CANONICAL BASES OF QUANTUM SCHUBERT CELLS AND THEIR SYMMETRIES

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1 CANONICAL BASES OF QUANTUM SCHUBERT CELLS AND THEIR SYMMETRIES ARKADY BERENSTEIN AND JACOB GREENSTEIN Abstract. The goal of ths work s to provde an elementary constructon of the canoncal bass B(w) n each quantum Schubert cell U q (w) and to establsh ts nvarance under modfed Lusztg s symmetres. To that effect, we obtan a drect characterzaton of the upper global bass B up n terms of a sutable blnear form and show that B(w) s contaned n B up and ts large part s preserved by modfed Lusztg s symmetres. Contents 1. Introducton and man results 2 2. Defnton and characterzaton of B up Prelmnares Modfed Lusztg symmetres Blnear forms Lattces and sgned bass n U q (n + ) Sgned bass s (K, µ)-orthonormal Choosng B up nsde the sgned bass 9 3. Decorated algebras and proof of Theorem Decorated algebras An somorphsm between A and A U q (n + ) as a decorated algebra A new formula for T Proof of Theorem Proofs of mans results Propertes of quantum Schubert cells Lusztg s Lemma and proof of Theorem Contanment of B() n B up and proof of Theorem Embeddngs of bases and proof of Theorem Examples Bases for repetton free elements Bases for elements wth a sngle repetton Bases for type A Bases for type C Bases for b-schubert algebras 32 References 33 Ths work was partally supported by the NSF grant DMS (A. B.), by the Smons foundaton collaboraton grant no (J. G.), and by the ERC grant MODFLAT and the NCCR SwssMAP of the Swss Natonal Scence Foundaton (A. B. and J. G.). 1

2 2 CANONICAL BASES OF QUANTUM SCHUBERT CELLS AND THEIR SYMMETRIES 1. Introducton and man results Let g = b n + be a symmetrzable Kac-Moody Le algebra. For any w n ts Weyl group W, defne the algebra U q (w) by U q (w) := T w (U q (b )) U q (n + ) (1.1) whch we refer to as a quantum Schubert cell (see 2.1 for notaton). Ths termnology s justfed n Remark 4.7. The defnton (1.1) for an nfnte (affne) type frst appeared n [1, Proposton 2.3]. In [2] we conjectured that ths defnton concdes wth Lusztg s one whch was proved for all Kac-Moody algebras by Tansak n [19, Proposton 2.10] and ndependently by Kmura ([8, Theorem 1.3]). In a remarkable paper [7] Kmura proved that each U q (w) s compatble wth the upper global bass B up of U q (n + ). The am of the present work s twofold: to construct the bass B(w) of U q (w) explctly usng a generalzaton of Lusztg s Lemma. to compute the acton of Lusztg symmetres on these bases, thus partally verfyng Conjecture 1.16 from [2]. To acheve the frst goal, frst we provde an ndependent defnton (see 2.4 and 2.6) of the global crystal bass B up (whch concdes wth the dual canoncal bass). For reader s convenence, we put all necessary defntons and results n Secton 2. Let be the ant-lnear ant-nvoluton of U q (g) whch maps q 1 2 to q 1 2 and fxes Chevalley generators. It should be noted that we use a slghtly dfferent presentaton of U q (g) (see [2] and 2.1) and accordngly modfed T w so that they commute wth ([2]). Let = ( 1,..., m ) R(w). Generalzng [6, 13], we show (see 4.1) that the A := Z[q 1 2, q 1 2 ]- subalgebra of U q (n + ) generated by the X,k := T s1 s k 1 (E k ), 1 k m of U q (n + ) s n fact ndependent of, hence s denoted U A (w), and by Lemma 4.1 has an A-bass {X a : a Z m 0} where X a = q,a X a 1,1 Xam,m and q,a q 1 2 Z s defned n (4.1). The mportance of ths choce of the q,a s hghlghted by the followng verson of Lusztg s Lemma. Theorem 1.1. Let w W and R(w). For every a Z m 0 there exsts a unque b a = b,a U A (w) such that b a = b a and b a X a a a q 1 Z[q 1 ]X a. We prove Theorem 1.1 n 4.2. In partcular, elements b,a, a Z m 0 form a bass B() of U A (w) whch a pror depends on. However, the followng result mples that ths s not the case. Theorem 1.2. Let w W and R(w). Then for all a Z l(w) 0 we have b,a B up. Theorem 1.2 s proved n 4.3. It mples that for any, R(w) we have B() = B( ) and thus can ntroduce B(w). As a consequence, we recover the man result (Theorem 4.22) of [7]. Corollary 1.3. B(w) = B up U q (w) for all w W. Remark 1.4. To obtan hs result, Kmura used a rather elaborate theory of global crystal bases. By contrast, our proofs of Theorems 1.1 and 1.2 are qute elementary and short. Now we turn our attenton to the second goal, that s, to the acton of Lusztg s symmetres on U q (w). Theorem 1.5 ([2, Conjecture 1.17]). Let w, w W be such that l(ww ) = l(w) + l(w ). Then B(w) B(ww ), T w (B(w )) B(ww ). Remark 1.6. (a) In [2] we constructed a bass B g of U q (g) contanng B up and conjectured ([2, Conjecture 1.16]) that T w (B up ) B g. Thus, Theorem 1.5 provdes supportng evdence for that conjecture.

3 CANONICAL BASES OF QUANTUM SCHUBERT CELLS AND THEIR SYMMETRIES 3 (b) It would be nterestng to compare the symmetres dscussed above wth the quantum twst computed n [9]. We deduce Theorem 1.5 from Theorem 1.1 n 4.4. All these results are obtaned usng the followng strkng property of B up whch s parallel to a hghly non-trval result of Lusztg ([17]). Theorem 1.7. T s (b) B up whenever b B up T 1 s (U q (n + )). We prove Theorem 1.7 n 3.5. Our proof, whch s qute elementary and short, reles on the noton of decorated algebras (Defnton 3.1) to whch we generalze T s and obtan an explct formula (Theorem 3.6) for t. We conclude ths secton wth the followng curous applcaton of the above constructons. It s well-known (see e.g. Remark 2.14) that the natural lnear ant-nvoluton on U q (n + ) fxng the Chevalley generators (see 2.1) preserves B up. Snce T w = T 1 w (cf. [2]) t follows that 1 U q (w) = T 1 w 1 (U q (b )) U q (n + ) and Corollary 1.3 mples that U q (w) has a bass B(w) = U q (w) B up. In partcular, one can consder the algebras U q (w, w ) := U q (w) U q (w ), w, w W whch s natural to call b-schubert algebras. The followng s mmedate. Corollary 1.8. For any w, w W, the b-schubert algebra U q (w, w ) has a bass B(w, w ) := B(w) B(w ) = U q (w, w ) B up. Remark 1.9. (a) Based on numerous examples (see 5.5) one can conjecture that b-schubert algebras are Poncaré-Brkhoff-Wtt (PBW),.e. have bases consstng of ordered monomals (smlarly to Lemma 4.1). (b) One can also consder ntersectons U q (w) U q (w ); however, n ths case t appears (and s probably well-known) that the correspondng algebra s always U q (w ) where w s less than both w and w n the weak rght Bruhat order and s maxmal wth that property. Acknowledgements. The man part of ths paper was wrtten whle both authors were vstng Unversté de Genève (Geneva, Swtzerland). We are happy to use ths opportunty to thank A. Alekseev for hs hosptalty. We also benefted from the hosptalty of the Max-Planck-Insttut für Mathematk (Bonn, Germany), whch we gratefully acknowledge. 2. Defnton and characterzaton of B up 2.1. Prelmnares. Let g be a symmetrzable Kac-Moody algebra wth the Cartan matrx A = (a j ),j I. Let {α } I be the standard bass of Q = Z I. Fx d Z >0, I such that the matrx (d a j ),j I s symmetrc and defne a symmetrc blnear form (, ) : Q Q Z by (α, α j ) = d a j ; clearly, (γ, γ) 2Z for any γ Q. We wll wrte (α, γ), γ Q as an abbrevaton for (α, γ)d 1. The quantzed envelopng algebra U q (g) s an assocatve algebra over k = Q(q 1 2 ) generated by the E, F, K ±1, I subject to the relatons [E, F j ] = δ j (q 1 q )(K K 1 ), K E j = q a j E j K, K F j = q a j F j K, K K j = K j K (2.1) ( 1) s E r E j E s = ( 1) s F r F j F s = 0, j (2.2) r,s 0, r+s=1 a j r,s 0, r+s=1 a j

