A Littlewood-Richardson rule for Koornwinder polynomials
aa r X i v : . [ m a t h . R T ] D ec A Littlewood-Richardson rule forKoornwinder polynomials
Kohei YAMAGUCHI ∗ Abstract
Koornwinder polynomials are q -orthogonal polynomials equipped with extra five parameters andthe BC n -type Weyl group symmetry, which were introduced by Koornwinder (1992) as multivariateanalogue of Askey-Wilson polynomials. They are now understood as the Macdonald polynomialsassociated to the affine root system of type ( C ∨ n , C n ) via the Macdonald-Cherednik theory of doubleaffine Hecke algebras. In this paper we give explicit formulas of Littlewood-Richardson coefficientsfor Koornwinder polynomials, i.e., the structure constants of the product as invariant polynomials.Our formulas are natural ( C ∨ n , C n )-analogue of Yip’s alcove-walk formulas (2012) which were givenin the case of reduced affine root systems. Contents
References 32 ∗ Graduate School of Mathematics, Nagoya University. Furocho, Chikusaku, Nagoya, Japan, 464-8602.e-mail address: [email protected] Introduction
Askey-Wilson polynomials [AW85] are q -orthogonal polynomials of one variable equipped with extraparameters ( a, b, c, d ), which recover various q -analogue of Jacobi polynomials by specialization of theparameters. In [K92], Koornwinder introduced n -variable analogue of Askey-Wilson polynomials, whichare today called Koornwinder polynomials . In the n = 1 case they coincide with Askey-Wilson polyno-mials, and in the case of n ≥ a, b, c, d, t ). By specializingthese parameters, one can recover Macdonald polynomials [M88, M03] of BC n -types.Let us give a brief explanation on Macdonald polynomials. Let S be an affine root system in thesense of [M03, Chap. 1]. If S is reduced, then S = S ( R ) or S = S ( R ) ∨ , where S ( R ) is the affine rootsystem associated to an irreducible finite root system R , and S ( R ) ∨ is the dual of S ( R ). In the reducedcase, if R is a finite root system of type X ( X = A n , B n , C n , D n , BC n , E , E , E , F or G ), then wecall S ( R ) an affine root system of type X and call S ( R ) ∨ an affine root system of type X ∨ . We also calla reduced S an untwisted affine root system . On the other hand, if S is non-reduced, then it is of theform S = S ∪ S , where S and S are reduced affine root systems. In the non-reduced case, if S and S are of type X and Y respectively, then we call S an affine root system of type ( X, Y ).The
Macdonald polynomial P λ ( x ) is a q -orthogonal polynomial which is a simultaneous eigenfunctionof a family of q -difference operators associated to an affine root system S . Today Macdonald polynomialsare formulated by the Macdonald-Cherednik theory , which is based on the representation theory of affineHecke algebras. This theory was first developed for untwisted affine root systems. Below we call P λ ( x ) Macdonald polynomial of type X if the corresponding untwisted affine root system S is of type X .Let us go back to Koornwinder polynomials. By the works of Noumi [N95], Sahi [Sa99], Stokman[S00] and others, it is clarified that one can apply Macdonald-Cherednik theory to the non-reduced affineroot system of type ( C ∨ n , C n ) in the sense of [M03, Chap. 1], and that one can recover Koornwinderpolynomials as Macdonald polynomials of type ( C ∨ n , C n ). As a result, Koornwinder polynomials arecharacterized as the ones having most parameters in the family of Macdonald polynomials.For the convenience of the following explanation, let us give a brief account on the notations used inthis paper. First, we introduce the notations for the root system R of type C n . See § h ∗ Z := L ni =1 Z ǫ i be a lattice of rank n . We denote the set of roots by R := {± ǫ i ± ǫ j | i = j } ∪{± ǫ i | i = 1 , . . . , n } ⊂ h ∗ Z and denote simple roots by α i ∈ R ( i = 1 , . . . , n ). We define the innerproduct on h ∗ Z by h ǫ i , ǫ j i := δ i,j , and define the fundamental weights ω i ∈ h ∗ Z by h ω i , α j i = δ i,j . Notethat the weight lattice is P := Z ω ⊕ · · · ⊕ Z ω n = h ∗ Z . We denote the set of dominant weights by( h ∗ Z ) + := { µ ∈ h ∗ Z | h α ∨ i , µ i ≥ , i = 1 , . . . , n } ⊂ h ∗ Z . Here α ∨ i is the coroot corresponding to α i . We alsodenote by W the finite Weyl group.Next we introduce the notations for the affine root system S of type ( C ∨ n , C n ), and explain theparameters of Koornwinder polynomials. See § § e h ∗ Z := h ∗ Z ⊕ Z δ of the lattice h ∗ Z , we have the affine root system S := {± ǫ i + k δ, ± ǫ i + kδ | k ∈ Z , i =1 , . . . , n } ∪ {± ǫ i ± ǫ j + kδ | k ∈ Z , ≤ i < j ≤ n } ⊂ e h ∗ Z ⊗ Q of type ( C ∨ n , C n ) and the extended affineWeyl group W . By using the group ring t ( P ) of the weight lattice P = h ∗ Z , we can present the group W as W = t ( P ) ⋊ W . In the case of rank n ≥
2, there are five orbits for the action of W on S , andwe consider the parameters associated to these orbits, denoting them by ( t , t, t n , u , u n ). By adding theparameter q and the square root of each parameter, we define the base field K by K := Q ( q , t , t , t n , u , u n ) . Koornwinder polynomials have these five plus one parameters ( q, t , t, t n , u , u n ). In the case of rank n = 1, there are four W -orbits, and the parameters are ( q, t , t n , u , u n ). In this case Koornwinderpolynomials are equivalent to Askey-Wilson polynomials as mentioned before. By [N95, §
3] and [S00,(5.2)], we have the following correspondence to the original parameters ( q, a, b, c, d ) of Askey-Wilsonpolynomials. ( q, a, b, c, d ) = ( q, q t u , − q t u − , t n u n , − t n u − n ) . (1.1.1)For a family x = ( x , . . . , x n ) of commutative variables, we denote the Laurent polynomial ring of x i ’s by K [ x ± ]. The finite Weyl group W acts on K [ x ± ] naturally, and we denote the invariant ring by2 [ x ± ] W . For a dominant weight λ ∈ ( h ∗ Z ) + , we denote the (monic) Koornwinder polynomials by P λ ( x ) ∈ K [ x ± ] W . We sometimes denote P λ := P λ ( x ) for simplicity. The definition of the Koornwinder polynomial P λ ( x )will be explained in § H ( W ) in § DH ( W ) in § The understanding of Macdonald polynomials has been rapidly advanced since the emergence of theMacdonald-Cherednik theory. Currently Macdonald polynomials, in particular those of type A , appearin various fields in mathematics, and have increasing importance. However, the study of Koornwinderpolynomials seems to be less advanced than the Macdonald polynomials of the other root systems, andthere are many pending problems for the ( C ∨ n , C n )-type.In this paper, we consider Littlewood-Richardson coefficients c νλ,µ of Koornwinder polynomials P λ ,that is the structure constants of the product in the invariant ring K [ x ± ] W : P λ P µ = X ν c νλ,µ P ν . Hereafter we call c νλ,µ LR coefficients for simplicity.Let us recall what is known in the case of type A . The classical LR coefficients are the structureconstants of the product s λ s µ = P ν c νλ,µ s ν of Schur polynomials s λ in the ring of symmetric polyno-mials. We have explicit formulas for the classical LR coefficients via Young tableaux. Regarding Schurpolynomials as the irreducible characters of the general linear group, we can interpret the coefficient c νλ,µ as the multiplicity of the irreducible decomposition of the tensor representation. For Hall-Littlewoodpolynomials, which are t -deformations of Schur polynomials, we can also consider the LR coefficients c νλ,µ , and some explicit formulas are known. See [M95, Chap. II, (4.11)] for example.Although Macdonald polynomial of type A is a q -deformation of Hall-Littlewood polynomial, noexplicit formula for the corresponding LR coefficient c νλ,µ had been unknown for a long time. In [M95,Chap. VI, § Pieri coefficients using arms and legsof Young diagrams. Here Pieri coefficients mean the LR coefficients c νλ,µ with λ the one-row type ( k ) orthe one-column type (1 l ), where the weights are identified with Young diagrams or the partitions.On the LR coefficients of Macdonald polynomials, Yip [Y12] made a great progress. Using alcovewalks , an explicit formula of c νλ,µ is given in [Y12, Theorem 4.4] for the Macdonald polynomials ofuntwisted affine root systems. Also a simplified formula [Y12, Corollary 4.7] is derived in the case λ isequal to a minuscule weight. In particular, this simplified formula recovers Macdonald’s formula for Piericoefficients of type A [Y12, Theorem 4.9]. In Yip’s study, the key ingredient is the notion of alcove walk,originally introduced by Ram [R06]. We will explain the relevant notations and terminology in § The main result of this paper is the following Theorem 1, which is a natural ( C ∨ n , C n )-type analogueof Yip’s alcove walk formulas for LR coefficients in [Y12, Theorem 4.4]. Let us prepare the necessarynotations and terminology for the explanation.Let A be the fundamental alcove of the extended affine Weyl group W (see (2.1.8)). Given an element w ∈ W , we take a reduced expression w = s i · · · s i r . Given a bit sequence b = ( b , . . . , b r ) ∈ { , } r andanother element z ∈ W , we call a sequence of alcoves of the form p = (cid:0) p := zA, p := zs b i A, p := zs b i s b i A, . . . , p r := zs b i · · · s b r i r A (cid:1) an alcove walk of type −→ w := ( i , . . . , i r ) beginning at zA . We denote by Γ( −→ w , z ) the set of such alcovewalks. See Example 2.1.1 for examples of alcove walks.For an alcove walk p , we call the transition p k − → p k the k -th step pf p . The k -th step of p is calleda folding if b k = 0 where the bit sequence b corresponds to the alcove walk p (see Table 2.1.1).In our main result, we use a colored alcove walk introduced by Yip [Y12]. It is an alcove walkequipped with the coloring of folding steps by either black or gray. We denote by Γ C ( −→ w , z ) the set ofcolored alcove walks whose steps belong to the dominant chamber C ⊂ h ∗ R := h ∗ Z ⊗ R .3 heorem 1 (Theorem 3.4.2) . Let λ, µ ∈ h ∗ Z be dominant weights, W µ be the stabilizer of µ in the finiteWeyl group W (see (2.2.29)), and W µ be the complete system of representatives of W /W µ such thatthe length of each element is shortest in W (see (3.2.3)). Let also W λ ( t ) be the Poincar´e polynomialof the stabilizer W λ (see (2.2.32)). We take a reduced expression of the element w ( λ ) ∈ W in (2.2.28).Then we have P λ P µ = 1 t − w λ W λ ( t ) X v ∈ W µ X p ∈ Γ C ( −−−→ w ( λ ) − , ( vw ( µ )) − ) A p B p C p P − w . wt( p ) . Here w ∈ W is the longest element, and the weight wt( p ) ∈ h ∗ Z is determined from the element e ( p ) ∈ W corresponding to the end of the colored alcove walks p as in (3.3.1). The coefficients A p , B p and C p arefactorized, and we have A p := Y α ∈ w ( µ ) − L ( v − ,v − µ ) ρ ( α ) , B p := Y α ∈L ( t (wt( p )) w ,e ( p )) ρ ( − α ) . Here the term ρ ( α ) is given by ρ ( α ) := t − t − q sh( − α ) t ht( − α ) − q sh( − α ) t ht( − α ) ( α W.α n ) t n (1 + t t − n q sh( − α ) t ht( − α ) )(1 − t − t − n q sh( − α ) t ht( − α ) )1 − q sh( − α ) t ht( − α ) ( α ∈ W.α n ) ,q sh( α ) := q − k , t ht( α ) := Q γ ∈ R s + t h γ ∨ ,β i Q γ ∈ R ℓ + ( t t n ) h γ ∨ ,β i ( α = β + kδ ∈ S ) , where we used R s + := { ǫ i ± ǫ j | ≤ i < j ≤ n } and R ℓ + := { ǫ i | ≤ i ≤ n } . For the notation L , see(2.1.10) in § C p is given by C p = Q rk =1 C p,k with the factor C p,k determined fromthe k -th step of the alcove walk p in Proposition 3.3.2. Here we display the relevant formulas for C p,k : ψ ± i ( z ) := ∓ t − t − − z ± ( i = 1 , . . . , n − ,ψ ± ( z ) := ∓ ( u n − u − n ) + z ± ( u − u − )1 − z ± , ψ ± n ( z ) := ∓ ( t n − t − n ) + z ± ( t − t − )1 − z ± ,n i ( z ) := 1 − tz − z − t − z − z ( β ∈ W.α i , i = 1 , . . . , n − ,n ( z ) := (1 − u n u z )(1 + u n u − z )1 − z (1 + u − n u z )(1 − u − n u − z )1 − z ( β ∈ W.α ) ,n n ( z ) := (1 − t n t z )(1 + t n t − z )1 − z (1 + t − n t z )(1 − t − n t − z )1 − z ( β ∈ W.α n ) . Note that the term A p actually depends only on v ∈ W µ , which corresponds to the beginning of thecolored alcove walk p .Let us explain the outline of proof of Theorem 1. We denote by E µ ( x ) ∈ K [ X ± ] the non-symmetricKoornwinder polynomials [Sa99, S00], which will be explained in § • { E µ ( x ) | µ ∈ h ∗ Z } is a K -basis of K [ x ± ]. • P µ ( x ) is obtained by symmetrizing E µ ( x ) (Fact 2.2.2). More precisely, using the symmetrizer U in(2.2.33), we have P µ ( x ) = 1 t − w µ W µ ( t ) U E µ ( x ) . The outline of proof is a straight ( C ∨ n , C n )-type analogue of Yip’s derivation in [Y12]. The argumentcan be divided into four steps, and below we explain them abbreviating some coefficients and ranges ofsummations.(i) For dominant weights λ, µ ∈ ( h ∗ Z ) + , we derive an expansion formula x µ E λ ( x ) = X p ∈ Γ C c p E ̟ ( p ) ( x )4f the product of the non-symmetric Koornwinder polynomial E λ ( x ) and the monomial x µ (Corol-lary 3.1.5). Here the index set Γ C consists of alcove walks belonging to the dominant chamber C .The symbol ̟ ( p ) ∈ ( h ∗ Z ) + will be given in (3.1.7).(ii) We use Ram-Yip type formula (Fact 3.3.1), an expansion formula for the non-symmetric Koorn-winder polynomials in terms of monomials: E µ ( x ) = X p ∈ Γ f p t d( p ) x wt( p ) . This formula was derived by Orr and Shimozono [OS18], based on the work of Ram and Yip [RY11]on the same type formula for the untwisted affine root systems.(iii) Using (i) and (ii), we can calculate the product of the non-symmetric Koornwinder polynomial E µ ( x ) and the Koornwinder polynomial P λ ( x ) in an extension g DH ( W ) of the double affine Heckealgebra DH ( W ), and express it as a sum over alcove walks (3.3.4). Then we can rewrite it as asum over colored alcove walks and have (Proposition 3.3.2): E µ ( x ) P λ ( x ) = X v ∈ W λ X p ∈ Γ C A p C p E ̟ ( p ) ( x ) . (iv) Theorem 1 is obtained by symmetrizing E µ ( x ) in (iii) and switching λ ↔ µ . Organization
