Affine Pieri rule for periodic Macdonald spherical functions and fusion rings

Let $\hat{\mathfrak{g}}$ be an untwisted affine Lie algebra or the twisted counterpart thereof (which excludes the affine Lie algebras of type $\widehat{BC}_n=A^{(2)}_{2n}$). We present an affine Pieri rule for a basis of periodic Macdonald spherical functions associated with $\hat{\mathfrak{g}}$. In type $\hat{A}_{n-1}=A^{(1)}_{n-1}$ the formula in question reproduces an affine Pieri rule for cylindric Hall-Littlewood polynomials due to Korff, which at $t=0$ specializes in turn to a well-known Pieri formula in the fusion ring of genus zero $\widehat{\mathfrak{sl}}(n)_c$-Wess-Zumino-Witten conformal field theories.


Introduction
The Hall polynomials form a generalization of the Littlewood-Richardson coefficients that provide the structure constants of the classical Hall algebra in the basis of Hall-Littlewood polynomials; these structure constants (which are polynomial in the Hall-Littlewood parameter) are known to enjoy a very intricate combinatorics [M95, Chapters II, III]. Indeed, the Hall algebra and its generalizations in terms of quivers turn out to encode a host of combinatorial, algebra-geometric, and representation-theoretic data [BS15,M95,S12,WZ18]. Recently, Korff introduced an affine analog of the Hall polynomials; these arise as structure constants of a t-deformation of the fusion ring (a.k.a. Verlinde algebra) for sl(n) c -Wess-Zumino-Witten conformal field theories with respect to a natural basis built from cylindric Hall-Littlewood polynomials [K13]. While to date the precise geometric and/or representation-theoretic interpretation of this t-deformed fusion ring has yet to be disclosed, indications of an intimate relation with the deformed Verlinde algebras in [T04,TW09] have been noticed [GP17, K13,OY14].
At t = 0 the Hall-Littlewood polynomials become Schur polynomials. The corresponding Littlewood-Richardson coefficients [M95,Chapter I.9] and their affine counterparts, which arise as fusion coefficients for sl(n) c -Wess-Zumino-Witten conformal field theories [DMS97, Chapter 16], have received massive attention across the mathematics literature because of their rich combinatorics and profound applications in representation theory and Schubert calculus, cf. e.g. [F97] and [GW90, G91, KS10, MS12] as well as further references therein. Korff's t-deformation is different from the q-deformed fusion ring in [FLOT96], which recovers the sl(n) c -Wess-Zumino-Witten fusion ring at the value q = 1. Of special interest is in this connection the well-known fact that the closely related gl(n) c -fusion ring amounts to a q = 1 degeneration of the small quantum cohomology ring of the Grassmannian of n-dimensional linear subspaces in C n+c [A95,BCF99]. The structure coefficients of this small quantum cohomology ring in the basis of Schubert classes, the genus zero 3-point Gromov-Witten invariants, can be computed as quantum counterparts of the Littlewood-Richardson coefficients for Schur polynomials [B97, G91, I91, R01,ST97,T05,V92,W95]. Various other combinatorial constructions related to the computation of genus zero 3-point Gromov-Witten invariants have been considered in the literature, e.g. via the structure constants of algebras of symmetric polynomials in bases of cylindric Schur polynomials [P05, M06], in bases of k-Schur polynomials [LM08, L-S10, L-Z14], or in bases of noncommutative Schur polynomials in variables from a plactic algebra [KS10], respectively.
If at least one of the two factors in the Littlewood-Richardson product consists of a Hall-Littlewood polynomial attached to a partition with only a single column, then the explicit form of the pertinent Hall polynomials is given by the Pieri formula [M95,Chapter III.3]. The affine analog of this Pieri formula for cylindric Hall-Littlewood polynomials can be found in [K13,Corollary 7.4]. The purpose of the present work is to generalize the affine Pieri formula in question from sl(n) c to the case of an arbitrary affine Lie algebraĝ [K90], excluding those of type BC n = A (2) 2n . In other words,ĝ is assumed to be untwisted or to be the twisted counterpart of an untwisted affine Lie algebra.
