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TWIST AUTOMORPHISMS ON QUANTUM UNIPOTENT CELLS AND DUAL CANONICAL BASES YOSHIYUKI KIMURA AND HIRONORI OYA 7 1 0 Abstract. Inthispaper,weconstructtwistautomorphismsonquantumunipotentcells, 2 which are quantum analogues of the Berenstein-Fomin-Zelevinsky twist automorphisms n on unipotent cells. We show that those quantum twist automorphisms preserve the dual a canonical bases of quantum unipotent cells. J We also prove that quantum twist automorphisms admit the quantum analogues of 0 Geiß-Leclerc-Schröer’sdescription of (non-quantum) twist automorphisms. Namely, they 1 are described by the syzygy functors for representations of preprojective algebras. Geiß- ] Leclerc-Schröer’s results are essentially used in our proof. In particular, we show that A quantum twist automorphisms are compatible with quantum cluster monomials. The Q 6-periodicity of specific quantum twist automorphisms is also discussed. . h t a m Contents [ 2 1. Introduction 1 v 2. Preliminaries 5 8 6 3. Quantum unipotent cell and the De Concini-Procesi isomorphism 23 2 4. Quantum twist isomorphisms 29 2 5. Twist automorphisms on quantum unipotent cells 34 0 . 6. Quantum twist automorphism and quantum cluster algebras 36 1 0 7. Finite type : periodicity 48 7 References 50 1 : v i X 1. Introduction r a 1.1. Canonical bases and cluster algebras. For simplicity, let G be a connected, simply-connected, complex simple algebraic group with a fixed maximal torus H, a pair of Borel subgroups B with B ∩ B = H, a Weyl group W = Norm (H)/H and the ± + − G maximal unipotent subgroups N ⊂ B . Let U (g) be the Drinfeld-Jimbo quantized en- ± ± q veloping algebra of the corresponding Lie algebra g, and U−(g) be its negative part which q arises fromthetriangulardecomposition ofg. In[Lus90], Lusztig constructed thecanonical bases B of U−(g) using perverse sheaves on the varieties of quiver representations when q The work of the first author is supported by JSPS Grant-in-Aid for Scientific Research (S) 24224001. The work of the second author is supported by Grant-in-Aid for JSPS Fellows (No. 15J09231) and the Programfor Leading Graduate Schools, MEXT, Japan. 1 TWIST AUTOMORPHISMS ON QUANTUM UNIPOTENT CELLS AND DUAL CANONICAL BASES 2 g is simply-laced. In [Kas91], Kashiwara constructed the lower global bases Glow(B(∞)) of U−(g) in general. Lusztig proved that the two bases of U−(g) coincide. In this paper, q q we call the bases the canonical bases. The canonical bases have interesting structures; one is positivity of structure constants of multiplications and (twisted) comultiplication, and another is the combinatorial structure which is called Kashiwara crystal structure. Using the positivity of the canonical bases, Lusztig generalized the notion of the total positivity for reductive groups and related algebraic varieties. Since U− has a natural non-degenerate Hopf pairing which makes it into a (twisted) q self-dual bialgebra, we can consider U− as a quantum analogue of the coordinate rings q C[N ]. The combinatorial structure of Blow and its dual basis Bup (with respect to the − non-degenerate Hopf pairing), called the dual canonical bases, has intensively studied by Lusztig and Berenstein-Zelevinsky (in the type A-case) and it became one of the origin of cluster algebras introduced by Fomin-Zelevinsky. 