We explain the organization of this paper.In §
2, we explain Koornwinder polynomials based on the Macdonald-Cherednik theory. In § R of type C n in § S of type ( C ∨ n , C n ) in § § § H ( W ) of type( C ∨ n , C n ) in § DH ( W ) of type ( C ∨ n , C n ) in § E λ (Fact 2.2.2). Finally we introduce Koornwinder polynomials P λ in § §
3, we derive our main Theorem 3.4.2. The outline of the discussion is given by the four steps(i)–(iv) previously explained, and the organization of § §
4, we derive several corollaries of the main Theorem 3.4.2. In § n = 1, that is the case of Askey-Wilson polynomials. In particular, we give a simplified formula for thePieri coefficient (Proposition 4.1.3), and recover the recurrence formula of Askey-Wilson polynomials in[AW85] from our Pieri formula (Remark 4.1.4). In § q →
0, andshow that LR coefficients are somewhat simplified (Proposition 4.2.1). In § n = 2. Notation and terminology
Here are the notations and terminology used throughout in this paper. • We denote by Z the ring of integers, by N = Z ≥ := { , , , . . . } the set of non-negative integers,by Q the field of rational numbers, and by R the field of real numbers. • We denote by e the unit of a group. • We denote an action of a group G on a set S by g.s for g ∈ G and s ∈ S , and denote the G -orbitof s by G.s or by Gs . • For a commutative ring k and a family of commutative variants x = ( x , x , . . . ), we denote by k [ x ± ] the Laurent polynomial ring k [ x ± , x ± , . . . ]. • We denote by δ i,j the Kronecker delta. 5 Koornwinder polynomials C n Let ( R, h ∗ Z , R ∨ , h Z ) be the root data of type C n . Thus h Z = L ni =1 Z ǫ ∨ i and h ∗ Z = L ni =1 Z ǫ i are lattices ofrank n , and we have the non-degenerate bilinear form h , i : h Z × h ∗ Z → Z , h ǫ ∨ i , ǫ j i = δ i,j . We identify h ∗ Z = h Z and ǫ i = ǫ ∨ i by this bilinear form h , i . The set R of roots and the set R ∨ of coroots are givenby R = {± ǫ i ± ǫ j | i = j } ∪ {± ǫ i | i = 1 , . . . , n } ⊂ h ∗ Z ,R ∨ = {± ǫ i ± ǫ j | i = j } ∪ {± ǫ i | i = 1 , . . . , n } ⊂ h Z . We use the following choice of the subset R + ⊂ R of positive roots and the subset R ∨ + ⊂ R ∨ of positivecoroots. R + := { ǫ i ± ǫ j | i < j } ∪ { ǫ i | i = 1 , . . . , n } , R ∨ + := { ǫ i ± ǫ j | i < j } ∪ { ǫ i | i = 1 , . . . , n } , We have R = R + ⊔ − R + and R ∨ = R ∨ + ⊔ − R ∨ + . The simple roots α i ∈ R ( i = 1 , . . . , n ) are given by α := ǫ − ǫ , . . . , α n − := ǫ n − − ǫ n , α n := 2 ǫ n . For each root α ∈ R , we denote the associated coroot by α ∨ := 2 α/ h α, α i ∈ h ∗ Z = h Z . The correspondence α α ∨ is a bijection, and we have h α ∨ , α i = 2. The coroots for simple roots are α ∨ = ǫ − ǫ , . . . , α ∨ n − = ǫ n − − ǫ n , and α ∨ n = ǫ n . We call α ∨ i simple coroots.For α ∈ R , we write s α the reflection by the hyperplane H α := { x ∈ h ∗ R | h α ∨ , x i = 0 } in h ∗ R = h ∗ Z ⊗ Z R .That is, s α .x := x − h α ∨ , x i α, x ∈ h ∗ R . We write s i := s α i for i = 1 , . . . , n . The finite Weyl group W is defined to be the subgroup of GL( h ∗ R )generated by s , . . . , s n . As an abstract group, W is a Coxeter group with generators s , . . . , s n andrelations s i = 1 ( i = 1 , . . . , n ) ,s i s j = s j s i ( | i − j | > ,s i s i +1 s i = s i +1 s i s i +1 ( i = 1 , . . . , n − ,s n − s n s n − s n = s n s n − s n s n − . Next we introduce notation for weights of the root system of type C n . For i = 1 , . . . , n , we define ω i := ǫ + · · · + ǫ i ∈ h ∗ Z , and call them the fundamental weights. Then we have h α ∨ i , ω j i = δ i,j for i, j = 1 , . . . , n . We define the root lattice Q and the weight lattice P by Q := Z α ⊕ · · · ⊕ Z α n ⊂ h ∗ Z = P := Z ω ⊕ · · · ⊕ Z ω n ⊂ h ∗ R . (2.1.1)The action of W ⊂ GL( h ∗ R ) on h ∗ R preserves the weight lattice P = h ∗ Z . We denote this action by λ w.λ for w ∈ W and λ ∈ P . ( C ∨ n , C n )Let t ( P ) be the group algebra of the weight lattice P = h ∗ Z . Denoting by t ( λ ) ∈ t ( P ) the elementassociated to λ ∈ P , we have t ( P ) = { t ( λ ) | λ ∈ h ∗ Z } and t ( λ ) t ( µ ) = t ( λ + µ ) ( λ, µ ∈ h ∗ Z ). Let us considerthe lattice extension e h ∗ Z := h ∗ Z ⊕ Z δ of h ∗ Z and the coefficient extension e h ∗ R := e h ∗ Z ⊗ Z R . We define theaction of t ( P ) on e h ∗ R by t ( λ ) . ( µ + mδ ) := µ + ( m − h µ, λ i ) δ, µ + mδ ∈ e h ∗ R = h ∗ R ⊕ R δ. The relation of w ∈ W and t ( λ ) ∈ t ( P ) in the group GL( e h ∗ R ) is then given by wt ( λ ) w − = t ( w.λ ). Thesubgroup W ⊂ GL( e h ∗ R ) generated by t ( P ) and W is called the extended affine Weyl group . That is, W := t ( P ) ⋊ W ⊂ GL( e h ∗ R ) . (2.1.2)6he action of the element s := t ( ǫ ) s ǫ ∈ W on P = h ∗ Z is given by s.ǫ = δ − ǫ and s.ǫ i = ǫ i ( i = 2 , . . . , n ), which is the same as the reflection s := s α with respect to the hyperplane H α := { x ∈ h ∗ R | h α ∨ , x i = 0 } for the affine root α := δ − ǫ ∈ e h ∗ Z . Here we set α ∨ := c − ǫ , where c is thebasis element in the one-dimensional extension e h R := h R ⊕ R c of h R := h Z ⊗ Z R . We also set h c, x i = 1for all x ∈ h ∗ R .As an abstract group, W is a Coxeter group with generators s , s , . . . , s n and relations s i = 1 ( i = 0 , . . . , n ) ,s i s j = s j s i ( | i − j | > ,s i s i +1 s i = s i +1 s i s i +1 ( i = 1 , . . . , n − ,s i s i +1 s i s i +1 = s i +1 s i s i +1 s i ( i = 0 , n − . We define the length ℓ ( w ) of an element w ∈ W to be the length of the reduced expression of w by thegenerators s , . . . , s n . We also denote by B the corresponding Bruhat order. The reduced expressionsof t ( ǫ i ) ( i = 1 , . . . , n ) are given by t ( ǫ ) = s s · · · s n − s n s n − · · · s s ,t ( ǫ ) = s s s · · · s n s n − · · · s ,t ( ǫ i ) = s i − · · · s s · · · s n s n − · · · s i ,t ( ǫ n ) = s n − · · · s s s · · · s n . (2.1.3)Now we define the affine root system S of type ( C ∨ n , C n ) in the sense of [M03, (1.3.18)] and[S00] by S := {± ǫ i + k δ, ± ǫ i + kδ | k ∈ Z , i = 1 , . . . , n } ∪ {± ǫ i ± ǫ j + kδ | k ∈ Z , ≤ i < j ≤ n } ⊂ e h ∗ R . (2.1.4)We also define the subset S + ⊂ S of positive roots by S + := { α + kδ, α ∨ + k δ | α ∈ R + , α ∨ ∈ R ∨ + , k ∈ N } ∪ { α + kδ, α ∨ + k δ | α ∈ R − , α ∨ ∈ R ∨− , k ∈ N } . (2.1.5)We then have S = S + ⊔ S − with S − := − S + . We also set e R := R ∪ R ∨ . Then any β ∈ S can be uniquelywritten as β = α + kδ ∈ S with α ∈ e R and k ∈ Z . We denote the corresponding projection S → e R by β := α ( β = α + kδ ∈ S, α ∈ e R, k ∈ Z ) . (2.1.6)We also denote e R + := R + ∪ R ∨ + , e R − := − e R + . (2.1.7)Hereafter the case of type ( C ∨ n , C n ) is also called the rank n case. Alcove walks are introduced by Ram [R06] as analogue of Littelmann paths for affine Hecke algebras.They are valuable combinatorial objects, and used in Ram-Yip type formula [RY11, OS18] for non-symmetric Macdonald-Koornwinder polynomials, and in Yip’s formula [Y12] for Littlewood-Richardsonrules of Macdonald polynomials in the untwisted affine root systems. In this part we introduce thenotation of alcove walks which will be used throughout in the text. Basically we follow the notations in[Y12, § β = α + kδ ∈ S ( α ∈ e R , k ∈ Z ) as a affine linear function on h ∗ R by β ( v ) = h α, v i + k ( v ∈ h ∗ R ) . An alcove is defined to be a connected component of the complement h ∗ R \ S α ∈ S H α of the hyperplanes H α := { x ∈ h ∗ R | α ( x ) = 0 } . The fundamental alcove A is the alcove given by A := { x ∈ h ∗ R | α i ( x ) > i = 0 , . . . , n ) } . (2.1.8)7ts boundary consists of the hyperplanes H α , H α , . . . , H α n . Note that the mapping W ∋ w wA ∈ π ( h ∗ R \ S α ∈ S H α )is a bijection. An alcove wA is surrounded by n + 1 hyperplanes, say H γ i ( i = 0 , . . . , n ). We call theintersection H γ i ∩ wA an edge of the alcove wA , where wA denotes the closure with respect to theEuclidean topology. Note that each hyperplane H γ i separates wA and another alcove vA , which can bewritten as v = ws j for some j = 0 , . . . , n . Then the edge H γ i ∩ wA is just the intersection wA ∩ ws j A ,and has two sides, which we call the wA -side and the ws j A -side .Given an alcove wA , we give a sign ± to each of the two sides on an edge of wA . Let H γ i ( i = 0 , . . . , n )be the hyperplanes surrounding wA . By renaming the indices i if necessary, we can assume that thehyperplane H γ i separates wA and ws i A . Then using the projection γ i γ i in (2.1.6) and the symbols e R ± in (2.1.7), we set the signs by the following rule. • If γ i ∈ e R + , then the wA -side of H γ i ∩ wA is assigned by + and the ws i A -side is by − . • If γ i ∈ e R − , then wA -side is assigned by − and the ws i A -side is by +.See Figure 2.1.1 for the assignment in the rank 2 case.Figure 2.1.1: Signs for the edges of the fundamental alcove A in the rank 2 caseGiven an element w ∈ W and a reduced expression w = s i · · · s i r , we define a subset L ( w ) ⊂ S by L ( w ) := (cid:8) α i , s i α i , . . . , s i · · · s i r − α i r (cid:9) . (2.1.9)The set { H β | β ∈ L ( w ) } consists of the hyperplanes separating A and wA . Given elements v, w ∈ W and their reduced expressions, we also set L ( v, w ) := ( L ( v ) ∪ L ( w )) \ ( L ( v ) ∩ L ( w )) . (2.1.10)The set { H β | β ∈ L ( v, w ) } consists of the hyperplanes separating vA and wA . If v B w , then we have L ( v, w ) = v. L ( e, v − w ) = v. L ( v − w ) . (2.1.11)Let us again given w ∈ W and a reduced expression w = s i · · · s i r . Then the mapping { , } r ∋ ( b , . . . , b r ) s b i · · · s b r i r ∈ { v ∈ W | v B w } is a bijection. Let us given extra z, w ∈ W such that v B w . We can write v = s b i · · · s b r i r with b = ( b , . . . , b r ) ∈ { , } r . We then consider the following sequence p of alcoves. p = (cid:0) p := zA, p := zs b i A, p := zs b i s b i A, . . . , p r := zs b i · · · s b r i r A (cid:1) . The sequence p is called an alcove walk of type −→ w = ( i , . . . , i r ) beginning at zA , and we denote byΓ( −→ w , z ) the set of alcove walks of this kind. The symbol −→ w emphasizes that we choose a reducedexpression w = s i · · · s i r . 8 xample 2.1.1 (Alcove walks in the rank 2 case) . For w = s s s s and z = e ∈ W , the two alcovewalks p := ( A, A, s A, s s A, s s s A ) , p := ( A, s A, s s A, s s s A, s s s s A ) ∈ Γ( −→ w , z )are shown in Figure 2.1.2, where the gray region is the fundamental alcove A , and the number i = 0 , , W -orbit of H α i . O ω ω O ω ω Figure 2.1.2: Alcove walks p and p For an alcove walk p ∈ Γ( −→ w , z ) and k = 1 , . . . , r , the transition p k − → p k is called the k -th step of p .The k -th step is called a crossing if b k = 1, and called a folding if b k = 0. The correspondence betweenthe bit b k and the k -th step is shown in Table 2.1.1, where we denote by v k − ∈ W the element suchthat p k − = v k − A . b k crossing folding p k (cid:1) p k p k (cid:0) = p k v k (cid:2) s i k A Table 2.1.1: Correspondence between bits and stepsLet us again given z, w ∈ W with a reduced expression w = s i · · · s i r . For an alcove walk p =( zA, . . . , zs b i · · · s b r i r A ) ∈ Γ( −→ w , z ), we define e ( p ) ∈ W by e ( p ) := zs b i · · · s b r i r . (2.1.12)Thus e ( p ) corresponds to the end of p . We also define h k ( p ) ∈ S for k = 1 , . . . , r by the following rule.Denote v := s b i · · · s b k − i k − for simplicity, so that we have p k − = vA . Then we define h k ( p ) := the affine root such that the corresponding hyperplane H h k ( p ) separates vA and vs i k A. (2.1.13)Furthermore, we call the k -th step of p ∈ Γ( −→ w , z ) an ascent if zs b i · · · s b k − i k − B zs b i · · · s b k i k , and call it a descent if zs b i · · · s b k − i k − < B zs b i · · · s b k i k . We denote the set of descent steps of p bydes( p ) := { k = 1 , . . . , r | the k -th step is a descent } . (2.1.14)Recalling the sign on an edge of an alcove (see Figure 2.1.1 for an example), we can classify each stepof an alcove walk p into four types as in Table 2.1.2, where we used the symbol v k − ∈ W such that p k − = v k − A . 9sing this classification, we define ϕ ± ( p ) ⊂ { , . . . , r } by ϕ + ( p ) := { k | the k -th step of p is a positive folding } ,ϕ − ( p ) := { k | the k -th step of p is a negative folding } , (2.1.15)and define ξ des ( p ) ⊂ { , . . . , r } by ξ des ( p ) := { k | the k -th step of p is a crossing and k ∈ des( p ) } . (2.1.16)Note that we fix a reduced expression w = s i · · · s i r in the definitions of ϕ ± ( p ) and ξ des ( p ). p o(cid:3)(cid:4)(cid:5)(cid:6)ve crossing n(cid:7)(cid:8)(cid:9)(cid:10)(cid:11)(cid:12)(cid:13) crossing p (cid:14)(cid:15)(cid:16)(cid:17)(cid:18)(cid:19)(cid:20) folding (cid:21)(cid:22)(cid:23)(cid:24)(cid:25)(cid:26)(cid:27)(cid:28) folding − + p k − p k (cid:29) − p k − p k (cid:30) − p k − = p k v k − s i k A − (cid:31) p k − = p k v k − s i k A Table 2.1.2: Classification of steps in alcove walks