Let us recall at this point that from the perspective of Lie algebras the Hall-Littlewood polynomials in n variables are associated with sl(n). The corresponding generalization of these polynomials to simple Lie algebras of arbitrary type is given by the Macdonald spherical functions [M01,NR03,P06,S06], which were constructed originally by Macdonald as spherical functions on p-adic symmetric spaces [M71]. With the aid of suitable representations of the affine Hecke algebra, the Pieri formula for the Hall-Littlewood polynomials was generalized to a Pieri formula for Macdonald spherical functions of arbitrary simple Lie type in [DE12].
The key to achieve an analogous generalization of the affine Pieri formula in [K13] is to connect with the work in [D06]. To this end, we will detail briefly how affine Pieri formulas arise in the context of [D06], while also emphasizing in which sense these differ from the usual Pieri formulas for the Hall-Littlewood polynomials in [M95].
In order to generalize the affine Pieri formula (1.2) from sl(n) c to other affine Lie algebras, we present an affine counterpart of the Pieri formula for Macdonald spherical functions of arbitrary simple Lie type from [DE12], which stems from the implementation of periodic boundary conditions. The underlying representations of the affine Hecke algebra that lead to this affine Pieri formula are inspired by previous constructions for the graded affine Hecke algebra that were developed in the context of the study of quantum integrable particle models, cf. [GS79, EOS06] and references therein. From this perspective, a partial construction for twisted affine Lie algebras can be found in [DE13]; here we apply these techniques to present a combinatorial model to compute the structure constants of deformed genus zero Wess-Zumino-Witten fusion rings for both twisted and untwisted affine Lie algebras (excluding those of type BC n = A (2) 2n , cf. [D20]). In line with was remarked at the end of the first paragraph for sl(n) c , we expect that these deformed fusion rings are isomorphic to deformed Verlinde algebras from [T04,TW09]; for sl(2) c this isomorphism is manifest from the explicit construction in [AGP16, Appendices A and B].
The material is organized as follows. Section 2 presents our deformation of the genus zero Wess-Zumino-Witten fusion ring, which is built from a basis of periodic Macdonald spherical functions. The main result is an affine Pieri rule that permits to compute the structure constants for the multiplication in the periodic Macdonald spherical basis by basis elements attached to weights that are either minuscule or quasi-minuscule. After setting up some further notational preliminaries concerning the affine Weyl group in Section 3, the pertinent structure constants are exhibited in Section 4. When the deformation parameter vanishes, one finds a corresponding Pieri formula and structure constants for the genus zero Wess-Zumino-Witten fusion ring itself. The bulk of the paper is devoted to the proof of our Pieri rule via a suitable representation of the Hecke algebra of the affine Weyl group. Specifically, the affine Hecke algebra is first employed in Section 5 to construct an affine intertwining operator acting in the space of complex functions over the weight lattice. Via a standard construction involving the idempotent associated with the trivial representation of the Hecke algebra of the finite Weyl group, the periodic Macdonald spherical functions arise in Section 6 upon acting with the affine intertwining operator. In Section 7 it is shown that the periodic Macdonald spherical functions give rise to a basis for a finite-dimensional algebra of functions supported on critical points of a 'fusion potential' of the type in Eq. (1.3). We apply the affine intertwining operator so as to derive a family of difference operators diagonalized by the basis of periodic Macdonald spherical functions. The action of these difference operators permits us to compute the corresponding structure constants associated with this basis. In Section 8 the computation in question is carried out explicitly for the particular case of the Pieri formula, and Section 9 outlines how to recover the structure constants more generally from the action of the difference operators.