1.2. Quantum unipotent subgroups and dual canonical bases. For a Weyl group element w ∈ W (and a lift w˙ ∈ Norm (H)), the unipotent root subgroups N (w) := G − N ∩w˙Nw˙−1 or Schubert cells B w˙B /B in the full flag varieties G/B has attracted − + + + + much attention in the development of the theory of total positivity for reductive groups. Geiß-Leclerc-Schroër [GLS11] introduced a cluster algebra structure on C[N (w)] using − representationtheoryofpreprojectivealgebras, calledanadditivecategorification,andthey proved that the dual semicanonical bases S∗ is compatible with C[N (w)], that is S∗ ∩ − C[N (w)]givesabasisofC[N (w)]andthesetofclustermonomials, whicharemonomials − − of cluster variables ina single cluster, is contained in the dual semicanonical bases S∗. Here we note we identify the coordinate rings C[N (w)] of the unipotent subgroups N (w) as − − invariant subalgebras C[N ]N−∩w˙N−w˙−1 fixing a splitting N ≃ (N ∩w˙N w˙−1)×N (w) − − − − − as varieties. For a nilpotent Lie algebra n (w) associated with the subgroup N (w), a quantum − − analogue U−(w) of the universal enveloping algebras U(n (w)) are introduced by De q − Concini-Kac-Procesi [DKP95] and also by Lusztig [Lus10] as subalgebras of the quantized enveloping algebras U− which are generated by quantum root vectors defined by Lusztig’s q braid group symmetry on the quantized enveloping algebras U (g) and they are the linear q spans of their Poincaré-Birkhoff-Witt type orthogonal monomials with respect to the non- degenerate pairing on U−. In [Kim12], the first author proved that the subalgebras U−(w) q q are compatible with the dual canonical bases, that is Bup ∩ U−(w) is a base of U−(w) q q and the specialization of U−(w) (using the dual canonical basis) at q = 1 is isomorphic to q the coordinate rings C[N (w)], hence U−(w) are also considered as a quantum analogue − q of the coordinate rings C[N (w)] of the unipotent subgroups. − Geiß-Leclerc-Schroër [GLS13] proved that U−(w) admits a quantum cluster algebra q structure in the sense of Berenstein-Zelevinsky if g is symmetric via the additive categorification and Goodearl-Yakimov [GY16b, GY14] proved using the frame work of quantum nilpotent algebras. Kang-Kashiwara-Kim-Oh [KKKO14, KKKO15] showed that the set of TWIST AUTOMORPHISMS ON QUANTUM UNIPOTENT CELLS AND DUAL CANONICAL BASES 3 quantum cluster monomials is contained in the dual canonical bases via symmetric quiver Hecke algebras if g is symmetric. 1.3. Unipotent cells and cluster structure. For a pair (w ,w ) of Weyl group ele- + − ments, the intersections Gw+,w− := B w˙ B ∩ B w˙ B are called double Bruhat cells + + + − − − and the maximal torus H acts Gw+,w− by left (or right) multiplication. For certain lift w ∈ G of w ∈ W, the intersection B w˙ B ∩ N w N is a section of the quotient − − + + + − − − Gw+,w− → H \ Gw+,w−. The unipotent cells N ∩ B w˙ B are special cases of reduced − + + + doubleBruhat cells with w istheunit. The (upper) cluster structure ofthedouble Bruhat − cells and unipotent cells are studied in details, see Berenstein-Fomin-Zelevinsky [BFZ05] (see also