In this subsection, we explain the realization of non-symmetric Koornwinder polynomials via the poly-nomial representation of the affine Hecke algebra type C n , and introduce Koornwinder polynomials bytheir symmetrization. ( C ∨ n , C n ) and polynomial representations Recall the affine root system S of type ( C ∨ n , C n ) and the extended affine Weyl group W explained in § { t α | α ∈ S } be parameters satisfying the condition t α = t β for β ∈ W.α . Since the W -orbitsin S are given by W.α i = W.α ∨ i ( i = 1 , . . . , n − , W.α n , W.α ∨ n , W.α , W.α ∨ , we can replace the family { t α } by( t α , t α i = t α ∨ i , t α n , t α ∨ , t α ∨ n ) = ( t , t, t n , u , u n ) . (2.2.1)We will also denote t , . . . , t n − := t . Now we set the base field K as K := Q ( q , t , t , t n , u , u n ) , (2.2.2)and all the linear spaces, their tensor products, and the algebras will be those over K unless otherwisestated.The affine Hecke algebra H ( W ) is the associative algebra generated by T , T , . . . , T n subject to thefollowing relations. ( T i − t i )( T i + t − i ) = 0 ( i = 0 , . . . , n ) ,T i T j = T j T i ( | i − j | > , ( i, j )
6∈ { ( n, , (0 , n ) } ) , (2.2.3) T i T i +1 T i = T i +1 T i T i +1 ( i = 1 , . . . , n − , (2.2.4) T i T i +1 T i T i +1 = T i +1 T i T i +1 T i ( i = 0 , n − . (2.2.5)The relations (2.2.3)–(2.2.5) are called the braid relations .Given an element w ∈ W together with a reduced expression w = s i · · · s i r , we consider the alcovewalk ( A, s i A, . . . , s i · · · s i r A = wA ) ∈ Γ( −→ w , e ), and define Y w ∈ H ( W ) by Y w := T ǫ i · · · T ǫ r i r , (2.2.6)10here we set ǫ k := 1 if the k -th step of p is a positive crossing, and set ǫ k := − k -th step isa negative crossing according to the classification in Figure 2.1.2. The decomposition of Y w by T i ’s isindependent of the choice of a reduced expression of w . By the relations of H ( W ), we find that thefamily { Y w | w ∈ W } is mutually commutative [N95, § § Y t ( ǫ i ) using the reduced expression of t ( ǫ i ) in (2.1.3).The result is Y t ( ǫ ) = T · · · T n T n − · · · T ,Y t ( ǫ ) = T − T · · · T n − T n T n − · · · T ,Y t ( ǫ i ) = T − i − · · · T − T · · · T n − T n T n − · · · T i ,Y t ( ǫ n ) = T − n − · · · T − T T · · · T n . (2.2.7)Now we denote by K [ Y ± ] = K [ Y ± , . . . , Y ± n ] ⊂ H ( W ) , Y i := Y t ( ǫ i ) ( i = 1 , . . . , n )the ring of Laurent polynomials in Y , . . . , Y n . Then we have an isomorphism H ( W ) ≃ H ( W ) ⊗ K [ Y ± ],where H ( W ) is the Hecke algebra of the finite Weyl group W . The latter is the subalgebra of H ( W )generated by T , . . . , T n .Next we review the basic representation of the affine Hecke algebra H ( W ) introduced by Noumi[N95]. Let K ( x ) = K ( x , . . . , x n ) be the field of rational functions with n variables. Then the mapping T i t i + t − i − t i x i /x i +1 − x i /x i +1 ( s i −
1) ( i = 1 , . . . , n − ,T t + t − (1 − u t q x − )(1 + u − t q x − )1 − qx − ( s − ,T n t n + t − n (1 − u n t n x n )(1 + u − n t n x n )1 − x n ( s n −
1) (2.2.8)defines a ring homomorphism ρ : H ( W ) → End( K ( x )). Moreover its image is contained in the en-domorphism algebra End K ( K [ x ± ]) ⊂ End K ( K ( x )) of the Laurent polynomials. We call ρ the basicrepresentation of H ( W ). Hereafter we identify H ( W ) and its image under ρ , and regard H ( W ) as asubalgebra of End K ( K [ x ± ]). The right hand sides of (2.2.8) are q -difference operators called Dunkloperators of type ( C ∨ n , C n ).Let us give a simplified description of (2.2.8). Using u i := i = 1 , . . . , n − u ( i = 0) u n ( i = n ) , x α i := x i /x i +1 ( i = 1 , . . . , n − qx − ( i = 0) x n ( i = n ) , we can rewrite T i ’s as T i = t i + t − i (1 − u i t i x αi )(1 + u − i t i x αi )1 − x α i ( s i − , (2.2.9)where we identified the left and right hand sides in (2.2.8) as claimed before. Let us further define therational functions c i ( z ) , d i ( z ) ∈ K ( z ) by c i ( z ) := t − i (1 − u i t i z )(1 + u − i t i z )1 − z , d i ( z ) := t i − c i ( z ) = ( t i − t − i ) + ( u i − u − i ) z − z . (2.2.10)Then we can rewrite (2.2.8) or (2.2.9) as T i = t i + c i ( x α i )( s i −
1) = t i s i + d i ( x α i )(1 − s i ) = c i ( x α i ) s i + d i ( x α i ) . (2.2.11)11or later use, we calculate the action of the element Y β on 1 in the basic representation for an affineroot β = α + kδ ∈ S ( α ∈ e R , k ∈ Z ). Let us define q sh( α + kδ ) := q − k , t ht( α + kδ ) := Q γ ∈ R s + t h γ ∨ ,α i Q γ ∈ R ℓ + ( t t n ) h γ ∨ ,α i . (2.2.12)Here R s + := { ǫ i ± ǫ j | ≤ i < j ≤ n } denotes the set of positive short roots, and R ℓ + := { ǫ i | ≤ i ≤ n } denotes the set of of positive long roots. Then we can check Y β q sh( β ) t ht( β ) . (2.2.13)See also [S00, Proposition 4.5] for a more general formula.Finally we recall the Lusztig relations in the basic representations of affine Hecke algebra. For eachweight λ = ( λ , . . . , λ n ) ∈ P = h ∗ Z , we define x λ ∈ K [ x ± ] by x λ := x λ · · · x λ n n ∈ K [ x ± ] . (2.2.14) Fact 2.2.1 (Lusztig relations, [L89, Proposition 3.6]) . For i = 0 , . . . , n and λ ∈ h ∗ Z , we have T i x λ − x s i .λ T i = d i ( x α i )( x λ − x s i .λ ) , where the rational function d i ( z ) is defined by (2.2.10). Next we review the double affine Hecke algebra DH ( W ) of type ( C ∨ n , C n ) and the non-symmetric Koorn-winder polynomials E λ ( x ), following [M03], [Sa99] and [S00].As in the previous § H ( W ) as a K -subalgebra of End K ( K [ x ± ]) by the basic represen-tation (2.2.8). We define the double affine Hecke algebra DH ( W ) ⊂ End K ( K [ x ± ]) as the K -subalgebragenerated by K [ x ± ], H ( W ) and K [ Y ± ]. Thus DH ( W ) := (cid:10) K [ x ± ] , H ( W ) , K [ Y ± ] (cid:11) ⊂ End K ( K [ x ± ]) . As in the case of untwisted affine root systems, the algebra DH ( W ) has the Cherednik anti-involution φ [Sa99, § φ ( x i ) = Y − i , φ ( Y i ) = x − i , φ ( T i ) = T i ( i = 1 , . . . , n ) ,φ ( u n ) = t , φ ( t ) = u n . (2.2.15)On the element T the anti-involution acts as φ ( T ) = T − s ǫ x − . In fact, we have T = Y T − s ǫ and T s ǫ = T · · · T n T n − · · · T by (2.2.7). Hereafter we denote T ∨ i := φ ( T i ) ( i = 0 , . . . , n ) . (2.2.16)Next we introduce the x - and Y -intertwiners for DH ( W ) following [M03, § g DH ( W ) be thecoefficient extension of DH ( W ) by rational functions of x ’s and Y ’s. In other words, we set g DH ( W ) := (cid:10) K ( x ) , H ( W ) , K ( Y ) (cid:11) ⊂ End K ( K ( x )) . (2.2.17)Here K ( x ) and K ( Y ) are the fields of rational functions of x i and Y i ( i = 1 , . . . , n ) respectively. For i = 0 , . . . , n , we define S xi ∈ g DH ( W ) by S xi := T i + ϕ + i ( x α i ) = T − i + ϕ − i ( x α i ) , (2.2.18)where ϕ ± i ( z ) := ∓ ( t i − t − i ) + z ± ( u i − u − i )1 − z ± ∈ K ( z ) . (2.2.19)We call S xi the x -intertwiners . 12et us explain some basic properties of x -intertwiners. Recalling the rational function d i ( z ) in (2.2.10)and the expression of T i in (2.2.11), we have ϕ + i ( z ) = d i ( z ) , S xi = T i − d i ( x α i ) = c i ( x α i ) s i . (2.2.20)For each weight λ ∈ h ∗ Z , we have S xi x λ = x s i ( λ ) S xi (2.2.21)by the Lusztig relations (Fact 2.2.1). Moreover, by [M03, (5.5.2)], the x -intertwiners S xi ( i = 0 , . . . , n )satisfy the same braid relations as (2.2.3)–(2.2.5): S xi S xj = S xj S xi ( | i − j | > ,S xi S xi +1 S xi = S xi +1 S xi S xi +1 ( i = 1 , . . . , n − ,S xi S xi +1 S xi S xi +1 = S xi +1 S xi S xi +1 S xi ( i = 0 , n − . (2.2.22)Given an element w ∈ W , choose a reduced expression w = s i · · · s i p , and set S xw := S xi · · · S xi p ∈ g DH ( W ) . (2.2.23)By the braid relations, S xw is independent of the choice of a reduced expression of w .Next we introduce Y -intertwiners . First, note that the anti-involution φ can be extended to g DH ( W ).In fact, g DH ( W ) is the Ore localization of the non-commutative algebra DH ( W ) by the commutativesubalgebras K [ x ± ] and K [ Y ± ], and φ is an isomorphism on these commutative subalgebras. We denotethe extension of φ to g DH ( W ) by same symbol φ . Now we define the Y -intertwiners S Yi ∈ g DH ( W ) by S Yi := φ ( S xi ) = T i + ψ + i ( Y − α i ) = T − i + ψ − i ( Y − α i ) ( i = 1 , . . . , n ) ,S Y := φ ( S x ) = T ∨ + ψ +0 ( qY ) = ( T ∨ ) − + ψ − ( qY ) , (2.2.24)where the symbols ψ ± i ( z ) denote the images of ϕ ± i ( z ) given in (2.2.19) under the extended anti-involution φ . That is, we have ψ ± i ( z ) := ϕ ± i ( z ) = ∓ t − t − − z ± ( i = 1 , . . . , n − ,ψ ± ( z ) := ∓ ( u n − u − n ) + z ± ( u − u − )1 − z ± ,ψ ± n ( z ) := ∓ ( t n − t − n ) + z ± ( t − t − )1 − z ± . (2.2.25)We can deduce properties of S Yi ’s from those of S xi ’s. For example, applying the anti-involution φ tothe relation (2.2.21), we have S Yi Y λ = Y s i λ S Yi (2.2.26)for each i = 0 , . . . , n and λ ∈ h ∗ Z . We can also see that S Yi ’s satisfy the same braid relations as (2.2.22).For an element w ∈ W , we can define S Yw ∈ g DH ( W ) by choosing a reduced expression w = s i · · · s i p and S Yw := S Yi · · · S Yi p ∈ g DH ( W ) . (2.2.27)It is well-defined by the braid relations of S Yi ’s.Finally we explain the non-symmetric Koornwinder polynomials. For each weight µ ∈ P = h ∗ Z , weregard t ( µ ) W ⊂ W by the decomposition W = t ( P ) ⋊ W in (2.1.2). Then we define w ( µ ) ∈ W by thefollowing description: w ( µ ) is the shortest element among t ( µ ) W ⊂ W . (2.2.28)Now we have: 13 act 2.2.2 ([Sa99, § . For µ ∈ h ∗ Z , the element E µ ( x ) := S Yw ( µ ) K [ x ± ]. We call it the non-symmetric Koornwinder polynomial associated to µ .By (2.2.26), E µ ( x ) is a simultaneous eigenfunction of the family (cid:8) Y λ | λ ∈ h ∗ Z (cid:9) of Dunkl operators.Note that our normalization of E µ ( x ) is different from that in [Sa99, S00]. In loc. cit., the coefficient of x µ is normalized to be 1. Now we introduce
Koornwinder polynomials by symmetrizing non-symmetric Koornwinder polynomials.First, we define the set ( h ∗ Z ) + ⊂ h ∗ Z of dominant weights by( h ∗ Z ) + := { µ ∈ h ∗ Z | h α ∨ i , µ i ≥ , i = 1 , . . . , n } . For a dominant weight µ ∈ ( h ∗ Z ) + , we denote the stabilizer of µ in the finite Weyl group W by W µ := { w ∈ W | w.µ = µ } ⊂ W , (2.2.29)and denote the longest element among W µ by w µ ∈ W µ . (2.2.30)Next, using the notations in § § t w ∈ K for each w ∈ W by t w := Y β ∈L ( w ) t β ∈ K . (2.2.31)Here { t α | α ∈ S } is the W -invariant family of parameters (2.2.1), K is the base field (2.2.2), and L ( w ) ⊂ S is given by (2.1.9). If w = s i · · · s i r ∈ W is the shortest element, then we have t w = t i · · · t i r .For a dominant weight µ ∈ ( h ∗ Z ) + , we define the Poincar´e polynomial W µ ( t ) ∈ K of the stabilizer W µ by W µ ( t ) := X u ∈ W µ t u . (2.2.32) Lemma 2.2.3.
