishing conditions and derived the corresponding generalized Macdonald-Ruijsenaars operators and their quantum integrals acting in the quotient module. In some particular cases the eigenfunctions of these operators can be given by Baker-Akhiezer functions ( [7], [9], [11]) but more generally the understanding of eigenfunctions is a major challenge in the area. It follows from our construction that the restriction of a Weyl invariant eigenfunction of the higher Macdonald operators M λ onto the plane D 0 is an eigenfunction of the corresponding operators M λ . However the question of finding the solutions is non-trivial as the specializations of the parameters so that the ideals I D are invariant are such that the Macdonald polynomials do not exist in general at these specializations. We also note that more generally, the non-symmetric generalized eigenfunctions for the Cherednik-Dunkl operators at the special parameters are studied in [21]. Further comparison with [21] may be important for extracting the information on the eigenfunctions of the operators M λ . At generic values of the parameters for the restricted operators the eigenfunctions might have different structure and description. Thus in type A a complete set of eigenfunctions (so called super Macdonald polynomials) is constructed in [10] at generic η, , these polynomials are symmetric in each set of variables x i , y j and span a subspace in the space of functions of x i , y j with simple description.
Another important question is on the composition series of the polynomial representation.
In the case of R = A N it is given in terms of the ideals defined by vanishing ("wheel") conditions ( [19], [22]). It would be interesting to see if the ideals I D can be used to define the composition factors in some other cases. Further, the vanishing conditions of the submodules in the polynomial representation for the rational Cherednik algebras were helpful to establish unitarity of certain submodules [32]. It would be interesting to see if analogous approach can be applied for the case of DAHA.

A Other affine root systems
The Macdonald-Ruijsenaars systems and their generalized versions can be defined starting from a general affine root system. The cases of affine root systems not considered in the main part of the paper lead to particular cases of the operators M λ from Section 5. Nevertheless we present these cases for completeness. Let S ⊂ V be an irreducible affine root system [25,2] and let W = W (S) be the group generated by {s α | α ∈ S} -affine Weyl group for the affine root system S. Let ∆ 1 = {a 0 , a 1 , . . . , a n } be a basis of S and let The sets S 1 = {α ∈ S | α/2 / ∈ S} and S 2 = {α ∈ S | 2α / ∈ S} are reduced affine root subsystems in V and ∆ 1 and ∆ 2 are bases of S 1 and S 2 respectively. The group W is generated by the affine simple reflections s i = s a i = sā i .
Let t : S 2 → C\{0} and u : S 2 → C\{0} be W -invariant functions such that u α = u(α) = 1 for all α ∈ S 1 ∩S 2 . Let t i = tā i , u i = uā i , i = 0, 1, . . . The Proposition follows from the established cases S = (C ∨ n , C n ) and S = R where R is a reduced root system. Now consider the affine root systems not covered in the previous sections. These are S = (B n , B ∨ n ), BC n , (BC n , C n ) and (C ∨ n , BC n ). In the above notations we have S 1 = B n , S 2 = B n ∨ for S = (B n , B ∨ n ), S 1 = S 2 = BC n for S = BC n ; S 1 = BC n , S 2 = C n for S = (BC n , C n ); S 1 = C n ∨ , S 2 = BC n for S = (C ∨ n , BC n ). We consider the standard realizations of these affine root systems, so in particular, S ⊃ D n = {±e i ± e j | 1 1 < j n}.
Define the lattice L Y = Z n = P ∨ Bn = Q ∨ Cn = P ∨ Dn , where P ∨ Bn , Q ∨ Cn and P ∨ Dn are the coweight lattice for R = B n , the coroot lattice for R = C n and the coweight lattice for R = D n respectively. Let H L Y Comparing the expressions Y e 1 Bn = π Bn 1 T 1 · · · T n−2 T n−1 T Bn n T n−1 · · · T 1 , (A.8) Y e 1 Dn = π Dn 1 T 1 · · · T n−2 T Dn n T n−1 · · · T 1 (A.9) we derive Y e 1 Bn = Y e 1 Dn . Due to the formula Y e i R = T −1 i−1 Y e i−1 R T −1 i−1 valid for both R = B n and R = D n we establish the equality (A.6) for all λ = e i , where i = 1, . . . , n, and, hence, for all λ ∈ L Y .
It follows that the operators M λ for the affine root system S coincide with the operators M λ from Section 5 at the corresponding specialization. In particular the specializations of the Koornwinder operator (5.8) give the operators M −e 1 determined by (A.3). The specializations of the generalized Koornwinder operator (5.12) give the generalized Macdonald-Ruijsenaars operators M −e 1 for the affine root system S. Each specialization of the operators M λ from Section 5 gives commutative algebra containing the operator M −e 1 . These algebras were constructed in Sections 4, 6 for S = B n , D n , B n ∨ . For other cases the invariant ideals and these operators can also be defined using the specialization of the invariant ideals for S = (C ∨ n , C n ). For example consider the case S = (B n , B ∨ n ), so S 2 = B n ∨ . Let η : B ∨ n → C be a Winvariant function and let S 0 be a subset of the set of simple roots of B n . Define the ideal I D by the formulae (6.7) and (6.8). Let S 0 = m ℓ=1 S ℓ be the decomposition to the irreducible components S 1 , . . . , S m ⊂ S 0 . Letȟ ℓ η be the corresponding generalized Coxeter number defined by (6.9). LetF be the space defined in Subsection 6 for R = B n . Then the following statement takes place.
Theorem A.4. Suppose that t 2 1 = e η 1 if n > 1, ǫ n t n u ǫn n = e ηn/2 for some ǫ n = ±1 anď h ℓ η = for all ℓ = 1, . . . , m. Then the ideal I D ⊂F is invariant under the actions of the operators (A.2) and π 1 . The corresponding generalized Macdonald-Ruijsenaars operator M −e 1 can be obtained by the specialization of the parameters in the generalized Koornwinder operator (5.12).