Geiß-Leclerc-Schroër [GLS11] and Williams [Wil13]). In fact, in [GLS11], it is shown that the coordinate ring of the unipotent subgroup has a cluster algebra structure with unlocalized frozen variables, and the coordinatering of the unipotent cell hasa cluster algebra structure with fully localized frozen variables. For a Weyl group element w ∈ W, Berenstein-Fomin-Zelevinsky [BFZ96] (in the type A-case) and Berenstein-Zelevinsky [BZ97] (in general) introduced twist automorphisms which are automorphisms on unipotent cells N ∩ B w˙B for solving the factorization − + + problems, called the Chamber Ansatz, which describe the inverse of the “toric chart” of the Schubert varieties. In [GLS11, GLS12], Geiß-Leclerc-Schroër studied the additive categorification of the twistautomorphismusingrepresentationtheoryofpreprojectivealgebras, infact, itisgiven by the syzygy on the Frobenius subcategory associated with w. In fact, they treated the coordinate ring of the unipotent cells as the localization of the coordinate rings unipotent subgroups with respect to the (unipotent) minors associated with Weyl group elements, using the “multiplicative property” of dual semicanonical bases, they introduced the “dual semicanonical bases” of the coordinate ring of the unipotent cells. In this paper, we study the construction of the quantum analogue of the twist automorphisms on the quantum unipotent cells, which is the “quantized coordinate rings of the unipotent cells” and its relation to the additive categorification. 1.4. Quantum unipotent cells. Quantum coordinateringsofdouble Bruhat cells, called quantum double Bruhat cells, are introduced by De Concini-Procesi [DP97] in the study of representation theory of quantum groups at root of unity and also intensively studied by Joseph [Jos95] in the study of prime spectra of quantized coordinate ring of G. Berenstein- Zelevinsky [BZ05] conjectured that quantum double Bruhat cells admit a structure of quantum cluster algebras via quantum minors. Goodearl-Yakimov [GY16a] proved the conjecture using a quantum analogue of the Fomin-Zelevinsky twist of the double Bruhat cells. In [DP97], De Concini-Procesi studied the relation between the quantum unipotent subgroups and the quantum unipotent cells in finite type case. In [Kim12], the injectivity result of De Concini-Procesi is generalized via the study of crystal bases. Berenstein-Rupel [BR15] studied the quantum unipotent cells via the Hall algebra tech- nique and they constructed quantum analogue of the twist maps under the conjecture concerning the quantum cluster algebra structure and they showed that the quantum twist TWIST AUTOMORPHISMS ON QUANTUM UNIPOTENT CELLS AND DUAL CANONICAL BASES 4 automorphisms preserve thetriangularbases (inthesense ofBerenstein-Zelevinsky [BZ14]) of the quantum unipotent cells when the Weyl group element w is the square of the acyclic Coxeter elements c with ℓ(w) = 2ℓ(c ), where Q is an acyclic quiver on the Dynkin Q Q diagram associated with G. We note that Qin [Qin16] proved that the triangular bases (=localized dual canonical bases) in the sense of [Qin15] coincide with the triangular bases in the sense of Berenstein-Zelevinsky when g is symmetric. 