For each element µ ∈ ( h ∗ Z ) + , we have X u ∈ W µ (cid:18) Y α ∈L (1 ,u ) t α − t ht( − α ) t − α − t ht( − α ) (cid:19)(cid:18) Y α ∈L ( u,w µ ) t − α − t ht( − α ) t α − t ht( − α ) (cid:19) = t − w µ W µ ( t ) . For a proof, see [Y12, Lemma 3.4].Next we define the symmetrizer U by U := X w ∈ W t − w w T w . (2.2.33)By [M03, (5.5.9)], we then have U T i = U t i , T i U = t i U ( i = 1 , . . . , n ) . (2.2.34)Hereafter we denote the ring of W -invariant Laurent polynomials by K [ x ± ] W := (cid:8) f ∈ K [ x ± ] | w.f = f, w ∈ W (cid:9) . Here W acts on x λ (2.2.14) by the action on the weight λ . Also recall that for each µ ∈ ( h ∗ Z ) + ⊂ h ∗ Z wedefined w ( µ ) ∈ t ( µ ) W ⊂ W by (2.2.28). Fact 2.2.4 ([S00, Theorem 6.6]) . For each dominant weight λ ∈ ( h ∗ Z ) + , the element P λ ( x ) := 1 t − w µ W λ ( t ) U S Yw ( λ ) t − w λ W µ ( t ) U E λ ( x ) ∈ g DH ( W )belongs to K [ x ± ] W . We call P λ ( x ) the ( monic ) Koornwinder polynomial associated to λ .Note that the coefficient of x λ in P λ ( x ) is 1 since the coefficient of the top term x λ in U S Yw ( λ ) t − w λ W λ ( t ). To emphasize the root system ( C ∨ n , C n ), we call P λ ( x ) the Koornwinder polynomial of rank n or of type ( C ∨ n , C n ). 14 Littlewood-Richardson coefficients
Yip [Y12, Theorem 4.4] derived a combinatorial explicit formula of LR coefficients for Macdonald poly-nomials P λ ( x ) in the case of untwisted affine root systems. In this section, we derive a ( C ∨ n , C n )-analogueof Yip’s formula. The outline of the derivation is quite similar to Yip’s proof [Y12, §§ In [Y12, Theorem 3.3], Yip derived an expansion formula for the product of the monomial x µ andthe non-symmetric Macdonald polynomial E λ ( x ) in the case of untwisted affine root systems. In thissubsection, we give its ( C ∨ n , C n )-type analogue (Corollary 3.1.5).We will use the notations in § g DH ( W ) is the extension (2.2.17) of the doubleaffine Hecke algebra DH ( W ) of type ( C ∨ n , C n ), S Yi ∈ g DH ( W ) is the Y -intertwiner (2.2.24), and S Yw for w ∈ W is the product of S Yi ’s (2.2.27). We also denote the Bruhat order in W by B .As a preparation of Proposition 3.1.3, we derive a product formula of the Y -intertwiners. Proposition 3.1.1.
For w ∈ W and i = 0 , . . . , n , we have the following relations in g DH ( W ).(i) If w B s i w , then S Yi S Yw = S Ys i w .(ii) If w < B s i w , then S Yi S Yw = n i ( Y − α i ) S Ys i w , where n ( Y β ) := (1 − u n u Y β )(1 + u n u − Y β )1 − Y β (1 + u − n u Y β )(1 − u − n u − Y β )1 − Y β ( β ∈ W.α ) ,n i ( Y β ) := 1 − tY β − Y β − t − Y β − Y β ( β ∈ W.α i , < i < n ) ,n n ( Y β ) := (1 − t n t Y β )(1 + t n t − Y β )1 − Y β (1 + t − n t Y β )(1 − t − n t − Y β )1 − Y β ( β ∈ W.α n ) . Proof.
Fix w ∈ W and choose a reduced expression w = s i · · · s i r . By the definitions (2.2.27), (2.2.24)and the equation (2.2.20), we have S Yw = S Yi · · · S Yi r = ( T ∨ i + ψ + i ( Y − α i )) · · · ( T ∨ i r + ψ + i r ( Y − α ir ))= c ∨ i ( Y − α i ) s i · · · c ∨ i r ( Y − α ir ) s i r = c ∨ i ( Y − β ) · · · c ∨ i r ( Y − β r ) w. Here we set β k := s i · · · s i k − ( α i k ) ( k = 1 , . . . , r ) and c ∨ i ( z ) := φ ( c i ( z )) = u − n (1 − u n u z )(1 + u n u − z )1 − z ( i = 0) ,t − − tz − z (0 < i < n ) ,t − n (1 − t n t z )(1 + t n t − z )1 − z ( i = n ) . Since w = s i · · · s i r is a reduced expression, we have β k ∈ S + for k = 1 , . . . , r , where S + ⊂ S denotesthe set of positive affine roots (2.1.5). The product S Yi ( i = 0 , . . . , n ) and S Yw is now calculated as S Yi S Yw = c ∨ i ( Y − α i ) s i c ∨ i ( Y − β ) · · · c ∨ i r ( Y − β r ) w. (3.1.1)If ℓ ( s i w ) = ℓ ( w ) + 1, then the equation (3.1.1) becomes S Yi S Yw = S Ys i w . If ℓ ( s i w ) = ℓ ( w ) −
1, then thereexists k ∈ { , . . . , r } such that s i ( β k − ) ∈ S + and s i ( β k ) ∈ S − . Since we have β k = α i , the equation(3.1.1) becomes S Yi S Yw = c ∨ i ( Y − α i ) s i c ∨ i ( Y − β ) · · · c ∨ i r ( Y − β r ) w = c ∨ i ( Y − α i ) c ∨ i ( Y − s i ( β ) ) · · · c ∨ i k − ( Y − s i ( β k − ) ) s i c ∨ i k ( Y − α i ) · · · c ∨ i r ( Y − β r ) w = c ∨ i ( Y − α i ) c ∨ i ( Y α i ) c i ( Y − s i ( β ) ) · · · \ c ∨ i k ( Y α i ) · · · c ∨ i r ( Y − s i ( β r ) ) s i w = c ∨ i ( Y − α i ) c ∨ i ( Y α i ) S Ys i w . Here the symbol b denotes skipping the term. Then the consequence follows from the equality c ∨ i ( Y − α i ) c ∨ i ( Y α i ) = n i ( Y − α i ), which can be checked by a direct calculation.15he same discussion shows the following statement. Corollary 3.1.2.
For w ∈ W and i = 0 , . . . , n . we have the following relations in g DH ( W )(i) If w B ws i , then S Yw S Yi = S Yws i .(ii) If w < B ws i , then S Yw S Yi = S Yws i n i ( Y − α i ), where n i ( Y − α i ) is given in Proposition 3.1.1.Next we recall the notations on alcove walks in § z, w ∈ W together with a reducedexpression z = s i r · · · s i , we defined the set Γ( −→ z , w ) of alcove walks of type −→ z = ( i r , . . . , i ) beginningat wA . For an alcove walk p = ( p , . . . , p r ) ∈ Γ( −→ z , w ), the k -th step means the the transition from p k − to p k , which is classified into the four types in Table 2.1.2.Now we define x z ∈ DH ( W ) for z ∈ W with a chosen reduced expression z = s i r · · · s i . Let q be thealcove walk given by q := ( zA, zs i A, zs i s i A, . . . , zs i · · · s i r A = A ) ∈ Γ( −→ z − , z ) . Here −→ z − := −−→ z − = ( i , . . . , i r ). Then we define x z by x z := ( T ∨ i r ) ǫ r · · · ( T ∨ i ) ǫ , (3.1.2)where T ∨ i := φ ( T i ) ∈ DH ( W ) as in (2.2.16), and we set ǫ k := 1 if the k -th step is a positive crossing,and ǫ k := − k -th step is a negative crossing according to the classification in Table 2.1.2. Proposition 3.1.3.
Given z, w ∈ W with a chosen reduced expression z = s i r · · · s i , we have x z S Yw = X p ∈ Γ( −→ z − ,w − ) S Ye ( p ) − g p ( Y ) n p ( Y )in g DH ( W ), where e ( p ) ∈ W is the element (2.1.12), and the terms g p ( Y ) and n p ( Y ) are given by g p ( Y ) := Y k ∈ ϕ − ( p ) (cid:0) − ψ − i k ( Y − h k ( p ) ) (cid:1) Y k ∈ ϕ + ( p ) (cid:0) − ψ + i k ( Y − h k ( p ) ) (cid:1) ,n p ( Y ) := Y k ∈ ξ des ( p ) n i k ( Y − h k ( p ) ) . Here h k ( p ) is given by (2.1.13), ϕ + ( p ) and ϕ − ( p ) are by (2.1.15), ξ des ( p ) is by (2.1.16), ψ ± i ( z ) = φ ( ϕ ± i ( z ))is by (2.2.25), and n i ( z ) is given in Proposition 3.1.1. Proof.