2. Affine Pieri Rule 2.1. Macdonald spherical functions. Let V be a real finite-dimensional Euclidean vector space with inner product ·, · spanned by an irreducible reduced crystallographic root system R 0 . We write Q, P , and W 0 , for the root lattice, the weight lattice, and the Weyl group associated with R 0 . The semigroup of the root lattice generated by a (fixed) choice of positive roots R + 0 is denoted by Q + whereas P + stands for the corresponding cone of dominant weights (see e.g. [B68, H90] for more details concerning root systems). The dual root system R ∨ 0 := {α ∨ | α ∈ R 0 } and its positive subsystem R ∨,+ 0 are obtained from R 0 and R + 0 by applying the involution (2.1) Definition 2.1. For λ ∈ P + , the Macdonald spherical function M λ : V → C is the W 0 -invariant trigonometric polynomial given explicitly by Here t : R 0 −→ C is a root multiplicity function such that t wα = t α for every w ∈ W 0 and α ∈ R 0 .
For our purposes the range of the root multiplicity function will be restricted such that t : R 0 → (−1, 1) \ {0}.

Basis of periodic Macdonald spherical functions.
Let ϕ and ϑ denote the highest root and the highest short root of R + 0 , respectively. We fix an admissible pair (R 0 ,R 0 ) withR 0 being equal either to R ∨ 0 or to u ϕ R 0 , where u ϕ = 2 ϕ,ϕ , and with the positive systemR + 0 obtained from R + 0 . In particular, for simply-laced R 0 We also writeQ,Q ∨ ,P andP ∨ for the root lattice, the co-root lattice, the weight lattice and the co-weight lattice ofR 0 , respectively. In this setup it will turn out natural to extend the domain of the root multiplicity function in a straightforward manner: t : Given a fixed positive integer c > 1, we consider two affine alcoves in P andP : and an associated set of nodes is defined as the unique global minimum of a radially unbounded strictly convex Let C(P c ) denote the algebra of functions f : P c → C. For any λ ∈ P c , the periodic Macdonald spherical function M (c) λ ∈ C(P c ) is given by the restriction of M λ to the nodes P c .
Theorem 2.2 (Basis). The periodic Macdonald spherical functions M (c) λ , λ ∈ P c , form a basis of C(P c ).
Remark 2.3. In Sect. 6 we will introduce the W 0 -invariant affine Macdonald spherical functions Φ ξ ∈ C(P ). It will be seen that, for ξ ∈ P c , the lattice function Φ ξ is periodic with respect to translations over elements in cQ ∨ ⊂ P and that M (c) λ (ξ) = Φ ξ (λ) for λ ∈ P c (see Rem. 6.5 for more details). 2.3. Structure constants. Theorem 2.2 gives rise to an affine analog of the (tdeformed) Littlewood-Richardson coefficients: When µ is minuscule or quasi-minuscule we have an explicit expression for the structure constants c ν,(c) λ,µ (t). Let us recall in this connection that a weight µ ∈ P is called minuscule if 0 ≤ µ, α ∨ ≤ 1 for all α ∈ R + 0 and quasi-minuscule if 0 ≤ µ, α ∨ ≤ 2 for all α ∈ R + 0 with the upper bound being realized only once (i.e., the quasi-minuscule weight is unique and equal to the highest short root ϑ).
To formulate this explicit expression for the corresponding structure constants let us put: cf. e.g. [M72]).
To state the exact expressions for the coefficient U λ,ω (t) in (2.10) and for c ν,(c) λ,µ (t) in Eq. (2.7) when µ is (quasi)-minuscule, some more notation regarding the underlying affine Weyl group and affine root system is required. (A more thorough discussion can be found e.g. in [B68, H90, M03]).

The affine Weyl group
The affine root system R associated with the admissible pair (R 0 ,R 0 ) is the set of all affine roots α ∨ + m α rc = m α (α + rc) (α ∈ R 0 , r ∈ Z). An affine root a = α ∨ + m α rc ∈ R will be regarded as an affine linear function a : V → R of the form a(x) = x, α ∨ + rc (x ∈ V , α ∈ R, r ∈ Z), and gives rise to an affine reflection s a : V → V across the hyperplane V a := {x ∈ V | a(x) = 0} given by s a (x) := x− a(x)α. The choice of positive roots R + 0 , with a simple basis α 1 , . . . , α n , determines the set of affine positive roots R + := R ∨,+ 0 ∪{α ∨ +m α rc | α ∈ R 0 , r ∈ N} and a corresponding basis of affine simple roots a 0 , . . . , a n of the form a 0 := α ∨ 0 + c and a j := α ∨ j for j = 1, . . . , n. Here n denotes the rank of R 0 (= dim V ). Notice that these conventions imply that the affine root system R is of twisted type if R 0 = u ϕ R 0 is not simply-laced and of untwisted type otherwise.
