樊赵兵-哈尔滨工程大学教师个人主页.pdf
樊赵兵-哈尔滨工程大学教师个人主页.pdf
DIFFERENTIAL OPERATOR REALIZATION OF BRAID GROUP ACTION ON ıQUANTUM GROUPS arXiv:2307.03501v1 [math.RT] 7 Jul 2023 ZHAOBING FAN, JICHENG GENG, AND SHAOLONG HAN Abstract. We construct a unique braid group action on modified q-Weyl algebra Aq (S). Under this action, we give a realization of the braid group action on quasisplit ıquantum groups ı U(S) of type AIII. Furthermore, we directly construct a unique braid group action on polynomial ring P which is compatible with the braid group action on Aq (S) and ı U(S). 1. Introduction The quantum groups, introduced independently by Drinfeld and Jimbo (cf. [Dr86, J86]), are certain families of Hopf algebras which are deformations of universal enveloping algebras of Kac-Moody algebras. In the theory of quantum groups, braid group action is an important research topic. These actions lead to the construction of PBW bases and canonical bases of quantum groups (cf. [Lus90a, Lus90b, Lus94]). When we focus on affine quantum groups, besides their Serre presentation, the Drinfeld presentation is another important presentation (cf. [Dr88]), which plays a fundamental role in the representation theory of affine quantum groups. The braid group action of affine quantum groups is an essential tool to study Drinfeld’s new realization (cf. [Da93, Be94]). Moreover, the braid group action can be applied widely to the field of geometric representation theory, knot theory and categorification. The ıquantum groups Uı arising from quantum symmetric pairs (U, Uı ) associated to Satake diagrams (cf. [Let99, Kolb14]) can be seen as non-trivial generalization of quantum groups. Therefore, it is natural to ask whether there exists braid group action on ıquantum groups. In [KP11], Kolb and Pellegrini used software to construct the braid group action on a class of ıquantum groups of finite types (including all quasisplit types and type AII). The braid group of type AIII/AIV was constructed by Dobson in [Dob20]. The Hall algebraic approach to braid group action of quasi-split ıquantum groups was studied in [LW21]. For general case, Wang and Zhang gave a conceptual construction of relative braid group action on ıquantum groups of arbitrary finite types (cf. [WZ22]). Soon after, Zhang generalized these constructions to Kac-Moody type (cf. [Z22]). The Drinfeld type presentation for ıquantum groups can be found in a series of papers [LW21b, LWZ22, Z21]. The aim of this paper is to give differential operator realization of braid group action on two typical quasi-split ıquantum groups of type AIII, i.e., U and Uı . In [FGH22], the authors introduced the modified q-Weyl algebra Aq (S) associated with Satake diagram S by using modified q-differential operators on polynomial ring. They constructed an Key words and phrases. ıquantum group, modified q-Weyl algebra, braid group action. 1 2 ZHAOBING FAN, JICHENG GENG, AND SHAOLONG HAN algebraic homomorphism ϕ from quasi-split ıquantum groups ı U(S) to Aq (S). Within this framework, we shall construct braid group action of type B on two classes of modified q-Weyl algebras. Subsequently, we show that the braid group operators on Aq (S) are intertwined with Kolb’s braid group operators via the homomorphism ϕ. This means that we give a realization of Kolb’s braid group action. In this realization, we show that the braid group action of Aq (S) is unique. It is worth mentioning that this realization can be seen as the generalization of Floreanini and Vinet’s work (cf. [FV91]). They defined the braid group action on q-Weyl algebra Wq (n) which is intertwined with the braid group action of quantum groups Uq (An−1 ) and Uq (Cn ) with respect to the oscillator representations. Since the modified q-Weyl algebra Aq (S) has a natural action on a polynomial ring P, the ı U(S)-module structure on P can be obtained naturally by the homomorphism ϕ. Using the braid group action on Aq (S), we further construct directly braid group action on P which is compatible with the braid group action on ı U(S). We also show that this compatible braid group action on P is unique. This is different from the approach by Wang and Zhang’s braid group action on ı U(S)-modules (cf. [WZ22, Section 10.2]), because they regarded the U-module M as a ı