We show the statement by induction on the length of z ∈ W . If ℓ ( z ) = 0, that is z = e , then theright hand side consists only of the term for p = ( p = wA ), so that it is equal to S Yw , and we have therelation.Next we assume z = e and that the result holds for any element w ∈ W such that w < ℓ ( z ).Fix a reduced expression of z , and write it as z = s i ζ , ζ = s i r · · · s i . By the hypothesis, we can write x z S Yw = ( T ∨ i ) ǫ x ζ S Yw = X p ∈ Γ( −→ ζ − ,w − ) ( T ∨ i ) ǫ S Ye ( p ) − g p ( Y ) n p ( Y ) . (3.1.3)Here ǫ ∈ {± } is the sign determined by z . Let us calculate the rightmost side. Take an element p = ( w − A, w − s ǫ i A, . . . , w − s ǫ i · · · s ǫ r i r A ) ∈ Γ( −→ ζ − , w − ) . Since we have ( T ∨ i ) ± = S Yi − ψ ± i ( Y − α i ) by the definition (2.2.16) of T ∨ i , the term contributed by p becomes( T ∨ i ) ǫ S Ye ( p ) − g p ( Y ) n p ( Y ) = ( S Yi − ψ ǫi ( Y − α i )) S Ye ( p ) − g p ( Y ) n p ( Y )= S Yi S Ye ( p ) − g p ( Y ) n p ( Y ) + ( − ψ ǫi ( Y − α i )) S Ye ( p ) − g p ( Y ) n p ( Y )= S Yi S Ye ( p ) − g p ( Y ) n p ( Y ) + S Ye ( p ) − ( − ψ ǫi ( Y − e ( p ) α i )) g p ( Y ) n p ( Y ) . In the last equality we used (2.2.26). We treat the two terms in the last line separately.For the first term S Yi S Ye ( p ) − g p ( Y ) n p ( Y ), we further divide the argument into two cases according tothe Bruhat order. 16i) The case e ( p ) − B s i e ( p ) − . By Proposition 3.1.1, we have S Yi S Ye ( p ) − = S Ys i e ( p ) − = S e ( p ) − ,where the alcove walk p = ( w − A, w − s ǫ i A, . . . , w − s ǫ i · · · s ǫ r i r A, w − s ǫ i · · · s ǫ r i r s ǫi A ) ∈ Γ( −→ z , w − ) (3.1.4)is an extension of p by a crossing (Table 2.1.2). By the hypothesis e ( p ) − B s i e ( p ) − , the laststep of p is an ascent, and we have ϕ + ( p ) = ϕ + ( p ), ϕ − ( p ) = ϕ − ( p ) and ξ des ( p ) = ξ des ( p ). Thuswe have g p ( Y ) n p ( Y ) = g p ( Y ) n p ( Y ) and S Yi S Ye ( p ) − g p ( Y ) n p ( Y ) = S e ( p ) − g p ( Y ) n p ( Y ).(ii) The case e ( p ) − < B s i e ( p ) − . By Propositions 3.1.1, we have S Yi S Ye ( p ) − = n i ( Y − α i ) S Ys i e ( p ) − = n i ( Y − α i ) S Ye ( p ) − = S Ye ( p ) − n i ( Y − e ( p ) α i ) . Here p ∈ Γ( −→ z , w − ) is the same as (3.1.4), but in this case the last step is a descent crossing, andthe hyperplane crossed by the last step is H e ( p ) α i since h r +1 ( p ) = − e ( p )( α i ) = − e ( p ) s i ( α i ) = e ( p ) α i . We then have ξ des ( p ) = ξ des ( p ) ∪ { r + 1 } and n p ( Y ) = n p ( Y ) n i ( Y − h r +1 ( p ) ). Combining themwith ϕ + ( p ) = ϕ + ( p ) and ϕ − ( p ) = ϕ − ( p ), we have n i ( Y − e ( p ) α i ) g p ( Y ) n p ( Y ) = g p ( Y ) n p ( Y ).Hence also in this case, we have S Yi S Ye ( p ) − g p ( Y ) n p ( Y ) = S e ( p ) − g p ( Y ) n p ( Y ).Taking the summation over p , we therefore have X p ∈ Γ( −→ ζ − ,w − ) S Yi S Ye ( p ) − g p ( Y ) n p ( Y ) = X p ∈ Γ( −→ z − ,w − ) , the last step is a crossing S Ye ( p ) − g p ( Y ) n p ( Y ) . (3.1.5)Next we consider the term S Ye ( p ) − ( − ψ ǫi ( Y − e ( p ) α i )) g p ( Y ) n p ( Y ). We make a similar argument asin the first term, and here we use the alcove walk p ∈ Γ( −→ z , w − ) which is an extension of p by afolding. We have e ( p ) = e ( p ), ϕ + ( p ) = ϕ + ( p ) ∪ { r + 1 } and ξ des ( p ) = ξ des ( p ). Using p we have S Ye ( p ) − ( − ψ ǫi ( Y − e ( p ) α i )) g p ( Y ) n p ( Y ) = S Ye ( p ) − g p ( Y ) n p ( Y ). We therefore have X p ∈ Γ( −→ ζ − ,w − ) S Ye ( p ) − ( − ψ ǫi ( Y − e ( p ) α i )) g p ( Y ) n p ( Y ) = X p ∈ Γ( −→ z − ,w − ) , the last step is a folding S Ye ( p ) − g p ( Y ) n p ( Y ) . (3.1.6)By (3.1.3) and (3.1.5), (3.1.6), we have x z S Yw = P p ∈ Γ( −→ z − ,w − ) S Ye ( p ) − g p ( Y ) n p ( Y ). Hence the induc-tion step is proved.The definition (3.1.2) of x z for z ∈ W and the definition (2.2.14) of x µ for µ ∈ P = h ∗ Z are consistentin the following sense. Recall that we denote by t ( µ ) ∈ t ( P ) ⊂ W the element associated to µ ∈ P = h ∗ Z . Lemma 3.1.4.
We have x t ( µ ) = x µ for µ ∈ P = h ∗ Z . In particular, we have x t ( ǫ i ) = x i for i = 1 , . . . , n . Proof.
It is enough to show the latter half. By (2.2.7), we have Y − i = T − i · · · T − n − T − n T − n − · · · T − T − T · · · T i − ( i = 1 , . . . , n ) . Applying the anti-involution φ (2.2.15) to these. x i = φ ( Y − i ) = T ∨ i − · · · T ∨ ( T ∨ ) − ( T ∨ ) − · · · ( T ∨ n − ) − ( T ∨ n ) − ( T ∨ n − ) − · · · ( T ∨ i ) − . On the other hand, we can calculate x t ( ǫ i ) directly by Definition (3.1.2), and can check x i = x t ( ǫ i ) .We denote the dominant chamber for the weight lattice by C := { x ∈ h ∗ R | h α ∨ , x i > , α ∈ R + } . As for the fundamental alcove A (2.1.8), we have A ⊂ C .17et v, w ∈ W , and choose a reduced expression v = s i · · · s i r of v . If an alcove walk p ∈ Γ( −→ v , w )satisfies e ( p ) − A ⊂ C , where e ( p ) ∈ W is the element (2.1.12), then using the W -valued function w ( ) in(2.2.28), we define ̟ ( p ) ∈ ( h ∗ Z ) + by the relation e ( p ) − = w ( ̟ ( p )) . (3.1.7)Also we define Γ C ( −→ v , w ) ⊂ Γ( −→ v , w ) byΓ C ( −→ v , w ) := { p = ( p , . . . , p r ) ∈ Γ( −→ v , w ) | p i ∈ C, ∀ i = 0 , . . . , r } . (3.1.8)Using these symbols, we have the following corollary of Proposition 3.1.3. Corollary 3.1.5 (c.f. [Y12, Corollary 4.1]) . Let λ, µ ∈ h ∗ Z , and fix a reduced expression t ( λ ) = s i r · · · s i .Then we have x λ E µ ( x ) = X p ∈ Γ C ( −−−→ t ( − λ ) ,w ( µ ) − ) g p n p E ̟ ( p ) ( x ) ,g p := Y k ∈ ϕ − ( p ) (cid:0) − ψ − i k ( q sh( − h k ( p )) t ht( − h k ( p )) ) (cid:1) Y k ∈ ϕ + ( p ) (cid:0) − ψ + i k ( q sh( − h k ( p )) t ht( − h k ( p )) ) (cid:1) ,n p := Y k ∈ ξ des ( p ) n i k ( q sh( − h k ( p )) t ht( − h k ( p )) ) . Proof.
We apply x z S Yw = P p ∈ Γ( −→ z − ,w − ) S Ye ( p ) − g p ( Y ) n p ( Y ) in Proposition 3.1.3 to z = t ( λ ) and w = w ( µ ). Since x t ( λ ) = x λ by Lemma 3.1.4, we have x λ S Yw ( µ ) = X p ∈ Γ( −−−→ t ( − λ ) ,w ( µ ) − ) S Ye ( p ) − g p ( Y ) n p ( Y ) . Taking the product of each side with 1 and using the definition of the non-symmetric Koornwinderpolynomial E µ ( x ) (Fact 2.2.2) and the equality Y β q sh( β ) t ht( β ) in (2.2.13), we have x λ E µ ( x ) = X p ∈ Γ( −−−→ t ( − λ ) ,w ( µ ) − ) g p n p S Ye ( p ) − . Next we consider the condition under which the factor n i k ( q sh( − h k ( p )) t ht( − h k ( p )) ) in n p vanishes. By thedefinition of the factor (Proposition 3.1.1), the condition is q sh( − h k ( p )) t ht( − h k ( p )) = t ± ( i k = 1 , . . . , n − q sh( − h k ( p )) t ht( − h k ( p )) = t ± t ± n ( i k = n ). Then by the definition (2.1.13) of h k ( p ), the alcove walk p that contributes to the summation is contained in the dominant chamber C . Now the consequencefollows from the definition of E µ ( x ) and and that (3.1.7) of ̟ ( p ). In this subsection we prepare some lemmas for the symmetrizer U and the Koornwinder polynomials P λ ( x ), which are ( C ∨ n , C n )-type analogue of [Y12, Proposition 3.6]. Lemma 3.2.1 (c.f. [Y12, Proposition 3.6 (a)]) . The symmetrizer U (2.2.33) has the following expression. U = X w ∈ W S Yw Y α ∈L ( w − ,w − ) b ( Y − α ) ,b ( Y − α ) := t − t − Y − α − Y − α ( α W .α n ) t n (1 + t t − n Y − α )(1 − t − t − n Y − α )1 − Y − α ( α ∈ W .α n ) . (3.2.1)Here L ( v, w ) ⊂ S is given by (2.1.10), and w ∈ W is the longest element (2.2.30).18 roof. By the definition of U and the definition (2.2.27) of the Y -intertwiner S Yw , we can expand U as U = X w ∈ W S Yw b w ( Y ) , b w ( Y ) ∈ K ( Y ) . For the longest element w ∈ W , the coefficient of T w in U is 1, and thus we have b w ( Y ) = 1.We calculate the term b w ( Y ) for w ∈ W \ { w } by induction on the length ℓ ( w ). Assume b v ( Y ) = Q α ∈L ( v − ,w − ) b ( Y − α ) for any element v ∈ W satisfying ℓ ( v ) > ℓ ( w ). By the equality U T i = U t i ( i = 1 , . . . , n ) in (2.2.34) and the definition (2.2.24) of S Yi , we have X w ∈ W S Yw b w ( Y ) t i = U t i = U T i = X w ∈ W S Yw b w ( Y ) T i = X w ∈ W S Yw b w ( Y )( S Yi − ψ + i ( Y − α i )) . (3.2.2)Now note that for w = w there exists an index i = 1 , . . . , n such that w B v := ws i . Taking this index i and comparing the coefficients of S Yw in the equality (3.2.2) with the help of (2.2.26) and Corollary3.1.2, we have b v ( Y ) t i = b w ( s i .Y ) − b v ( Y ) ψ + i ( Y − α i ). Here b w ( s i .Y ) is obtained from b w ( Y ) by replacing Y λ with Y s i .λ . Then by the definition (2.2.25) of ψ + i ( z ) we have b w ( Y ) /b v ( s i .Y ) = t i + ψ + i ( Y − s i .α i ) = t i + ψ + i ( Y α i )= t i − t − Y − αi − Y − αi (0 < i < n ) t n (1+ t t − n Y − αn )(1 − t − t − n Y − αn )1 − Y − αn ( i = n ) , so that it is equal to b ( Y − α i ). On the other hand, by (2.1.11) we have L ( w − , w − ) = s i . L ( v − , w − ) ⊔{ α i } . Thus we have b w ( Y ) = b v ( s i .Y ) b ( Y − α i ) = Q α ∈L ( w − ,w − ) b ( Y − α ).We can apply the argument of the proof to the stabilizer W µ ⊂ W for a dominant weight µ ∈ ( h ∗ Z ) + instead of W . As a result, we have the following claim. Corollary 3.2.2.
