A deformation of affine Hecke algebra and integrable stochastic particle system
aa r X i v : . [ m a t h - ph ] O c t A DEFORMATION OF AFFINE HECKE ALGEBRA ANDINTEGRABLE STOCHASTIC PARTICLE SYSTEM
YOSHIHIRO TAKEYAMA
Abstract.
We introduce a deformation of the affine Hecke algebra of type GL which describes the commutation relations of the divided difference operatorsfound by Lascoux and Sch¨utzenberger and the multiplication operators. Mak-ing use of its representation we construct an integrable stochastic particle system.It is a generalization of the q -Boson system due to Sasamoto and Wadati. Wealso construct eigenfunctions of its generator using the propagation operator. Asa result we get the same eigenfunctions for the ( q, µ, ν )-Boson process obtained byPovolotsky. Introduction
In this article we introduce a deformation of the affine Hecke algebra of type GL and construct an integrable stochastic particle system making use of its representa-tion.In a previous paper [11] we constructed a discrete analogue of the non-ideal Bosegas with delta-potential interactions on a circle, which we call the periodic deltaBose gas for short. While a discretization of the periodic delta Bose gas and itsgeneralization was studied by van Diejen [3, 4] from the viewpoint of the theoryof Macdonald’s spherical functions, our discretization is motivated by the desire tounderstand an algebraic structure of integrable stochastic models.The discrete model constructed in [11] contains two parameters. Specializingthe parameters suitably and taking the limit as the system size goes to infinity,its Hamiltonian H becomes the time evolution operator for the joint moment of theintegrable stochastic system called the O’Connell-Yor semi-discrete directed polymer[8]. We can construct eigenfunctions of H using the propagation operator G , whichsends an eigenfunction of (a half of) the discrete Laplacian to that of H . To define G we generalize the construction by by van Diejen and Emsiz [5] of the integral-reflection operators due to Yang [12] and Gutkin [6]. Then the discrete integral-reflection operators determine a representation of the affine Hecke algebra of type GL .Recently the author found that the Hamiltonian H is related to another stochasticsystem more closely. The operator H acts on the space of functions on the orthogo-nal lattice Z k , where k is the number of particles, and leaves the space of symmetricfunctions invariant. Identify the space of symmetric functions with the space of func-tions on the fundamental chamber W k = { ( m , . . . , m k ) ∈ Z k | m ≥ · · · ≥ m k } . We assign to each element ( m , . . . , m k ) of W k the configuration of k bosonic particleson Z such that the particles are on the sites m , . . . , m k . Then, by specializing thetwo parameters of H in another way and adding a constant, we obtain the transitionrate matrix of the q -Boson system introduced by Sasamoto and Wadati [10].In this paper we generalize the above construction of an integrable stochasticparticle system. Our ingredient is a deformation of the affine Hecke algebra oftype GL . In [7] Lascoux and Sch¨utzenberger characterize the difference operatorsacting on polynomials which satisfy the braid relations. The operators contain fourparameters and at a spacial point they turn into the Demazure-Lustzig operatorswhich give a polynomial representation of the affine Hecke algebra. Our deformedalgebra arises from the commutation relations between the difference operators dueto Lascoux and Sch¨utzenberger and the multiplication operators. By definition it hasa polynomial representation. Making use of it we can construct the discrete integral-reflection operators with more parameters and define