4 4 CANONICAL BASES OF QUANTUM SCHUBERT CELLS AND THEIR SYMMETRIES for all, j I, where q = q d, X k (n) v = n v 1 v, n v! = := ( k ) 1X k s q s=1 n t v, (n) v! = n v!, 1 n v t=1 and s v = v s v s. We also set ( ) n = k v k 1 t=0 (n t) v (k) v! = k 1 t=0 n t v k v! and X (n) := X n /(n) q!. We denote by U q (n + ) (respectvely, U q (n )) the subalgebra of U q (g) generated by the E (respectvely, the F ), I. Let K be the subalgebra of U q (g) generated by the K ±1, I and set U q (b ± ) = KU q (n ± ). It s easy to see from the presentaton that U q (g) admts ant-nvolutons t and, where t nterchanges E and F for each I and preserves the K ±1 whle preserves the E and F whle K = K 1. Furthermore, U q (g) admts an ant-lnear ant-nvoluton whch preserves all generators and maps q 1 2 to q 1 2. The algebra U q (n + ) s naturally graded by Q + := I Z 0α va deg E = α. We denote the homogeneous component of U q (n + ) of degree γ Q + by U q (n + ) γ. Ths can be extended to a Q-gradng on U q (g) va deg F = α, deg K = Modfed Lusztg symmetres. Let W be the Weyl group of g. It s generated by the smple reflectons s, I whch act on Q va s (α j ) = α j a j α. Gven w W, denote by R(w) the set of reduced words for w, that s, the set of = ( 1,..., m ) I m of mnmal length m := l(w) such that w = s 1 s m. It s well-known that the form (, ) s W -nvarant. The followng essentally concdes wth Theorem 1.13 from [2]. Lemma 2.1. (a) For each I there exsts a unque automorphsm T of U q (g) whch satsfes T (K j ) = K j K a j and q 1 K 1 F, = j T (E j ) = ( 1) r q s+ 1 2 a j E r E j E s, j r+s= a j q 1 K E, = j T (F j ) = ( 1) r q s+ 1 2 a j F r F j F s, j r+s= a j (b) For all x U q (g), T (x) = T (x), (T (x)) = T 1 (x ) and (T (x)) t = T 1 (x t ). (c) The T, I satsfy the brad relatons on U q (g), that s, they defne a representaton of the Artn brad group Br g of g on U q (g) Blnear forms. Followng [2, 16], we defne a symmetrc blnear form, on U q (n + ). Let V = I ke and let, be the blnear form on V defned by E, E j = δ j (q q 1 ). Extend t naturally to T (V ) va k v 1 v k, v 1 v l = δ kl v r, v r, v r, v r V, 1 r k. r=1 Defne a lnear map Ψ : V V V V by Ψ(E E j ) = q (α,α j ) E j E. Fnally, defne, Ψ va u, v Ψ = δ k,l [k] Ψ!(u), v = δ k,l u, [k] Ψ!(v), u V k, v V l and [k] Ψ! End k V k s the standard notaton for the braded k-factoral (see e.g. [2, A.1]). It s well-known (see e.g. [16]) that the kernel J of the canoncal map T (V ) U q (n + ), E E s the radcal of, Ψ. Thus, we have a well-defned non-degenerate symmetrc blnear form,

5 CANONICAL BASES OF QUANTUM SCHUBERT CELLS AND THEIR SYMMETRIES 5 on U q (n + ) gven by u + J, v + J = u, v Ψ. Ths form n fact concdes wth the form, we ntroduced n [2, A.3] f we dentfy U q (n ) wth U q (n + ) va t. We wll often use the followng obvous Lemma 2.2. Let x, x U q (n + ) be homogeneous. Then x, x 0 mples that deg x = deg x. Defne, : U q (n + ) U q (n + ) k by where x, y = µ(γ)q 1 2 (γ,γ) x, y, x, y U q (n + ) γ µ(γ) = q 1 4 (γ,γ)+ 1 2 η(γ), γ Q (2.3) and η Hom Z (Q, Z) s defned by η(α ) = d. Note the followng propertes of µ whch wll be often used n the sequel µ(rα ) = q (r+1 2 ), µ(s γ) = µ(γ)q 1 2 (α,γ), µ(γ + γ ) = µ(γ)µ(γ )q 1 2 (γ,γ ) Defne an ant-lnear automorphsm of U q (g) by x = (sgn γ) x, x U q (g) γ where sgn : Q {±1} s the homomorphsm of abelan groups defned by sgn(α ) = 1. Then (cf. [2]) x, y = x, ỹ = x, ȳ. (2.5) 2.4. Lattces and sgned bass n U q (n + ). Let A = Z[q 1 2, q 1 2 ] whch s a subrng of Q(q 1 2 ). Denote A 0 = Z[q, q 1 ] and A 1 = q 1 2 A 0 ; clearly, A = A 0 A 1 as an A 0 -module. Followng [2, 3.1], for any J I, let U Z (n + ) J (respectvely, U Z (n ) J ) be the A 0 -subalgebra of U q (n + ) (respectvely, U q (n )) generated by the E n U Z (n + ) I. Set Clearly, U Z (n + ) s an A 0 -submodule of U q (n + ). (2.4) (respectvely, F n ), J, n Z 0. We abbrevate U Z (n ± ) := U Z (n + ) = {x U q (n + ) : x, U Z (n + ) A 0 }. Lemma 2.3. We have q 1 2 (γ,γ ) xx U Z (n + ) for all x U Z (n + ) γ, x U Z (n + ) γ. In partcular, all powers of a homogeneous element of U Z (n + ) are n U Z (n + ) and U A (n + ) := U Z (n + ) A0 A s an A-algebra. Proof. Followng [16, 1.2], let : U q (n + ) U q (n + ) U q (n + ) be the braded co-multplcaton defned by (E ) = E E, where U q (n + ) U q (n + ) = U q (n + ) U q (n + ) endowed wth an algebra structure va (x y)(x y ) = q (γ,γ ) xx yy for all x, y U q (n + ), y U q (n + ) γ, x U q (n + ) γ. Then xx, y = x, y (1) x, y (2) and so xx, y = q 1 2 (γ,γ ) x, y (1) x, y (2), x U q (n + ) γ, x U q (n + ) γ, where (y) = y (1) y (2) n Sweedler-lke notaton. It follows from [16, Lemma 1.4.2] that (U Z (n + )) U Z (n + ) A0 U Z (n + ), hence we can assume that y (1), y (2) U Z (n + ) provded that y U Z (n + ). All assertons are now mmedate. Defne, for any γ Q + and set B ±up = γ Q + B ±up γ. B ±up γ = {b U Z (n + ) γ : b = b, µ(γ) 1 b, b 1 + q 1 Z[[q 1 ]])} (2.6)

6 6 CANONICAL BASES OF QUANTUM SCHUBERT CELLS AND THEIR SYMMETRIES 2.5. Sgned bass s (K, µ)-orthonormal. Let R be a commutatve untal subrng of a feld k. Followng [16, ], a subset B ± of a free R-module L s a sgned bass of L f B ± = B ( B) for some bass B of L. Let K be a subrng of k not contanng 1. We say that B s (K, µ)-orthonormal for some µ R wth respect to a fxed symmetrc blnear parng, : L R L k f µ b, b δ b,b + K, b, b B. Accordngly, we say that a sgned bass B ± s (K, µ)-orthonormal f t contans a (K, µ)- orthonormal bass of L. The followng result s parallel to [16, Theorem ]. Theorem 2.4. B ±up s a sgned bass of U Z (n + ) wth R = A 0. Moreover, for each γ Q +, B ±up γ s a (K, µ(γ) 1 )-orthonormal bass where µ s defned by (2.3) and K = q 1 Z[[q 1 ]] Q(q). Proof. We need the followng general setup. We say that a doman R 0 s strongly ntegral f a sum of squares of ts non-zero elements s never zero and f c c 2 n = 1, c r R 0, mples that for all 1 n, c = ±δ j for some 1 j n. Let R be a doman wth a subdoman R 0. Gven a totally ordered addtve monod Γ, a map ν : R Γ { } s called an R 0 -lnear valuaton f the followng hold for all f, g R (V 1 ) ν(f) = f and only f f = 0 (V 2 ) ν(r 0 \ {0}) = 0, (V 3 ) ν(fg) = ν(f) + ν(g); (V 4 ) ν(f + g) max(ν(f), ν(g)). It follows that ν(f) ν(g) = ν(f + g) = max(ν(f), ν(g)). (2.7) Furthermore, for each a Γ, set R a = {r R : ν(r) a} and R <a = {r R : ν(r) < a}. Clearly, R a and R <a are R 0 -submodules of R and R <a R a. In the sprt of [12, 2.1], we call the R 0 -module R a /R <a the leaf of ν at a; we say that ν has one-dmensonal leaves f for each a ν(r), the leaf of ν at a s a non-zero cyclc R 0 -module. Let M be a free R-module wth a bass B. Then we can defne ν B : M Γ { } by ν B ( b B c b b) = max ν(c b ). (2.8) b Clearly (V 4 ) holds and we also have ν B (fx) = ν(f) + ν B (x), f R, x M. We wll need the followng Lemma. Lemma 2.5. Suppose that ν : R Γ { } has one-dmensonal leaves and let M be a free R module wth a bass B. Then every x M wth ν B (x) > 0 can be wrtten as x = fx 0 + x 1 where f R wth ν(f) = ν(x), 0 x 0 b B R 0b and x 1 M satsfes ν B (x 1 ) < ν B (x). Proof. Let x M wth a = ν B (x) > 0 and wrte x = x b b = b B b B : ν(x b )=a x b b + b B : ν(x b )<a Snce ν has one-dmensonal leaves and R a R <a, R a /R <a s a non-zero cyclc R 0 -module. Let f R a be any element whose mage generates R a /R <a as an R 0 -module. Then ν(f) = a and for every b B wth ν(x b ) = a there exsts r b R 0 such that ν(x b r b f) < a. Set x 0 = r b b, x 1 = x fx 0. Clearly, ν B (x 1 ) < a, whence x 0 0. b B : ν(x b )=a x b b.