For each µ ∈ ( h ∗ Z ) + , we have X u ∈ W µ t − w µ u T u = X w ∈ W µ S Yw Y α ∈L ( w − ,w − µ ) b ( Y − α ) . Here b ( Y − α ) ∈ K ( Y ) is given by (3.2.1).For a dominant weight µ ∈ ( h ∗ Z ) + , we denote by W µ ⊂ W (3.2.3)the complete system of representatives of the quotient set W /W µ consisting of the shortest elements.We also denote by v µ ∈ W µ its longest element.Now let us recall the element w ( µ ) ∈ t ( µ ) W ⊂ W in the (2.2.28). We then have the following lemmafor the Koornwinder polynomial P µ ( x ) (Fact 2.2.4) and the non-symmetric Koornwinder polynomial E µ ( x ) (Fact 2.2.2). Lemma 3.2.3 (c.f. [Y12, Proposition 3.6 (b)]) . For λ ∈ ( h ∗ Z ) + we have P λ ( x ) = X v ∈ W λ h Y α ∈ w ( λ ) − L ( v − ,v − λ ) ρ ( α ) i E v.λ ( x ) ,ρ ( α ) := t − t − q sh( − α ) t ht( − α ) − q sh( − α ) t ht( − α ) ( α W.α n ) t n (1 + t t − n q sh( − α ) t ht( − α ) )(1 − t − t − n q sh( − α ) t ht( − α ) )1 − q sh( − α ) t ht( − α ) ( α ∈ W.α n ) , where sh( β ) and ht( β ) for β ∈ S are given by (2.2.12).19 roof. We write Lemma 3.2.1 as U = X w ∈ W S Yw b ( w − ,w − ) ( Y ) , b ( w − ,w − ) ( Y ) := Y α ∈L ( w − ,w − ) b ( Y − α ) . Since W λ consists of representatives of W /W λ , there exist v ∈ W λ and u ∈ W λ uniquely such that w = vu . Using Corollary 3.2.2, we have U = X w ∈ W S Yw b ( w − ,w − ) ( Y ) = h X v ∈ W λ S Yv b ( v − ,v − λ ) ( Y ) ih X u ∈ W λ S Yu b ( u − ,w − λ ) ( Y ) i = h X v ∈ W λ S Yv b ( v − ,v − λ ) ( Y ) ih X u ∈ W λ t − w λ u T u i . The product with S Yw ( λ ) U S Yw ( λ ) h X v ∈ W λ S Yv b ( v − ,v − λ ) ( Y ) ih X u ∈ W λ t − w λ u T u i S Yw ( λ ) t − w λ W λ ( t ) X v ∈ W λ S Yv b ( v − ,v − λ ) ( Y ) S Yw ( λ ) , (3.2.4)where in the second equality we used the Poincar´e polynomial (2.2.32) and the relation ( T u f )1 = t u f for u ∈ W and f ∈ K [ x ± ] satisfying u ( f ) = f . The latter relation is shown as follows. If s i f = f forsome i = 1 , . . . , n , then we have ( T i − t i ) f = c i ( x α i )( s i − f = 0, and so T i f = t i f . Now the relationfollows by induction on the length of u ∈ W .Let us continue the calculation (3.2.4). Note that we have vw ( λ ) = w ( v.λ ) for v ∈ W λ . By thisrelation and (2.2.26), each term in the right hand side of (3.2.4) becomes S Yv b ( v − ,v − λ ) ( Y ) S Yw ( λ ) S Yv S Yw ( λ ) b ( v − ,v − λ ) ( w ( λ ) − .Y )1 = (cid:0) S Yw ( v.λ ) (cid:1)(cid:0) b ( v − ,v − λ ) ( w ( λ ) − .Y )1 (cid:1) . Here b ( v − ,v − λ ) ( w ( λ ) − .Y ) is obtained from b ( v − ,v − λ ) ( Y ) by replacing Y µ with Y w ( λ ) − .µ . Now let usrecall the equality Y α q sh( α ) t ht( α ) in (2.2.13). Then we have b ( Y − α )1 = ρ ( α ), and therefore b ( v − ,v − λ ) ( w ( λ ) − .Y )1 = Y α ∈L ( v − ,v − λ ) (cid:0) b ( Y − w ( λ ) − .α )1 (cid:1) = Y α ∈ w ( λ ) − L ( v − ,v − λ ) ρ ( α ) . By summing over v ∈ W λ we have U S Yw ( λ ) t − w λ W λ ( t ) X v ∈ W λ h Y α ∈ w ( λ ) − L ( v − ,v − λ ) ρ ( α ) i E v.λ ( x ) , Now the result follows from the definition of P λ ( x ) (Fact 2.2.4). In [Y12, Theorem 4.2], Yip derived an expansion formula E µ ( x ) P λ ( x ) = P ν a νλ,µ E ν ( x ) for the productof the non-symmetric Macdonald polynomial E µ ( x ) and the Macdonald polynomial P λ ( x ) in the case ofuntwisted affine root systems. In this subsection, we give its ( C ∨ n , C n )-type analogue, i.e., an expansionformula for the product of the non-symmetric Koornwinder polynomial and the Koornwinder polynomial(Proposition 3.3.2).As a preparation, we cite the explicit formula of the non-symmetric Koornwinder polynomial viaalcove walks derived by Orr and Shimozono [OS18]. It is a ( C ∨ n , C n )-analogue of the explicit formula ofthe non-symmetric Macdonald polynomial in the untwisted affine root systems derived by Ram and Yip[RY11]. Let us call these formulas Ram-Yip type formulas .We prepare the necessary notations for the explanation. Let us given v, w ∈ W and a reducedexpression of w . For an alcove walk p ∈ Γ( −→ w , z ), we denote the decomposition of the element e ( p ) ∈ W (2.1.12) with respect to the presentation W = t ( P ) ⋊ W by e ( p ) = t (wt( p )) d( p ) , d( p ) ∈ W , wt( p ) ∈ h ∗ Z . (3.3.1)20 act 3.3.1 ([RY11, Theorem 3.1], [OS18, Theorem 3.13]) . For µ ∈ h ∗ Z , let w ( µ ) be the shortest elementamong t ( µ ) W ⊂ W (2.2.28), and fix its reduced expression w ( µ ) = s i · · · s i r . Then we have E µ ( x ) = X p ∈ Γ( −−−→ w ( µ ) ,e ) f p t d( p ) x wt( p ) ,f p := Y k ∈ ϕ + ( p ) ψ + i k ( q sh( − β k ) t ht( − β k )) ) Y k ∈ ϕ − ( p ) ψ − i k ( q sh( − β k )) t ht( − β k ) ) , where we set β k := s i r · · · s i k +1 ( α i r ) for k = 1 , . . . , r .Next we introduce some notations necessary for Proposition 3.3.2, which are basically the ones in[Y12, § v, w ∈ W and a reduced expression v = s i · · · s i r . Recall the set Γ C ( −→ v , w ) ofalcove walks belonging to the dominant chamber C as in (3.1.8). Consider an alcove walk in Γ C ( −→ v , w )together with coloring of all the folding steps by either black or gray. We call such a data a colored alcovewalk , and denote by Γ C ( −→ v , w ) (3.3.2)the set of colored alcove walks arising from alcove walks in Γ C ( −→ v , w ).For a colored alcove walk p ∈ Γ C ( −→ v , w ), we denote by p ∗ ∈ Γ( −→ v − , w − e ( p )) (3.3.3)the uncolored alcove walk obtained by straightening all the gray foldings steps of p and by translationso that it ends at e ( p ∗ ) = e ∈ W . More explicitly, for a colored positive walk p ∈ Γ C ( −→ v , w ) with p = ( wA, ws b i A, . . . , ws b i · · · s b r i r A ) , we define e p k for k = 1 , . . . , r as follows, according to whether the k -th step p k − = ws b i · · · s b k − i k − A → p k = ws b i · · · s b k i k A is a gray folding step or not: e p k := ( ws b i · · · s b k − i k − s i k A ( p k − → p k is a gray folding step) p k (otherwise) . Thus we obtain a new uncolored alcove walk e p = ( e p , . . . , e p r ) ∈ Γ( −→ v , w ), which was called the oneobtained “by straightening all the gray foldings”. Next we denote by ( c , . . . , c r ) ∈ { , } r the bitsequence corresponding to e p . In other words, we have e p = ( wA, . . . , ws c i · · · s c r i r A ). Now the alcove walk p ∗ is obtained by reversing the order of e p and translating the start to w − e ( e p ). Explicitly, we have p ∗ := ( s c i · · · s c r i r A, s c i · · · s c r − i r − A, . . . , s c i A, A ) . Proposition 3.3.2 (c.f. [Y12, Theorem 4.2]) . For a weight µ ∈ h ∗ Z , we take a reduced expression w ( µ ) = s i r · · · s i of w ( µ ) ∈ t ( µ ) W ⊂ W . Then for any dominant weight λ ∈ ( h ∗ Z ) + we have E µ ( x ) P λ ( x ) = X v ∈ W λ X p ∈ Γ C ( −−−→ w ( µ ) − , ( vw ( λ )) − ) A p C p E ̟ ( p ) ( x ) . Here W λ is given by (3.2.3), and the term A p is given with the help of ρ ( α ) in Lemma 3.2.3 by A p := Y α ∈ w ( λ ) − L ( v − ,v − λ ) ρ ( α ) ,ρ ( α ) := t − t − q sh( − α ) t ht( − α ) − q sh( − α ) t ht( − α ) ( α W.α n ) t n (1 + t t − n q sh( − α ) t ht( − α ) )(1 − t − t − n q sh( − α ) t ht( − α ) )1 − q sh( − α ) t ht( − α ) ( α ∈ W.α n ) . C p is given by C p := Q rk =1 C p,k , whose factor C p,k is determined by the k -th step of p asfollows. C p,k := k -th step of p is a positive crossing Q k ∈ ξ des ( p ) n i k ( q sh( − h k ( p )) t ht( − h k ( p )) ) a negative crossing − ψ + i k ( q sh( − h k ( p )) t ht( − h k ( p )) ) a gray positive folding − ψ − i k ( q sh( − h k ( p )) t ht( − h k ( p )) ) a gray negative folding ψ + i k ( q sh( − β k ) t ht( − β k ) ) a black folding and the k -th step of p ∗ is positive ψ − i k ( q sh( − β k ) t ht( − β k ) ) a black folding and the k -th step of p ∗ is negative , where n i ( Y β ) is given by Proposition 3.1.1, ψ ± i k ( z ) is given by (2.2.25) and h k ( p ) is given by (2.1.13).We also used β k := s i · · · s i r − ( α i r ) for k = 1 , . . . r . Finally ̟ ( p ) is given by (3.1.7).Note that the term A p actually depends only on v ∈ W µ , which corresponds to the beginning of thecolored alcove walk p . Proof.
On the Ram-Yip type formula E µ ( x ) = P h ∈ Γ( −−−→ w ( µ ) ,e ) f h t d( h ) x wt( h ) (Fact 3.3.1), let us act U S Yw ( λ ) E µ ( x ) U S Yw ( λ ) (cid:2) X h ∈ Γ( −−−→ w ( µ ) ,e ) f h t d( h ) x wt( h ) (cid:3) U S Yw ( λ ) X h ∈ Γ( −−−→ w ( µ ) ,e ) f h x e ( h ) U S Yw ( λ ) . Here the second equality follows from the definition (3.3.1) of wt( h ) and d( h ), as well as from the relation T i U = t i U in (2.2.34). Moreover, by Lemma 3.2.3 and using the notation in its proof, we have E µ ( x ) U S Yw ( λ ) X h ∈ Γ( −−−→ w ( µ ) ,e ) f h x e ( h ) (cid:2) t − w λ W λ ( t ) X v ∈ W λ S Yv b ( v − ,v − λ ) ( Y ) S Yw ( λ ) (cid:3) = t − w λ W λ ( t ) X v ∈ W λ X h ∈ Γ( −−−→ w ( µ ) ,e ) f h S Yv S Yw ( λ ) b ( v − ,v − λ ) ( w ( λ ) − .Y )1= t − w λ W λ ( t ) X v ∈ W λ A p X h ∈ Γ( −−−→ w ( µ ) ,e ) f h x e ( h ) S Yvw ( λ ) . (3.3.4)Here we set A p := b (( vw ( λ )) − ,t ( − w λ )) = Q α ∈ w ( λ ) − L ( v − ,v − λ ) ρ ( α ). As for the factor f h x e ( h ) S Yvw ( λ ) z := ( vw ( λ )) − and using Proposition 3.1.3 and Corollary 3.1.5, wehave f h x e ( h ) S Yvw ( λ ) f h X q ∈ Γ( −−→ e ( h ) − ,z ) S Ye ( q ) − n q ( Y ) g q ( Y )1 = f h X q ∈ Γ C ( −−→ e ( h ) − ,z ) n q g q E ̟ ( q ) ( x ) . (3.3.5)We will rewrite this sum over uncolored alcove walks in Γ C ( −−→ e ( h ) − , z ) as a sum over colored alcove walksin Γ C ( −−−→ w ( µ ) − , z ).Let us given an uncolored alcove walk q ∈ Γ C ( −−→ e ( h ) − , z ). Since q is an alcove walk of type −−→ e ( h ) − = −−−→ w ( µ ) − , we can compare the bit sequence of q with the bit sequence of h . In this comparison, if the k -thstep of q is a folding and the k -th step of h is a crossing, then we color the k -th folding step of q by gray.Otherwise we color it by black. Thus we obtain a colored alcove walk, which is denoted by p . Note thatwe have p ∈ Γ C ( −−−→ w ( µ ) − , z ). Then each term of the right hand side in (3.3.5) is equal to f h n q g q E ̟ ( q ) ( x ) = f p ∗ n p g p E ̟ ( p ) ( x ) , where p ∗ is given by (3.3.3). We can also express f p ∗ using β k = s i · · · s i k − ( α i k ) as f p ∗ = Y k ∈ ϕ + ( p ∗ ) ψ + i k ( q sh( − β k ) t ht( − β k ) ) Y k ∈ ϕ − ( p ∗ ) ψ − i k ( q sh( − β k ) t ht( − β k ) ) .
22s a result, the last line of (3.3.4) is rewritten by a sum over p ∈ Γ C ( −−−→ w ( µ ) − , z ).Divided by the factor t − w λ W λ ( t ), the left hand side of (3.3.4) is equal to P λ ( x ). Thus we have E µ ( x ) P λ ( x ) = X v ∈ W λ A p X h ∈ Γ( −−−→ w ( µ ) ,e ) f h X q ∈ Γ C ( −−→ e ( h ) − ,z ) n q g q E ̟ ( q ) ( x )= X v ∈ W λ A p X p ∈ Γ C ( −−−→ w ( µ ) − ,z ) f p ∗ g p n p E ̟ ( p ) ( x ) . We obtain the result by collecting the terms from f p ∗ , g p and n p which depend only on the k -th step of p ∈ Γ C ( −−−→ w ( µ ) − , z ) and denoting them by C p,k . In this subsection, we derive our main Theorem 3.4.2 on LR coefficients of Koornwinder polynomials.We start with a preliminary lemma. Recall the complete system W λ of representatives of W /W λ in(3.2.3) and the element w ( λ ) ∈ t ( λ ) W in (2.2.28). Lemma 3.4.1 (c.f. [Y12, Proposition 3.7]) . Let λ ∈ ( h ∗ Z ) + . If v ∈ W λ satisfies vw ( λ ) < B w ( λ ), then wehave U S Yv S Yw ( λ ) h Y α ∈L ( w ( λ ) − , ( vw ( λ )) − ) ρ ( − α ) i U S Yw ( λ ) , where ρ ( α ) is defined in Lemma 3.2.3. Proof.