the propagation operator G as before. One of the main results of this article is construction of the discreteHamiltonian H satisfying the commutation relation HG = G ∆, where ∆ is thediscrete Laplacian, in this generalized setting.Our operator H also leaves the space of symmetric functions invariant, and byspecializing the four parameters suitably we obtain a transition rate matrix of acontinuous time Markov chain on W k . The resulting model is described as follows.It is a stochastic particle system on the one-dimensional lattice Z controlled by twoparameters s and q . The particles can occupy the same site simultaneously. Someparticles may move from site i to i − i ∈ Z . The rate atwhich r particles move to the left from a cluster with c particles is given by s r − [ r ] r − Y p =0 [ c − p ]1 + s [ c − − p ] ( c ≥ r ≥ , where [ n ] := (1 − q n ) / (1 − q ) is the q -integer. In the case of s = 0, the rate is equal tozero unless r = 1 and hence only one particle may move with the rate proportionalto 1 − q c . Thus we recover the q -Boson system.Using the propagation operator G we can construct symmetric eigenfunctions ofthe operator H by means of the Bethe ansatz method, which we call the Bethe wavefunctions. After the specialization of the parameters we obtain the eigenfunctionsof the transition rate matrix. They are parameterized by a tuple z = ( z , . . . , z k ) ofdistinct constants and are given by X σ ∈ S k Y ≤ i The paper is organized as follows. In Section 2 we define the deformation of theaffine Hecke algebra and introduce its representations which are the origin of thepropagation operator. In Section 3 we define the discrete Hamiltonian H and thepropagation operator G , and prove the commutation relation HG = G ∆. Usingthe operator G we construct the Bethe wave functions. In Section 4 we describethe construction of the stochastic particle system arising from the Hamiltonian H .We prove some polynomial identities which we use to rewrite the operator H inAppendix A.2. A Deformation of Affine Hecke Algebra and Its Representation Preliminaries. Throughout this paper we fix an integer k ≥ 2. Let V := ⊕ ki =1 R v i be the k -dimensional Euclidean space with an orthonormal basis { v i } ki =1 ,and V ∗ the linear dual of V . We let { ǫ i } ki =1 denote the dual basis of V ∗ correspondingto { v i } ki =1 . Set α ij := ǫ i − ǫ j for i, j = 1 , . . . , k . The subset R := { α ij | i = j } of V ∗ forms the root system of type A k − with the simple roots a i := α i,i +1 (1 ≤ i < k ).Denote by R ± the set of the associated positive and negative roots. For v ∈ V , set I ( v ) := { a ∈ R + | a ( v ) < } . The Weyl group W of type A k − is generated by the orthogonal reflections s i : V → V (1 ≤ i < k ) defined by s i ( v ) := v − a i ( v ) a ∨ i , where a ∨ i := v i − v i +1 is thesimple coroot. Denote the length of w ∈ W by ℓ ( w ).For any v ∈ V , the orbit W v intersects the closure of the fundamental chamber C + := { v ∈ V | a i ( v ) ≥ i = 1 , . . . , k − } at one point. Take a shortest element w ∈ W such that wv ∈ C + . Then I ( v ) = R + ∩ w − R − and hence the shortest element is uniquely determined. Denote it by w v .We will make use of the following proposition. Proposition 2.1. Suppose that v, v ′ ∈ V satisfy I ( v ) ⊂ I ( v ′ ) . Then w v ′ = w w v v ′ w v and ℓ ( w v ′ ) = ℓ ( w w v v ′ ) + ℓ ( w v ) . A deformation of affine Hecke algebra. Let us define a deformation of theaffine Hecke algebra of type GL k . Definition 2.2. Let α, β, γ, δ be complex constants and set q := 1 + βγ − αδ. We define the algebra A k to be the unital associative C -algebra with the generators X ± i (1 ≤ i ≤ k ) and T i (1 ≤ i < k ) satisfying the following relations:( T i − T