U(S)-module by restriction. Motivated by this work, in the forthcoming paper, we shall study the braid group action on the modified q-weyl algebra corresponding to affine quasi-split ıquantum groups, and then obtain the braid group action on these affine ıquantum groups. The paper is organized as follows. In Section 2, we recall the definition of ıquantum groups ı U(S) associated to two Satake diagrams and the braid group action on these ′ ıquantum groups. In Section 3, we define four variants braid group operators Ti,e ′′ and Ti,−e (e ∈ {1, −1}) on modified q-Weyl algebra Aq (S) and show that they satisfy braid group relations of type B. Moreover, we show that braid group action on Aq (S) coincides with the action on ı U(S) by the homomorphism ϕ : ı U(S) → Aq (S). In Section 4, we define the braid group action on polynomial ring P and further prove that the braid group action on Aq (S) is compatible with the action on P. This compatible action induce compatible braid group action of ı U(S) on P. Acknowledgements. Z. Fan was partially supported by the NSF of China grant 12271120, the NSF of Heilongjiang Province grant JQ2020A001, and the Fundamental Research Funds for the central universities. 2. Braid group action on ıquantum groups We consider the Dynkin diagram of type An−1 . There are n nodes {1, 2, · · · , n} on this Dynkin diagram. We define an involution ρn between these nodes by ρn (i) = n + 1 − i (1 ≤ i ≤ n). Let n = 2r or 2r + 1. The definition of ρn can be described as the following Satake diagrams. DIFFERENTIAL OPERATOR REALIZATION OF BRAID GROUP ACTION 3 Satake diagram I 1 2 2r 2r − 1 ··· ··· r−1 r r+2 r+1 Satake diagram II 1 2 2r + 1 2r r ··· ··· r+1 r+2 Let I = {1, 2, · · · , n}. We denote by S the above Satake diagrams uniformly. Definition 2.1. [KP11, Proposition 4.1] The ıquantum group ı U(S) associated with above Satake diagrams is generated by elements {Bi | i ∈ I} and {Ki | i ∈ I \{j}, ρn (j) = j} subject to the following relations Ki Bj = q j·ρn(i)−j·i Bj Ki , for all i, j ∈ I, Ki − Kρn (i) , if i · j = 0, Bj Bi − Bi Bj = δρn (i),j q − q −1 Bi2 Bj − (q + q −1 )Bi Bj Bi + Bj Bi2 = δi,ρn (i) Bj Ki Kρn (i) = 1, − δj,ρn (i) (q + q −1 )Bi (q −1 ςi Ki + q 2 ςρn (i) Kρn (i) ), if i · j = −1, where ςi = δi,r + q −1 δi,r+1 for i ∈ I. Proposition 2.2. There exist two algebra anti-automorphisms Ω and Ψ on ı U(S) defined by Ω(Bi ) = Bρn (i) , Ψ(Bi ) = Bi , Ω(Ki ) = Kρn (i) , Ω(q) = q −1 , Ψ(Ki ) = q −δi,r δr·ρn (r),−1 Kρn (i) , Proof. It is straightforward to verify by direct computation. Ψ(q) = q. We denote by U (resp. Uı ) the ıquantum group corresponding to Satake diagram I (resp. II). Let e ∈ {1, −1}. We set [x, y]e = xy − q e yx. The following Proposition is slight modification of coefficients in [KP11, Theorem 4.3, Theorem 4.4, Theorem 4.6, Theorem 4.7]. The proof is exactly the same as that in [KP11, Section 4.5]. Proposition 2.3 ([KP11, Theorem 4.3–4.4, Theorem 4.6–4.7]). There exist unique ′ ′′ automorphisms τi,e and τi,−e (i ∈ {1, 2, · · · , r}) on U satisfying the following: if j = i 6= r, Kρn (i) , ′ ′′ τi,e (Kj ) = τi,−e (Kj ) = Ki Kj , if |i − j| = 1, i 6= r, K , otherwise. j 4 ZHAOBING FAN, JICHENG GENG, AND SHAOLONG HAN If 1 ≤ i ≤ r − 1, If i = r, −Bρn (i) Kρen (i) , e −Ki Bi , ′ τi,e (Bj ) = [Bi , Bj ]−e , [Bj , Bρn (i) ]e , B , j −Kie Bρn (i) , e −Bi Kρn (i) , ′′ τi,−e (Bj ) = [Bj , Bi ]−e , [Bρn (i) , Bj ]e , B , j if i = j, if i = ρn (j), if i · j = −1, if i · ρn (j) = −1, otherwise, if i = j, if i = ρn (j), if i · j = −1, if i · ρn (j) = −1, otherwise. q −e [[Br−1 , Br ]e , Br+1 ]e − Kre Br−1 , e Kr Br , ′ τr,e (Bj ) = Br+1 Kρen (r) , q −e [[Br+2 , Br+1 ]e , Br ]e − Br+2 Kρen (r) , B , j q −e [Br+1 , [Br , Br−1 ]e ]e − Kρen (r) Br−1 , e Kρn (r) Br , ′′ τr,−e (Bj ) = Br+1 Kre , q −e [Br , [Br+1 , Br+2 ]e ]e − Br+2 Kre , B , j if j = r − 1, if j = r, if j = r + 1, if j = r + 2, otherwise, if j = r − 1, if j = r, if j = r + 1, if j = r + 2, otherwise. ′ ′′ There exist unique automorphisms τi,e and τi,−e (i ∈ {1, 2, · · · , r+1}) on Uı satisfying the following: if i = j, Kρn (i) , ′ ′′ τi,e (Kj ) = τi,−e (Kj ) = Ki Kj , if |i − j| = 1, i 6= r + 1, K , otherwise. j If 1 ≤ i ≤ r, −Bρn (i) Kρen (i) , −Kie Bi , [B , B ] , i j −e ′ τi,e (Bj ) = [Bj , Bρn (i) ]e , [Br , [Br+1 , Br+2 ]e ]−e + Br+1 Kρen (r) , Bj , if i = j, if i = ρn (j), if i · j = −1, ρn (i) · j 6= −1, if i · j 6= −1, ρn (i) · j = −1, if i · j = −1, ρn (i) · j = −1, otherwise, DIFFERENTIAL OPERATOR REALIZATION OF BRAID GROUP ACTION −Kie Bρn (i) , −Bi Kρen (i) , [B , B ] , j i −e ′′ τi,−e (Bj ) = [Bρn (i) , Bj ]e , [[Br , Br+1 ]e , Br+2 ]−e + Br+1 Kρen (r) , Bj , If i = r + 1, [Br+1 , Br ]−e , ′ τr+1,e (Bj ) = [Br+2 , Br+1 ]e , B , j [Br+1 , Br ]e , ′′ τr+1,−e (Bj ) = [Br+2 , Br+1 ]−e , B , j 5 if i = j, if i = ρn (j), if i · j = −1, ρn (i) · j 6= −1, if i · j 6= −1, ρn (i) · j = −1, if i · j = −1, ρn (i) · j = −1, otherwise. if j = r, if j = r + 2, otherwise, if j = r, if j = r + 2, otherwise. ′ ′′ Let I = {1, 2, · · · , [ n+1 ]}. The automorphisms {τi,e }i∈I and {τi,−e }i∈I are inverse 2 ′ ′′ ′′ ′ ′ ′′ each other, i.e., τi,e τi,−e = τi,−e τi,e = id. Moreover, {τi,e }i∈I and {τi,−e }i∈I satisfy braid relations of type Br or Br+1 , i.e., the following relations hold: n+1 ] − 1, τi−1 τi τi−1 = τi τi−1 τi , if 2 ≤ i ≤ [ 2 n+1 ], τi−1 τi τi−1 τi = τi τi−1 τi τi−1 , if i = [ 2 τi τj = τj τj , if |i − j| = 6 1, ′ ′′ where τi := τi,e or τi,−e . Proposition 2.4. The automorphism τi commutes with Ω, i.e., τi ◦ Ω = Ω ◦ τi . ′ ′ Proof. We only prove the formula τr,e ◦ Ω(Br+2 ) = Ω ◦ τr,e (Br+2 ) for U . By Proposition 2.2 and Proposition 2.3, we have ′ ′ τr,e ◦ Ω(Br+2 ) = τr,e (Br−1 ) =q −e [[Br−1 , Br ]e , Br+1 ]e − Kre Br−1 =q −e ((Br−1 Br − q e Br Br−1 )Br+1 − q e Br+1 (Br−1 Br − q e Br Br−1 ) − Kre Br−1 =q −e Br−1 Br Br+1 − Br Br−1 Br+1 − Br+1 Br−1 Br + q e Br+1 Br Br−1 − Kre Br−1 =Ω(q −e (Br+2 Br+1 − q e Br+1 Br+2 )Br − Br (Br+2 Br+1 − q e Br+1 Br+2 ) − Br+2 Kr−e ) =Ω(q −e [[Br+2 , Br+1 ]e , Br ]e − Br+2 Kr−e ) ′ =Ω ◦ τr,e (Br+2 ). The proof of other cases is similar. 3. Braid group action on modified q-Weyl algebra Definition 3.1 ([FGH22, Definition 3.1]). The modified q-Weyl algebra Aq (S) associated with Satake diagram I (resp. II) is generated by di , xi , mi (i ∈ {1, 2, · · · , r, r + 1}) 6 ZHAOBING FAN, JICHENG GENG, AND SHAOLONG HAN over Q(q) subject to the following relations: −1 mi m−1 i = mi mi = 1, mi mj = mj mi , (3.1) di mj = mj di , xi mj = mj xi , di xj = xj di , if i 6= j, (3.2) di dj = dj di , xi xj = xj xi , (3.3) di mi = q 1+δi,ρn (r) δi,r+1 mi di , xi mi = q −1−δi,ρn (r) δi,r+1 mi xi , (3.4) mi − m−1 q 1+δi,ρn (r) δi,r+1 mi − q 1+δi,ρn (r) δi,r+1 m−1 i i , xi di = . (3.5) di xi = q − q −1 q − q −1 Proposition 3.2. There exist two anti-automorphisms ω, ψ on Aq (S) defined by ω(xi ) = di , ω(di ) = xi , ω(mi ) = m−1 i , ω(q) = q −1 , ψ(xi ) = (−1)i+1 xi , ψ(di ) = (−1)i di , ψ(mi ) = q δi,r+1 (i·ρn (i)−2δi,ρn (i) )−1 m−1 i , ψ(q) = q. Proof. The verifications of anti-automorphisms ω, ψ are easy and will be skipped. Proposition 3.3 ([FGH22, Theorem 4.1]). There exists a Q(q)-algebra homomorphism ϕ : ı U(S) → Aq (S) which is given by if 1 ≤ i ≤ r, xi+1 di , Bi = xρn (i) d1+ρn (i) , if ρn (r) ≤ i ≤ ρn (1), x d , if i = r + 1, ρn (r + 1) = r + 1, r+1 r+1 −δi,r mi m−1 if 1 ≤ i ≤ r = ρn (r + 1), q i+1 , δi,r+1 −1 Ki = q mρn (i) m1+ρn (i) , if r + 1 = ρn (r) ≤ i ≤ ρn (1), mr m−1 if i = r = ρn (r + 2). r+1 , Let Aq (resp. Aıq ) denote the modified q-Weyl algebra corresponding to Satake diagram I (resp. II). ′ ′′ Theorem 3.4. There exist automorphisms Ti,e and Ti,−e (i ∈ {1, 2, · · · , r}) on Aq satisfying the following: if i 6= r, j = i + 1, mi , ′ ′′ Ti,e (mj ) = Ti,−e (mj ) = mi+1 , if i 6= r, j = i, m , otherwise, j e −2e q mr dr+1 , if i = r, j = i + 1, −e −2e −e q mr mr+1 dr , if i = j = r, ′ Ti,e (dj ) = −q −e m−e if i 6= r, j = i + 1, i+1 di , −e m di+1 , if i = j 6= r, i dj , otherwise, −e 2e q mr xr+1 , if i = r, j = i + 1, e e e q mr mr+1 xr , if i = j = r, ′ Ti,e (xj ) = −q e mei+1 xi , if i 6= r, j = i + 1, e m xi+1 , if i = j 6= r, i xj , otherwise, DIFFERENTIAL OPERATOR REALIZATION OF BRAID GROUP ACTION 7 −e 2e q m dr+1 , if i = r, j = i + 1, 2e er e q mr mr+1 dr , if i = j = r, ′′ Ti,−e (dj ) = mei+1 di , if i 6= r, j = i + 1, −q e mei di+1 , if i = j 6= r, dj , otherwise, e −2e q mr xr+1 , if i = r, j = i + 1, −e −e −e q mr mr+1 xr , if i = j = r, ′′ Ti,−e (xj ) = m−e if i 6= r, j = i + 1, i+1 xi , −e −e −q mi xi+1 , if i = j 6= r, xj , otherwise. ′ ′′ There exist automorphisms Ti,e and Ti,−e (i ∈ {1, 