1.5. Quantum unipotent cells and the dual canonical bases. Our main results in this paper are as follows: (1) WegeneralizetheDeConcini-ProcesiisomorphismsforthelocalizationsA [N (w)∩w˙G ] q − 0 of the quantum unipotent subgroups A [N (w)] and the quantum unipotent cells q − A Nw for arbitrary symmetrizable Kac-Moody Lie algebras (see Theorem 3.17). q − The quantum cluster structure on the quantum unipotent cells can be proved as a (cid:2) (cid:3) corollary of the De Concini-Procesi isomorphism (see Corollary 6.14). (2) We introduce a quantum analogue γ of the twist isomorphism between the unipo- w tent cells Nw and N (w)∩ w˙G which is defined using the Gauss decomposition − − 0 (See Theorem 4.21). (3) We introduce a quantum analogue of the twist automorphism of unipotent cells on the quantum coordinate ring A Nw of the unipotent cells (without referring the q − quantum cluster algebra structure) and show that the quantum twist preserves the (cid:2) (cid:3) dual canonical bases (see Theorem 5.4). In fact, we introduce a quantum analogue of the twist automorphism as a composite of the De Concini-Procesi isomorphism and the quantum twist isomorphism. The result that the dual canonical bases are preserved under the twist automorphism is proved as a consequence of the properties of two isomorphisms and the dual canonical bases. We note that our construction is independent of the construction by Berenstein-Rupel [BR15]. (4) We relatethe quantum twist automorphisms and the quantum cluster structure un- dertheadditivecategorification(seeTheorem6.19). Wealsostudythe6-periodicity of the twist automorphisms in finite type cases (see Theorem 7.1). 1.6. Outline of the paper. The paper is organized as follows. In Section 2, we fix the convention in this paper. In particular, we give a brief review of the quantum unipotent subgroups and the dual canonical bases. In Section 3, we define the dual canonical bases of the localized quantum coordinate rings and prove De Concini-Procesi isomorphisms in general. In Section 4, a quantum analogue of the twist isomorphism is introduced. In Section 5, we define a quantum analogue of the twist automorphism as a composite of quantum twist isomorphism and the De Concini-Procesi isomorphism. In Section 6, we relate the quantum twist automorphisms and the quantum cluster algebra structures on the quantum unipotent cells via Geiß-Leclerc-Schroër’s additive categorification and the De Concini-Procesi isomorphisms. In Section 7, we study the periodicity of the specific twist automorphisms in finite type. 1.7. Furtherwork. ThecomparisonbetweentheconstructionbyBerenstein-Rupel[BR15] and the quantum analogue of the Chamber Ansatz will be discussed in another paper. TWIST AUTOMORPHISMS ON QUANTUM UNIPOTENT CELLS AND DUAL CANONICAL BASES 5 There is another type of “quantum twist maps” which is not an automorphism, introduced by Lenagan-Yakimov, which are a quantum analogue of the Fomin-Zelevinsky twist isomorphism and the authors showed that they also preserve the dual canonical bases of A [N (w)] [KO16]. q − Acknowledgement. The authors are grateful to Bernard Leclerc for his enlightening advice concerning cluster algebras and their categorifications. The authors would like to express our sincere gratitude to Yoshihisa Saito, the supervisor of the second author, for his helpful comments. TheauthorswishestothankRyoSatoandBeaSchumannforseveralinteresting comments and discussions. The second author thanks the University of Caen Normandy, where a part of this paper was written, for the hospitality. 