7 Henceforth CANONICAL BASES OF QUANTUM SCHUBERT CELLS AND THEIR SYMMETRIES 7 R 0 s a strongly ntegral doman k s a feld contanng R 0 ; R 0 R k as subrngs ν : k Γ { } s an R 0 -lnear valuaton; K s an R 0 -subalgebra of k such that ν(f) < 0 for all f K (note that ths mples that K R 0 = ) and (1 + K ) K. There s a feld nvoluton of k whch restrcts to R and s dentty on R 0, whle K K = ν(r \ R 0 ) > 0 where R = {f R : f = f}; The restrcton of ν to R s a valuaton ν : R Γ { } wth one-dmensonal leaves; For an R-module L, an endomorphsm of Z-modules ϕ : L L s called ant-lnear f for all r R, x L we have r x = r x. Ant-lnear endomorphsms of a k-vector space V are defned smlarly. Let V be a k-vector space wth a non-degenerate symmetrc blnear form,. Suppose that ϕ, ϕ are ant-lnear nvolutons on V satsfyng x, y = ϕ(x), ϕ (y), x, y V. Let L be a free R-module such that V = k R L. Denote L = {x V : x, L R}. Clearly, L s a free R-module and V = k R L. Gven µ R defne B ± (µ) = {b L : ϕ (b) = b, µ b, b 1 + K } B ±(µ) = {b L : ϕ(b) = b, µ b, b 1 + K }. Proposton 2.6. Suppose that dm k V <. The followng are equvalent (a) B ± (µ) s a (K, µ)-orthonormal sgned R-bass of L. (b) B ±(µ 1 ) s a (K, µ 1 )-orthonormal sgned R-bass of L. In that case, B ± (µ) and B ±(µ) are dual to each other wth respect to,. Proof. (a) = (b) Let B ± (µ) be any bass of L contaned n B ± (µ). Snce, s non-degenerate, for each b B ± (µ) there exsts a unque δ b L such that δ b, b = δ b,b. Clearly, the set B ± (µ) := {δ b : b B ± (µ)} s a bass of L. Note that ϕ(δ b ) = δ b. Lemma 2.7. B ± (µ) s a (K, µ 1 )-orthonormal bass of L. In partcular, ν(µ 1 δ b, δ b ) 0 wth the equalty holdng f and only f b = b. Proof. We need the followng Lemma 2.8. Let G = (G r,s ) 1 r,s n be a matrx over k such that µg rs δ rs + K. Then G s nvertble and M = (M rs ) 1 r,s n = G 1 satsfes µ 1 M rs δ rs + K. Proof. Let r,s (G) be the mnor of G obtaned by removng the rth row and the sth column. Then t s easy to see that µ n 1 r,s (G) δ rs + K. Smlarly, µ n det G 1 + K hence G s nvertble. Moreover, µ n (det G) K. Snce M rs = ( 1) r+s (det G) 1 s,r (G), the asserton follows. Snce B ± (µ) s (K, µ)-orthogonal, the above Lemma apples to the (fnte) Gram matrx G = ( b, b ) b,b B ± (µ) of, wth respect to the bass B ± (µ) and hence µ 1 M b,b δ b,b + K where M = G 1. Snce µ 1 δ b = b B ± (µ) µ 1 M b,b b, we have µ 1 δ b b + K B ± (µ). Ths mples that µ 1 δ b, δ b = µ 1 δ b, δ b b, δ b + K B ± (µ), δ b = δ b,b + K, b, b B ± (µ) Ths proves Lemma 2.7.

8 8 CANONICAL BASES OF QUANTUM SCHUBERT CELLS AND THEIR SYMMETRIES Note that for any x = b x bδ b, y = b y b δ b n L we have µ 1 x, y = b,b x b y b µ 1 δ b, δ b. Defne ν = ν B ± (µ) : L Γ { } as n (2.8). Snce ν(µ 1 δ b, δ b ) 0 for all b, b B ± (µ) by Lemma 2.7, t follows from (V 3 ) and (V 4 ) that Clearly, for x L we have ν(µ 1 x, y ) ν(x) + ν(y), x, y L. (2.9) ϕ(x) = x x b B ± (µ) R δ b. (2.10) Thus, the set (L ) ϕ of ϕ-nvarant elements n L s a free R -module wth a bass B ± (µ). The followng Lemma s the crucal pont of our argument. Lemma 2.9. Let x L and suppose that ϕ(x) = x. Then (a) If ν(x) = 0, that s, x = b x bδ b wth x b R 0, then µ 1 x, x b x2 b K and ν(µ 1 x, x ) = 0. (b) If ν(x) > 0 then ν(µ 1 x, x ) > 0 Proof. Wrte x = b x bδ b, x b R. To prove (a), note that ν(x) = 0 and ϕ(x) = x mply that x b R 0 for all b B ± (µ). We have µ 1 x, x = b x 2 bµ 1 δ b, δ b + b b x b x b µ 1 δ b, δ b. By Lemma 2.7, the frst sum belongs to b x2 b + K whle the second sum belongs to K. Snce R 0 s strongly ntegral, b x2 b 0. Thus, ν(µ 1 x, x ) = 0. To prove (b), let a = ν(x) > 0. Applyng Lemma 2.5 to M = (L ) ϕ and the rng R, we can wrte x = fx 0 + x 1 where f R, ν(f) = a, ϕ(x 0 ) = x 0 (and so ϕ(x 1 ) = x 1 ), ν(x 0 ) = 0 and ν(x 1 ) < a. Then ν(µ 1 x, x ) = ν(f 2 µ 1 x 0, x 0 + 2fµ 1 x 0, x 1 + µ 1 x 1, x 1 ) = ν(f 2 µ 1 x 0, x 0 ) = 2a > 0 snce ν(µ 1 x 1, x 1 ), ν(fµ 1 x 0, x 1 ) < 2a by (2.9) and (V 3 ), (V 4 ) whle ν(µ 1 x 0, x 0 ) = 0 by part (a). Ths proves (b). It follows from Lemma 2.9(a,b) that f x L s fxed by ϕ and µ 1 x, x 1+K then x = ±δ b for some b B ± (µ) by the strong ntegralty of R 0. Thus, B ±(µ) = B ± (µ) ( B ± (µ) ). Ths completes the proof of the mplcaton (a) = (b) and the last asserton. The opposte mplcaton follows by the symmetry between L and L and ϕ and ϕ. We now apply Proposton 2.6 wth L = U Z (n + ) γ, v = q 1 2, ϕ =, ϕ =, R = A 0 = Z[v 2, v 2 ], k = Q(v) and K = v 2 Z[[v 2 ]] Q(v). We defne ν : Q(v) Z { } va ( ν cv n 1 + f ) = n 1 + g where c Q, n Z and f, v v 1 Z[v 1 ]. Note that R = Z[q + q 1 ] and ν has one-dmensonal leaves on R snce ν((v + v 1 ) n ) = n. By [16, Theorem ], B ±can U Z (n + ) s a (K, µ(γ))- orthonormal sgned bass of L. Snce B ±up γ = (B can U Z (n + )) ± n the notaton of Proposton 2.6, t s a sgned (K, µ(γ) 1 )-orthonormal bass of L = U Z (n + ) γ. Ths completes the proof of Theorem 2.4.