2, · · · , r + 1}) on Aıq satisfying the following: if j = i + 1, mi , ′ ′′ Ti,e (mj ) = Ti,−e (mj ) = mi+1 , if j = i 6= r + 1, m , otherwise, j −e mr+1 dr+1 , if i = j = r + 1, −q −e m−e d , if j = i + 1, ′ i+1 i Ti,e (dj ) = −e mi di+1 , if i = j 6= r + 1, dj , otherwise, q −e mer+1 xr+1 , if i = j = r + 1, −q e me x , if j = i + 1, i+1 i ′ Ti,e (xj ) = e mi xi+1 , if i = j 6= r + 1, xj , otherwise, mer+1 dr+1 , if i = j = r + 1, me d , if j = i + 1, i+1 i ′′ Ti,−e (dj ) = e e −q mi di+1 , if i = j 6= r + 1, dj , otherwise, q e m−e if i = j = r + 1, r+1 xr+1 , m−e x , if j = i + 1, ′′ i+1 i Ti,−e (xj ) = −e −e −q mi xi+1 , if i = j 6= r + 1, xj , otherwise. ′ ′′ ′′ ′ Moreover, we have Ti,e Ti,−e = Ti,−e Ti,e = id. ′ and Proof. We need to show relations (3.1)–(3.5) hold under the action of operators Ti,e ′′ ′ Ti,−e . We only consider Tr,e acting on dj1 xj2 (j1 , j2 ∈ {r, r + 1}) for Aq . If j1 = j2 = r, we have ′ −e e e Tr,e (dr xr ) = q −e m−e r mr+1 dr mr mr+1 xr = dr xr 8 ZHAOBING FAN, JICHENG GENG, AND SHAOLONG HAN qmr − q −1 m−1 qmr − q −1 m−1 r r ′ = = Tr,e ( ). −1 −1 q−q q−q If j1 = j2 = r + 1, we have ′ 2e Tr,e (dr+1 xr+1) = m−2e r dr+1 mr xr+1 = dr+1 xr+1 q 2 mr+1 − q −2 m−1 q 2 mr+1 − q −2 m−1 r+1 r+1 ′ = = Tr,e ( ). −1 −1 q−q q−q If j1 = r + 1, j2 = r, we have ′ e e Tr,e (dr+1 xr ) = q 2e m−2e r dr+1 mr mr+1 xr ′ = q 2e mer mer+1 xr m−2e r dr+1 = Tr,e (xr dr+1 ). If j1 = r, j2 = r + 1, we have ′ −e 2e Tr,e (dr xr+1) = q −3e m−e r mr+1 dr mr xr+1 −e −e ′ = q −3e m2e r xr+1 mr mr+1 dr = Tr,e (xr+1 dr ). The verifications for the relations in Aıq follows a similar approach. ′ ′′ Proposition 3.5. The automorphism Ti := Ti,e or Ti,−e commutes with ω, i.e., ω ◦ Ti′ = Ti′ ◦ ω. ′ ′ Proof. For Aq , we give the proof only for the formula ω ◦ Tr,e (dj ) = Tr,e ◦ ω(dj ) such that j = r, r + 1. If j = r, we have ′ −e 2e e e ω ◦ Tr,e (dr ) = ω(q −2e m−e r mr+1 dr ) =q xr mr mr+1 ′ ′ =q e mer mer+1 xr = Tr,e (xr ) = Tr,e ◦ ω(dr ). If j = r + 1, we have ′ ′ −e 2e ′ ω ◦ Tr,e (dr+1) = ω(q em−2e r dr+1 ) = q xr+1 mr = Tr,e (xr+1 ) = Tr,e ◦ ω(dr+1 ). The proof for Aıq is similar. Theorem 3.6. The following braid relations hold. Ti−1 Ti Ti−1 = Ti Ti−1 Ti , Ti−1 Ti Ti−1 Ti = Ti Ti−1 Ti Ti−1 , Ti Tj = Tj Tj , n+1 ] − 1, 2 n+1 ], if i = [ 2 if 2 ≤ i ≤ [ if |i − j| = 6 1, (3.6) (3.7) (3.8) ′ ′′ where Ti := Ti,e or Ti,−e . Proof. For Aq , we only show the proof for the relation in (3.7), the proof for other ′ ′ ′′ relations is similar. It is enough to consider the case of Ti = Ti,e due to Ti,e Ti,−e = ′′ ′ Ti,−e Ti,e = id. We only need to show the following relations hold for j ∈ {r, r + 1, r − 1}. Tr−1 Tr Tr−1 Tr (dj ) = Tr Tr−1 Tr Tr−1 (dj ), (3.9) Tr−1 Tr Tr−1 Tr (xj ) = Tr Tr−1 Tr Tr−1 (xj ), (3.10) Tr−1 Tr Tr−1 Tr (mj ) = Tr Tr−1 Tr Tr−1 (mj ). (3.11) DIFFERENTIAL OPERATOR REALIZATION OF BRAID GROUP ACTION The verification of (3.11) is easy, we omit it here. For (3.9), if j = r, we have Tr−1 Tr Tr−1 Tr (dr ) − Tr Tr−1 Tr Tr−1 (dr ) −e −e −e =Tr−1 Tr Tr−1 (q −2e m−e r mr+1 dr ) − Tr Tr−1 Tr (−q mr dr−1 ) −e −e −e −e =Tr−1 Tr (−q −3e m−e r−1 mr+1 mr dr−1 ) − Tr Tr−1 (−q mr dr−1 ) −e −e −e −2e =Tr−1 (−q −3e m−e r−1 mr+1 mr dr−1 ) − Tr (−q mr−1 dr ) −2e −e −3e −e −2e −e = − q −3e m−e mr mr−1 mr+1 dr = 0. r mr−1 mr+1 dr + q If j = r + 1, we have Tr−1 Tr Tr−1 Tr (dr+1 ) − Tr Tr−1 Tr Tr−1 (dr+1 ) =Tr−1 Tr Tr−1 (q e mr−2e dr+1 ) − Tr Tr−1 Tr (dr+1 ) e −2e =Tr−1 Tr (q e m−2e r−1 dr+1 ) − Tr Tr−1 (q mr dr+1 ) −2e e −2e =Tr−1 (q 2e m−2e r−1 mr dr+1 ) − Tr (q mr−1 dr+1 ) −2e 2e −2e −2e =q 2e m−2e r mr−1 dr+1 − q mr mr−1 dr+1 = 0. If j = r − 1, we have Tr−1 Tr Tr−1 Tr (dr−1) − Tr Tr−1 Tr Tr−1 (dr−1 ) =Tr−1 Tr Tr−1 (dr−1) − Tr Tr−1 Tr (m−e r−1 dr ) −2e −e −e =Tr−1 Tr (m−e mr−1 m−e r−1 dr ) − Tr Tr−1 (q r mr+1 dr ) −e −e −3e −2e −e =Tr−1 (q −2e m−e mr mr−1 m−e r−1 mr mr+1 dr ) − Tr (−q r+1 dr−1 ) −e −e −3e −2e −e = − q −3e m−2e mr mr−1 m−e r mr−1 mr+1 dr−1 + q r+1 dr−1 = 0. Hence (3.9) holds for j = r, r + 1, r − 1. For (3.10), if j = r, we have Tr−1 Tr Tr−1 Tr (xr ) − Tr Tr−1 Tr Tr−1 (xr ) =Tr−1 Tr Tr−1 (q e mer mer+1 xr ) − Tr Tr−1 Tr (−q e mer xr−1 ) =Tr−1 Tr (−q 2e mer mer−1 mer+1 xr−1 ) − Tr Tr−1 (−q e mer xr−1 ) =Tr−1 (−q 2e mer mer−1 mer+1 xr−1 ) − Tr (−q e m2e r−1 xr ) 2e 2e e e e e = − q 2e m2e r−1 mr mr+1 xr + q mr−1 mr mr+1 xr = 0. If j = r + 1, we have Tr−1 Tr Tr−1 Tr (xr+1) − Tr Tr−1 Tr Tr−1 (xr+1 ) =Tr−1 Tr Tr−1 (q −e m2e r xr+1 ) − Tr Tr−1 Tr (xr+1 ) −e 2e =Tr−1 Tr (q −e m2e r−1 xr+1 ) − Tr Tr−1 (q mr xr+1 ) 2e −e 2e =Tr−1 (q −2e m2e r−1 mr xr+1 ) − Tr (q mr−1 xr+1 ) 2e −2e 2e 2e =q −2e m2e mr mr−1 xr+1 = 0. r mr−1 xr+1 − q 9 10 ZHAOBING FAN, JICHENG GENG, AND SHAOLONG HAN If j = r − 1, we have Tr−1 Tr Tr−1 Tr (xr−1 ) − Tr Tr−1 Tr Tr−1 (xr−1 ) =Tr−1 Tr Tr−1 (xr−1 ) − Tr Tr−1 Tr (mer−1 xr ) =Tr−1 Tr (mer−1 xr ) − Tr Tr−1 (q e mer−1 mer mer+1 xr ) e e =Tr−1 (q e mer−1 mer mer+1 xr ) − Tr (−q 2e m2e r mr−1 mr+1 xr−1 ) e e 2e 2e e e = − q 2e m2e r mr−1 mr+1 xr−1 + q mr mr−1 mr+1 xr−1 = 0. Hence (3.10) holds for j ∈ {r, r + 1, r − 1}. We can similarly verify the braid relations (3.6)–(3.8) for Aıq , which gives the desired result. Theorem 3.7. We have the following intertwining relations ω ◦ ϕ = ϕ ◦ Ω, Ti ◦ ϕ = ϕ ◦ τi . (3.12) ′ ′ Proof. For U and Aq , we only prove the formula Ti,e ◦ ϕ(Bj ) = ϕ ◦ τi,e (Bj ), where 1 ≤ i, j ≤ r. The following calculations stem from Proposition 2.3, Proposition 3.3 and Theorem 3.4. If i = j = r, we have ′ ′ e ′ Tr,e ◦ ϕ(Br ) = Tr,e (xr+1 dr ) = q −e mer m−e r+1 xr+1 dr = ϕ(Kr Br ) = ϕ ◦ τr,e (Br ). If i = r, j = r − 1, we have ′ ′ Tr,e ◦ ϕ(Br−1 ) = Tr,e (xr dr−1 ) = q e mer mer+1 xr dr−1 , and the right-hand side, ′ ϕ ◦ τr,e (Br−1 ) = ϕ(q −e [[Br−1 , Br ]e , Br+1 ]e − Kre Br−1 ) =ϕ(q −e Br−1 Br Br+1 − Br Br−1 Br+1 − Br+1 Br−1 Br + q e Br+1 Br Br−1 − Kre Br−1 ) =q −e xr dr−1 xr+1 dr xr dr+1 − xr+1dr xr dr−1xr dr+1 − xr dr+1 xr dr−1 xr+1 dr + q e xr dr+1 xr+1 dr xr dr−1 − q −e mer m−e r+1 xr dr−1 −1 −1 −1 =(q − q −1 )−2 (q −e (mr − m−1 r ) − (qmr − q mr ))(mr+1 − mr+1 ) e −1 2 −2 −1 −e e −e − ((q −1 mr − qm−1 ) − q (m − m ))(q m − q m ) r r+1 r r r+1 xr dr−1 − q mr mr+1 xr dr−1 ( qmr mr+1 xr dr−1 , if e = 1, = −1 −1 −1 q mr mr+1 xr dr−1, if e = −1. ′ ′ Hence Tr,e ◦ ϕ(Br−1 ) = ϕ ◦ τr,e (Br−1 ). If i 6= r, j = i + 1, we have ′ ′ (xi+2 di+1 ) = −q −e m−e ◦ ϕ(Bi+1 ) = Ti,e Ti,e i+1 di xi+2 , and the right-hand side, ′ ϕ ◦ τi,e (Bi+1 ) =ϕ([Bi , Bi+1 ]−e ) = xi+1 di xi+2 di+1 − q −e xi+2 di+1 xi+1 di =(q − q −1 )−1 di xi+2 ((1 − q −e+1 )mi+1 + (q −e−1 − 1)m−1 i+1 ) ( −q −1 di xi+2 m−1 i+1 , if e = 1, = −qdi xi+2 mi+1 , if e = −1. DIFFERENTIAL OPERATOR REALIZATION OF BRAID GROUP ACTION 11 ′ ′ Hence Ti,e ◦ ϕ(Bi+1 ) = ϕ ◦ τi,e (Bi+1 ). If i = j 6= r, we have ′ ′ ′ Ti,e ◦ ϕ(Bi ) = Ti,e (xi+1 di ) = −xi di+1 mei+1 m−e i = ϕ ◦ τi,e (Bi ). If i 6= r and j = i − 1, we have ′ ′ Ti,e ◦ ϕ(Bi−1 ) = Ti,e (xi di−1 ) = mei xi+1 di−1 , and the right-hand side, ′ ϕ ◦ τi,e (Bi−1 ) = ϕ([Bi , Bi−1 ]−e ) = xi+1 di xi di−1 − q −e xi di−1 xi+1 di = (q − q −1 )−1 ((q − q −e )mi + (q −e − q −1 )m−1 i )xi+1 di−1 ( mi xi+1 di−1 , if e = 1, = −1 mi xi+1 di−1 , if e = −1. ′ ′ Hence Ti,e ◦ ϕ(Bi−1 ) = ϕ ◦ τi,e (Bi−1 ). The proof of other cases is similar and we will skip it. Theorem 3.8. The automorphism Ti is unique such that the intertwining relation Ti ◦ ϕ = ϕ ◦ τi and braid relations (3.6)–(3.8) hold. Proof. We consider Aq , while skipping the similar proof for Aıq . For 1 ≤ i ≤ r, we denote by fi = Bi , ei = Bρn (i) , ki = Ki , ki−1 = Kρn (i) . By Proposition 2.3, we have −ei ki−e if j = i 6= r, e if j = i = r, ki fi ′ τi,e (fj ) = [fi , fj ]−e (3.13) if |i − j| = 1, i 6= r e q −e [[fj , fj+1 ]e , ej+1]e − kj+1 fj if i = j + 1 = r, fj otherwise, e −ki fi if j = i 6= r, −e if j = i = r, ei ki ′ τi,e (ej ) = [ej , ei ]e (3.14) if |i − j| = 1, i 6= r −e −e q [[ej , ej+1 ]e , fj+1]e − ej kj+1 if i = j + 1 = r, ej otherwise, −1 if j = i 6= r, ki ′ τi,e (kj ) = ki kj (3.15) if |i − j| = 1, i 6= r k otherwise. j ′ We assume that there exists another T′i,e : Aq → Aq satisfying T′i,e ◦ ϕ = ϕ ◦ τi,e and the braid relations (3.6)–(3.8). We first consider (3.13), if i = j + 1 = r, we have ′ T′r,e ◦ ϕ(fr−1 ) = ϕ ◦ τr,e (fr−1 ). It follows that T′r,e (xr dr−1) =ϕ(q −e [[fr−1 , fr ]e , er ]e − kre fr−1 ) 12 ZHAOBING FAN, JICHENG GENG, AND SHAOLONG HAN =ϕ(q −e fr−1 fr er − fr fr−1 er − er fr−1 fr + q e er fr fr−1 − kre fr−1 ) =q −e xr dr−1 xr+1 dr xr dr+1 − xr+1 dr xr dr−1 xr dr+1 − xr dr+1xr dr−1 xr+1 dr e + q e xr dr+1 xr+1 dr xr dr−1 − (q −1 mr m−1 r+1 ) xr dr−1 −1 −1 −1 qmr − q −1 m−1 r mr+1 − mr+1 −e mr − mr mr+1 − mr+1 − =(q q − q −1 q − q −1 q − q −1 q − q −1 −1 −1 q 2 mr+1 − q −2 m−1 r+1 q mr − qmr q − q −1 q − q −1 −2 −1 −1 2 e e mr − mr q mr+1 − q mr+1 − (q −1 mr m−1 +q r+1 ) )xr dr−1 −1 −1 q−q q−q ( −1 q −1 m−1 if e = −1 r mr+1 xr dr−1 , = qmr mr+1 xr dr−1 , if e = 1 − =q e mer mer+1 xr dr−1 . If i = j + 1, i 6= r, we have T′i,e (xi di−1 ) =ϕ(fi fi−1 − q −e fi−1 fi ) =xi+1 di xi di−1 − q −e xi di−1 xi+1 di (q − q −e )mi + (q −e − q −1 )m−1 i xi+1 di−1 q − q −1 ( m−1 if e = −1 i xi+1 di−1 , = mi xi+1 di−1 , if e = 1 = =mei xi+1 di−1 . If j = i + 1, i 6= r, we have T′i,e (xi+2 di+1 ) =ϕ(fi fi+1 − q −e fi+1 fi ) =xi+1 di xi+2 di+1 − q −e xi+2 di+1 xi+1 di (1 − q 1−e )mi+1 + (q −1−e − 1)m−1 i+1 = di xi+2 −1 q−q ( −qmi+1 di xi+2 , if e = −1 = −1 −1 −q mi+1 di xi+2 , if e = 1 = − q −e m−e i+1 di xi+2 . It follows from the above calculations and (3.13) that −e −xi di+1 (mi m−1 if j = i 6= r, i+1 ) −1 e −1 (q mi mi+1 ) xi+1 di if j = i = r, me x d if i = j + 1, i 6= r i i+1 i−1 T′i,e (xj+1dj ) = −e −e −q mi+1 di xi+2 if j = i + 1, i 6= r e e e q mr mr+1 xr dr−1 if i = j + 1 = r, x d otherwise, i+1 j (3.16) DIFFERENTIAL OPERATOR REALIZATION OF BRAID GROUP ACTION Using similar calculations on (3.14) and (3.15), we obtain e −(mi m−1 if j = i 6= r, i+1 ) xi+1 di −1 −e −1 xi di+1 (q mi mi+1 ) if j = i = r, m−e x d if i = j + 1, i 6= r i−1 i+1 i T′i,e (xj dj+1) = e e −q mi+1 xi di+2 if j = i + 1, i 6= r −2e −e −e q mr mr+1 xr−1dr if i = j + 1 = r, x d otherwise, j i+1 and −1 (mi m−1 i+1 ) xi+1 di q −δj,r m m−1 m m−1 r−1 r j j+1 −1 ′ −δj,r Ti,e (q mj mj+1 ) = −1 −1 mi mi+1 mj mj+1 −δj,r q mj m−1 j+1 if j = i 6= r, if j = i + 1 = r, if |i − j| = 1, i 6= r, j 6= r, otherwise. 13 (3.17) (3.18) For i = r, by (3.16) and (3.17), we have the following system of equations e (T′r,e xr+1)(T′r,e dr ) = (q −1 mi m−1 i+1 ) xr+1 dr , (3.19) (T′r,e xr )(T′r,e dr−1 ) = q e mer mer+1 xr dr−1, (3.20) −e (T′r,e xr )(T′r,e dr+1 ) = xr dr+1 (q −1 mr m−1 r+1 ) , (3.21) −e (T′r,e xr−1)(T′r,e dr ) = q −2e m−e r mr+1 xr−1 dr , (3.22) (T′r,e xr−1)(T′r,e dr−2) = xr−1dr−2 , (3.23) (T′r,e xr−2)(T′r,e dr−1) = xr−2dr−1 . (3.24) For i 6= r, by (3.16) and (3.17) again, we have the following system of equations −e (3.25) (T′i,e xi )(T′i,e di+1 ) = −(mi m−1 i+1 ) xi+1 di , e (3.26) (T′i,e xi )(T′i,e di−1 ) = mei xi+1 di−1 , (T′i,e xi−1 )(T′i,e di ) = m−e i xi−1 di+1 , (3.27) (3.28) (T′i,e xi+2 )(T′i,e di+1 ) = −q −e m−e i+1 di xi+2 , (3.29) (T′i,e xi+1 )(T′i,e di+2 ) = −q e mei+1 xi di+2 . (3.30) (T′i,e xi+1 )(T′i,e di ) = −xi di+1 (mi m−1 i+1 ) , From the braid relations in (3.6)–(3.8), one can show that T′r,e dr−2 = dr−2 , T′r,e xr−2 = xr−2 , T′i,e xi+2 = xi+2 , T′i,e di+2 = di+2 . Therefore, by solving the system of equations (3.19)–(3.24) and (3.25)–(3.30), we obtain e −2e q mr dr+1 , if i = r, j = i + 1, −2e −e −e q mr mr+1 dr , if i = j = r, T′i,e (dj ) = −q −e m−e if i 6= r, j = i + 1, i+1 di , −e m di+1 , if i = j 6= r, i dj , otherwise, 14 ZHAOBING FAN, JICHENG GENG, AND SHAOLONG HAN −e 2e q m xr+1 , e er e q mr mr+1 xr , ′ Ti,e (xj ) = −q e mei+1 xi , me xi+1 , i xj , if i = r, j = i + 1, if i = j = r, if i 6= r, j = i + 1, if i = j 6= r, otherwise. Similarly, by (3.18), we have if i 6= r, j = i + 1, mi , ′ Ti,e (mj ) = mi+1 , if i 6= r, j = i, m , otherwise. j ′ Hence T′i,e coincide with the definition of Ti,e . By a similar approach, one can con′′ ′′ struct Ti,−e : Aq (S) → Aq (S) which is the same as Ti,−e . Summarizing the above, the theorem is proved. 4. Braid group action on polynomial ring Let P := Q(q)[X1 , · · · , Xr , Xr+1 ] be a polynomial ring over Q(q). Theorem 4.1 ([FGH22, Theorem 3.2]). The polynomial ring P is an irreducible Aq (S)module with the following actions. di Xa = [(1 + δi,ρn (r) δi,r+1 )ai ]Xa−ei , xi Xa = Xa+ei , mi Xa = q (1+δi,ρn (r) δi,r+1 )ai Xa , a r+1 where Xa = X1a1 · · · Xr+1 for (a1 , · · · , ar+1 ) ∈ Zr+1 ≥0 and ei is the tuple such that the i-th element is 1 and the other elements are 0. ′ ′′ ]) on Recall that n = 2r or 2r + 1. We define linear operators Ti,e , Ti,−e (1 ≤ i ≤ [ n+1 2 P as follows. −e −ai+1 eai ai+1 q (si Xa ), if i 6= r + 1, (−q ) 1 2 + 3 ea −ea ′ a ea +2ea a a r r r+1 r+1 Ti,e (X ) = q 2 r 2 X , if i = r, ρn (r) = r + 1, q 21 a2r+1 e− 21 ar+1 e Xa , if i = r + 1, ρn (r + 1) = r + 1, −e ai −ea a a if i 6= r + 