i + q ) = 0 (1 ≤ i < k ) , T i T i +1 T i = T i T i +1 T i (1 ≤ i ≤ k − ,T i T j = T j T i ( | i − j | > , X i X j = X j X i ( i, j = 1 , . . . , k ) ,X i +1 T i − T i X i = T i X i +1 − X i T i = ( α + βX i )( γ + δX i +1 ) (1 ≤ i < k ) ,X i T j = T j X i ( i = j, j + 1) . YOSHIHIRO TAKEYAMA When β = γ = 0, the algebra A k is isomorphic to the affine Hecke algebra oftype GL k . We will use the property that any symmetric polynomial in X , . . . , X k commutes with T i (1 ≤ i < k ) in A k .Set L := k M i =1 Z v i and denote by F ( L ) the vector space of C -valued functions on L . The Weyl groupacts on F ( L ) by ( wf )( x ) := f ( w − x ). Set F ( L ) W := { f ∈ F ( L ) | wf = f for all w ∈ W } . Now we introduce a right action of A k on the group algebra C [ L ] due to Las-coux and Sch¨utzenberger [7]. In the following we identify C [ L ] with the Laurentpolynomial ring C [ e ± v , . . . , e ± v k ]. Proposition 2.3. [7] Define the C -linear operators ˇ X i (1 ≤ i ≤ k ) and ˇ T i (1 ≤ i The assignment X i b X i and T i b T i extends uniquely to a leftrepresentation of the algebra A k on F ( L ) . The action is explicitly given as follows: ( b X i f )( x ) = f ( x − v i ) . If a i ( x ) > then ( b T i f )( x ) = αδf ( x ) + (1 + βγ ) f ( s i x ) + αγ a i ( x ) X j =1 f ( s i x + ja ∨ i + v i +1 )+ ( αδ + βγ ) a i ( x ) − X j =1 f ( s i x + ja ∨ i ) + βδ a i ( x ) − X j =0 f ( s i x + ja ∨ i − v i +1 ) . EFORMED AFFINE HECKE ALGEBRA AND INTEGRABLE STOCHASTIC SYSTEM 5 When a i ( x ) = 0 , we have ( b T i f )( x ) = f ( x ) . If a i ( x ) < then ( b T i f )( x ) = − βγf ( x ) + (1 − αδ ) f ( s i x ) − αγ − a i ( x ) − X j =0 f ( s i x − ja ∨ i + v i +1 ) − ( αδ + βγ ) − a i ( x ) − X j =1 f ( s i x − ja ∨ i ) − βδ − a i ( x ) X j =1 f ( s i x − ja ∨ i − v i +1 ) . We will often use the fact that ( b T i f )( x ) = 0 for any f ∈ F ( L ) if a i ( x ) = 0.3. Discrete Hamiltonian and Propagation Operator Discrete Hamiltonian. Hereafter we assume that1 + βγ [ n ] = 0for any positive integer n , where [ n ] := 1 − q n − q is the q -integer.We define the functions d ± i (1 ≤ i ≤ k ) and δ j ,j ,...,j r (1 ≤ j < j < · · · < j r ≤ k )on L by d + i ( x ) := { p | i < p ≤ k, α ip ( x ) = 0 } , (3.1) d − i ( x ) := { p | ≤ p < i, α pi ( x ) = 0 } . and δ j ,j ,...,j r ( x ) := (cid:26) ǫ j ( x ) = · · · = ǫ j r ( x )) , . If r = 1 we set δ j ≡ H by H : = − αγ k X j =1 [ d + j ]1 + βγ [ d + j ]+ k X r =1 ( − βδ ) r − [ r − q − r ( r − / X ≤ j < ··· 0) and [0]! := 1. Note that the index j in the factor1+ βγ [ d + j + d − j − p ] may be replaced with any j p because d + i ( x )+ d − i ( x ) = d + j ( x )+ d − j ( x )if α ij ( x ) = 0.Using the equality k X j =1 [ d + j ] = k X j =1 q d − j d + j , YOSHIHIRO TAKEYAMA we see that when β = 0 it holds that H = k X j =1 q d − j ( b X j − αγd + j ) . This operator is introduced in [11] as a discrete analogue of the Hamiltonian of thedelta Bose gas under periodic boundary condition .For convenience we write down the action of H more explicitly. For a non-emptysubset J = { j , . . . , j m } ( j < · · · < j m ) of { , , . . . , k } , we define the operator H J acting on F ( L ) by H J := − αγ m − X d =1 [ d ]1 + βγ [ d ](3.2) + m X r =1 ( − βδ ) r − [ r − q − r ( r − / Q r − p =0 (1 + βγ [ m − − p ]) e r ( b X j , q b X j , . . . , q m − b X j m ) , where e r is the elementary symmetric polynomial of degree r . Then the value( Hf )( x ) is written as follows. Lemma 3.1. For x ∈ L , decompose the set { , , . . . , k } into a direct sum ⊔ Nn =1 J xn so that i and j belong to the same subset J xn if and only if α ij ( x ) = 0 . Then for any f ∈ F ( L ) it holds that ( Hf )( x ) = N X n =1 ( H J xn f )( x ) . (3.3)From the expression (3.3) we