The affine Weyl group W is defined as the group generated by the affine reflections s a , a ∈ R and contains the finite Weyl group W 0 as the subgroup fixing the origin. It is an infinite Coxeter group with the simple affine reflections s j := s aj (j = 0, 1, . . . , n) as generators and subject to the relations Here m jk = 1 if j = k and m jk ∈ {2, 3, 4, 6} if j = k (and the provision that for n = 1 the order m 10 = m 01 = ∞). In particular, any w ∈ W can be decomposed as Furthermore, since our positive scale parameter c is integral-valued the weight lattice P ⊂ V is stable with respect to the action of W and P c (2.4) is a fundamental domain for this restriction. Given x ∈ V , we will also write w x ∈ W for the unique shortest affine Weyl group element such that where (α ∨ + m α rc) ′ := α ∨ denotes the differential and The action of w ∈ W on V induces a dual action on the space C(V ) of functions f : V → C given by In Section 4 we will use that C(P ) is an invariant subspace under this action.

Affine Littlewood-Richardson coefficients and fusion rules
We are now in the position to make the coefficient U λ,ω (t) in Theorem 2.4 explicit: (with the convention that sign(0) := 0) and (4.2b) (cf. Eq. (2.9)). Here θ : P → N ∪ {0} denotes the function Remark 4.1. Observe that d λ,ν is also a Laurent polynomial in t α , α ∈ R 0 . We will also see in Lemma 8.2 that θ(λ + ν) = 0 if ν is in the orbit of a minuscule weight and therefore it is also possible to write d λ, is called a length multiplicative function. We compatibilize this function with the root multiplicative function by setting t j := t sj = t αj for j = 0, 1, . . . , n. For any finite subgroup G ⊂ W we consider the generalized Poincaré series of G associated with the length multiplicative function t.
When R 0 is of type A n−1 and ω is minuscule, Corollary 4.2 reproduces the affine Pieri rule in Eq. (1.2).
The corresponding parameter degeneration of the structure constants in C(P c ), model the fusion rules of the genus-zero Wess-Zumino-Witten conformal field theories associated with the affine Lie algebraĝ of type H18, K90] (cf. also Remark 4.3 below). Corollary 4.2 gives rise to the following Pieri rule in the fusion ring for µ = ω ∈ P + minuscule or quasi-minuscule: When R 0 is of type A n−1 and ω is minuscule, this Pieri rule amounts to Eq. (1.5).
To infer the boxed Pieri rule, one first observes that lim t→0 V λ,ν−λ (t) = 1, lim t→0 W 0;ω (t) = 1, and is a simple root and 0 otherwise, this shows that lim t→0 U λ,ω (t) = −N λ,ω in view of Lemma 8.2 below. The asserted Pieri rule in the fusion ring is now immediate from Corollary 4.2.

Affine intertwining operator
The remainder of the paper is devoted to the proofs of Theorems 2.2 and 2.4 with the aid of the affine Hecke algebra H. By definition, H is the unital associative algebra over C with invertible generators T 0 , T 1 . . . , T n such that the following relations are satisfied where the number of factors m jk on both sides of the braid relation (5.2) is the same as the order of the corresponding braid relation (3.1) for W (see e.g. [H90, M03]). For a reduced expression w = s j1 · · · s j ℓ , let T w := T j1 . . . T j ℓ (which does not depend on the choice of the reduced expression by virtue of the braid relations). It is known that the elements T w , with w ∈ W , form a basis for H over C.
The subalgebra of H generated by T 1 , . . . T n is referred to as the finite Hecke algebra H 0 (associated with W 0 and t).