9 CANONICAL BASES OF QUANTUM SCHUBERT CELLS AND THEIR SYMMETRIES Choosng B up nsde the sgned bass. It remans to descrbe a canoncal way to choose B up nsde B ±up. Needless to say, t can be taken as the dual bass of B can wth respect to,. However, t more nstructve to provde an ntrnsc defnton. To that effect, followng [2, 3.5] and also [16, Proposton 3.1.6], defne k-lnear endomorphsms, op, I of U q (n + ) by [F, x] = (q q 1 )(q 1 2 (α,γ α ) K (x) q 1 2 (α,γ α ) K 1 op (x)), x U q (n + ) γ. (2.11) We need the followng propertes of these operators (cf. [2, Lemmata 3.18 and 3.20]). Lemma For all x U q (n + ) γ and I we have (a) (x) = (x), op (x) = op (x), (x ) = op (x) and op (x) = op (x). (b) for all y U q (n + ), n Z 0 where f (n) (c), op = (q q 1 x, ye n = (n) ) n f n. (x), y, x, E n y = ( op ) (n) (x), y, are quas-dervatons. Namely, for x U q (n + ) γ, y U q (n + ) γ we have It s easy to see that whence op (xy) = q 1 2 (α,γ ) (x)y + q 1 2 (α,γ) x (y), (xy) = q 1 2 (α,γ ) op (x)y + q 1 2 (α,γ) x op (y). (n) (E r ) = (2.12) ( ) r E r n = ( op ) (n) (E r ) (2.13) n q ((q q 1 The followng s an mmedate consequence of ths dentty and Lemma ) ) n (E r ) = E r n = ((q q 1 Corollary For all I we have (a) f x U Z (n + ) γ then 1 q (x), 1 q op (x) q 1 2 (α,γ) U Z (n + ); (b) the (n), ( op ) (n), n Z 0 restrct to operators on U Z (n + ). ) op ) n (E r ) (2.14) By degree consderatons t s clear that, op are locally nlpotent, that s, for any x U q (n + ) we have k (x) = ( op ) k (x) = 0 for k 0. Thus, for each x U q (n + ) \ {0} we can defne l (x) as the maxmal k > 0 such that k (x) 0. Defne (top), ( op ) (top) : U q (n + ) \ {0} U q (n + ) \ {0} by (top) (x) = (l (x)) (x) and ( op ) (top) (x) = ( (top) (x )) = ( op ) (l (x )) (x). Smlar notaton wll be used for other locally nlpotent operators n the sequel. For any sequence = ( 1,..., m ) I m set (top) = (top) m (top) 1. Proposton For every b B ±up there exsts = ( 1,..., m ) such that (top) (b) {±1}. Moreover, f = ( 1,..., m ) also satsfes (top) (b) {±1} then (top) (b) = (top) (b) {±1}. Thus, we can defne B up to be the set of all b B ±up such that (top) (b) = 1 for some = ( 1,..., m ). Proof. By Proposton 2.6, B ±up contans the dual bass B of B can. Our goal s to prove that B up = B. We need the followng result. Lemma (top) (b) B for all b B, I. Moreover, f (top) (b) = (top) (b ) and l (b) = l (b ) for some b B then b = b.

10 10 CANONICAL BASES OF QUANTUM SCHUBERT CELLS AND THEIR SYMMETRIES Proof. Followng [16, 14.3], denote B can ; r = B can E r U q (n + ) and B can ;r = B can ; r \ B can ; r+1. It follows from [16, 14.3] that for all I, B can = r 0 B can ;r. (2.15) Let b B can and let n = l (δ b ), u = (top) (δ b ) = (n) (δ b ), where δ b s the element of B satsfyng δ b, b = δ b,b. Then u ker whch, by Lemma 2.10(c), s orthogonal to B can ;s, s > 0. Thus, we can wrte u = b B can ;0 u, b δ b = b B can ;0 δ b, E n b δ b. By [16, Theorem ], for each b B can ;0 there exsts a unque π,n (b ) B can ;n such that E n b π ;n (b ) r>n Z[q, q 1 ]B can ;r. Usng Lemma 2.10(c) agan, we conclude that for any b B can ;r wth r > n, δ b, b δ b, E r U q (n + ) = (r) (δ b ), U q (n + ) = 0. Thus, u = b B can ;0 δ b, π ;n (b ) δ b. Note that, snce u 0, we cannot have δ b, π ;n (b ) = 0 for all b B can ;0. Snce δ b, b = δ b,b, we conclude that there exsts a unque b B can ;0 such that π ;n (b ) = b and then u = (top) (δ b ) = δ b. Snce π ;n : B can ;0 B can ;n s a bjecton by [16, Theorem ], the frst asserton follows. The second asserton s mmedate from (2.15). Ths mples that for every element b B, there exsts = ( 1,..., m ) such that (top) (b) = 1. Snce 1 s the unque element of B of degree 0, for any sequence such that (top) (b) = c k, one has c = 1. Ths completes the proof of Proposton Remark Snce B can s preserved by by [16, Theorem ] and s self-adjont wth respect to,, t follows that B up s preserved by. In partcular, we can replace by op n Lemma 2.13 and Proposton Note that Lemma 2.13 and Remark 2.14 mmedately yeld the followng well-known fact. Corollary Let x U q (n + ) and wrte x = b B c up b (x)b. Then c b (x) 0 mples that l (b) l (x) and l (b ) l (x ) and (top) (x) = c b (x) (top) (b), ( op ) (top) (x) = c b (x)( op ) (top) (b) b B up : l (b)=l (x) b B up : l (b )=l (x ) are the decompostons of ( ) (top) (x) and ( op ) (top) (x), respectvely, n the bass B up. 3. Decorated algebras and proof of Theorem Decorated algebras. Let A be an assocatve Z-graded algebra over k = Q(v 1 2 ). Denote the degree of a homogeneous u A by u. Defnton 3.1. We say that A = A(E, F +, F ) s decorated f t contans an element E wth E = 2 and admts mutually commutng locally nlpotent k-lnear endomorphsms F +, F : A A of degree 2 satsfyng F ± (E) = 1 and for x, y A homogeneous. F ± (xy) = v ± 1 2 y F ± (x)y + v 1 2 x xf ± (y) (3.1)

11 CANONICAL BASES OF QUANTUM SCHUBERT CELLS AND THEIR SYMMETRIES 11 Denote A ± = ker F and set A 0 = A + A. Clearly, F ± restrcts to an endomorphsm of A ± whch wll also be denoted by F ±. Snce F ± are skew dervatons, A ± are subalgebras of A. The followng s a basc example of a decorated algebra. Let F m,n = k E, x, y wth the Z-gradng defned by x = m, y = n, E = 2. The followng s mmedate Lemma 3.2. There exsts unque operators F ± End k F m,n such that F ± (E) = 1, F ± (x) = F ± (y) = 0 and (3.1) holds. In partcular, F m,n s a decorated algebra and for any decorated algebra A and any x, y A 0 homogeneous the natural homomorphsm of graded assocatve algebras φ x, y : F x, y A, x x, y y s a homomorphsm of decorated algebras, that s, t commutes wth F ±. Defne K 1 2, E± End k A by K (x) = v 2 x u, E ± (x) = ± 1 1 (v ± 1 2 x Ex v 1 2 x xe), for x A homogeneous. Clearly E ± are of degree 2. The followng s easly checked. Lemma 3.3. (a) For x, y A homogeneous we have E (r) + (x) = ( 1) r v 1 2 (r+ x 1)(r r ) E r xe r E (r) (x) = r +r =r r +r =r E (r) ± (xy) = r +r =r v ( 1) r v 1 2 (r+ x 1)(r r ) E r xe r (3.2) v ± 1 2 (r y r x ) E (r ) ± (x)e (r ) ± (y). (3.3) and [E ±, F ± ] = 1 1 v (K K 1 ), [E ±, F ] = 0. In partcular, E ±, F ± and K provde actons of Chevalley generators of U v (sl 2 ) n ts standard presentaton on A; (b) A ± s a U v (sl 2 ) ± -module subalgebra of A (c) Any homomorphsm of decorated algebras A A commutes wth each of U v (sl 2 ) + - and U v (sl 2 ) -actons. Remark 3.4. Suppose that y A 0. The followng s rather standard an s an obvous consequence of say [16, Corollary 3.1.9]. ( ) a b y F (a) ± E (b) E (b a) ± (y), 0 a b ± (y) = a v (3.4) 0, a > b Suppose that E ± are locally nlpotent on A ±. Then A ± are drect sums of fnte dmensonal U v (sl 2 ) ± -modules and f x A 0 s homogeneous then x 0. We need followng Lemma 3.5. (a) There exsts unque somorphsms of U v (sl 2 )-modules σ ± : A ± A such that σ ± A0 = d A0, where A ± s regarded as a U v (sl 2 ) ± -module. In partcular, σ ± σ = d A. (b) There exsts unque k-lnear nvoluton η ± : A ± A ± such that η ± E ± = F ± η ±, η ± F ± = E ± η ±, η ± K = K 1 η ± (3.5) and η ± (x) = E (top) ± (x) = E ( x ) ± (x) for x A 0 homogeneous. Proof. Part (a) s mmedate from the sem-smplcty of A ± as U v (sl 2 ) ± -modules and the fact that any endomorphsm of any lowest weght U v (sl 2 )-module fxng all lowest weght vectors s dentty on that module. To prove (b) recall that every smple fnte dmensonal U v (sl 2 )-module V λ of type 1 has a bass {z k } 0 k such that E(x k ) = (k) v z k 1, F (z k ) = (λ k) v z k+1, K(z k ) = v λ 2k z k. Then t s easy to see that η λ End k V λ defned by η(z k ) = z λ k s the unque lnear map satsfyng (3.5)