1, (−q ) q i i+1 (si X ), 3 1 2 ′′ a ea − ea +ea −2ea a a − r r r+1 r+1 Ti,−e (X ) = q 2 r 2 X , if i = r, ρn (r) = r + 1, q − 12 a2r+1 e+ 21 ar+1 e Xa , if i = r + 1, ρn (r + 1) = r + 1, where si switch Xi and Xi+1 in Xa . ′ ′′ Lemma 4.2. The operators Ti,e and Ti,−e are inverse of each other, i.e., ′ ′′ ′′ ′ Ti,e Ti,−e = Ti,−e Ti,e = id. Proof. The assertion can be obtained by direct calculation. Theorem 4.3. For any k ∈ Aq (S) and f (X1 , · · · , Xr+1 ) ∈ P, we have Ti (kf (X1, · · · , Xr+1)) = Ti (k)Ti (f (X1 , · · · , Xr+1 )), (4.1) ′ ′′ . Moreover, we have the following braid relations. where Ti := Ti,e or Ti,−e Ti−1 Ti Ti−1 = Ti Ti−1 Ti , if 2 ≤ i ≤ [ n+1 ] − 1, 2 (4.2) DIFFERENTIAL OPERATOR REALIZATION OF BRAID GROUP ACTION Ti−1 Ti Ti−1 Ti = Ti Ti−1 Ti Ti−1 , Ti Tj = Tj Ti , if i = [ n+1 ], 2 if |i − j| = 6 1. 15 (4.3) (4.4) ′ , then by Lemma Proof. We only show the proof for Aq . If relation (4.1) holds for Ti,e 4.2, we have ′′ ′′ ′′ ′ ′′ ′′ ′ ′ Ti,−e (kXa ) = Ti,−e Ti,−e Ti,e (kXa ) = Ti,−e Ti,−e (Ti,e (k)Ti,e (Xa )) ′′ ′′ ′ ′ ′′ ′ ′ ′′ = Ti,−e Ti,−e (Ti,e Ti,e Ti,−e (k)Ti,e Ti,e Ti,−e (Xa )) ′′ ′′ ′ ′ ′′ ′′ ′′ ′′ = Ti,−e Ti,−e Ti,e Ti,e (Ti,−e (k)Ti,−e (Xa )) = Ti,−e (k)Ti,−e (Xa ). Hence, for (4.1), it is enough to show the following relations hold. ′ ′ ′ Ti,e (dj Xa ) = Ti,e (dj )Ti,e (Xa ), (4.5) ′ ′ ′ Ti,e (xj Xa ) = Ti,e (xj )Ti,e (Xa ), (4.6) ′ ′ ′ Ti,e (mj Xa ) = Ti,e (mj )Ti,e (Xa ). (4.7) For (4.5), if i = r, j = r + 1, we have ′ ′ ′ Tr,e (dr+1Xa ) − Tr,e (dr+1 )Tr,e (Xa ) 1 3 2 =[2ar+1 ]q 2 ear + 2 ear −e(ar+1 −1)+2ear (ar+1 −1) (X a−er+1 ) 1 2 3 ear + 2 ear −ear+1 +2ear ar+1 2 − q e m−2e (Xa ) = 0 r dr+1 q If i = j = r, we have ′ ′ ′ Tr,e (dr Xa ) − Tr,e (dr )Tr,e (Xa ) 2 1 3 =[ar ]q 2 e(ar −1) + 2 e(ar −1)−ear+1 +2e(ar −1)ar+1 (Xa−er ) 1 3 2 ear + 2 ear −ear+1 +2ear ar+1 −e 2 (Xa ) = 0. − q −2e m−e r mr+1 dr q If i = r, j 6= r, j 6= r + 1, we have ′ ′ ′ Tr,e (dj Xa ) − Tr,e (dj )Tr,e (Xa ) 1 3 2 =[aj ]q 2 ear + 2 ear −ear+1 +2ear ar+1 (Xa−ej ) 1 2 3 − dj q 2 ear + 2 ear −ear+1 +2ear ar+1 (Xa ) = 0. If i 6= r, j = i + 1, we have ′ ′ ′ Ti,e (di+1 Xa ) − Ti,e (di+1 )Ti,e (Xa ) =[ai+1 ](−q −e ) −(ai+1 −1) eai (ai+1 −1) q −e + q −e m−e i+1 di (−q ) ((si Xa )X−ei ) −ai+1 eai ai+1 q (si Xa ) = 0. If i 6= r, i = j, we have ′ ′ ′ Ti,e (di Xa ) − Ti,e (di )Ti,e (Xa ) =[ai ](−q −e ) −ai+1 eai+1 (ai −1) q −e − m−e i di+1 (−q ) ((si Xa )X−ei+1 ) −ai+1 eai ai+1 q (si Xa ) = 0. 16 ZHAOBING FAN, JICHENG GENG, AND SHAOLONG HAN If i 6= r, j 6= i, j 6= i + 1, we have ′ ′ ′ Ti,e (dj Xa ) − Ti,e (dj )Ti,e (Xa ) =[aj ](−q −e ) − (−q −e ) −ai+1 eai+1 ai q −ai+1 eai+1 ai q (Xa−ej ) [aj ](Xa−ej ) = 0. For (4.6), if i = r, j = r + 1, we have ′ ′ ′ Tr,e (xr+1Xa ) − Tr,e (xr+1 )Tr,e (Xa ) 1 2 3 1 2 =q 2 ear + 2 ear −e(ar+1 +1)+2ear (ar+1 +1) (Xa+er+1 ) 7 − q 2 ear + 2 ear −ear+1 +2ear ar+1 −e (Xa+er+1 ) = 0. If i = j = r, we have ′ ′ ′ Tr,e (xr Xa ) − Tr,e (xr )Tr,e (Xa ) 2 1 3 =q 2 e(ar +1) + 2 e(ar +1)−ear+1 +2e(ar +1)ar+1 (Xa+er ) 1 5 2 − q 2 ear + 2 ear +ear+1 +2ear ar+1 +2e (Xa+er ) = 0. If i = r, j 6= r, j 6= r + 1 we have ′ ′ ′ Tr,e (xj Xa ) − Tr,e (xr )Tr,e (Xa ) 2 1 3 =q 2 e(ar +1) + 2 e(ar +1)−ear+1 +2e(ar +1)ar+1 (Xa+er ) 1 5 2 − q 2 ear + 2 ear +ear+1 +2ear ar+1 +2e (Xa+er ) = 0. If i 6= r, j = i + 1, we have ′ ′ ′ Ti,e (xi+1 Xa ) − Ti,e (xi+1 )Ti,e (Xa ) =(−q −e ) −ai+1 −1 eai (ai+1 +1) q + q e mei+1 xi (−q −e ) ((si Xa )Xei ) −ai+1 eai ai+1 q (si Xa ) = 0. If i 6= r, j = i, we have ′ ′ ′ (xi )Ti,e (Xa ) Ti,e (xi Xa ) − Ti,e =(−q −e ) −ai+1 eai+1 (ai +1) q − mei xi+1 (−q −e ) ((si Xa )Xei+1 ) −ai+1 eai ai+1 q (si Xa ) = 0. If i 6= r, j 6= i, j 6= i + 1 we have ′ ′ ′ Ti,e (xj Xa ) − Ti,e (xj )Ti,e (Xa ) =(−q −e ) −ai+1 eai+1 ai q (si Xa+ej ) − xj (−q −e ) −ai+1 eai ai+1 q (si Xa ) = 0. To sum up the above calculations, for all cases, the relations (4.5) and (4.6) hold. Similar to the proof above, we can prove the formula (4.7) is true. For the braid relations (4.2)–(4.4), if 2 ≤ i ≤ r − 1, we have ′ Ti,e Xa = (−q −e ) ′ ′ Ti−1,e Ti,e Xa = (−q −e ) −ai+1 eai ai+1 q (si Xa ), −2ai+1 eai ai+1 +eai−1 ai+1 q (si−1 si Xa ), DIFFERENTIAL OPERATOR REALIZATION OF BRAID GROUP ACTION ′ ′ ′ Ti,e Ti−1,e Ti,e Xa = (−q −e ) ′ Ti−1,e Xa = (−q −e ) ′ ′ Ti,e Ti−1,e Xa = (−q −e ) ′ ′ ′ Ti−1,e Xa = (−q −e ) Ti−1,e Ti,e −2ai+1 −ai eai ai+1 +eai−1 ai+1 +eai−1 ai q −ai eai−1 ai q 17 (si si−1 si Xa ), (si−1 Xa ), −ai −ai+1 eai ai−1 +eai−1 ai+1 q (si si−1 Xa ), −2ai+1 −ai eai ai+1 +eai−1 ai+1 +eai−1 ai q (si−1 si si−1 Xa ). ′ ′ ′ ′ ′ ′ Since si si−1 si Xa = si−1 si si−1 Xa , the relation Ti,e Ti−1,e Ti,e = Ti−1,e Ti,e Ti−1,e holds. ′ ′ ′ If i = r, we compute Tr−1,e Tr,e Tr−1,e (Xa ) as follows. ′ ′ ′ Tr−1,e Tr,e Tr−1,e (Xa ) ′ ′ (−q −e ) =Tr−1,e Tr,e −ar ear−1 ar q (sr−1 Xa ) 1 2 3 −ar ear−1 ar 1 2 3 a =q 2 ear−1 + 2 ear−1 −ear+1 +2ear−1 ar+1 (−q −e ) q =q 2 ear−1 + 2 ear−1 −ear+1 +2ear−1 ar+1 (−q −e ) r−1 ′ Tr−1,e (sr−1 Xa ) −ar 2ear−1 ar q (4.8) (Xa ). By (4.8), we have ′ ′ ′ ′ Tr,e Tr−1,e Tr,e Tr−1,e (Xa ) 1 3 2 a −ar 2ear−1 ar a −ar 2ear−1 ar =q 2 ear−1 + 2 ear−1 −ear+1+2ear−1 ar+1 (−q −e ) r−1 q ′ Tr,e (Xa ) and ′ ′ ′ ′ Tr−1,e Tr,e Tr−1,e (Tr,e Xa ) 1 2 3 =q 2 ear−1 + 2 ear−1 −ear+1 +2ear−1 ar+1 (−q −e ) r−1 q ′ (Tr,e Xa ). ′ ′ ′ ′ ′ ′ ′ ′ Hence, we have Tr,e Tr−1,e Tr,e Tr−1,e = Tr−1,e Tr,e Tr−1,e Tr,e . The proof of (4.4) is trivial, so we omit it. Theorem 4.4. The operator Ti satisfying (4.1) is unique. Proof. We show the proof for the action of Aq on P. We assume that there exists another linear operators T′i,e such that (4.1) holds. For 1 ≤ i ≤ r and a = (a1 , a2 , · · · , ar+1 ) ∈ Zr+1 ≥0 , we consider the following action. a a r+1 r+1 T′i,e ((da11 · · · dr+1 )Xa ) = Ti,e (da11 · · · dr+1 )T′i,e (Xa ). (4.9) If i = r, then the formula in (4.9) will be written in the following form. r r−1 Y Y ar+1 ′ ar e −2e −e Tr,e (Xa ) ( [ai ]!)[ar+1 ]!! = ( dai i )(q −2e m−e r mr+1 dr ) (q mr dr+1 ) i=1 i=1 =( r−1 Y a a r ear+1 r+1 ′ −e r r+1 m−2ea dr+1 Tr,e (Xa ) dai i )q −2ear m−ea r r+1 (mr dr ) q i=1 =q ear+1 −2ear −e ar (a2r −1) −2ear ar+1 −ear −2ear+1 r ( m−ea r+1 mr r+1 Y dai i )T′r,e (Xa ) i=1 1 2 3 −ear −2ear+1 r ( = q − 2 ear − 2 ear +ear+1−2ear ar+1 m−ea r+1 mr r+1 Y dai i )T′r,e (Xa ) i=1 (4.10) 18 ZHAOBING FAN, JICHENG GENG, AND SHAOLONG HAN where [ar+1 ]!! = [2ar+1 ][2ar+1 − 2] · · · [2]. On the other hand, we have the following formula. q − 21 ea2r − 23 ear +ear+1 −2ear ar+1 −ear −2ear+1 r m−ea ( r+1 mr r+1 Y dai i )Xa i=1 (4.11) T′r,e (Xa ) = q 2 ear + 2 ear −ear+1 +2ear ar+1 Xa . (4.12) r Y − 21 ea2r − 23 ear +ear+1 −2ear ar+1 =q ( [ai ]!)[ar+1 ]!! i=1 Comparing (4.10) and (4.11), we have 1 3 2 If i 6= r, the formula in (4.9) will be written in the following form. r i−1 r+1 Y Y Y ai ai+1 a aj −e −e −e ( [aj ]!)[ar+1 ]!! = ( dj )(mi di+1 ) (−q mi+1 di ) ( dj j )T′i,e (Xa ) j=1 j=1 j=i+2 −e ai+1 −eai ai+1 = (−q ) q −ea i m−ea mi+1 i+1 ( i i−1 r+1 Y Y aj ai+1 ai a dj )di di+1 ( dj j )T′i,e (Xa ) j=1 j=i+2 (4.13) On the other hand, we have −e ai+1 −eai ai+1 (−q ) q −ea i m−ea mi+1 i+1 ( i r+1 i−1 Y Y a aj ai+1 ai dj j )(si Xa ) dj )di di+1 ( j=i+2 (4.14) (si Xa ). (4.15) j=1 a =(−q −e ) i+1 q −eai ai+1 ( r Y [ai ]!)[ar+1 ]!! i=1 Comparing (4.13) and (4.14), we have T′i,e (Xa ) = (−q −e ) −ai+1 eai ai+1 q ′ Combining the formulas in (4.12) and (4.15), we can see that T′i,e coincide with Ti,e . ′′ By a similar approach, we can also prove that the linear operator Ti,−e is unique. By Proposition 3.3 and Theorem 4.1, the polynomial ring P is a ı U(S)-module with the following actions. Bi Xa = ϕ(Bi )Xa , Ki Xa = ϕ(Ki )Xa . Theorem 4.5. For any u ∈ ı U(S) and f (X1 , · · · , Xr+1 ) ∈ P, we have Ti (uf (X1 , · · · , Xr+1 )) = τi (u)Ti (f (X1 , · · · , Xr+1 )), ′ ′′ where Ti := Ti,e or Ti,−e . Proof. It is enough to show the following relations hold. ′ ′ ′ Ti,e (Bj Xa ) = τi,e (Bj )Ti,e (Xa ), (4.16) ′ ′ ′ Ti,e (Kj Xa ) = τi,e (Kj )Ti,e (Xa ). (4.17) DIFFERENTIAL OPERATOR REALIZATION OF BRAID GROUP ACTION 19 We only show the proof of (4.16) for U . If 1 ≤ j ≤ r, by Theorem 4.3 and Theorem 3.7, we have ′ ′ ′ ′ ′ Ti,e (Bj Xa ) = Ti,e (ϕ(Bj )Xa ) = Ti,e (xj dj+1 Xa ) = Ti,e (xj dj+1 )Ti,e (Xa ) ′ ′ ′ ′ ′ ′ = Ti,e (ϕ(Bj ))Ti,e (Xa ) = ϕ(τi,e (Bj ))Ti,e (Xa ) = τi,e (Bj )Ti,e (Xa ). The proof for the case ρn (r) ≤ j ≤ ρn (1) is similar. 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Harbin Engineering University, Harbin, China Email address: fanzhaobing@hrbeu.edu.cn Harbin Engineering University, Harbin, China Email address: jcgeng@hrbeu.edu.cn Harbin Engineering University, Harbin, China Email address: algebra@hrbeu.edu.cn