To define the affine intertwining operator we need the following integral-reflection representation of H.
Proposition 5.1. The following defines an action of H on C(P ): where J j : C(P ) → C(P ) denotes the operator given by

4)
Proof. For any k ∈ {0, 1, . . . , n} consider the (finite dimensional) parabolic subalgebra H k of H generated by T 0 , T 1 , . . . , T k−1 , T k+1 , . . . , T n . The idea of the proof is to show that, for any (fixed) k, T j → I j , j = k extends to a representation of H k on C(P ). Here I j := t j s j + (t j − 1)J j denote the operator on the right-hand side of Eq. (5.3). The fact that H is generated by T 0 , . . . , T n subjected to the braid relations and quadratic relations implies then the proposition. For this let us introduce the vertices v 0 = 0, v 1 , . . . , v n of the alcove A c (3.3), so in particular We fix a k ∈ {0, 1, . . . , n} and consider the finite subsystem obtained from R by considering the vertex v k as the "new origin": Then a → a ′ defines a root system isomorphism from the parabolic subsystem {a ∈ R | a(v k ) = 0} of R onto R ∨ k . The root system R k is a finite root system of rank n in V , although not necessarily irreducible. A basis of simple roots for R k is given by α j , j = k. The map s j → s ′ j = s α ∨ j , j = k defines a group isomorphism from the parabolic subgroup W k = s j | j = k of W to the finite Weyl group W 0 (R k ) of R k . This isomorphism induces a natural isomorphism from the parabolic subalgebra H k = T j | j = k of H to the finite Hecke algebra H 0 (W 0 (R k )) associated with W 0 (R k ) and t j , j = k.
From R k ⊂ R 0 follows that P ⊂ P (R k ), where P (R k ) denotes the weight lattice of R k . We will also need the (−v k )-translation P ′ k := −v k + P ⊂ P (R k ) of P .
For any j = k consider the integral-reflection operator I ′ j : C(P (R k )) → C(P (R k )) associated to the finite root system R k , i.e. I ′ j = t j s ′ j + (t j − 1)J ′ j where J ′ j is given by the same expression as (5.4) but with a j (λ) replaced mechanically by α ∨ j , λ . Observe that C(P ′ k ) is invariant under the operators J ′ j and I ′ j , j = k. The linear isomorphism ℓ k : By [DE12,Lem. 4.2], applied to the finite root system R k and restricted to the finite Hecke algebra part, it follows that T j → I ′ j , j = k, defines a representation of H 0 (W 0 (R k )) on C(P (R k )) (see also Remark 5.2 below). Since C(P ′ k ) is an invariant subspace under this representation it follows that T j → I ′ j , j = k extends to a representation of H 0 (W 0 (R k )) on C(P ′ k ). Using the above mentioned isomorphism from H k to H 0 (W 0 (R k )) we deduce that T j → I ′ j , j = k extends to a representation of H k on C(P ′ k ). By taking the pullback of the linear isomorphism ℓ k we conclude that T j → I j , j = k extends to representation of H k on C(P ). Since k was arbitrary this finishes the proof, as indicated in the beginning.
Alternatively, the proof of [DE12,Lem. 4.2] works verbatim if the requirement that R be irreducible is dropped.
The affine intertwining operator J : C(P ) → C(P ) is now defined as follows: (5.6) Proposition 5.3. The operator J is invertible.
Proposition 5.3 is a direct consequence of Lemma 5.4 below. Given v, w ∈ W we recall that v ≤ w in the Bruhat partial order on W if v may be obtained by deleting simple reflections from the reduced expression of w (see [M03,Sec. 2.3]). For x ∈ V , we denote by [x] the finite set {y ∈ V | y + = x + and w y ≤ w x } and by Conv [x] the convex hull of [x]. Now we consider the following partial order on P : (5.7) Lemma 5.4. The action of J is triangular with respect to the above partial order: for some coefficients J λ,µ ∈ C and with J λ,λ = t[λ] −1 .