see that Proposition 3.2. H (cid:0) F ( L ) W (cid:1) ⊂ F ( L ) W .Proof. Suppose that f ∈ F ( L ) W . It suffices to show that ( Hf )( s i x ) = Hf ( x ) forany x ∈ L and 1 ≤ i < k . If a i ( x ) = 0 it is trivial. Let us consider the case of a i ( x ) = 0. Denote the transposition ( i, i +1) ∈ S k by τ . Consider the decomposition { , , . . . , k } = ⊔ Nn =1 J s i xn given in Lemma 3.1 with x replaced by s i x . Then it holdsthat J s i xn = τ ( J xn ). Note that { i, i + 1 } 6⊂ τ ( J xn ) for any n because a i ( x ) = 0. For1 ≤ j < · · · < j m ≤ k satisfying { i, i + 1 } 6⊂ { j , . . . , j m } and 0 ≤ r ≤ m , it holdsthat (cid:16) e r ( b X τ ( j ) , q b X τ ( j ) , . . . , q m − b X τ ( j m ) ) f (cid:17) ( s i x ) = (cid:16) e r ( b X j , q b X j , . . . , q m − b X j m ) f (cid:17) ( x )because f ∈ F ( L ) W . Therefore ( H τ ( J xn ) f )( s i x ) = ( H J xn f )( x ). From Lemma 3.1 wefind that ( Hf )( s i x ) = ( Hf )( x ). (cid:3) For later use we rewrite the operator H J using the two equalities below. SeeAppendix A for the proof. The parameters α and β in [11] are equal to αγ and q = 1 − αδ , respectively. EFORMED AFFINE HECKE ALGEBRA AND INTEGRABLE STOCHASTIC SYSTEM 7 Lemma 3.3. Let m be a positive integer and z , . . . , z m commutative indeterminates.Then the following equality holds. m X r =1 ( − βδ ) r − [ r − q − r ( r − / Q r − p =0 (1 + βγ [ m − − p ]) e r ( z , qz , . . . , q m − z m )= 1 β m X r =1 ( − δ ) r − [ r − Q rp =1 (1 + βγ [ p − X ≤ b < ···
Let m be a positive integer. Then the following equality holds. β m X r =1 ( − δ ) r − [ r − Q rp =1 (1 + βγ [ p − α r X ≤ b < ···
11 + βγ [ d ]Lemma 3.3 and Lemma 3.4 imply the following formula. Proposition 3.5. Let J = { j , . . . , j m } ( j < · · · < j m ) be a non-empty subset of { , , . . . , k } . Then it holds that H J = − αβ m + 1 β m X r =1 ( − δ ) r − [ r − Q rp =1 (1 + βγ [ p − X ≤ b < ···
Let w be an element of the Weyl group W and w = s i · · · s i r ∈ W a reduced expression. Then we set b T w := b T i · · · b T i r . It does notdepend on the choice of the reduced expression of w . Definition 3.6. We define the propagation operator G : F ( L ) → F ( L ) by G ( f )( x ) := ( b T w x f )( w x x ) . Hereafter, for x ∈ L , we denote by σ x the element of the symmetric group S k determined by w x ( v i ) = v σ x ( i ) (1 ≤ i ≤ k ) . Then ǫ i ( x ) = ǫ σ x ( i ) ( w x x ) for 1 ≤ i ≤ k .In the rest of this subsection we prove the following proposition. Proposition 3.7. Suppose that ≤ t < · · · < t r ≤ k and that x ∈ L satisfies ǫ t ( x ) = · · · = ǫ t r ( x ) . Then for any f ∈ F ( X ) it holds that ( b X t · · · b X t r G ( f ))( x )= (cid:16) b X l − r +1 · · · b X l ( b T l − r · · · b T σ x ( t ) ) · · · ( b T l − · · · b T σ x ( t r ) ) b T w x ( f ) (cid:17) ( w x x ) , where l = σ x ( t ) + d + t ( x ) . First we note that the functions d ± i have the following properties. YOSHIHIRO TAKEYAMA Lemma 3.8. (1) For x ∈ L, ≤ i ≤ k and ≤ j < k , it holds that d ± i ( s j x ) = d ± i ( x ) ( j = i − , i ) ,d ± i − ( x ) ∓ θ ( a i − ( x ) = 0) ( j = i − ,d ± i +1 ( x ) ± θ ( a i ( x ) = 0) ( j = i ) , where θ ( P ) = 1 or if P is true or false, respectively.(2) For any x ∈ L and ≤ i ≤ k , it holds that d ± i ( x ) = d ± σ x ( i ) ( w x x ) .Proof. The proof of (1) is straightforward. Let w x = s i · · · s i ℓ be a reduced expres-sion. Then a i p ( s i p · · · s i ℓ x ) = 0 for all 1 ≤ p ≤ ℓ , and hence d ± i ( x ) = d ± σ x ( i ) ( w x x ). (cid:3) Lemma 3.9. Suppose that ≤ t < · · · < t r ≤ k and that x ∈ L satisfies ǫ t ( x ) = · · · = ǫ t r ( x ) .(1) The values σ x ( t p )+ d + t p ( x ) and σ x ( t p ) − d − t p ( x ) are independent of p = 1 , , . . . , r .(2) Set l ± = σ x ( t ) ± d ± t ( x ) . Then a l − − ( w x x ) > , a i ( w x x ) = 0 ( l − ≤ i Lemma 3.10. Suppose that ≤ t < · · · < t r ≤ k and that x ∈ L satisfies ǫ t ( x ) = · · · = ǫ t r ( x ) . Set y = x − P rp =1 v t p , l = σ x ( t ) + d + t ( x ) , m = σ y ( t ) − d − t ( y ) and u := ( s l − r · · · s σ x ( t ) )( s l − r +1 · · · s σ x ( t ) ) · · · ( s l − · · · s σ x ( t r ) ) , (3.4) u := ( s m + r − · · · s σ y ( t r ) − )( s m + r − · · · s σ y ( t r − ) − ) · · · ( s m · · · s σ y ( t ) − ) . Then u w x = u w y and ℓ ( u w x ) = ℓ ( u w y ) = ℓ ( u ) + ℓ ( w x ) = ℓ ( u ) + ℓ ( w y ) . (Notethat the right hand sides of (3.4) are reduced expressions of u and u .)Proof. Set z = x − P rp =1 v t p / 2. Since | P rp =1 a ( v t p ) | ≤ a ∈ R + , I ( x ) and I ( y )are contained in I ( z ). Hence Proposition 2.1 implies that w z = w w x z w x = w w y z w y and ℓ ( w w x z ) + ℓ ( w x ) = ℓ ( w w y z ) + ℓ ( w y ). Thus it suffices to show that w w x z = u and w w y z = u . Here we prove that w w x z = u . The proof for w w y z = u is similar.Let us write down the set I ( w x z ). It consists of all the positive roots a ∈ R + suchthat a ( w x z ) = a ( w x x ) − P rp =1 a ( v σ x ( t p ) ) / < 0. Since w x x ∈ C + it is equivalent to EFORMED AFFINE HECKE ALGEBRA AND INTEGRABLE STOCHASTIC SYSTEM 9 requiring that a ( w x x ) = 0 and there exists p such that a ( v σ x ( t p ) ) = 1 and a ( v σ x ( t q ) ) =0 if q = p . Therefore, from Lemma 3.9, we find that I ( w x z ) = r G p =1 { α σ x ( p ) ,i | σ x ( t p ) < i ≤ l, i = σ x ( t p +1 ) , . . . , σ x ( t r ) } . It is equal to R + ∩ u − R − , and hence w w x z = u . (cid:3) Now let us prove Proposition 3.7. We use the notation of Lemma 3.10. Since a i ( w y y ) = 0 for m ≤ i < σ y ( t r ) and a i ( w x x ) = 0 for σ x ( t ) ≤ i < l , it holds that w y y = u w y y = u w x ( x − r X p =1 v t p ) = u ( w x x − r X p =1 v σ x ( t p ) ) = w x x − l X j = l − r +1 v j and ( b T − u g )( w y y ) = g ( w y y ) for any g ∈ F ( L ). Moreover b T w y = b T − u b T u b T w x . There-fore ( b X t · · · b X t r G ( f ))( x ) = ( b T w y f )( w y y ) = ( b T u b T w x f )( w x x − l X j = l − r +1 v j )= ( b X l − r +1 · · · b X l b T u b T w x ( f ))( w x x ) . This completes the proof of Proposition 3.7.3.3. Commutation relation of H and G . In this subsection we prove the follow-ing theorem. Theorem 3.11. It holds that HG = G ( P ki =1 b X i ) . Therefore, if f ∈ F ( L ) is aneigenfunction of the difference operator P ki =1 b X i , then G ( f ) is an eigenfunction ofthe discrete Hamiltonian H with the same eigenvalue. Hereafter we set b V i := α + β b X i (1 ≤ i ≤ k ) . Lemma 3.12. Suppose that ≤ i < k and that x ∈ L satisfies a i ( x ) = 0 . Then forany g ∈ F ( L ) and p ≥ it holds that (cid:16) ( q p − + δ [ p − b V i )( b T i + β ( γ + δ b X i +1 ))( g ) (cid:17) ( x ) = (cid:16) ( q p + δ [ p ] b V i +1 )( g ) (cid:17) ( x ) . Proof. For simplicity we write P ≡ P if two operators P , P acting on F ( L ) satisfy( P ( g ))( x ) = ( P ( g ))( x ). Since a i ( x ) = 0 it holds that b T i Q ≡ Q for any operator Q acting on F ( L ). Using β ( γ + δ b X i +1 ) = q − δ b V i +1 we obtain( q p − + δ [ p − b V i )( b T i + β ( γ + δ b X i +1 )) ≡ q p − ( q + δ b V i +1 ) + δ [ p − b V i b T i + δ [ p − b V i ( q − δ b V i +1 ) . Since b V i b T i = b T i b V i +1 − b V i ( q − δ b V i +1 ), it is equivalent to q p − ( q + δ b V i +1 ) + δ [ p − b V i +1 = q p + δ [ p ] b V i +1 . This completes the proof. (cid:3) Lemma 3.13. Suppose that r, l and ν , . . . , ν r are positive integers satisfying ≤ ν < · · · < ν r ≤ l ≤ k . For a subset I = { i , . . . , i d } ( i < · · · < i d ) of { , , . . . , r } ,set c ( I ) := ( β/qα ) d q P dp =1 i p , b Q I := b X l − d +1 · · · b X l ( b T l − d · · · b T ν i ) · · · ( b T l − · · · b T ν id ) . For ≤ p ≤ r define the operator b S p : F ( L ) → F ( L ) by b S p := X I ⊂{ p,p +1 ,...,r } α r − p +1 c ( I ) l −| I | Y i = ν p ( q p − + δ [ p − b V i ) · b Q I . (3.5) If x ∈ L satisfies a i ( x ) = 0 for ν ≤ i < l , then it holds that ( b S p ( g ))( x ) = (1 + βγ [ p − b S p +1 b V ν p Y ν p