12 12 CANONICAL BASES OF QUANTUM SCHUBERT CELLS AND THEIR SYMMETRIES and such that η λ (z) = E (λ) (z) for any lowest weght vector z of V λ. It remans to observe that η λ can be extended unquely to any sem-smple U v (sl 2 )-module An somorphsm between A and A +. The followng s qute surprsng. Theorem 3.6. Let A be a decorated algebra such that the operators E ± are locally nlpotent on A ±. Then the map τ := η + σ : A A + s an somorphsm of algebras. Proof. Gven m, n Z 0, defne, for all 0 r mn(m, n), t, t Z 0 C r;t,t (v) = v 1 2 (mt nt ) ( 1) l v lt kt (k l)(1+m+n r) (n l) ( ) ( ) v!(m k) v! t t (n r) v!(m r) v! k l k,l Z 0, k+l=r In partcular, for t, t Z 0 such that t + t mn(m, n) we have C t +t ;t,t (v) = ( 1)t v 1 2 (mt nt +(t +t 1)(t t )) (m t ) v!(n t ) v! (n t t ) v!(m t t ) v!. v. (3.6) v These elements of Z[v, v 1 ] are essentally Clebsch-Gordan coeffcents or 3j-symbols (cf. [11, Chapter VII]) and play an mportant role n the representaton theory of U v (sl 2 ) due to the followng fact, whch s easy to check by drect computatons. Lemma 3.7. Let m, n Z 0 and let V, V be U v (sl 2 )-modules and let x V, y V be lowest weght vectors of type 1 of respectve lowest weghts m and n,.e. K(x) = v m x and K(y) = v m y. For a Z 0 and 0 r mn(m, n) set u (a,r,±) (x, y) := (±1) r C r;t,t (v±1 )E (t ) (x) E (t ) (y) V V. (3.7) t +t =a+r Then (a) u (0,r,±) (x, y) s a lowest weght vector n V ± V whch s V V endowed wth a U v (sl 2 )- module structure va the comultplcaton ± : U v (sl 2 ) U v (sl 2 ) U v (sl 2 ) defned by ± (E) = E K ± K 1 2 E, ± (F ) = F K ± K 1 2 F and ± (K 1 2 ) = K 1 2 K 1 2 (b) u (a,r,±) (x, y) = E (a) (u (0,r,±) (x, y)). We now establsh some propertes of the C r;t,t (v). Proposton 3.8. There exst C ± r;k,l (v) k, 0 k m, 0 l n, 0 r mn(m, n, k + l) such that mn(m,n,k+l) r=0 (±1) r C± r;k,l (v)c r;t,t (v±1 ) = δ k,t δ l,t. (3.8) Proof. Let V m denote the smple (m + 1)-dmensonal U v (sl 2 )-module of type 1 and let u m be ts lowest weght vector. Then the u (l) m := E (l) (u m ), 0 l m, form a bass of V m. For 0 r mn(m, n), 0 a m+n 2r, defne u (a,r,±) m, n V m ± V n by u (a,r,±) m, n := u (a,r,±) (u m, u n ) n the notaton of Lemma 3.7. Then for any 0 k m, 0 l n, 0 r mn(m, n, k + l) we have by Lemma 3.7 u (k+l r,r,±) m, n = (±1) r C r;t,t (v±1 )u (t ) m u (t ) n. (3.9) On the other hand, snce t,t Z 0, t +t =k+l V m ± V n = 0 r mn(m,n) V m n+2r, (3.10)

13 CANONICAL BASES OF QUANTUM SCHUBERT CELLS AND THEIR SYMMETRIES 13 t follows from Lemma 3.7 that {u (a,r,±) m, n : 0 r mn(m, n), 0 a m + n 2r} s a bass of V m ± V n compatble wth the decomposton (3.10). Then for all 0 k m, 0 l n, and 0 r mn(m, n, k + l) there exst C ± r;k,l (v) k such that u (k) m u (l) m = 0 r mn(k+l,m,n) C ± r;k,l (v)u(k+l r,r,±) m, n. (3.11) Snce {u (t ) m u (t ) n : 0 t m, 0 t n} s also a bass of V m ± V n, (3.9) and (3.11) yeld (3.8). We also need the followng result whch s probably well-known. Proposton 3.9. For all 0 t m, 0 t n, 0 r mn(m, n) we have C r;m t,n t (v) = ( 1)r C r;t,t (v 1 ). (3.12) Proof. The followng Lemma s readly obtaned by applyng E (respectvely, F ) to both sdes of (3.7) and usng Lemma 3.7(b). Lemma The C r;t,t (v) wth 0 t m, 0 t n and 0 r mn(m, n) are determned by the recurrence relaton (t + t r) v C r;t,t (v) = vt 1 2 n (t ) v C r;t 1,t (v) + v 1 2 m t (t ) v C r;t,t 1(v) (3.13) wth the ntal condtons ( C r;t,0(v) = v 1 2 (r(1+m+n r) nt ) (n) v! t (n r) v! r Furthermore, C r;0,t (v) = ( 1) r v 1 2 (mt r(1+m+n r)) (m) v! (m r) v! ), v ( ) t (m + n r t t ) v C r;t,t (v) = vt 1 2 n (m t ) v C r;t +1,t (v) r. v + v 1 2 m t (n t ) v C r;t,t +1(v). We argue by nducton on n t. Usng (3.15) wth t = n we obtan whence by (3.14) C r;t,n(v) = v 1 2 t n (m r) v!(m t ) v! (m) v!(m r t ) v! C r;0,n(v) C r;t +1,n(v) = (m r t ) v v 1 (m t 2 n C r;t ),n(v), v = ( 1) r v 1 2 (n(m t ) r(1+m+n r)) (n) v! (n r) v! ( ) m t r v (3.14) (3.15) = ( 1) r C r;m t,0(v 1 ). Thus, (3.12) holds for all 0 t m and for t = n. Suppose that (3.12) was establshed for all 0 t m and for all s + 1 t n. To prove the nductve step, we use nducton on m t. We have by (3.14) ( 1) r (n r s) v C r;m,s (v 1 ) = ( 1) r C r;m,s+1 (v 1 )(n s) v v 1 2 m = v 1 2 m (n s) v C r;0,n s 1 (v) = ( 1) r v 1 2 (m(n s) r(1+m+n r)) (m) ( ) v!(n s) v n s 1 = (n r s) v C r;0,n s (v). (m r) v! r v