Proof. We observe that t[λ] = t w λ and proceed inductively in the length of w λ . For ℓ(w λ ) = 0 clearly (J f )(λ) = f (λ). Next, assuming ℓ(w λ ) > 0 let j be such that where the step IH hinges on the induction hypothesis and the last equality is due to the fact that the convex hull of Conv [s j λ] and corresponds to term with µ = s j λ and the coefficient of f (λ) in (T j f )(s j λ) is equal to 1 when s j λ ≺ λ (by Eqs. (5.3), (5.4)). Hence, upon comparing the coefficients of f (λ) on both sides of (5.9) it is seen that J λ,λ = t −1 j J sj λ,sj λ , which proves the lemma.
To prepare for the next section, we finish with a convenient characterization of the W -invariant subspace of C(P ) in terms of H and J .
Lemma 5.5. The W -invariant subspace consists of the functions f : C(P ) → C that satisfy Proof. For any f ∈ C(P ), j ∈ {0, . . . n} and λ ∈ P we have Here χ denotes the characteristic function of [0, ∞) and we also used that Hence, f is W -invariant if and only if J T j J −1 f =t j f .

Periodic Macdonald spherical functions
For a ξ ∈ V we define the affine Macdonald spherical functions function in C(P ): where e iξ denotes the plane wave function e iξ (λ) := e i λ,ξ (λ ∈ P ). The plane waves decomposition for φ ξ (6.1) in the next theorem is a known result, see [M68,Thm. 1] and (with more details) [M71, (4.1.2)] or also [NR03,Thm. 2.9(a)] and [P06,Thm. 6.9]). To keep our presentation self contained we include a brief verification based on the representation from Proposition 5.1. For any w ∈ W we define the finite set Let us observe that the cardinality of R(w) is equal to the length of w and that for any λ ∈ P we have that R(w λ ) = R[λ] (the reader may consult [M03, Section 2.2] and [M03, (2.4.4)], respectively). It is also clear that for decomposes as the following linear combination of plane waves In particular, φ ξ (λ) = M λ (ξ) (λ ∈ P + , ξ ∈ V reg ).
Proof. From the action of T j (j = 1, . . . , n) (5.3) we have that for any ξ ∈ V reg Since the stabilizer of ξ ∈ V reg for the action of W 0 ⋉ 2πQ ∨ is trivial, all the vectors wξ, for w ∈ W 0 , are different to each other modulo 2πQ ∨ . Then, for any ξ ∈ V reg the plane waves e iwξ , w ∈ W 0 , are linearly independent in C(P ). Therefore, the function φ ξ may be written as for some unique coefficients C w (ξ) ∈ C. It follows from Eq. (6.5a) that for any reduced expression w = s j ℓ . . . s j1 ∈ W 0 the action of T w on e iξ is of the form c j k (s j k · · · s j2 s j1 ξ)   e iwξ + l.o., (6.5c) for some coefficients c j k and l.o. is a linear combination of plane waves e ivξ with v < w in the Bruhat partial order on W 0 . Let w 0 be the longest element of W 0 . Applying the above identity to a reduced expression w 0 = s j ℓ . . . s j1 (so ℓ = #R + 0 ) we conclude that In the last equality we have used that R + 0 = R(w 0 ) and in the third that (see e.g. [M03, (2.2.9)]) R + 0 = R(w 0 ) = {s j1 s j2 · · · s j k−1 α j k | k = 1, 2, . . . , ℓ}. Let us denote the trivial idempotent having then φ ξ = ı 0 e iξ . Since T j ı 0 = t j ı 0 we have that T j φ ξ = t j φ ξ for j = 1, . . . , n.
It follows from Eq. (6.5a) and the linear independence of the plane waves that for On the other hand, from the product formula in Eq. (6.4) it follows that for any ξ ∈ V reg C(s j ξ)c j (ξ) = C(ξ)c j (−ξ) for all j ∈ {1, . . . , n}.