Proposition 3.14. Suppose that ≤ t < · · · < t r ≤ k and that x ∈ L satisfies ǫ t ( x ) = · · · = ǫ t r ( x ) . Then for any f ∈ F ( L ) it holds that r Y p =1 ( α + βq p − b X t p ) G ( f ) ! ( x )= r Y p =1 (1 + βγ [ p − r Y p =1 b V σ x ( t p ) Y σ x ( t p )
Bethe wave functions. Using Theorem 3.11 we can construct symmetriceigenfunctions of H , which we call the Bethe wave functions . Set L + := L ∩ C + . Proposition 3.15. For a tuple p = ( p , . . . , p k ) of distinct complex parameters,define the function Φ p ∈ F ( L ) W by Φ p | L + = X σ ∈ S k Y ≤ i Hereafter we identify the space of symmetric functions F ( L ) W with the space offunctions on L + . Denote it by F ( L + ). A linear operator Q on F ( L + ) is said tobe stochastic if it is given in the form ( Qf )( x ) = P y = x c ( y, x )( f ( y ) − f ( x )) where c ( y, x ) ≥ F ( L + ) determines a stochastic one-dimensional particlesystem with continuous time as follows. Denote by S k the set of configurations of k bosonic particles on the one-dimensional lattice Z . For x = P kj =1 m j v j , denote by ν ( x ) the configuration of k particles on Z such that the particles are on the sites m , . . . , m k . For example, if k = 6 and x = 3 v + 3 v + 3 v + v − v − v , ν ( x ) isthe configuration where three particles are located on the site 3, one particle on thesite 1 and two particles on the site − 2. Then the map ν : L + → S k is bijection. Weidentify F ( L + ) and the set of functions on S k through the map ν . Then the stochastic EFORMED AFFINE HECKE ALGEBRA AND INTEGRABLE STOCHASTIC SYSTEM 13 operator Q is regarded as the backward generator of the stochastic process on S k with continuous time, where c ( y, x ) gives the rate at which the state ν ( x ) changesto ν ( y ).Now we give a sufficient condition for H | F ( L ) W = H | F ( L + ) to be stochastic up toconstant. Proposition 4.1. Let λ be a constant. The operator ˜ H := ( H + λ ) | F ( L + ) is stochasticonly if λ = − k and ( α + β )( γ + δ ) = 0 . To prove Proposition 4.1, we introduce the cluster coordinate of a point in L + following [1]. For x ∈ L + we determine a set of positive integers M and c i (1 ≤ i ≤ M ) by the property that P Mi =1 c i = k , ǫ c ( x ) > ǫ c + c ( x ) > · · · > ǫ c + ··· + c M ( x ), and ǫ j ( x ) = ǫ c + ··· + c i ( x ) if c + · · · + c i − < j ≤ c + · · · + c i . We call the tuple ( c , . . . , c M )the cluster coordinate of x ∈ L + . It describes the number of particles in each clusterin the configuration ν ( x ).In terms of the cluster coordinate the action of H for f ∈ F ( L ) W is written asfollows. Fix x ∈ L + and let ( c , . . . , c M ) be its cluster coordinate. Then( Hf )( x ) = M X i =1 n − αγ c i − X d =1 [ d ]1 + βγ [ d ] f ( x )(4.1) + c i X r =1 ( − βδ ) r − [ r − q − r ( r − Q r − p =0 (1 + βγ [ c i − − p ]) e r (1 , q, . . . , q c i − ) f ( x − r − X p =0 v c + ··· + c i − p ) o . We use this formula in the proof below. Proof of Proposition 4.1. If c , . . . , c M are all equal to one, then then ( Hf )( x ) = P kj =1 f ( x − v j ). Hence λ should be equal to − k so that ˜ H is stochastic.In general, set K m := − m − αγ m − X d =1 [ d ]1 + βγ [ d ](4.2) + m X r =1 ( − βδ ) r − [ r − q − r ( r − Q r − p =0 (1 + βγ [ m − − p ]) e r (1 , q, . . . , q m − ) . The operator ˜ H is stochastic only if P Mi =1 K c i = 0 for