Recall the equality
U T i = U t i for i = 1 , . . . , n in (2.2.34). Therefore we have U S Yi S Yw U (cid:0) T ∨ i + ψ + i ( Y − α i ) (cid:1) S Yw U S Yw (cid:0) t i + ψ + i ( q sh( − w − α i ) t ht( − w − α i ) ) (cid:1) . Assume that v ∈ W λ satisfies vw ( λ ) < B w ( λ ), and take a reduced expression v = s i · · · s i r . Using theabove relation, we expand the product U S Yv S Yw ( λ ) U S Yi · · · S Yi r S Yw ( λ ) U S Yv S Yw ( λ ) U ( t i + ψ + i ( Y − α i )) S Ys i v S Yw ( λ ) U S Ys i v ( t i + ψ + i ( Y − s ir ··· s i ( α i ) )) S Yw ( λ )
1= ( t i + ψ + i ( Y − s ir ··· s i ( α i ) )) U ( t i + ψ + i ( Y − α i )) S Ys i s i v S Yw ( λ ) · · · = U h r Y j =1 (cid:0) t i j + ψ + i j ( Y − w ( λ ) − s ir ··· s ij +1 α ij ) (cid:1)i S Yw ( λ ) U S Yw ( λ ) h r Y j =1 (cid:0) t i j + ψ + i j ( Y − w ( λ ) − s ir ··· s ij +1 α ij ) (cid:1)i U S Yw ( λ ) h Y α ∈L ( w ( λ ) − , ( v.w ( λ )) − ) ρ ( − α ) i . Therefore the claim is obtained.We prepare some symbols to state the main theorem. For µ ∈ h ∗ Z , the orbit W .µ contains a uniquedominant weight. We denote it by µ + ∈ W .µ ∩ ( h ∗ Z ) + . (3.4.1)Let us also recall the set Γ C ( −→ v , w ) of colored alcove walks defined in (3.3.2). Theorem 3.4.2.
Let us given dominant weights λ, µ ∈ ( h ∗ Z ) + . Choose a reduced expression w ( λ ) = s i r · · · s i of the shortest element w ( λ ) ∈ t ( λ ) W in (2.2.28). Then we have P λ ( x ) P µ ( x ) = 1 t − w λ W λ ( t ) X v ∈ W µ X p ∈ Γ C ( −−−→ w ( λ ) − , ( vw ( µ )) − ) A p B p C p P − w . wt( p ) ( x ) . A p := Q α ∈ w ( µ ) − L ( v − ,v − µ ) ρ ( α ) with ρ ( α ) given in Proposition 3.3.2. The term C p is the same asthat in Proposition 3.3.2, and wt( p ) ∈ h ∗ Z is defined by (3.3.1). The term B p is defined by B p := Y α ∈L ( t (wt( p )) w ,e ( p )) ρ ( − α ) . Proof.
The strategy is to calculate the product of Koornwinder polynomials by acting the symmetrizer U to each side of the equation in Proposition 3.3.2.For a colored alcove walk p ∈ Γ C ( −−→ w ( λ ) − , ( vw ( µ )) − ), let z ∈ W be the shortest element among { z ∈ W | z.̟ ( p ) + = ̟ ( p ) } . Note that we have w ( ̟ ( p ) + ) − = w ( − w ̟ ( p ) + ). Since e ( p ) ∈ t (wt( p )) W by the definition of wt( p ), w ( − w ̟ ( p ) + ) is the shortest element among t (wt( p )) W . By Lemma 3.4.1,we then have U E ̟ ( p ) ( x )1 = U S Yz S Yw ( ̟ ( p ) + ) h Y α ∈L ( t (wt( p )) w ,e ( p )) ρ ( − α ) i P − w . wt( p ) ( x ) . By Proposition 3.3.2 and this equality, we have P λ ( x ) P µ ( x ) = 1 t − w λ W λ ( t ) U E λ ( x ) P µ ( x )1= 1 t − w λ W λ ( t ) X v ∈ W µ X p ∈ Γ C ( −−−→ w ( λ ) − , ( vw ( µ )) − ) A p B p C p P − w . wt( p ) ( x ) . Hence the claim is obtained.
In Theorem 3.4.2, we derived an explicit formula of the LR coefficient c νλ,µ in the product P λ ( x ) P µ ( x ) = P ν c νλ,µ P ν ( x ) of Koornwinder polynomials using alcove walks. In this section, we discuss several special-izations of the formula. As mentioned in § V = R ǫ ∨ of dimension 1 and its dual space V ∗ = R ǫ . The root system of type C is R = {± ǫ } ⊂ V ∗ , thesimple root is α = 2 ǫ , and the fundamental weight is ω = ǫ . The weight lattice is P = h ∗ Z = Z ǫ ⊂ V ∗ ,and the set of dominant weights is ( h ∗ Z ) + = N ǫ . The finite Weyl group W is the group of order twogenerated by s := s α , and the longest element of W is w = s . The affine root system of rank 1 is S = (cid:8) ± ǫ + kδ, ± ǫ + k δ | k ∈ Z (cid:9) with α = δ − ǫ , and the extended affine Weyl group W is the groupgenerated by s and s := s α . The decomposition W = t ( P ) ⋊ W (2.1.2) is the semi-direct product of t ( P ) = h t ( ǫ ) = s s i ≃ Z and W = h s i ≃ Z / Z .We denote by P l ( x ) = P l ( x ; q, t , t , u , u )the Askey-Wilson polynomial associated to the dominant weight λ = lω = lǫ ( l ∈ N ). Note that it hasfive parameters.First, we consider the simplest case. Following the case of type A (see § c νλ,µ with λ or µ equal to a minuscule weight Pieri coefficients . Since the weight ω is the unique minusculeweight in the root system of type C n , we consider the case λ = ω for the rank one case.Let us write down explicitly the Askey-Wilson polynomial P ( x ) = P ω ( x ). In the following calcula-tion, we need an explicit form of the term ρ ( α ) ( α ∈ S ) in Proposition 3.3.2 and Theorem 3.4.2. Theresult is: ρ ( α ) := t (1 + q k t t − ( t t ) − j )(1 − q k t − t − ( t t ) − j )1 − q k ( t t ) − j ( α = 2 jǫ + kδ ∈ S ) . (4.1.1)24 emma 4.1.1. The Askey-Wilson polynomial associated to the minuscule weight ω is P ( x ) = x + x − + ρ (2 δ − α ) ψ − ( qt t ) + t ψ +0 ( qt t ) + ψ +1 ( q t t ) ψ − ( qt t ) . Here ψ ± k ( z ) ( k = 0 ,
1) is given by (2.2.25) with n = 1. Explicitly, we have ψ ± ( z ) := ∓ ( u − u − ) + z ± ( u − u − )1 − z ± , ψ ± ( z ) := ∓ ( t − t − ) + z ± ( t − t − )1 − z ± . (4.1.2) Proof.
Below we use the word non-symmetric Askey-Wilson polynomials to mean non-symmetric Koorn-winder polynomials (Fact 2.2.2) in the rank 1 case. By Lemma 3.2.3, we can rewrite P ( x ) as a linearcombination of non-symmetric Askey-Wilson polynomials E k ( x ) = E kω ( x ), k ∈ Z . The result is P ( x ) = ρ (2 δ − α ) E ( x ) + E − ( x ) . Next, using the Ram-Yip type formula (Fact 3.3.1), we can expand E ( x ) and E − ( x ) by monomials.The results are E ( x ) = t x + ψ − ( qt t ) , E − ( x ) = x − + t ψ +0 ( qt t ) + t ψ +1 ( q t t ) x + ψ +1 ( q t t ) ψ − ( qt t ) . By these formulas, we have P ( x ) = x − +( t ρ (2 δ − α )+ t ψ +1 ( q t t )) x + ψ − ( qt t ) ρ (2 δ − α )+ t ψ +0 ( qt t )+ ψ +1 ( q t t ) ψ − ( qt t ) . By a direct calculation, the coefficients of x is shown to be t ρ (2 δ − α ) + t ψ +1 ( q t t ) = 1 . Therefore the claim is obtained.
Remark 4.1.2.
Let us replace the parameters ( q, t , t , u , u ) with the original parameters ( q, a, b, c, d )of Askey-Wilson polynomials in [AW85]. The correspondence (1.1.1) of parameters can be rewritten as( q, t , t , u , u ) = ( q, − q − ab, − cd, − a/b, − c/d ) . Using this correspondence and the relation abcd = qt t n , we can rewrite P ( x ) as P ( x ) = x + x − + πs − s ′ − π , π := abcd, s := a + b + c + d, s ′ := a − + b − + c − + d − . We can then compare P ( x ) with the original Askey-Wilson polynomials p n ( z ) in [AW85, p.5]. By loc.cit., we have p ( z ) = 2(1 − π ) z + πs − s ′ , and thus(1 − π ) P ( x ) = p (cid:0) ( x + x − ) / (cid:1) . Therefore they coincide up to the normalization factor.
Proposition 4.1.3.
For a dominant weight λ = lω ∈ ( h ∗ Z ) + , l ∈ N , we have P ( x ) P l ( x ) = P l +1 ( x ) + F l P l ( x ) + G l P l − ( x ) ,F l := ρ ( − lδ + α )( − ψ − ( q l +1 t t ) + ψ − ( qt t )) + ρ (2 lδ − α )( − ψ +0 ( q l − t t ) + ψ +0 ( qt t )) ,G l := ρ (2 lδ − α ) ρ ( − l − δ + α ) n ( q l − t t ) . Here ρ ( α ) is given by (4.1.1), ψ ± ( z ) is given by (4.1.2), and n ( z ) is given in Proposition 3.1.1 with n = 1. Explicitly, the last one is given by n ( z ) := (1 − u u z )(1 + u u − z )1 − z (1 + u − u z )(1 − u − u − z )1 − z . roof. By Theorem 3.4.2, we have P λ ( x ) P µ ( x ) = 1 t − w λ W λ ( t ) X v ∈ W µ X p ∈ Γ C ( −−−→ w ( λ ) − , ( vw ( µ )) − ) A p B p C p P − w . wt( p ) ( x )for dominant weights λ, µ ∈ ( h ∗ Z ) + . We apply this equation to the case λ = ω and µ = lω . In this casethe stabilizer W µ ⊂ W in (2.2.29) is W µ = { e } , and the complete system W µ (3.2.3) of representativesof W /W λ is W lω = { e, s } . As for the shortest element w ( ν ) ∈ t ( ν ) W given in (2.2.28), we have by t ( ω ) = s s that w ( ω ) = s and w ( lω ) = ( s s ) l − s .First, we calculate the denominator t − w ω W ω ( t ). As for the longest element w λ ∈ W λ in (2.2.30), wehave w ω = e . Thus, by recalling the definition (2.2.31) of t w ( w ∈ W ), we have t − w ω W ω ( t ) = t − e t e = 1.Next, as for the sum in the right hand side, we calculate the case v = s . The set of alcove walksis then Γ C ( −−→ w ( λ ) − , ( vw ( µ )) − ) = Γ C ( −→ s , t ( lω )). In the upper half of Table 4.1.1, we display the alcovewalks p therein together with the corresponding terms A p , B p and C p . In the table we denote by H and H the hyperplanes in the W -orbits of H α and H α respectively. We also denote a black folding bya solid line, and a gray folding by a dotted line.Next we study the case v = e . The set of alcove walks is Γ C ( −−→ w ( λ ) − , ( vw ( µ )) − ) = Γ C ( −→ s , w ( lω )),and in the lower half of Table 4.1.1 we display the alcove walks p therein together with the correspondingterms A p , B p and C p . Table 4.1.1: Colored alcove walks in Proposition 4.1.3The claim is obtained by to summaries the above calculation.26 emark 4.1.4. Continuing Remark 4.1.2, we rewrite the result in Proposition 4.1.3 in terms of theoriginal parameters ( q, a, b, c, d ) of Askey-Wilson polynomials. The result is P ( x ) P l ( x ) = P l +1 ( x ) + F l P l ( x ) + G l − P l − ( x ) , (4.1.3)where the factors F l and G l are given by F l := f l + ( πs ′ − s )1 − π , f l := q l − (1 + q l − π )( qs + πs ′ ) − q l − (1 + q ) π ( s + qs ′ )(1 − q l − π )(1 − q l π ) ,G l := g l γ l − γ γ l , g l := (1 − q l ) (1 − q l − ab )(1 − q l − ac )(1 − q l − ad )(1 − q l − bc )(1 − q l − bd )(1 − q l − cd )(1 − q l − π )(1 − q l − π ) ,π := abcd, s := a + b + c + d, s ′ := a − + b − + c − + d − ,γ l := ( q l − π ; q ) l = (1 − q l − π )(1 − q l π ) · · · (1 − q l − π ) . In the case l = 0, we have ρ ( − α ) = 0, and thus F = 0. If we define p l ( z ) by the relation P l ( x ) = γ − l p l (( x + x − ) / zp l ( z ) = h l p l +1 ( z ) + f l p l ( z ) + g l p l − ( z ) , h l := 1 − q l − π (1 − q l − π )(1 − q l π ) , p ( z ) = 1 , p − ( z ) = 0 . This recurrence formula is nothing but the one in [AW85, (1.24)–(1.27)]. Thus p l coincides with theoriginal Askey-Wilson polynomial in [AW85], and in particular, it can be expressed as a q -hypergeometricseries.So far we studied Pieri coefficients. Next we study the general LR coefficients for Askey-Wilsonpolynomials. Corollary 4.1.5.
For dominant weights lω and mω in ( h ∗ Z ) + , l, m ∈ N , we have P lω ( x ) P mω ( x ) = X v ∈ W X p ∈ Γ C ( −−−−−−−−→ t (( l − ω ) s ,t ( mω ) s v ) A AWp B AWp C p P wt( p ) ( x ) , where the terms A AWp and B AWp are given by A AWp := ( ρ (2 mδ − α ) ( v = e )1 ( v = s ) , B AWp := ( ρ ( − ℓ ( e ( p )) δ + α ) ( ℓ ( e ( p )) ∈ Z )1 ( ℓ ( e ( p )) / ∈ Z )with ρ ( α ) in Proposition 4.1.3, and C p is given in Theorem 3.4.2. Proof.