Hence, C(wξ) also satisfies the recurrence relation in Eq. (6.7). Finally, by downward induction with respect to the Bruhat order starting from the initial condition C w0 (ξ) = C(w 0 ξ) (and using that c j (ξ) = 0), we conclude that C w (ξ) = C(wξ) for all w ∈ W 0 and any ξ ∈ V reg .
Before stating the next results, let us recall that the nodes P c are given by the unique global minima stemming from the strictly convex Morse functions V µ (2.6), µ ∈P c (2.5). Given µ, the existence of the global minimum is guaranteed because V µ (ξ) is smooth and V µ (ξ) → +∞ for ξ → ∞. Since ξ µ is a minimum of V µ , it is a solution for ∇V µ = 0: Lemma 6.2. The critical points ξ µ , µ ∈P c are all distinct and belong to the open alcove (with respect to the affine action of W 0 ⋉ 2πQ ∨ on V ) Moreover, the position of ξ µ depends analytically on the parameters t α ∈ (−1, 1).
Proof. From Eq. (6.8) it is clear that one can recover µ from the value of ξ µ , thus ξ µ = ξ λ if µ = λ. Also, for any β ∈ R + 0 we have that c ξ µ , β + ρ v (ξ µ ), β = 2π ρ + µ, β . (6.10a) Since v α (x) is an odd function it follows moreover that From Eqs. (6.10a), (6.10b) one deduces that ξ µ , β > 0 for µ ∈P c . Here one exploits that v α (x) is strictly monotonously increasing and that Moreover, from Eqs. (6.10b), (6.10c) with β = ϕ and the quasi-periodicity of the function v α (x) one deduces that for ξ µ , ϕ ≥ 2π we would have where in the last connection we used that for any root multiplicity function t : R 0 → C and root β ∈ R 0 we have Now by combining the Eq. (6.10d) with Eq. (6.10a) for β = ϕ, we would have c + ρ, ϕ + 1 ≤ ρ + µ, ϕ = µ, ϕ + ρ, ϕ , which contradicts our assumption that µ ∈P c . Hence, one must have that ξ µ , ϕ < 2π, i.e. ξ µ ∈ A. Finally, it is clear that the critical equation (6.8) is analytic in the parameters t α ∈ (−1, 1). Since the Jacobian of the critical equation is invertible, the implicit function theorem now ensures that the dependence of the critical point ξ µ is also analytic in these parameters. Proof. Since T j ı 0 = t j ı 0 for j = 1, . . . , n (see Eq. (6.6)), we have that (6.11) Hence, by Lemma 5.5, we only need to prove that Now, by comparing with the corresponding decomposition of t 0 φ ξ , we have that By substituting the product expansion for C(·) over R + 0 (cf. Eq. (6.4)) we have The relation now may be written as ∀v ∈ W 0 , (6.12) by using that an overall flip of the signs in the factors at the right-hand side cancels out because α∈R + 0 (−1) β ∨ ,α = (−1) β ∨ ,2ρ = 1 for all β ∈R 0 . To finish this proof, let us observe that if we multiply Eq. (6.10a) by the imaginary unit and exponentiate both sides, using that v α (x) = 2 arctan then it follows that ξ µ is indeed a solution for Eq. (6.12).
Remark 6.4. In [DE12,Eqn. (5.12a)] a Macdonald spherical function Φ ξ ∈ C(P ) W0 was introduced in terms of an intertwiner operator built up (essentially) from an integral-reflection representation of the finite Hecke algebra H 0 . In contrast, the affine Macdonald spherical function Φ ξ ∈ C(P ) W0 (6.1) is based on the affine intertwiner operator J (5.6), built up from the integral-reflection representation of the affine Hecke algebra H.
Remark 6.5. For y ∈ V , let us denote by τ y : V → V the translation determined by the action τ y (x) := x + y. Then the affine Weyl group admits the alternative Because of the above proposition it follows that Φ ξµ (µ ∈P c ) is W 0 -invariant and cQ ∨ -periodic, explaining the name periodic Macdonald spherical function for Φ ξµ .
Remark 6.6. From the proof of Proposition 6.3, it is clear that for every µ ∈P the vector ξ = ξ µ solves the following algebraic system of equations of Bethe type Indeed, at ξ = ξ µ Eq. (6.12) is satisfied and the short roots ofR ∨ 0 generateR ∨ 0 over Z.

Proof of Theorem 2.2 (Basis)
For any ω ∈ P + we consider the free operator L ω;1 : C(P ) → C(P ) given by and the operator L ω : C(P ) → C(P ) given by For the isotropy group W λ of λ ∈ P c in W , Macdonald's product formula for the generalized Poincaré series of the Coxeter group associated with the length multiplicative function t [M72] tells us that withĥ t andê t given by (2.8), (2.9). Armed with this identity, we can readily infer that for any µ ∈P c the function Φ ξµ is nonzero in C(P c ) ∼ = C(P ) W . Indeed, from Proposition 6.1, Lemma 6.2, Proposition 6.3 and the trivial action of J on C(P c ), it follows that λ (ξ) for any ξ ∈ P c and λ ∈ P c . (7.4) In particular, at λ = 0 this yields that where we used Macdonald's identity from Ref. [M72,Thm. (2.8)] for the ⋆ equality.
Proposition 7.1 (Completeness of the Periodic Macdonald Spherical Functions).
The restriction of the functions Φ ξµ , µ ∈P c , constitutes a basis for C(P c ) that diagonalizes the commuting operators L ω simultaneously: Proof. For any ω ∈ P + the action of L ω;1 on a plane wave yields Hence, given µ ∈P c it follows that at ξ = ξ µ : The upshot is that the nontrivial eigensolutions Φ ξµ , µ ∈P c in Eq. (7.5) must be linearly independent in C(P c ), in view of Lemma 6.2 and the well-known fact that the W 0 -invariant trigonometric polynomials m ω (e iξ ), ω ∈ P + , separate the points of the fundamental alcove A.
To finish the proof, it suffices to verify that dim C(P c ) = |P c | = |P c |, for this confirms that the eigenfunctions Φ ξµ (µ ∈P c ) form a basis of C(P c ). To this end we first observe that P c consists of all nonnegative integral combinations c 1 ω 1 + · · · + c n ω n of the fundamental weights of R 0 satisfying c 1 m 1 + · · · + c n m n ≤ c, where the positive integers m 1 , . . . , m n refer to the coefficients of the highest root −α ∨ 0 of R 0 in the simple basis α ∨ 1 , . . . , α ∨ n of R ∨ 0 . Similarly,P c consists of all nonnegative integral combinations c 1ω1 + · · · + c nωn of the fundamental weights ofR 0 satisfying c 1m1 + · · · + c nmn ≤ c and where the positive integersm 1 , . . . ,m n now refer to the coefficients of ϕ in the simple basisα ∨ 1 , . . . ,α ∨ n ofR ∨ 0 . IfR 0 = u ϕ R 0 then clearlym j = m j for all j and ifR 0 = R ∨ 0 then them 1 , . . . ,m n are a permutation of the m 1 , . . . , m n . So in both cases |P c | = |P c |, which completes the proof of the proposition.
Proposition 7.1 guarantees that the square matrix M For any function f ∈ C(P ) W , ω ∈ P + (quasi)-minuscule and λ ∈ P c we have that The last equality hinges on the following lemma (whose proof is delayed until subsection 8.2): Lemma 8.1. For any f ∈ C(P ), λ ∈ P c and ν ∈ P ⋆ ϑ := {wη | w ∈ W 0 and η ∈ P is a minuscule or quasi-minuscule weight}, one has that The action of L ω on f is therefore of the form The computation of the coefficients U λ,ω (t) and V λ,ν (t) hinges on the following lemma (whose proof is relegated in turn to subsection 8.1): Lemma 8.2. For λ ∈ P c and ν ∈ P ⋆ ϑ , we are in either one of the following two situations: i) When (λ + ν) + = λ, then w ′ λ+ν ν = α j for some j ∈ {0, . . . , n} with t j = t 0 and θ(λ + ν) = 0.