any tuple ( c , . . . , c M ) ofpositive integers such that P Mi =1 c i = k . Since K = 0 and K = − ( α + β )( γ + δ ) / (1 + βγ ), we see that ( α + β )( γ + δ ) should be zero. (cid:3) Moreover, we have the following property. Lemma 4.2. If ( α + β )( γ + δ ) = 0 , the constant K m defined by (4.2) is equal tozero for any m ≥ . Proof. From Lemma 3.3 and Lemma 3.4 it holds that K m = − (1 + βα ) m + 1 β m X r =1 ( − δ ) r − [ r − q − ( m − r e r (1 , q, . . . , q m − ) r Y p =1 α + βq p − βγ [ p − . Using this expression we find that K m − K m − = − ( α + β )( γ + δ )1 + βγ m − X r =1 ( − δ ) r − [ r ]! q − ( m − r e r (1 , q, . . . , q m − ) r Y p =2 α + βq p − βγ [ p ]for m ≥ 2. This completes the proof because K = 0. (cid:3) Now we define the stochastic operator H ( s, q ) on F ( L + ) by( H ( s, q ) f )( x ) = M X i =1 c i X r =1 s r − [ r ] r − Y p =0 [ c i − p ]1 + s [ c i − − p ] f ( x − r − X p =0 v c + ··· + c i − p ) − f ( x ) ! , where ( c , . . . , c M ) is the cluster coordinate of x . It determines the stochastic particlesystem on Z described as follows. In continuous time some particles may move fromsite i to i − i ∈ Z . The rate at which r particles move tothe left from a cluster with c particles is given by s r − [ r ] r − Y p =0 [ c − p ]1 + s [ c − − p ] ( c ≥ r ≥ . It is non-negative if, for example, s ≥ < q < Proposition 4.3. When ( α + β )( γ + δ ) = 0 , it holds that ( H − k ) | F ( L + ) = (cid:26) H ( q − αδ, q − ) ( α + β = 0) , H ( βγ, q ) ( γ + δ = 0) . Proof. Use the equality q − r ( r − / e r (1 , q, . . . , q m − ) = Q r − p =0 [ m − p ][ r ]! , and we obtain the desired formula. (cid:3) Moreover, using Proposition 3.15, we obtain eigenfunctions of H ( s, q ): Proposition 4.4. Let z = ( z , . . . , z k ) be a tuple of distinct complex parameters,and set ν := s − q + s . (4.3) EFORMED AFFINE HECKE ALGEBRA AND INTEGRABLE STOCHASTIC SYSTEM 15 Then the function Ψ z on L + defined by Ψ z := X σ ∈ S k Y ≤ i We use Proposition 3.15 in the case where δ = − γ . Note that q = 1 + βγ − αδ = 1 + ( α + β ) γ . Setting p i = (1 − z i ) / (1 + βz i /α ), we have1 + ( α + βp j )( γ + δp i ) p j − p i = qz i − z j z i − z j . Set s = βγ . Then β/α is equal to − ν . Thus we see that Ψ z is an eigenfunction of H ( s, q ) = ( H − k ) | F ( L + ) with eigenvalue k X i =1 p i − k = k X i =1 − z i − νz i − k = ( ν − k X i =1 z i − νz i . This completes the proof. (cid:3) It should be noted that the function Ψ z is equal to the eigenfunction for the( q, µ, ν )-Boson process constructed by means of the coordinate Bethe ansatz [9]. Appendix A.Here we prove Lemma 3.3 and Lemma 3.4. For that purpose we show the followingequality. Lemma A.1. Let m be a positive integer and x, y, z , . . . , z m commutative indeter-minates. For ≤ s ≤ m , set I m,s ( x, y ) := m − s X r =0 [ r + s − q − x − y ) r Q r + sa =1 ( x + [ a − y )(A.1) × X ≤ b < ···
First we prove X ≤ b < ··· 1. Since the equality holds trivially when r = 0 or s = 0,we assume that r > s > 0. Denote the left hand side by K m,r,s . Using e s ( z b , qz b , . . . , q r + s − z b r + s ) = e s ( z b , qz b , . . . , q r + s − z b r + s − )+ q r + s − z b r + s e s − ( z b , qz b , . . . , q r + s − z b r + s − ) , we see that K m,r,s = m X b = r + s q b (cid:0) K b − ,r − ,s + q r + s − z b K b − ,r,s − (cid:1) . From the hypothesis of the induction it is equal to q s ( s − / m X b = r + s q b (cid:8) e r − ( q s +1 , . . . , q b − ) e s ( qz , . . . , q b − z b − )+ z b e r ( q s +1 , . . . , q b ) e s − ( qz , . . . , q b − z b − ) (cid:9) . Use q b z b e s − ( qz , . . . , q b − z b − ) = e s ( qz , . . . , q b z b ) − e s ( qz , . . . , q b − z b − )and q b e r − ( q s +1 , . . . , q b − ) − e r ( q