14 14 CANONICAL BASES OF QUANTUM SCHUBERT CELLS AND THEIR SYMMETRIES Ths proves (3.12) for t = m and t = s. Assume that (3.12) s establshed for k + 1 t m and for s t m. Then usng (3.15) and (3.13) we obtan ( 1) r (m+n r k s) v C r;k,s (v 1 ) = ( 1) r C r;k+1,s (v 1 )(m k) v v s+ 1 2 n + ( 1) r C r;k,s+1 (v 1 )(n s) v v 1 2 m+k = C r;m k 1,n s (v)(m k) v v s+ 1 2 n + C r;m k,n s 1 (v)(n s) v v 1 2 m+k = (m + n r k s) v C r;m k,n s (v). Thus, (3.12) holds for all 0 t m and for all s t m, whch completes the proof of Proposton 3.9. Let x, y A 0 be homogeneous and let m = x, n = y. For r 0, defne x r y A by x r y = r,r 0 r +r r r r ( 1) r +r (n r+t) v (m r+t) v t=1 t=1 r t=r +r +1 (m+n 2r+t+1) v E r xe r r r ye r. Clearly x r y s homogeneous of degree 2r m n. Let V m,n be the subspace of F m,n wth the bass {E a xe b ye c : a, b, c Z 0 }. Clearly, E ± (V m,n ), F ± (V m,n ) V m,n and x r y V m,n. Proposton (F m,n ) 0 V m,n s spanned by the x r y, r 0 as a k-vector space. Proof. It s easy to check that x r y (F m,n ) 0. Conversely, let u (F m,n ) 0 V m,n be homogeneous of degree 2r m n and wrte u = c r,r E r xe r r r ye r. Then It follows that and Thus, 1 v F ± (u) = = r 1, r 0, r +r r + + r,r 0 r +r r r,r 0,r +r r 1 r 0, r 1, r +r r r,r 0, r +r r 1 c r,r v± 1 2 ( x + y +2(r r )) E r 1 xe r r r ye r c r,r v± 1 2 ( y x +2(r r )) E r xe r r r 1 ye r c r,r v 1 2 ( x + y +2(r r )) E r xe r r r ye r 1 (c r +1,r v± 1 2 ( x + y +2(r r 1)) + c r,r v± 1 2 ( y x +2(r r )) + c r,r +1v 1 2 ( x + y +2(r r 1)) )E r xe r r r ye r. c r +1,r x + y + 2r r r 2 v + c r,r y r + r 1 v = 0 c r,r +1 x + y + 2r r r 2 v + c r,r x r + r 1 v = 0. c r,r = c r 0,r t=1 (n r + t) r v ( 1)r r t=1 (m + n 2r + = ( 1) r +r t=1 c (n r + t) v t=1 (m r + t) v 0,0 r + t) r. +r v t=1 (m + n 2r + t + 1) v Therefore, u = c 0,0 r t=1 (m + n 2r t + 1) v 1 x r y. r

15 CANONICAL BASES OF QUANTUM SCHUBERT CELLS AND THEIR SYMMETRIES 15 Proposton Let A be a decorated algebra and let x, y A 0 be homogeneous wth x = m, y = n. For all r 0 we have x r y A 0 and x r y = ) C r;t,t (v)e(t + (x)e (t ) + (y) = ( 1) r C r;t,t (v 1 )E (t ) (x)e (t ) (y). (3.16) t +t =r t +t =r Proof. Denote M m the lowest weght Verma module of type 1 wth a fxed lowest weght vector 1 m of lowest weght m. The followng s mmedate. Lemma Let m, n Z 0. (a) For any U v (sl 2 )-module V and a lowest weght vector x V of type 1 and of lowest weght m the assgnment 1 m x defnes a homomorphsm of U v (sl 2 )-modules j x : M m V. (b) Let A be a U v (sl 2 )-module algebra va the co-multplcaton ± : U v (sl 2 ) U v (sl 2 ) U v (sl 2 ). Then for any lowest weght x, y A of lowest weght m and n, respectvely, j ±,x,y := µ A (j x j y ) : M m ± M n A n the notaton of Lemma 3.7, where µ A : A ± A A s the multplcaton map, s a homomorphsm of U v (sl 2 )-modules. We use Lemma 3.13(b) wth A = F m,n regarded as a U v (sl 2 ) ± -module n the notaton of 3.1. Let x ±r y := j ±,x,y (u (0,r,±) (1 m, 1 n )) F m,n n the notaton of Lemmata 3.7 and 3.13(b). By Lemmata 3.7 and 3.13(b), x ±r y s a lowest weght vector for U v (sl 2 ) ± and x +r y (respectvely, x r y) s equal to the frst (respectvely the second) sum n (3.16). Ths, together wth (3.2) and Lemma 3.3(b) mples that x ±r y (F m,n ) 0 V m,n. Snce x ±r y = x + y + 2r = x r y, t now follows from Proposton 3.11 that x ±r y = c ± x r y for some c ± k. Snce F m,n s free, usng (3.2) to compare the coeffcents of E r xy we conclude that c ± = 1. The asserton for any decorated algebra A follows from Lemmata 3.2 and 3.3(c). We can now complete the proof of Theorem 3.6. By constructon, τ s an somorphsm of U v (sl 2 )-modules. Explctly, f z A 0 then τ(e (r) (z)) = E ( z r) + (z). It suffces to prove that for any x, y A 0 homogeneous wth x = m, y = n we have τ(e (k) (x))τ(e (l) (y)) = E (m k) + (x)e (n k) + (y) = τ(e (k) (x)e (l) (y)). By Lemmata 3.7 and 3.13, Propostons 3.8 and (3.16) we have for all 0 r mn(m, n), a Z 0, 0 k m and 0 x n and Therefore, E (a) + (x r y) = E (a) (x r y) = t +t =a+r t +t =a+r mn(m,n,k+l) E (k) (x)e (l) (y) = mn(m,n,k+l) τ(e (k) (x)e (l) (y)) = = mn(m,n,k+l) r=0 r=0 C r;k,l (v) C r;t,t (v)e(t ) + (x)e (t ) + (y), ( 1) r C r;t,t (v 1 )E (t ) (x)e (t ) (y) r=0 C r;k,l (v)e(k+l r) (x r y). C r;k,l (v)e(m+n r k l) + (x r y) s +s =m+n k l C r;s,s (v)e(s ) + (x)e (s ) + (y) (3.17)

16 16 CANONICAL BASES OF QUANTUM SCHUBERT CELLS AND THEIR SYMMETRIES = ( mn(m,n,k+l) t +t =k+l r=0 ( mn(m,n,k+l) = t +t =k+l where we used (3.8) and (3.12). r=0 C r;k,l (v)c r;m t,n t (v) )E (m t ) + (x)e (n t ) + (y) ) ( 1) r C r;k,l (v)c r;t,t (v 1 ) = E (m k) + (x)e (n l) + (y) = τ(e (k) (x))τ(e (l) (y)), E (m t ) + (x)e (n t ) + (y) Note that for x A homogeneous, τ(x) can be calculated explctly n the followng way. Frst, f y A 0 s homogeneous and x = E (r) (y) then τ(x) = E ( y r) + (y) = E (r x ) + (y) = ( 2r x r ) 1 v E (r x ) + F (r) (x). (3.18) By lnearty, t remans to observe that any homogeneous element of A can be wrtten, unquely, as x = r max(0, x ) E(r) (x r ) where x r A 0 and x r = x 2r. We wll also need the followng property of τ. Lemma F (top) + τ = F (top). Proof. Gven x A, wrte x = r max(0, x ) E(r) (x r ) where x r A 0 and x r = x 2r. Then τ(x) = r max(0, x ) E(r x ) + (x r ). Let r 0 = max{r max(0, x ) : x r 0}. Then by (3.4) ( ) F (top) (x) = F (r 0) (x) = F (r 0) E (r 0) 2r0 x (x r0 ) = x r0. r 0 v On the other hand, F (top) + τ(x) = F (r 0 x ) + τ(x) = F (r 0 x ) + E (r 0 x ) + (x r0 ) = ( ) 2r0 x 3.3. U q (n + ) as a decorated algebra. Defne a comultplcaton on U q (g) by Then (cf. [16, 3.1.5]) (E r ) = r 0 x r0. v (E ) = E 1 + K 1 E, (F ) = 1 F + F K, I. q r r E r r +r =r K r E r, (F r ) = r +r =r q r r F r K r F r. (3.19) For any J, J I denote U Z (g) J,J the A 0 -subalgebra of U q (g) generated by U Z (n ) J, U Z (n + ) J, the K ±1 and the ( K ) ;c a q, a Z 0, c Z and J J, where ( ) K; c a v = a 1 k=0 K 1 v k c Kv c k v k+1 v k 1. We also abbrevate U Z (g) J = U Z (g) J,I and U Z (g) := U Z (g) I,I. The correspondng k-subalgebras of U q (g) wll be denoted U q (g) J,J. It follows from (3.19) that U Z (g) s a Hopf A 0 -algebra.

17 CANONICAL BASES OF QUANTUM SCHUBERT CELLS AND THEIR SYMMETRIES 17 Let ad be the correspondng adjont acton of U q (g) on tself. Consder the extenson Ũq(g) of U q (g) obtaned by adjonng K ± 1 2, I. Defne operators E, F on Ũq(g) va E (x) = (ad E 1 K 1 2 ) = E K xk 2 K 1 2 xk 1 2 E q 1, q F (x) = (ad K 1 2 F 1 )(x) = (x) K 1 Clearly, E and F restrct to operators on U q (g) and we have op (x)k 1. (3.20) [E, F ] = 1 q 2 ad([e, F ]) = 1 q 1 (K K 1 ), (3.21) where K (x) = K xk 1. We wll also need operators E op, F op defned by E op (x) = (E (x )), We collect some propertes of these operators n the followng Lemma. F op (x) = (F (x )). (3.22) Lemma (a) U q (n + ) s a decorated algebra wth E = E, F + =, F = op, v = q and x = (α, γ) for x U q (n + ) γ ; n partcular, E + = E and E = E op. (b) E, F commute wth. (c) If x U Z (g) γ then E (r) (x), F (r) (x) q 1 2 (rα,γ) U Z (g) for all r Z 0 ; (d) For all x U q (n + ) γ, y U q (n + ) we have E (r) (x), y = ( 1) r q 1 2 (r+(α,γ) 1)(r r ) x, (r ) ( op ) (r ) (y) (e) T E op r +r =r = F T, T F op = E T. Proof. Parts (a) and (b) are obvous from the defntons. Snce U Z (g) s a Hopf A 0 -algebra, the frst asserton n (c) follows from E (r) = ( 1) r q (r 2) ad(e r K r 2 ), F (r) = q (r 2) (ad K r 2 F r ), whle the second s mmedate from the above formulae and (3.19). Part (d) s mmedate from part (a), (3.2) and Lemma 2.10(b). Part (e) s easy to check usng Lemma A new formula for T. Usng the notaton from [2], denote by U := T 1 (U q (n + )) U q (n + ) and U := T (U q (n + )) U q (n + ). It follows from [16, Proposton ] that U = ker and U = ker op. Let U Z = U U Z (n + ) and U Z = U U Z (n + ). Lemma Let I. (a) For all x U q (n + ) γ, y U, z U and r 0 we have E (r) (x), y = q 1 2 (r+(α,γ) 1)r (b) E, F (respectvely, E op, F op (c) E (n) ( U Z ), F (n) x, F (r) (y), (E op ) (r) (x), z = q 1 2 (r+(α,γ) 1)r x, (F op ) (r) (z). ) restrct to locally nlpotent operators on U (respectvely, on U ). ( U Z ) U Z (respectvely, (E op ) (n) (U Z ), (F op ) (n) (U Z ) U Z ) for all n 0. Proof. We only prove the asserton for E and F. The asserton for E op and F op s proved smlarly usng (3.22) and the fact that s self-adjont wth respect to,. Snce by (3.20) F U = U, n partcular, F s a locally nlpotent operator on U. Part (a) s now mmedate from Lemma 3.15(d). Suppose that x U q (n + ) γ and E (n) (x) 0 for all n 0. Then T (E (n) (x)) s homogeneous of degree s (γ + nα ) = γ ((α, γ) + n)α / Q + for n 0. Snce T ( U) U q (n + ), ths s a contradcton. Ths proves (b).

18 18 CANONICAL BASES OF QUANTUM SCHUBERT CELLS AND THEIR SYMMETRIES To prove (c), note that the asserton for F follows from Corollary Snce U Z (n + ) = U Z (E U q (n + ) U Z (n + )), t suffces to prove that for x U Z U q (n + ) γ, y U Z U q (n + ) γ+nα we have E (n) (x), y A 0. But for such y we have by part (a) and (2.4) E (n) (x), y = q 1 2 n(n+(α,γ) 1) x, F (n) (y) = q (n 2) x, q 1 2 n(α,γ+nα ) F (n) (y) A 0 and q 1 2 n(α,γ+nα ) F (n) (y) U Z (n + ) by Lemma 3.15(c). Thus, gven x U U q (n + ) γ, y U U q (n + ) γ we can wrte unquely x = (E op ) (r) (x r ), y = E (r) (y r ), x r, y r U U U q (n + ) γ rα, (3.23) r max(0,(α,γ)) r max(0,(α,γ)) and x r, y r = 0 for r 0. Corollary Let x, x U U q (n + ) γ, y, y U U q (n + ) γ and wrte x, x and y, y as n (3.23). Then ( ) x, x 1 2 = q r(r+1 (α,γ)) 2r (α, γ) x r, x r r, r max(0,(α,γ)) q ( ) (3.24) y, y 1 2 = r(r+1 (α,γ)) 2r (α, γ) y r, y r r. q r max(0,(α,γ)) q Proof. Let z, z U U, z U q (n + ) γ. Let a b 0. Then we have by Lemma 3.16(a) and (3.4) E (a) (z), E (b) (z ) = q 1 2 b(b+(α,γ ) 1) F (b) E (a) (z), z ( ) ( ) = q 1 2 b(b+(α,γ ) 1) b a (α, γ ) E (a b) (z), z = δ a,b q 1 2 a(a+(α,γ ) 1) (α, γ ) z, z. b q a q Ths yelds the second dentty n (3.24). The frst follows from the second one by applyng. We now establsh a formula for the acton of T on U n terms of E op and E. Theorem Wrte x U U q (n + ) γ as n (3.23). Then T (x) =,γ)) (x r ). In partcular, (top) T (x) = ( op ) (top) (x). r max(0,(α,γ)) E (r (α Proof. We apply Proposton 3.6 to A = U q (n + ) whch s a decorated algebra by Lemma 3.15(a) wth locally nlpotent E ± on A ± by Lemma 3.16(b). We clam that T = τ. Snce both τ and T are somorphsms of algebras U U, t s enough to check that they concde on generators of U. Let j and defne for all 0 l a j E j l = Clearly, E j l = E j l ( ) 1 aj l q (E op ) (l) (E j ), E l j = (E j l) = and t s mmedate from (3.2) that E j l = ( ) 1 aj l q ( 1) r q 1 2 (r s)(l+a j 1) r+s=l Lemma Let j I, 0 m a j. Then (a) T (E j m) = E a j m j = τ(e j m) ( ) 1 aj l q E (l) (E j ). (3.25) E s E j E r. (3.26)

19 CANONICAL BASES OF QUANTUM SCHUBERT CELLS AND THEIR SYMMETRIES 19 (b) The elements E j l (respectvely, E j), j, 0 l a l j generate the algebra U (respectvely U). Proof. To prove (a), note that by Lemma 2.1 and (3.26) we have T (E j ) = E a j j. On the other hand, τ(e j ) = E ( a j) (E j ) = E a j j. Then by Lemma 3.15(e) and (3.4) T (E j l) = ( ) 1 aj l q ( F (l) (E a j j ) = aj l ) 1 q F (l) E ( a j) (E j ) = E a j l j. Snce by constructon τ also satsfes Lemma 3.15(e), t follows that τ(e j l) = T (E j l). Part (b) can be easly deduced from [16, ]. Ths mples that τ = T on U. The second asserton of Theorem 3.18 follows from Lemma We now prove the followng Proposton For all I, T (U Z ) = U Z. Proof. We need the followng Lemma Any element x U Z (n + ) can be wrtten as x = r,s 0 E r x rs E s, where x rs U U U Z (n + ) and only fntely many of them are non-zero. Proof. We need the followng elementary fact. Lemma Let V be a fnte dmensonal k-vector space wth a non-degenerate blnear form, : V V k. Assume that we have two orthogonal drect sum decompostons V = V 1 W 1 = V 2 W 2 wth respect to that form. Then V = (V 1 V 2 ) (W 1 + W 2 ) = (W 1 W 2 ) (V 1 + V 2 ) (orthogonal drect sum decompostons). Proof. Clearly (V 1 V 2 ) s orthogonal to W 1 +W 2 and (W 1 W 2 ) s orthogonal to V 1 +V 2. Note that for any v V, v, v = 0 f and only f v = 0. Ths mples that the sums U 1 = (V 1 V 2 )+(W 1 +W 2 ), U 2 = (W 1 W 2 ) + (V 1 + V 2 ) are drect. It remans to prove that dm U 1 = dm U 2 = dm V. Snce dm U 1 + dm U 2 = (dm W 1 + dm W 2 dm W 1 W 2 ) + dm V 1 V 2 + (dm V 1 + dm V 2 dm V 1 V 2 ) + dm W 1 W 2 = 2 dm V, and dm U 1, dm U 2 dm V the asserton follows. Gven γ Q +, let n (γ) be the coeffcent of α n γ. For any γ Q + we have two orthogonal drect sum decompostons U q (n + ) γ = (ker Uq(n + ) γ U q (n + ) γ α E ) = (ker op Uq(n + ) γ E U q (n + ) γ α ). Snce U q (n + ) γ s fnte dmensonal, t follows from Lemma 3.22 that U q (n + ) γ = ( U U U q (n + ) γ ) (U q (n + ) γ α E + E U q (n + ) γ α ). Then an obvous nducton on n (γ) mples that every x U q (n + ) can be wrtten n x = r,s 0 E r x rs E s, where x rs U U and only fntely many of the x rs are non-zero. We now prove by nducton on n (γ) that f x U Z (n + ) γ then x rs U Z (n + ) U U. If n (γ) = 0 then x = x 00 and there s nothng to prove. For the nductve step, we have E r (q r 1 x r+1,s + q (α,γ)+s+1 x r,s+1 )E s = q 1 2 (α,γ) 1 q (x) U Z (n + ), r,s 0 where we used Lemma 2.10(c) and Corollary 2.11(a). Then x r+1,s + q (α,γ)+r+s+2 x r,s+1 U Z (n + ) by the nducton hypothess. Let s 0 be such that x rs = 0 for all r and for all s > s 0. It follows then that x r,s0 U Z (n + ) for all r 0. Suppose now that x rt U Z (n + ) for all r and for all s+1 t s 0. Snce x r,s = q (α,γ)+s+r+1 x r 1,s+1 t follows that x r,s U Z (n + ) for all r, s 0 wth r + s > 0. It remans to observe that x 00 = x r,s 0,r+s>0 E r x rs E s.

20 20 CANONICAL BASES OF QUANTUM SCHUBERT CELLS AND THEIR SYMMETRIES Lemma Let x U Z U q (n + ) γ and wrte x = r max(0,(α,γ))(eop ) (r) (x r ) where x r U U U q (n + ) γ rα. Then ( ) (α,γ rα ) r q x r U Z. Proof. The argument s by nducton on l (x ). If l (x ) = 0, that s op (x) = 0, then x = x 0 and there s nothng to do. If l (x ) = n then x r = 0 for all r > n. We have ( op ) (top) (x) = ( op ) (n) (x) U Z by Corollary 2.11(b). On the other hand, ( op ) (n) (x) = ( op ) (n) (E op ) (n) (x n ) = ( ) 2n (α,γ) n q x n by (3.4). Thus, ( ) 2n (α,γ) n q x n U Z U Z. It remans to observe that the nducton hypothess apples to x (E op ) (n) (x n ). Thus, s suffces to consder x = (E op ) (r) (z) U Z where z U U U q (n + ) γ. We clam that T (x) = E ( (α,γ) r) (z) U Z. Gven y U Z (n + ) γ+( (α,γ) r)α, use Lemma 3.21 to wrte y = s,s 0 E s y s se s wth y s s U U U Z (n + ). Then by Lemma 2.10(b) and (3.4) E ( (α,γ) r) (z), y = E ( (α,γ) r) (z), E s y s se s = (s) E ( (α,γ) r) (z), y 0s s,s 0 s 0 = s 0 F (s) E ( (α,γ) r) (z), y 0s = snce U, E U q (n + ) = 0 and E (a) (α,γ) r s=0 ( ) s + r r E ( (α,γ) r s) q (z), y 0s ( ) (α =, γ) z, y 0, (α r,γ) r, q (z), y 0s = 0 f a > 0 by Lemma 3.15(d). Snce ( ) (α,γ) r q z U Z (n + ) by Lemma 3.23, t follows that E ( (α,γ) r) (z), y A Proof of Theorem 1.7. We need the followng result whch can also be deduced from [16, Proposton ]. However, our argument s much shorter. Lemma For all x, x U q (n + ) γ U, a, a Z 0 we have E a T (x), E a T (x ) = q 1 2 (a 1)(α,γ) δ a,a a q! x, x = q 1 2 (a 1)(α,γ) δ a,a µ(aα ) where µ s defned as n Theorem 2.4. a (1 q 2t ) x, x, Proof. It follows mmedately from Lemma 2.10(b) and (2.13) that f y, y ker op, y U q (n + ) γ and a, a Z 0 then E a y, E a y = δ a,a a q!q 1 2 a(α,γ ) y, y = δ a,a q (a+1 2 ) 1 2 a(α,γ ) a t=1 t=1 (1 q 2t ) y, y. Let γ = (α, γ). Snce T (x), T (x ) ker op U q (n + ) s γ, t remans to prove the asserton for a = a = 0, x = (E op ) (a) (z) and x = (E op ) (b) (z ) where z, z U U and a b max(0, γ ). Snce z U q (n + ) γ aα, z U q (n + ) γ bα we have, by Corollary 3.17 x, x = δ a,b q 1 2 a(1+a γ ) ( ) 2a γ a q z, z. On the other hand, usng Theorem 3.18 and Corollary 3.17 we obtan ( ) T (x), T (y) = E (a γ ) (z), E (b γ ) (z ) = q 1 2 (a γ )(a+1) 2a γ δ a,b z, z = q 1 2 γ x, y. a γ q

21 CANONICAL BASES OF QUANTUM SCHUBERT CELLS AND THEIR SYMMETRIES 21 Let b B up γ T 1 (U q (n + )). Snce T commutes wth we have T (b) = T (b) = T (b). By Proposton 3.20 we have T (b) U Z (n + ). Furthermore, by Lemma 3.24 and (2.4) µ(s γ) 1 T (b), T (b) = µ(γ) 1 q 1 2 (α,γ) T (b), T (b) = µ(γ) 1 b, b 1 + K. Thus, T (b) B ±up by (2.6). It remans to prove that T (b) B up. Snce ( op ) (top) (b) B up by Remark 2.14, there exsts a (( op ) (top) (b)) = 1. Let = (, 1,..., m ). Then T (b) = (top) (( op ) (top) (b)) = 1 by Theorem Thus, T (b) B up. sequence = ( 1,..., m ) I m such that (top) (T (b)) = (top) (top) (top) 4. Proofs of mans results 4.1. Propertes of quantum Schubert cells. Let w W and = ( 1,..., m ) R(w). Set X,k = T 1 T k 1 (E k ), 1 k m, and let U q () be the subalgebra of U q (n + ) generated by the X,k, 1 k m and set U Z () = U q () U Z (n + ), U Z () = U q () U Z (n + ). The followng s well-known. Lemma 4.1 ([16, Propostons and ]). The elements X a := X a 1,1 X am,m, a Zm 0 form an A 0 -bass of U Z () and a k-bass of U q (). Set α (k) = α (k) := s 1 s k 1 (α k ) = deg X,k and gven a = (a 1,..., a m ) Z m denote a = a = m k=1 a k deg X,k = m k=1 a kα (k). Defne X a = q,a X a 1,1 Xam,m, a = (a 1,..., a m ) Z m 0, where q,a = q k<l m (α(k),α (l) )a k a l. (4.1) Ths choce s justfed by the followng Proposton 4.2. For all R(w), a, a Z m 0 we have X a U Z (n + ) and m a m µ( a ) 1 X a, X a = δ a,a (1 q 2t r ). (4.2) Thus, the set {X a Proof. We need the followng r=1 t=1 : a = γ} s a (K, µ(γ) 1 )-orthonormal bass of U Z () γ and X a, X a = δ a,a, a, a Z m 0. (4.3) Lemma 4.3. For all w W, j I such that l(ws j ) = l(w) + 1 we have T w (E j ) U Z (n + ). Proof. The argument s by nducton on l(w). If l(w) = 0 there s nothng to prove. Suppose that w = s w wth l(w) = l(w ) + 1. Clearly, l(w s j ) = l(w ) + 1. Then T w (E j ) ker by [16, Lemma ] and also T w (E j ) U Z (n + ) by the nducton hypothess. Then by Proposton 3.20, T w (E j ) = T (T w (E j )) U Z (n + ). Ths mples that X,k U Z (n + ) and hence X a U Z (n + ) by Lemma 2.3. To prove (4.2) we use nducton on l(w). The case l(w) = 0 s trval. For the nductve step, assume that l(s w) = l(w) + 1 and note that we have X (a,a) (,) = q 1 2 a(α,s a ) E a T (X a ) = q 1 2 a(α, a ) E a T (X a ). Snce T (X a ) U, we have by Lemmata 2.10, 3.24 and (2.4) X (a,a),a ) (,),X(a (,) = q 1 2 (a(α, a )+a (α, a ) E a T (X a ), E a T (X a )