We apply the formula P λ ( x ) P µ ( x ) = 1 t − w λ W λ ( t ) X v ∈ W µ X p ∈ Γ C ( −−−→ w ( λ ) − , ( vw ( µ )) − ) A p B p C p P − w . wt( p ) ( x )in Theorem 3.4.2 to the case λ = lω and µ = mω . Similarly as in Proposition 4.1.3, we have W lω = { e } and W mω = { e, s } = W . Using t ( lω ) = ( s s ) l and t ( mω ) = ( s s ) m , we have w ( lω ) = t ( lω ) s =( s s ) l − s and w ( mω ) = ( s s ) m − s . Therefore the range of the sum of alcove walks in the right handside becomes Γ C ( −−→ w ( λ ) − , ( vw ( µ )) − ) = Γ C ( −−−−−−−−−→ t (( l − ω ) s , t ( mω ) s v ) ( v ∈ W ) . As for the denominator t − w lω W lω ( t ), we have by w lω = e that t − w lω W lω ( t ) = t − e t e = 1.Now we study the factors A p and B p , and want to reduce the ranges of the products. First, as forthe product A p = Q α ∈L (( vw ( µ )) − ,t ( − w µ )) ρ ( α ), the longest element w ∈ W is s and t ( µ ) = ( s s ) l = w ( µ ) − s . Thus, in the case v = e , we have L (( vw ( µ )) − , t ( − w µ )) = L ( w ( µ ) − , t ( µ )) = { mδ − α } .
27n the case v = s , we have L (( vw ( µ )) − , t ( − w µ )) = L ( w ( µ ) s , t ( µ )) = L ( t ( µ ) , t ( µ )) = ∅ . Hence A p is equal to A AWp in the claim.Next we consider the product B p = Q α ∈L ( t (wt( p )) w ,e ( p )) ρ ( − α ). we separate the argument accordingto whether the length ℓ ( e ( p )) of e ( p ) is even or odd. In the case ℓ ( e ( p )) is even, there is k ∈ N such that e ( p ) = ( s s ) n = t ( kω ), 0 ≤ k ≤ m . In this case, the range of the product is L ( t (wt( p )) w , e ( p )) = L ( t ( kω ) s , t ( kω )) = { kδ − α } . In the case ℓ ( e ( p )) is odd, there is k ∈ N such that we can write e ( p ) = ( s s ) k − s = t ( kω ) s , 1 ≤ k ≤ m .Thus the range of the product is L ( t (wt( p )) w , e ( p )) = L ( t ( kω ) s , t ( kω ) s ) = ∅ . Therefore B p is equal to B AWp in the claim.
In the case of type A n , the specialized Macdonald polynomials P A n λ ( x ; q = 0 , t ) coincide with Hall-Littlewood polynomials. Motivated by this fact, Yip calls in [Y12, § q = 0 Hall-Littlewood polynomials , and derived a simplified formulaof LR coefficients. Following Yip’s terminology, let us call the specialized Koornwinder polynomials P λ ( x ; t ) := P λ ( x ; q = 0 , t , t, t n , u , u n ) the Hall-Littlewood limit . Proposition 4.2.1 (c.f. [Y12, Corollary 4.13]) . Let us given dominant weights λ, µ ∈ ( h ∗ Z ) + and areduced expression w ( λ ) = s i r · · · s i of the shortest element w ( λ ) (2.2.28). Then we have P λ ( x ; t ) P µ ( x ; t ) = 1 t − w λ W λ ( t ) X v ∈ W µ X p ∈ Γ C + ( −−−→ w ( λ ) − , ( vw ( µ )) − ) F p ( t ) P − w . wt( p ) ( x ; t ) ,F p ( t ) := Y α ∈L (( vw ( µ )) − ,t ( − w µ )) t α Y α ∈L ( t (wt( p )) w ,e ( p )) t − α × Y k ∈ ϕ + ( p ) , α ik W.α ( t − α ik − t α ik ) Y k ∈ ϕ + ( p ) , α ik ∈ W.α ( u − n − u n ) . Here Γ C + ( −−→ w ( λ ) − , ( vw ( µ )) − ) is the subset of Γ C ( −−→ w ( λ ) − , ( vw ( µ )) − ) consisting of alcove walks whosefoldings are positive. Proof.
We denote the coefficient in Theorem 3.4.2 by a p ( q, t ) := A p B p C p . First, we show that if a p (0 , t ) = 0 for a colored alcove walk p ∈ Γ C ( −−→ w ( λ ) − , ( v.w ( µ )) − ), then all thefoldings of p are gray and positive. We assume that the k -th step of p is a gray negative folding. Then,as for the factor C p,k = − ψ − i k ( q sh( − h k ( p )) t ht( − h k ( p )) ) we have C p,k | q =0 = 0. In fact, we have ψ − i k ( z ) = ( t i k − t − i k ) + z − ( u i k − u − i k )1 − z − = z ( t i k − t − i k ) + z ( u i k − u − i k )1 − z , and by substituting z = q sh( − h k ( p )) t ht( − h k ( p )) and q = 0 we have C p,k | q =0 = 0. Thus we showed that nogray negative folding contributes to a p (0 , t ).Next we show that black foldings of p don’t contribute to a p (0 , t ). Note that there exists an alcovewalk l ∈ Γ( −−→ w ( λ ) , e ) whose steps are crossings since we fixed a reduced expression of w ( λ ). Moreover all28he steps of l are positive. Then we find that any alcove walk in Γ C ( −−→ w ( λ ) , e ) \ { l } has a negative folding.In other words, if an alcove walk p ∈ Γ C ( −−→ w ( λ ) − , ( vw ( µ )) − ) has a black folding, then p ∗ in (3.3.3) has atleast one negative folding. Then, as for the factor C p,k = − ψ − i k ( q sh( − β k ) t ht( − β k ) ), we have C p,k | q =0 = 0by a direct calculation. Thus, no black folding contributes to a p (0 , t ).By the discussion so far, we find that neither colored folding contributes to a p (0 , t ). Thus, the set ofalcove walks effective to the sum is (cid:8) p ∈ Γ C ( −−−→ w ( µ ) − , ( v.w ( λ )) − ) | ϕ ( p ) = ϕ + ( p ) (cid:9) .Specializing q = 0 in A p , B p and C p , we have A p | q =0 = Y α ∈L (( vw ( µ )) − ,t ( − w µ )) t α , B p | q =0 = Y α ∈L ( t (wt( p )) w ,e ( p )) t − α ,C p | q =0 = Y k ∈ ϕ + ( p ) , α ik W.α ( t − α ik − t α ik ) Y k ∈ ϕ + ( p ) , α ik ∈ W.α ( u − n − u n ) . Therefore the claim is obtained. Finally, as explicit examples of LR coefficients in Theorem 3.4.2, we calculate the product P λ ( x ) P µ ( x )of Koornwinder polynomials of rank 2.We write down the root system of rank 2. The root system of type C is R := {± ǫ ± ǫ } ∪ {± ǫ , ± ǫ } ⊂ V ∗ := R ǫ ⊕ R ǫ , the simple roots are α = ǫ − ǫ and α = 2 ǫ , and the fundamental weights are ω = ǫ and ω = ǫ + ǫ . The weight lattice is P = h ∗ Z = Z ǫ ⊕ Z ǫ ⊂ V ∗ , and the set of dominant weights is ( h ∗ Z ) + = { λ ǫ + λ ǫ ∈ h ∗ Z | λ ≥ λ ≥ } . The finite Weyl group W is the hyper-octahedral group of order 8generated by s := s α and s := s α . The longest element of W is w = s s s s = s s s s .The affine root system of type ( C ∨ , C ) is S = (cid:8) ± ǫ i + kδ, ± ǫ i + 12 kδ | k ∈ Z , i = 1 , (cid:9) ∪ {± ǫ ± ǫ + kδ | k ∈ Z } , and the affine simple root is α = δ − ǫ . The extended affine Weyl group W is generated by s , s and s := s α , and the decomposition W = t ( P ) ⋊ W (2.1.2) is a semi-direct product of t ( P ) = h t ( ǫ ) , t ( ǫ ) i ≃ Z and W = h s , s i ≃ {± } ⋊ S . The elements t ( ǫ ) and t ( ǫ ) have reduced expressions t ( ǫ ) = s s s s and t ( ǫ ) = s s s s respectively.In this setting we apply Theorem 3.4.2 to the case λ = ω and µ = ω . The result is as follows. Proposition 4.3.1.
For Koornwinder polynomials of rank 2, we have P ω ( x ) P ω ( x ) = P ω + ω ( x ) + F P ω ( x ) + GP ω ( x ) ,F := ρ ( − δ + ( ǫ + ǫ )) ρ ( − δ + 2 ǫ ) ρ ( − ( ǫ − ǫ ))( − ψ − ( q t t ) + ψ − ( qt t )) G := ρ (2 δ − ( ǫ + ǫ )) ρ (2 δ − ǫ ) ρ ( − ǫ ) ρ ( − δ + ( ǫ + ǫ )) n ( qt t ) Proof.
Applying Theorem 3.4.2 to n = 2, λ = ω and µ = ω , we have P ω ( x ) P ω ( x ) = 1 t − w ω W ω ( t ) X v ∈ W ω X p ∈ Γ C ( −−−−→ w ( ω ) − , ( vw ( ω )) − ) A p B p C p P − w . wt( p ) ( x ) . We have W ω = { e, s } , W ω = { e, s , s s , s s s } and w ( ω ) = s , w ( ω ) = s s s . The denominator t − w ω W ω ( t ) can be calculated with the help of w ω = s as t − w ω W ω ( t ) = t − s ( t e + t s ) = t − + t .Next we consider the term A p B p C p . The alcove walk p ∗ associated to p ∈ Γ C ( −−−→ w ( ω ) − , ( vw ( ω )) − )is given by either p ∗ or p ∗ in Table 4.3.1.Let us calculate the term A p = Q α ρ ( α ). The range of the product is w ( µ ) − L ( v − , v − µ ) = L (cid:0) ( vw ( ω )) − , t ( − w ω ) (cid:1) , ∗ p ∗ O ω ω ω O ω ω ω Table 4.3.1: Classification of p ∗ and according to v ∈ W ω = { e, s , s s , s s s } it is given by L (cid:0) ( vw ( ω )) − , t ( − w ω ) (cid:1) = { δ − ǫ , δ − ( ǫ + ǫ ) , δ − ǫ } ( v = e ) { δ − ( ǫ + ǫ ) , δ − ǫ } ( v = s ) { δ − ǫ } ( v = s s ) ∅ ( v = s s s ) . Then we have A p = ρ (2 δ − ǫ ) ρ (2 δ − ( ǫ + ǫ )) ρ (2 δ − ǫ ) ( v = e ) ρ (2 δ − ( ǫ + ǫ )) ρ (2 δ − ǫ ) ( v = s ) ρ (2 δ − ǫ ) ( v = s s )1 ( v = s s s ) . For each v ∈ W ω and the corresponding colored alcove walks p , we calculate B p and C p . The resultsare shown in Tables 4.3.2–4.3.5. The symbol in the column of p such as X and X refers to thecorresponding picture in Figure 4.3.1. p ∗ p B p C p − w wt( p ) p ∗ X ρ ( − ǫ ) ρ ( − δ + ( ǫ + ǫ )) n ( qt t ) ω X ρ ( − ( ǫ − ǫ )) − ψ +0 ( qt t ) ω p ∗ X ρ ( − ( ǫ − ǫ )) ψ +0 ( qt t ) ω Table 4.3.2: Colored lcove walks in the case v = ep ∗ p B p C p − w wt( p ) p ∗ Y ρ ( − ǫ ) ρ ( − δ + 2 ǫ ) ρ ( − δ + ( ǫ + ǫ )) n ( qt t ) ω Y ρ ( − ( ǫ − ǫ )) ρ ( − δ + 2 ǫ ) − ψ +0 ( qt t ) ω p ∗ Y ρ ( − ( ǫ − ǫ )) ρ ( − δ + 2 ǫ ) ψ +0 ( qt t ) ω Table 4.3.3: Colored alcove walks in the case v = s The claim is now obtained by summing the terms A p B p C p P − w wt( p ) ( x ).30 ∗ p B p C p − w wt( p ) p ∗ Z ω + ω Z ρ ( − δ + ( ǫ + ǫ )) ρ ( − δ + 2 ǫ ) ρ ( − ( ǫ − ǫ )) − ψ − ( q t t ) ω p ∗ Z ρ ( − δ + ( ǫ + ǫ )) ρ ( − δ + 2 ǫ ) ρ ( − ( ǫ − ǫ )) ψ − ( qt t ) ω Table 4.3.4: Colored alcove walks in the case v = s s p ∗ p B p C p − w wt( p ) p ∗ W ρ ( − δ + 2 ǫ ) 1 ω + ω W ρ ( − δ + 2 ǫ ) ρ ( − δ + ( ǫ + ǫ )) ρ ( − δ + 2 ǫ ) ρ ( − ( ǫ − ǫ )) − ψ − ( q t t ) ω p ∗ W ρ ( − δ + 2 ǫ ) ρ ( − δ + ( ǫ + ǫ )) ρ ( − δ + 2 ǫ ) ρ ( − ( ǫ − ǫ )) ψ − ( qt t ) ω Table 4.3.5: Colored alcove walks in the case v = s s s X X X
211 2 ω ω ω ω ω ω ω ω ω Y Y Y ω ω ω ω ω ω Z Z Z ω ω + ω ω + ω ω ω + ω ω + ω ω ω + ω ω + ω W W W ω ω + ω ω ω ω + ω ω ω ω + ω ω Figure 4.3.1: Colored alcove walks in Proposition 4.3.131 .4 Future works
As a concluding remark, let us give some problems which we would like to explore in future works. • Specialization of Theorem 3.4.2 to Macdonald polynomials of type B n , C n and BC n . • Simplified Pieri formulas and specialization to B n , C n and BC n types. • Tableau-type formulas for Koornwinder polynomials from Theorem 3.4.2.
Acknowledgements
The author would like to thank the thesis adviser Shintaro Yanagida for the guidance and careful readingof the manuscript. He would also like to thank Masatoshi Noumi for the explanation on the Macdonald-Cherednik theory and Koornwinder polynomials through the master course in Kobe University, and alsothank Satoshi Naito for his important comments.
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