s +1 , . . . , q b ) = e r ( q s +1 , . . . , q b − )(A.4)successively. Then we get the right hand side of (A.3).Now let us prove (A.2). Using (A.3) we see that I m,s ( x, y ) = q s ( s +1) / − ms e s ( z , qz , . . . , q m − z m ) J m,s ( x, y ) , where J m,s ( x, y ) is given by J m,s ( x, y ) := m − s X r =0 [ r + s − q − x + y ) r q − mr Q r + sa =1 ( x + [ a − y ) e r ( q s +1 , q s +2 , . . . , q m ) . It suffices to show that J m,s ( x, y ) = q − s + ms [ s − Q s − a =0 ( x + [ m − − a ] y )(A.5)for 1 ≤ s ≤ m . From the equality (A.4) with b replaced by m and x + [ n ] y = x + y + q [ n − y for n ≥ 1, we find J m,s ( x, y ) = q s J m − ,s ( x + y, qy ) for m > s . Nowthe desired equality (A.5) can be proved by induction on m . (cid:3) EFORMED AFFINE HECKE ALGEBRA AND INTEGRABLE STOCHASTIC SYSTEM 17 Proof of Lemma 3.3. We rewrite the right hand side. Expand the product r Y a =1 ( α + βq a − z b a ) − α r = r X s =1 α r − s β s − e s ( z b , qz b , . . . , q r − z b r ) . and exchange the order of the summation with respect to r and s . Using( − δ ) r + s − α r β s − = ( − βδ ) s − ( q − − βγ ) r , we see that the right hand side is equal to P m − s =1 ( − βδ ) s − I m,s (1 , βγ ), where I m,s ( x, y )is defined by (A.1). It is equal to the left hand side because of Lemma A.1. (cid:3) Proof of Lemma 3.4. Set K m ( x, y ) := m X r =1 [ r − q − x − y ) r − q − mr Q ra =1 ( x + [ a − y ) e r ( q, q , . . . , q m ) . Then the left hand side is equal to αK m (1 , βγ ) /β . Hence it suffices to show that K m ( x, y ) = m − X a =0 x + [ a ] y . In the same way as the proof of (A.5), we find the recurrence relation K m ( x, y ) =1 /x + K m − ( x + y, qy ) for m > 1. Now the equality above can be proved by inductionon m . (cid:3) Acknowledgments The research of the author is supported by JSPS KAKENHI Grant Number26400106. The author is grateful to I. Corwin, L. Petrov, A. Povolotsky and T.Sasamoto for valuable discussions. References [1] Borodin, A., Corwin, I., Petrov, L. and Sasamoto, T., Spectral theory for the q-Boson particlesystem, preprint, arXiv:1308.3475 .[2] Corwin, I., The ( q, µ, ν )-Boson process and ( q, µ, ν )-TASEP, preprint, arXiv:1401.3321 .[3] van Diejen, J. F., On the Plancherel formula for the (discrete) Laplacian in a Weyl chamberwith repulsive boundary conditions at the walls, Ann. Henri Poincare (2004), no. 1, 135–168.[4] van Diejen, J. F., Diagonalization of an integrable discretization of the repulsive delta Bosegas on the circle, Comm. Math. Phys. (2006), no. 2, 451–476.[5] van Diejen, J. F. and Emsiz, E., Unitary representations of affine Hecke algebras related toMacdonald spherical functions, J. Algebra (2012), 180–210.[6] Gutkin, E., Integrable systems with delta-potential, Duke Math. J. (1982), no. 1, 1–21.[7] Lascoux, A. and Sch¨utzenberger, M. P., Symmetrization operators on polynomial rings, Funct.Anal. Appl. (1987), no. 4, 324–326.[8] O’Connell, N. and Yor, M., Brownian analogues of Burke’s theorem, Stochastic Process. Appl. (2001), no. 2, 285–304.[9] Povolotsky, A. M., On the integrability of zero-range chipping models with factorized steadystates, J. Phys. A (2013), no. 46. [10] Sasamoto, T. and Wadati, M., Exact results for one-dimensional totally asymmetric diffusionmodels, J. Phys. A (1998), no. 28, 6057–6071.[11] Takeyama, Y., A discrete analogue of periodic delta Bose gas and affine Hecke algebra, Funk-cialaj Ekvacioj , (2014), 107–118.[12] Yang, C. N., Some exact results for the many-body problem in one dimension with repulsivedelta-function interaction, Phys. Rev. Lett. (1967) 1312–1315. Division of Mathematics, Faculty of Pure and Applied Sciences, University ofTsukuba, Tsukuba, Ibaraki 305-8571, Japan E-mail address ::