2. Preliminaries Notation 2.1. (1) Let k be a field. For a k-vector space V, set V∗ := Hom (V,k). Denote k by h , i: V∗ ×V → k, (f,v) 7→ hf,vi the canonical pairing. (2) For a k-algebra A, we set [a ,a ] := a a −a a for a ,a ∈ A. An Ore set M of A 1 2 1 2 2 1 1 2 stands for a left and right Ore set consisting of non-zero divisors. Denote by A[M−1] the algebra of fractions with respect to the Oreset M. In this case, A is naturally a subalgebra of A[M−1]. See [GW89, Chapter 6]. (3) An A-module V means a left A-module. The action of A on V is denoted by a.v fora ∈ A and v ∈ V. In this case, V∗ is regarded as a right A-module by hf.a,vi = hf,a.vi for f ∈ V∗,a ∈ A and v ∈ V. (4) For two letters i,j, the symbol δ stands for the Kronecker delta. ij 2.1. Kac-Moody Lie algebras and associated flag schemes. Inthissubsection, wefix thenotationson(symmetrizable) Kac-MoodyLiealgebrasgandassociated(notnecessarily a group) schemes G . Since we do not need the group structure on the whole G itself, we use the scheme introduced by Kashiwara [Kas89] (see also Kashiwara-Tanisaki [KT95]). Definition 2.2. A root datum consists of the following data; (1) I : a finite index set, (2) h : a finite dimensional Q-vector space, (3) P ⊂ h∗ : a lattice, called weight lattice, (4) P∗ = {h ∈ h | hh,Pi ⊂ Z} with the canonical pairing h−,−i : P∗⊗ P → Z, called Z the coweight lattice, (5) {α } ⊂ P : a subset, called the set of simple roots, i i∈I (6) {h } ⊂ P∗ : a subset, called the set of simple coroots, i i∈I (7) (−,−) : P ×P → Q : a Q-valued symmetric Z-bilinear form on P. satisfying the following conditions: (a) (α ,α ) ∈ 2Z for i ∈ I, i i >0 (b) hh ,λi = 2(α ,λ)/(α ,α ) for λ ∈ P and i ∈ I, i i i i (c) A = (hh ,α i) isasymmetrizablegeneralizedCartanmatrix, thatishh ,α i = 2, i j i,j∈I i i hh ,α i ∈ Z for i 6= j and hh ,α i = 0 is equivalent to hh ,α i = 0, i j ≤0 i j j i TWIST AUTOMORPHISMS ON QUANTUM UNIPOTENT CELLS AND DUAL CANONICAL BASES 6 (d) {α } ⊂ h∗, {h } ⊂ h are linearly independent subsets. i i∈I i i∈I The Z-submodule Q = Zα ⊂ P is called the root lattice, Q∨ = Zh ⊂ P∗ is i∈I i i∈I i called the coroot lattice. We set Q = Z α ⊂ Q and Q = −Q . For ξ = ξ α ∈ P + ≥0 i − + P i∈I i i Q, we set ht(ξ) = ξ ∈ Z. Let P := {λ ∈ P | hh ,λi ∈ Z for all i ∈ I} and we i∈I i P+ i ≥0 P assume that there exists {̟ } ⊂ P such that hh ,̟ i = δ . Set ρ := ̟ ∈ P . P i i∈I + i j ij i∈I i + A triple (h,Π,Π∨,(−,−)) is called a realization of A. Let g = n P⊕ h ⊕ n be an − + associated Kac-Moody Lie algebra and its triangular decomposition. Let g = g be α∈h∗ α its root space decomposition and ∆ = {α ∈ h∗ | g 6= 0}\{0} be the set of roots, and ∆ α ± L be the subsets of positive and negative roots. Let us fix a root datum (A,P,P∨,Π,Π∨) which gives a realization of A and H = Spec(C[P]) be the algebraic torus whose character lattice is P and C-valued point is Hom (P,C∗). Z For k ∈ Z , we set ∆≥k := {α ∈ ∆ | ±ht(α) ≥ k} and n≥k := g . Then we ≥0 ± ± ± α∈∆≥k α ± have ∆≥k +∆ ∩ ∆ ⊂ ∆≥k and n≥k is an ideal and n /n≥k isLa finite dimensional ± ± ± ± ± ± ± nilpot(cid:16)ent Lie alge(cid:17)bra. We set nˆ = limn /n≥k = g . ± ± ± α ←− αY∈∆± Let N be the pro-unipotent group scheme whose pro-nilpotent pro-Lie algebra is nˆ ± ± that is defined by N = limexp n /n≥k = Spec U (n )∗ , ± ± ± ± gr ←− that is we have C[N ] = U(n )∗ . (cid:16) (cid:17) (cid:16) (cid:17) ± ± gr Let B = H ⋉ N be the Borel subgroup. For i ∈ I, let P±, G , G± and N± be the ± ± i i i i group schemes associated with Lie P± = nˆ ⊕h⊕g , i ± ∓αi Lie(G ) = g ⊕h⊕g , (cid:0) i(cid:1) −αi αi Lie G± = h⊕g . i ±αi We introduce the scheme G and(cid:0) its(cid:1)open subscheme G. ∞ Definition 2.3. (1) For λ ∈ P , let V (λ) be the integrable highest weight g-module with + highest weight vector u of highest weight λ. λ (2) Let O (g) be the category of P-weighted integrable g-modules satisfying the fol- int lowing condition. (1) M = M with M = {m ∈ M | h.m = hh,µim for all h ∈ h} and dimM < µ∈M µ µ µ ∞ for µ ∈ P, L (2) thereexistsfinitelymanyλ ,··· ,λ ∈ P suchthatP (M) := {µ ∈ P | M 6= 0} ⊂ 1 m + µ λ +Q . 1≤j≤m j − S TWIST AUTOMORPHISMS ON QUANTUM UNIPOTENT CELLS AND DUAL CANONICAL BASES 7 It is well-known that O (g) is semisimple with its simple object being isomorphic int to the integrable highest weight modules {V (λ) | λ ∈ P }. By definition, M ∈ O (g) + int is equivalent that M = M with M = {m ∈ M | h.m = hh,µim for all h ∈ h} , µ∈M µ µ dimM < ∞ for µ ∈ P and dim U(p)m < ∞ for i ∈ I and m ∈ M. µ C i L Let ϕ: g → g be the anti-involution defined by ϕ(e ) = f ,ϕ(f ) = e ,ϕ(h) = h for i ∈ I i i i i and h ∈ h. For M ∈ O (g), we denote D M be the g-module on Hom(M ,C) int ϕ µ∈M µ defined by L hx.f,mi = hf,ϕ(x).mi for x ∈ g and m ∈ M. We note that D M ∈ O (g). For a g-module M, we denote Mr be the gop-module ϕ int {mr | m ∈ M} with the action of gop given by x.(mr) = (ϕ(x).m)r for x ∈ g and m ∈ M. Wedenote byOr (g) bethecategoryofintegrable gop-modulesMr such that M ∈ O (g). int int We interpret the category of gop-modules as the category of right U(g)-modules. We define the scheme G := Spec(R (g)) by as the spectrum of the ring of “strongly ∞ C regular functions” introduced by Kac-Peterson, that is R (g) := f ∈ Hom (U(g),C) U(g)f∈Oint(g) , C C fU(g)∈Or (g) int n (cid:12) o where we consider bimodule structure on Hom (U(g)(cid:12),C) defined by C (cid:12) hx.f.y,ui = hf,y.u.xi (x,y ∈ g,f ∈ Hom (U(g),C),u ∈ U(g)). C Let Φ = Φ : V (λ)⊗V (λ)r −→∼ R (g) λ C λX∈P+ be the map defined by hΦ (v ⊗v r),ui = (v ,uv ) for v ,v ∈ V (λ) and u ∈ U(g), λ 1 2 2 1 λ 1 2 where (−,−) : V (λ) ⊗ V (λ) → C be the symmetric bilinear form on V (λ) such that λ (u ,u ) = 1 and (x.v ,v ) = (v ,ϕ(x).v ) for v ,v ∈ V (λ) and x ∈ g. It is known that λ λ λ 1 2 λ 1 2 λ 1 2 it is an isomorphism of bimodules, called the Peter-Weyl isomorphism for Kac-Moody Lie algebras. ThemultiplicationsU p− ⊗U(g) → U(g)andU(g)⊗U p+ → U(g)inducecoaction i i morphisms R (g) → C P− ⊗ R (g) and R (g) → R (g) ⊗ C P+ , where C P+ is C (cid:0) i (cid:1) C C C (cid:0) (cid:1) i i the coordinate ring of P±. Hence we have the morphisms of schemes P−×G → G and i(cid:2) (cid:3) (cid:2) i(cid:3) ∞ (cid:2)∞ (cid:3) G ×P+ → G which give rise to the left action of P− and the right action of P+. ∞ i ∞ i i The scheme G contains a canonical point e and is endowed with a left action of P− ∞ i and a right action of P+. i Definition 2.4. LetGbetheopensubsetofG whichisgivenbytheunionofP−···P−eP+···P+, ∞ i1 im j1 jn that is G = P−···P−eP+···P+. i1 im j1 jn i1,···,im[,j1,···,jn∈I TWIST AUTOMORPHISMS ON QUANTUM UNIPOTENT CELLS AND DUAL CANONICAL BASES 8 Proposition 2.5. (1) The left P−action on G is free and the right P+ action on G is free. i i (2) The restricted left B action on G and the restricted right B action on G are − + independent of i ∈ I. (3) The left and right action of s and s on G coincides. i i Let N ×H×N → G betheopenimmersion defined by the“multiplication” (x,y,z) 7→ − + xyz and its image is denoted by G . Let 0 [ ] ×[ ] ×[ ] : G → N ×H ×N − 0 + 0 − + be the inverse morphism of the “multiplication”. We note that we use only the left B - − action and the right B -action on G. + For the Lie algebra anti-involution ϕ: g → g, let ϕ: U(g) → U(g) be the induced anti- involution as a C-algebra. We note that ϕ induces the anti-isomorphism of group schemes P± −→∼ P∓ for i ∈ I and we have the following diagram i i U p− ⊗U(g) // U(g) , i (cid:0) (cid:1) (cid:15)(cid:15) (cid:15)(cid:15) U(g)⊗U p+ // U(g) i where the horizontal homomorphisms are(cid:0)mul(cid:1)tiplications, the left vertical homomorphism is given by flip ◦ (ϕ⊗ϕ) and the right vertical homomorphism is ϕ. Hence we have the “anti-involution” ( )T on G , that is a morphism which intertwines the left P−-action ∞ i into the right P+-action and vice versa. It is clear that ( )T preserves G and G by its i 0 construction. Definition 2.6. The flag scheme X is defined as a quotient scheme G/B . + It is known that G/B is an essentially smooth and separated (in general, not quasi- + compact) scheme over C. Let e = B /B ∈ X be the image of e ∈ G. X + + 2.2. Unipotent subgroups and Schubert varieties. Definition 2.7. Let W be the Weyl group associated with the above root datum, that is, the group generated by {s } with the defining relations s2 = e for i ∈ I and (s s )mij = e i i∈I i i j for i,j ∈ I, i 6= j. Here e is the unit of W, m = 2 (resp. 3,4,6,∞) if a a = 0 ij ij ji (resp. 1,2,3,≥ 4), and w∞ := e for any w ∈ W. We have the group homomorphisms W → Auth and W → Auth∗ given by s (h) = h−hh,α ih , s (µ) = µ−hh ,µiα i i i i i i for i ∈ I, h ∈ h andµ ∈ h∗. For an element w of W, ℓ(w) denotes the length of w, that is, the smallest integer ℓ such that there exist i ,...,i ∈ I with w = s ···s . For w ∈ W, set 1 ℓ i1 iℓ I(w) := {i = (i ,...,i ) ∈ Iℓ(w) | w = s ···s }. 1 ℓ(w) i1 iℓ(w) An element of I(w) is called a reduced word of w. TWIST AUTOMORPHISMS ON QUANTUM UNIPOTENT CELLS AND DUAL CANONICAL BASES 9 For a Weyl group element w ∈ W, we specify two lifts w,w ∈ G of w ∈ W. For a simple reflection s , set i 0 −1 s = γ = exp(−e )exp(f )exp(−e ), i i 1 0 i i i (cid:18) (cid:19) 0 1 s = γ = exp(e )exp(−f )exp(e ). i i −1 0 i i i (cid:18) (cid:19) It is known that {s } and s satisfy braid relations. It follows that the lifts w and i i∈I i i∈I w can be uniquely defined by the condition (cid:8) (cid:9) w′w′′ = w′·w′′, w′w′′ = w′·w′′ for w′,w′′ ∈ W with ℓ(w′w′′) = ℓ(w′)+ℓ(w′′). Proposition 2.8 ([FZ99, Proposition 2.1]). For w ∈ W, we have the following properties: w−1 = wT = w−1, T −1 w−1 = w = w . Notation 2.9. For a H-stable subset Y of G and w ∈ W, we denote wY (resp. Yw) by wY (resp. Yw). For w ∈ W, let us denote by N (w) = N ∩wN w−1 the group scheme associated with ± ± ∓ n (w) := g . ± α α∈∆M±∩w∆∓ In fact, N (w) are unipotent subgroups of N with dim(N (w)) = ℓ(w). ± ± ± We have the following isomorphisms N ≃ N ∩wN w−1 ×(N (w)) ± ± ± ± ≃ (N (w))× N ∩wN w−1 , (cid:0) ± ±(cid:1) ± as schemes, where N± ∩ wN±w−1 is the p(cid:0)ro-unipotent gr(cid:1)oup scheme associated with g . α∈∆±∩w∆± α NLotation 2.10. For a H-stable subset Y of X and w ∈ W˙ , we denote wY by wY. Definition 2.11. (1) For w ∈ W, we set X˚ = w(N ∩w−1N w)e ⊂ G/B be the w − + X + ˚ locally closed subscheme of X. Let X be the Zariski closure of X endowed with the w w ˚ reduced scheme structure. X (resp. X ) are called (finite) Schubert varieties (resp. w w cells). (2) For w ∈ W, we set X˚w := B we = N we ⊂ G/B be the locally closed sub- − X − X + scheme of X. Let Xw be the Zariski closure of X˚w endowed with the reduced scheme structure. Xw(resp. X˚w) are called cofinite Schubert schemes (resp. cells). (3) For w ∈ W, we set U := wB e = wN e ⊂ G/B . w − X − X + TWIST AUTOMORPHISMS ON QUANTUM UNIPOTENT CELLS AND DUAL CANONICAL BASES10 Proposition 2.12 ([KT95, Proposition 1.3.2]). (1) X is the smallest subscheme of X w that is invariant by G+’s and contains we . i X (2) There is an isomorphism N ∩wN w−1 → X˚ + − w given by x 7→ xwe . In particular X˚ is isomorphic to the affine space Aℓ(w). X w ˚ (3) We have X = X , where ≤ is the Bruhat order on W. w y≤w y We note that the mForphism (N ∩wN w−1)×B → B wB given by (x,y) 7→ xTwy − + + + + is an isomorphism in G. Remark 2.13. We note that the union X ⊂ X has a structure of an ind-scheme w∈W w over C and it is also called the flag variety. S Proposition 2.14 ([KT95, Proposition 1.3.1]). (1) There is an isomorphism N ∩wN w−1 → X˚w − − given by x 7→ xwe . In particular, X˚ is affine space with codimXw = ℓ(w). X w (2) We have X = X˚ . w∈W w (3) We have X = X˚ for w ∈ W. w F y≥w y Corollary 2.15 (BirkFhoff decomposition). We have G = w∈W B−wB+. Proposition 2.16 ([Kas89]). U is an affine open subset of X and there is an isomorphism w F N ∩wN w−1 × N ∩wN w−1 −→∼ U − − + − w which is given by (x,y) 7→ xywB . In particular, N → G/B given by n 7→ n e is an (cid:0) + (cid:1) (cid:0) − (cid:1) + − − X open immersion. 2.3. Quantized enveloping algebras. In this subsection, we present the definitions of quantized enveloping algebras. Notation 2.17. Set qn −q−n q := q(αi,αi), [n] := for n ∈ Z, i 2 q −q−1 [n][n−1]···[n−k +1] n if n ∈ Z,k ∈ Z , >0 := [k][k −1]···[1] k  (cid:20) (cid:21) 1 if n ∈ Z,k = 0,  [n]! := [n][n−1]···[1] for n ∈ Z ,[0]! := 1. >0  For a rational function R ∈ Q(q), we define R as the rational function obtained from X i by substituting q by q (i ∈ I). i Definition 2.18. ThequantizedenvelopingalgebraU associatedwith P,I,{α } ,{h } ,( , ) q i i∈I i i∈I is the unital associative Q(q)-algebra defined by the generators (cid:0) (cid:1) e ,f (i ∈ I),qh (h ∈ P∗), i i and the relations (i)-(iv) below: