EXISTENCE OF KIRILLOV RESHETIKHIN CRYSTALS FOR NONEXCEPTIONAL TYPES

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1 REPRESENTATION THEORY An Electronc Journal of the Amercan Mathematcal Socety Volume 12, Pages (Aprl 14, 28) S (8)329-4 EXISTENCE OF KIRILLOV RESHETIKHIN CRYSTALS FOR NONEXCEPTIONAL TYPES MASATO OKADO AND ANNE SCHILLING Dedcated to Professor Masak Kashwara on hs sxteth brthday Abstract. Usng the methods of Kang et al. and recent results on the characters of Krllov Reshetkhn modules by Nakajma and Hernandez, the exstence of Krllov Reshetkhn crystals B r,s s establshed for all nonexceptonal affne types. We also prove that the crystals B r,s of type B n (1), D n (1), and A (2) 2n 1 are somorphc to recently constructed combnatoral crystals for r not aspnnode. 1. Introducton The theory of crystal bases by Kashwara [15] provdes a remarkably powerful tool to study the representatons of quantum algebras U q (g). For nstance, the calculaton of tensor product multplctes reduces to countng the number of crystal elements havng certan propertes. Although crystal bases are bases at q =, one can melt them to get actual bases, called global crystal bases, for ntegrable hghest weght representatons of U q (g). It turns out that the global crystal bass agrees wth Lusztg s canoncal bass [22], and t has many applcatons n representaton theory. The man focus of ths paper s affne fnte crystals, that s, crystal bases of fnte-dmensonal modules for quantum groups correspondng to affne Kac Moody algebras g. These crystal bases were frst developed by Kang et al. [13, 14], where t was also shown that ntegrable hghest-weght U q (g)-modules of arbtrary level can be realzed as sem-nfnte tensor products of perfect crystals. Ths s known as the path realzaton. Many perfect crystals were proven to exst and explctly constructed n [14]. Irreducble fnte-dmensonal U q(g)-modules were classfed by Char and Pressley [4, 5] n terms of Drnfeld polynomals. It was conjectured by Hatayama et al. [7, 8] that a certan subset of such modules known as Krllov Reshetkhn (KR) modules W s (r) have a crystal bass B r,s. Here the ndex r corresponds to a node of the Dynkn dagram of g except the prescrbed and s s an arbtrary postve nteger. Ths conjecture was confrmed n many nstances [2, 13, 14, 17, 19, 26, 34], but a proof for general r and s has not been avalable except type A (1) n n [14]. Only recently the exstence proof was completed n [27] for type D n (1). Usng the methods of [14] and recent results on the characters of KR modules [9, 1, 24], we Receved by the edtors August 8, 27 and, n revsed form, February 26, Mathematcs Subject Classfcaton. Prmary 17B37, 81R5; Secondary 5E15, 81R c 28 Amercan Mathematcal Socety

2 EXISTENCE OF KR CRYSTALS 187 establsh the exstence of Krllov Reshetkhn crystals B r,s for all nonexceptonal affne types n ths paper: Theorem 1.1. The Krllov-Reshetkhn module W s (r) assocated to any nonexceptonal affne Kac Moody algebra has a crystal bass B r,s. In addton we prove that for type B n (1), D n (1),andA (2) 2n 1 these crystals concde wth the combnatoral crystals of [3, 32]. Throughout the paper we denote by B r,s the KR crystal assocated wth the KR module W s (r). The combnatoral crystal of [3] s called B r,s. Our second man result s the followng theorem: Theorem 1.2. For 1 r n 2 for type D (1) n, 1 r n 1 for type B (1) n, 1 r n for type A (2) 2n 1 and s Z >, thecrystalsb r,s and B r,s are somorphc. The key to the proof of Theorem 1.1 s Proposton 2.1 below, whch s due to Kang et al. [14] and states that a fnte-dmensonal U q(g)-module havng a prepolarzaton and certan Z-form has a crystal bass f the dmensons of some partcular weght spaces are not greater than the weght multplctes of a fxed module and the values of the prepolarzaton of certan vectors n the module have some specal propertes. Usng the fuson constructon t s establshed that the KR modules have a prepolarzaton and Z-form. The requrements on the dmensons follow from recent results by Nakajma [24] and Hernandez [9, 1]. Necessary values of the prepolarzaton are calculated explctly n Propostons 4.1, 4.4, and 4.6. The somorphsm between the KR crystal B r,s and the combnatoral crystal Br,s s establshed by showng that somorphsms as crystals wth ndex sets {1, 2, 3,...,n} and {, 2, 3,...,n} already unquely determne the whole crystal. Before presentng our results, let us offer some speculatons on combnatoral the crystals B r,s were constructed combnatorally by Shmozono [31] usng the promoton operator. The promoton operator pr s the crystal analogue of the Dynkn dagram automorphsm that maps node to node + 1 modulo n + 1. The affne crystal operator f s then gven by f = pr 1 f 1 pr. Smlarly, the man tool used n [3] to construct the realzatons for the KR crystals. For type A (1) n combnatoral crystals B r,s of type B n (1), D n (1),andA (2) 2n 1 s the crystal analogue of the Dynkn dagram automorphsm that nterchanges nodes and 1. For type C (1) n and D (2) n+1, there exsts a Dynkn dagram automorphsm n. It s our ntenton to explot ths symmetry to construct B r,s of type C n (1) and D (2) n+1 explctly n a future publcaton. For type A (2) 2n no Dynkn dagram automorphsm exsts. However, t should stll be possble to construct these crystals by lookng at the {1, 2,...,n} and {, 1, 2,...,n 1} subcrystals as was done for r =1n[14]. Realzatons of B r,s as vrtual crystals were gven n [28, 29]. The paper s organzed as follows. In Secton 2 we revew necessary background on the quantum algebra U q(g) and the fundamental representatons. In partcular, we revew Proposton 2.1 of [14] whch provdes a crteron for the exstence of a crystal pseudobase. In Secton 3 we defne KR modules by the fuson constructon and show that these modules have a prepolarzaton. Ths reduces the exstence proof for KR crystals to condtons stated n Proposton 3.7. These condtons are checked explctly n Secton 4 for the varous types to prove Theorem 1.1. In Secton 5 we revew the combnatoral constructon of the crystals B r,s of types

3 188 MASATO OKADO AND ANNE SCHILLING B (1) n, D (1) n,anda (2) 2n 1 and prove n Secton 6 that they are somorphc to Br,s, thereby establshng Theorem Quantum affne algebra U q(g) and fundamental representatons 2.1. Quantum affne algebra. Let g be an affne Kac-Moody algebra and U q (g) the quantum affne algebra assocated to g. In ths secton g can be any affne algebra. For the notaton of g or U q (g) we follow [17]. For nstance, P s the weght lattce, I s the ndex set of smple roots, and {α } I (resp. {h } I )sthesetof smple roots (resp. coroots). Let (, ) be the nner product on P normalzed by (δ, λ) = c, λ for any λ P as n [12], where c s the canoncal central element and δ s the generator of null roots. We choose a postve nteger d such that (α,α )/2 Zd 1 for any I and set q s = q 1/d. Then U q (g) s the assocatve algebra over Q(q s )wth1generatedbye,f ( I), q h (h d 1 P,P =Hom Z (P, Z)) wth certan relatons. By conventon, we set q = q (α,α )/2,t = q h, [m] =(q m q m )/(q q 1 ), [n]!= n m=1 [m],e (n) = e n /[n]!,f (n) = f n/[n]!. Let {Λ } I be the set of fundamental weghts. Then we have P = ZΛ Zδ. We set P cl = P/Zδ. Smlar to the quantum algebra U q (g) whch s assocated wth P, we can also consder U q(g), whch s assocated wth P cl, namely, the subalgebra of U q (g) generated by e,f,q h (h d 1 (P cl ) ). Next we ntroduce two subalgebras ( Z-forms ) U q (g) KZ and U q (g) Z of U q (g). Let A be the subrng of Q(q s ) consstng of ratonal functons wthout poles at q s =. We ntroduce the subalgebras A Z and K Z of Q(q s )by A Z = {f(q s )/g(q s ) f(q s ),g(q s ) Z[q s ],g() = 1}, K Z = A Z [qs 1 ]. Then we have K Z A = A Z, A Z /q s A Z Z. We then defne U q (g) KZ as the K Z -subalgebra of U q (g) generated by e,f,q h ( I,h d 1 P ). U q (g) Z s defned as the Z[q s,qs 1 ]-subalgebra of U q (g) generated by e (n),f (n), { t n } ( I,n Z >) andq h (h d 1 P ). Here we have set { } x n = n k=1 (q1 k x q k 1 x 1 )/[n]!. U q (g) Z s a Z[q s,qs 1 ]-subalgebra of U q (g) KZ. We can also ntroduce subalgebras U q(g) KZ and U q(g) Z by replacng q h (h d 1 P ) wth q h (h d 1 (P cl ) ) n the generators. We defne a total order on Q(q s )by f>gf and only f f g n Z{q n s (c + q s A) c>} and f g f f>gor f = g. Let M and N be U q (g)(or U q(g))-modules. A blnear form (, ):M Q(qs ) N Q(q s ) s called an admssble parng f t satsfes (q h u, v) =(u, q h v), (2.1) (e u, v) =(u, q 1 t 1 f v), (f u, v) =(u, q 1 t e v),

4 EXISTENCE OF KR CRYSTALS 189 for all u M and v N. Equaton (2.1) mples (2.2) (e (n) u, v) =(u, q n2 t n f (n) v), (f (n) u, v) =(u, q n2 t n e (n) v). A symmetrc blnear form (, )onm s called a prepolarzaton of M f t satsfes (2.1) for u, v M. A prepolarzaton s called a polarzaton f t s postve defnte wth respectve to the order on Q(q s ) Crteron for the exstence of a crystal pseudobase. Here we recall the crteron for the exstence of a crystal pseudobase gven n [14]. We do not revew the noton of crystal bases, but refer the reader to [15]. We only note that q n the defnton of crystal base n [15] should be replaced by q s accordng to the normalzaton of the nner product (, )onp. We say (L, B) sacrystal pseudobase of an ntegrable U q (g) (oru q(g))-module M, f()l s a crystal lattce of M, () B = B ( B )whereb s a Q-base of L/q s L, () B = λ P B λ where B λ = B (L λ /q s L λ ), (v) ẽ B B {}, f B B {}, and(v)forb, b B, b = f b f and only f b =ẽ b. Note that only the condton () s replaced from the defnton of the crystal base. Let g be the fnte-dmensonal smple Le algebra whose Dynkn dagram s obtaned by removng the -vertex from that of g. In ths paper we specfy the -vertex as n [12] and set I = I \{}. Let P + be the set of domnant ntegral weghts of g and V (λ) be the rreducble hghest weght U q (g )-module of hghest weght λ for λ P +. The followng proposton s easly obtaned by combnng Proposton and of [14]. Proposton 2.1. Let M be a fnte-dmensonal ntegrable U q(g)-module. Let (, ) be a prepolarzaton on M, andm KZ a U q(g) KZ -submodule of M such that (M KZ,M KZ ) K Z.Letλ 1,...,λ m P +, and assume that the followng condtons hold: (2.3) (2.4) m dm M λk dm V (λ j ) λk for k =1,...,m. j=1 There exst u j (M KZ ) λj (j =1,...,m) such that (u j,u k ) δ jk + q s A, and (e u j,e u j ) q s q 2(1+ h,λ j ) A for any I. Set L = {u M (u, u) A} and set B = {b M KZ L/M KZ q s L (b, b) =1}. Here (, ) s the Q-valued symmetrc blnear form on L/q s L nduced by (, ). Then we have the followng: () (, ) s a polarzaton on M. () M j V (λ j) as U q (g )-modules. () (L, B) s a crystal pseudobase of M Fundamental representatons. For any λ P, Kashwara defned a U q (g)- module V (λ) called extremal weght module [16]. We brefly recall ts defnton. Let W be the Weyl group assocated to g and s the smple reflecton for α.letm be an ntegrable U q (g)-module. A vector u λ of weght λ P s called an extremal

5 19 MASATO OKADO AND ANNE SCHILLING vector f there exsts a set of vectors {u wλ } w W satsfyng (2.5) u wλ = u λ for w = e, f h,wλ, then e u wλ =andf ( h,wλ ) (2.6) u wλ = u s wλ, f h,wλ, then f u wλ =ande ( h,wλ ) (2.7) u wλ = u s wλ. Then V (λ) s defned to be the U q (g)-module generated by u λ wth the defnng relatons that u λ s an extremal vector. For our purpose, we only need V (λ) when λ = ϖ r for r I,whereϖ r s a level fundamental weght (2.8) ϖ r =Λ r c, Λ r Λ. Then the followng facts are known. Proposton 2.2 ([17, Proposton 5.16]). () V (ϖ r ) s an rreducble ntegrable U q (g)-module. () dm V (ϖ r ) µ < for any µ P. () dm V (ϖ r ) µ =1for any µ Wϖ r. (v) wt V (ϖ r ) s contaned n the ntersecton of ϖ r + I Zα and the convex hull of Wϖ r. (v) V (ϖ r ) has a global crystal base (L(ϖ r ),B(ϖ r )). (v) Any ntegrable U q (g)-module generated by an extremal weght vector of weght ϖ r s somorphc to V (ϖ r ). Let λ P = {λ P c, λ =}. V (λ) hasau q (g) Z -submodule V (λ) Z. Let {G(b)} b B(λ) stand for the global base of V (λ). The followng result was shown n [33] for g smply laced and λ = ϖ r, n [23] for g smply laced and λ s arbtrary, andn[1]forg and λ arbtrary. Proposton 2.3. () There exsts a prepolarzaton (, ) on V (λ). () {G(b)} b B(λ) s almost orthonormal wth respect to (, ), that s, (G(b),G(b )) δ bb mod q s Z[q s ]. Let d r be a postve nteger such that {k Z ϖ r + kδ Wϖ r } = Zd r. We note that d r =max(1, (α r,α r )/2) except n the case d r =1wheng = A (2) 2n and r = n. Then there exsts a U q(g)-lnear automorphsm z r of V (ϖ r )ofweghtd r δ sendng u ϖr to u ϖr +d r δ. Hence we can defne a U q(g)-module W (ϖ r )by W (ϖ r )=V(ϖ r )/(z r 1)V (ϖ r ). Ths module s called a fundamental representaton. For a U q(g)-module M let M aff denote the U q(g)-module Q(q s )[z, z 1 ] M wth the actons of e and f by z δ e and z δ f. For a Q(q s ) we defne the U q(g)-module M a by M aff /(z a)m aff. Proposton 2.4 ([17, Proposton 5.17]). () W (ϖ r ) s a fnte-dmensonal rreducble ntegrable U q(g)-module. () For any µ wt V (ϖ r ), W (ϖ r ) cl(µ) V (ϖ r ) µ.herethemapcl stands for the canoncal projecton P P cl. () dm W (ϖ r ) cl(µ) =1for any µ Wϖ r.

6 EXISTENCE OF KR CRYSTALS 191 (v) wt W (ϖ r ) s contaned n the ntersecton of cl(ϖ r + I Zα ) and the convex hull of Wcl(ϖ r ). (v) W (ϖ r ) has a global crystal base. (v) Any rreducble fnte-dmensonal ntegrable U q (g)-module wth cl(ϖ r) as an extremal weght s somorphc to W (ϖ r ) a for some a Q(q s ). We also need the followng lemma that ensures the exstence of the prepolarzaton on W (ϖ r ). Lemma 2.5 ([33, 23]). (z r u, z r v)=(u, v) for u, v V (ϖ r ). Remark 2.1. Ths lemma s gven as Proposton 7.3 of [33] and also as Lemma 4.7 of [23]. The lemmas or propertes used to prove t hold for any affne algebra g. Summng up the above dscussons we have Proposton 2.6. The fundamental representaton W (ϖ r ) has the followng propertes: () W (ϖ r ) has a polarzaton (, ). () There exsts a U q(g) Z -submodule W (ϖ r ) Z of W (ϖ r ) such that (W (ϖ r ) Z,W(ϖ r ) Z ) Z[q s,qs 1 ]. Before fnshng ths secton, let us menton the Drnfeld polynomals. It s known that rreducble fnte-dmensonal U q(g)-modules are classfed by I -tuple of polynomals {P j (u)} j I whose constant terms are 1. See e.g. [4]. The degree of P j s gven by λ, h j where λ s the hghest weght of the correspondng module. Hencewehave Lemma 2.7. W (ϖ r ) has the followng Drnfeld polynomals P r (u) =1 a ru, P j (u) =1for j r wth some a r Q(q s ). For types A (1) n,d n (1),E (1) 6,7,8 the explct value of a r s known [23, Remark 3.3]. 3. KR modules and the exstence of crystal bases 3.1. Fuson constructon. Let V be a U q(g)-module. An R-matrx, denoted by R(x, y), s an element of Hom U q (g)[x ±1,y ±1 ](V x V y,v y V x ). For V we assume the followng: (3.1) V V s rreducble. (3.2) There exsts λ P cl such that wt V λ + I Z α and dm V λ =1. Under these assumptons t s known (see e.g. [13]) that there exsts a unque R- matrx up to multple of a scalar functon of x, y. Take a nonzero vector u from V λ. We normalze R(x, y) n such a way that R(x, y)(u u )=u u. The normalzed R-matrx s known to depend only on x/y. Because of the normalzaton, some matrx elements of R(x, y) may have zeros or poles as a functon of x/y. At the ponts x/y = x /y Q(q s ) where there s no zero or pole, R(x,y )san somorphsm. Next we revew the fuson constructon followng secton 3 of [14]. Let s be a postve nteger and S s the s-th symmetrc group. Let s be the smple reflecton

7 192 MASATO OKADO AND ANNE SCHILLING whch nterchanges and +1,andletl(w) be the length of w S s.letr(x, y) denote the R-matrx for V x V y. For any w S s we can construct a well-defned map R w (x 1,...,x s ):V x1 V xs V xw(1) V xw(s) by R 1 (x 1,...,x s )=1, R s (x 1,...,x s )= d Vxj R(x,x +1 ) d Vxj, j< j>+1 R ww (x 1,...,x s )=R w (x w(1),...,x w(s) ) R w (x 1,...,x s ) for w, w such that l(ww )=l(w)+l(w ). Fx k d 1 Z \{}. Let us assume that (3.3) the normalzed R-matrx R(x, y) does not have a pole at x/y = q 2k. For each s Z >, we put R s =R w (q k(s 1),q k(s 3),...,q k(s 1) ): V q k(s 1) V q k(s 3) V q k(s 1) V q k(s 1) V q k(s 3) V q k(s 1), where w s the longest element of S s.thenr s s a U q(g)-lnear homomorphsm. Defne V s =ImR s. Let us denote by W the mage of R(q k,q k ):V q k V q k V q k V q k and by N ts kernel. Then we have (3.4) V s consdered as a submodule of V s = V q k(s 1) V q k(s 1) s contaned n s 2 = V W V (s 2 ). Smlarly, we have s 2 (3.5) V s s a quotent of V s / V N V (s 2 ). = In the sequel, followng [14] we defne a prepolarzaton on V s and study necessary propertes. Frst we recall the followng lemma. Lemma 3.1 ([14, Lemma 3.4.1]). Let M j and N j be U q(g)-modules and let (, ) j be an admssble parng between M j and N j (j =1, 2). Then the parng (, )between M 1 M 2 and N 1 N 2 defned by (u 1 u 2,v 1 v 2 )=(u 1,v 1 ) 1 (u 2,v 2 ) 2 for all u j M j and v j N j s admssble. Let V be a fnte-dmensonal U q(g)-module satsfyng (3.1) and (3.2). Suppose V has a polarzaton. The polarzaton on V gves an admssble parng between V x and V x 1. Hence t nduces an admssble parng between V x1 V xs V x 1. s V x 1 1 Lemma 3.2 ([14, Lemma 3.4.2]). If x j = x 1 s+1 j for j =1,...,s, then for any u, u V x1 V xs, we have (u, R w (x 1,...,x s )u )=(u,r w (x 1,...,x s )u). and

8 EXISTENCE OF KR CRYSTALS 193 By takng x = q k(s 2+1), we obtan the admssble parng (, ) between W = V q k(s 1) V q k(s 3) V q k(s 1) and W = V q k(s 1) V q k(s 3) V q k(s 1) that satsfes (3.6) (w, R s w )=(w,r s w) for any w, w W. Ths allows us to defne a prepolarzaton (, ) s on V s by (R s u, R s u ) s =(u, R s u ) for u, u V q k(s 1) V q k(s 3) V q k(s 1). Assume (3.7) V admts a U q(g) KZ -submodule V KZ such that (V KZ ) λ = K Z u. Let us further set (V s ) KZ = R s ((V KZ ) s ) (V KZ ) s. Then [14, Proposton3.4.3] follows: Proposton 3.3. () (, ) s s a nondegenerate prepolarzaton on V s. () (R s (u s ),R s(u s )) s =1. () ((V s ) KZ, (V s ) KZ ) s K Z KR modules. We want to apply the fuson constructon wth V beng the fundamental representaton W (ϖ r ). Let us take k to be (α r,α r )/2 except n the case k =1wheng = A (2) 2n and r = n. Proposton 3.4. Assumptons (3.1), (3.2), (3.3), and(3.7) hold for the fundamental representatons. Proof. (3.1) s a consequence of Proposton 2.4(v) and the fact that B(ϖ r )sa smple crystal (see [17]). (3.2) s vald by Proposton 2.4(v) wth λ = cl(ϖ r ). Notng that W (ϖ r ) s a good U q(g)-module, (3.3) s the consequence of Proposton 9.3 of [17]. (3.7) s vald, snce W (ϖ r )admtsau q(g) Z -submodule W (ϖ r ) Z nduced from V (ϖ r ) Z such that (W (ϖ r ) Z ) cl(ϖr ) = Z[q s,q 1 s ]u ϖr. For r I and s Z > we defne the U q(g)-module W s (r) to be the module constructed by the fuson constructon n secton 3.1 wth V = W (ϖ r )andk = (α r,α r )/2 except n the case k =1wheng = A (2) 2n and r = n. Proposton 3.5. () There exsts a prepolarzaton (, ) on W s (r). () There exsts a U q(g) KZ -submodule (W s (r) ) KZ of W s (r) such that ((W s (r) ) KZ, (W s (r) ) KZ ) K Z. () There exsts a vector u of weght sϖ r n (W s (r) ) KZ such that (u,u )=1. Proof. The results follow from Propostons 3.3 and 3.4. The followng proposton s an easy consequence of the man result of Kashwara [17]. Note also that hs result can be appled not only to KR modules but also to any rreducble modules.

9 194 MASATO OKADO AND ANNE SCHILLING Proposton 3.6. W s (r) s rreducble and ts Drnfeld polynomals are gven by { (1 a P j (u) = r qr 1 s u)(1 a rqr 3 s u) (1 a rqr s 1 u) (j = r), 1 (j r), except when g = A (2) 2n and r = n. Ifg = A(2) 2n q r wth q n the above formula. and r = n, they are gven by replacng Proof. Let V be a nonzero submodule of V s = W s (r). To show the rreducblty, t suffces to show that any vector v n V s s contaned n V. By defnton there exsts a vector u W (ϖ r ) s such that v = R s u. From Theorem 9.2() of [17] we have u s V. From Theorem 9.2() of loc. ct. there exsts x U q(g) such that u = (s) (x)u s,where (s) s the coproduct U q(g) U q(g) s. Hence we have v = R s (s) (x)u s = (s) (x)r s u s = (s) (x)u s V. Snce W s (r) s the rreducble module n (W (r) 1 ) q 1 s (W (s) r 1 ) qr 3 s (W (r) 1 ) q s 1 r generated by u s, the latter statement s clear from [4, Corollary 3.5], Lemma 2.7 and the fact that f V corresponds to {P j (u)}, thenv a corresponds to {P j (au)}. Ths rreducble U q(g)-module W s (r) s called Krllov Reshetkhn (KR) module: Snce the KR module W s (r) s also a U q (g )-module by restrcton, we have the followng drect sum decomposton as a U q (g )-module: (3.8) W (r) s N s (r) λ P + (λ) V (λ) Namely, N s (r) (λ) s the multplcty of the rreducble U q (g )-module V (λ) nw (r) Then we have a crteron that the KR module has a crystal pseudobase. Proposton 3.7. Suppose for any λ P + such that N s (r) (λ) > there exst u(λ) j (W s (r) ) KZ of weght λ for j =1,...,N s (r) (λ). If we have (u(λ) j,u(λ) k ) δ jk + q s A and (e j u(λ) k,e j u(λ) k ) q s q 2(1+ h j,λ ) j A for any j I,then(, ) on W s (r) s a polarzaton, and W s (r) has a crystal pseudobase. Proof. We use Proposton 2.1. All the assumptons except (2.4) are satsfed by Propostons 3.5. Note that (u(λ) j,u(µ) k )=fλ µ. Remark 3.1. From the prevous proposton t mmedately follows that f W s (r) s rreducble as a U q (g )-module, then t has a crystal pseudobase (see also [14, Proposton 3.4.4]). There s another case n whch the exstence of crystal pseudobase s proven for any l and any g except A (1) n as n [14, Proposton 3.4.5]. It corresponds to r =2wheng = B n (1),D n (1),A (1) 2n 1, r =6wheng = E(1) 6,andr =1 n all other cases. Here we follow the labelng of vertces of the Dynkn dagram by [12]. We remark that the crystal base of W (r) 1 for such r s treated n [2]. There s an explct formula of N s (r) (λ) called the (q = 1) fermonc formula. We have [3, 7, 8, 9, 1, 2, 24, 25] for references. To explan t, we ntroduce t and t for I by { 2 t = (α,α ) f g s untwsted, 1 fg s twsted, s.

10 EXISTENCE OF KR CRYSTALS 195 and t =(t for g ), where g s the dual Kac-Moody algebra to g. For p Z and m Z let ( ) ( p+m m stand for the bnomal coeffcent,.e., p+m ) m m = k=1 Then, for r I,s Z > and λ P + we have N s (r) (λ) = ( (a) p j + m (a) ) j, m a I,j 1 m (a) j p+k k. where p (a) j = δ a mn(j, s) 1 t a b I,k 1 (α a,α b )mn(t b j, t a k)m (b) k and the sum m s taken over all (m(a) j Z a I,j 1) satsfyng jm (a) j α a = sϖ r λ. a I,j 1 The proof of ths formula goes as follows. Set Q (r) s that Q (r) s = λ P + N (r) s. It suffces to show (λ)ch V (λ). By Theorem 8.1 of [8] (see also Theorem 6.3 of =chw (r) s [7] ncludng the twsted cases), t suffces to show that {Q (r) s } satsfes the condtons (A), (B), (C) n the theorem. (A) s evdent by the constructon of W s (r), and (B), (C) were verfed n [24, 9, 1] for the smply-laced, untwsted and twsted cases, respectvely. Note that condton (C) s replaced wth another convergence property (4.15) of [21]. Note also that there s an earler result by Char [3] for untwsted cases. It should also be noted that there s another explct formula M s (r) (λ) for the multplctes ) N s (r) (λ) whch nvolves unsgned bnomal coeffcents, that s, =fp< [8, 7]. It was recently shown by D Francesco and Kedem [6] ( p+m m that M (r) s (λ) =N s (r) (λ) n the untwsted cases. For nonexceptonal types, the explct value of N s (r) (λ) can be found n Secton 7 of [8] for untwsted cases, and n secton 6.2 of [7] for twsted cases. See (4.1). 4. Exstence of crystal pseudobases for nonexceptonal types In ths secton we show that any KR module for nonexceptonal type has a crystal pseudobase. For type A (1) n ths fact s establshed n [14]. So we do not deal wth the A (1) n case Dynkn data. Frst we lst the Dynkn dagrams of all nonexceptonal affne algebras except A (1) n n Table 1. We also lst the par (ν, g ) n the table wth a partton ν =,, and a smple Le algebra g whose Dynkn dagram s the one obtaned by removng the -vertex. Note that the dfference of ν comes from the dagram near the -vertex. ThesmplerootsfortypeB n,c n,d n are α = ɛ ɛ +1 for 1 <n, ɛ n 1 + ɛ n for type D n, α n = ɛ n for type B n, 2ɛ n for type C n,

11 196 MASATO OKADO AND ANNE SCHILLING B n (1) 2 n 1 1 C (1) n 1 D n (1) 2 n 2 A (2) 2n 1 1 A (2) 2n 1 2 n 1 1 D (2) n+1 1 n (,B n ) n 1 n (,C n ) n 1 (,D n ) n n 1 n (,C n ) n (,C n ) n 1 n (,B n ) Table 1. Dynkn dagrams and the fundamental weghts are Type D n : ϖ = ɛ ɛ for 1 n 2, ϖ n 1 =(ɛ ɛ n 1 ɛ n )/2, ϖ n =(ɛ ɛ n 1 + ɛ n )/2; Type B n : ϖ = ɛ ɛ for 1 n 1, ϖ n =(ɛ ɛ n 1 + ɛ n )/2; Type C n : ϖ = ɛ ɛ for 1 n, where ɛ ( =1,...,n) are vectors n the weght space of each smple Le algebra. (By conventon we set ϖ =.) These elements can be vewed as those of the weght lattce P of the affne algebra n Table 1. On P we defned the nner product (, ) normalzed as (δ, λ) = c, λ for λ P. Ths normalzaton s equvalent to settng (ɛ,ɛ j )=κδ j wth κ = 1 2 for C(1) n,=2ford (2) n+1, and = 1 for the other types. However, n ths secton we renormalze t by (ɛ,ɛ j )=δ j. Thssequvalentto settng (α,α )/2 =1for not an end node of the Dynkn dagram. We also note that α = δ ɛ 1 ɛ 2 f ν =, δ 2ɛ 1 f ν =, δ ɛ 1 f ν = Exstence of crystal pseudobases for KR modules. We frst present the branchng rule of KR modules of affne type lsted n Table 1 wth respect to the subalgebra U q (g ). They can be found n [8, Theorems 7.1 and 8.1] and

12 EXISTENCE OF KR CRYSTALS 197 [7, Theorems 6.2 and 6.3]. For I for g we say s a spn node f the vertex s flled n Table 1. If r I s a spn node, then the KR module W s (r) s rreducble as a U q (g )-module: W s (r) V (sϖ r ). Suppose now that r I s not a spn node. Let ω be a domnant ntegral weght of the form of ω = c ϖ. Assume c =for a spn node. In the standard way we represent ω by the partton that has exactly c columns of heght. Then the KR module W s (r) decomposes nto (4.1) W s (r) V (ω) ω as a U q (g )-module, where ω runs over all parttons that can be obtaned from the r s rectangle by removng peces of shape ν (wth ν as n Table 1). If r I s a spn node, the KR module W s (r) has a crystal pseudobase by Remark 3.1. Suppose r s not a spn node. As we have seen, we have N s (r) (λ) 1. Hence, by Proposton 3.7, n order to show the exstence of crystal pseudobase, t suffces to defne a vector u(λ) (W s (r) ) KZ of weght λ for any λ such that N s (r) =1,and show (u(λ),u(λ)) 1+q s A and (e j u(λ),e j u(λ)) q s q 2(1+ h j,λ ) j A for j I.In the subsequent subsectons, we do ths task by dvdng t nto 3 cases accordng to the shape of ν Calculaton of prepolarzaton: D n (1),B n (1),A (2) 2n 1 cases. We assume 1 r n 2forD n (1),1 r n 1forB n (1) and 1 r n for A (2) 2n 1.Letr =[r/2]. Let c =(c 1,c 2,...,c r ) be a sequence of ntegers such that s c 1 c 2 c r. For such c we defne a vector u m ( m r )nw s (r) nductvely by u m =(e (c m) r 2m e(c m) 2 e (c m) 1 )(e (c m) r 2m+1 e(c m) 3 e (c m) 2 )e (c m) u m 1, where u s the vector n () of Proposton 3.5. Set u(c) =u r. The weght of u(c) sgvenby r λ(c) = (c j c j+1 )ϖ r 2j, j= wherewehavesetc = s, c r +1 =,andϖ should be understood as. λ(c) represents all ω n (4.1) when c runs over all possble sequences. For l, m Z such that m l we defne the q-bnomal coeffcent by [ ] l [l]! (4.2) = m [m]![l m]!. The followng proposton calculates values of the prepolarzaton (, )onw s (r). Proposton 4.1. r (1) (u(c),u(c)) = j=1 q c j(2s c j ) [ 2s c j ], (2) (e j u(c),e j u(c)) = unless r j 2Z. If r j 2Z, then settng p =(r j)/2+1, (e j u(c),e j u(c)) s gven by r [ ] q 2s cp 1 1 [2s c p 1 ] q (c j δ j,p )(2s c j ) 2s δj,p. c j δ j,p j=1

13 198 MASATO OKADO AND ANNE SCHILLING For type D n (1) ths proposton s proven n [27]. The proof goes completely parallel also for type B n (1) and A (2) 2n 1. Note that q = q for n, q n = q, q 1/2,q 2 for D n (1),B n (1),A (2) 2n 1, respectvely, and q s = q 1/2 for B n (1),=q for D n (1),A (2) 2n 1.Snce q m 1 [m],q n(m n) 1+qA and h j,λ(c) = c p 1 c p, we have (u(c),u(c)) 1+q s A and (e j u(c),e j u(c)) q s q 2(1+ h j,λ(c) ) j A, forj I. Ths establshes the condtons of Proposton 3.7 and hence proves Theorem 1.1 that W s (r) has a crystal pseudobase. We denote the crystal of W s (r) by B r,s. Smlar to g one can consder g 1,whchs another (mutually somorphc) smple Le algebra obtaned by removng the vertex 1 from the Dynkn dagram of g. The followng proposton wll be used to show that B r,s s somorphc to B r,s, whch s gven combnatorally n the next secton. Proposton 4.2. Let 1 r n 2 for g = D n (1), 1 r n 1 for g = B n (1), 1 r n for g = A (2) 2n 1,ands Z >. Then for =, 1, B r,s decomposes as U q (g )-crystals nto B r,s B g (σ (ϖ r 2m1 + + ϖ r 2ms )). m 1 m s [r/2] Here B g (λ) s the crystal base of the hghest weght U q (g )-module of hghest weght λ, andσ s the automorphsm on P such that σ(λ )=Λ 1,σ(Λ 1 )=Λ,σ(Λ j )=Λ j (j >1) and extended lnearly. Proof. If =, the clam s a drect consequence of (4.1). For = 1 note that the Weyl group of g contans an element w whch sends ϖ j to σ(ϖ j ) for any j such that j r, where by conventon ϖ =. (Usng the orthogonal bass {ɛ } of secton 4.1 of the weght space of g, we can take an element w such that w(ɛ )=( 1) δ() ɛ,whereδ() =1f =1,n for g = D n (1), =1forg = B n (1) and (r),andδ() = otherwse.) Snce W s a drect sum also as a U q (g 1 )-module, A (2) 2n 1 t s enough to show the followng equalty of characters. (4.3) ch W s (r) = ch V g 1 (σ(ϖ r 2m1 + + ϖ r 2ms )) m 1 m s [r/2] s Here V g 1 (λ) denotes the hghest weght U q (g 1 )-module of hghest weght λ. But notng w(α )=α 1,w(α 1 )=α,w(α j )=α j (j>1) on P cl, (4.3) s shown from ch W s (r) = ch V g (ϖ r 2m1 + + ϖ r 2ms ) m 1 m s [r/2] snce w preserves the weght multplcty Calculaton of prepolarzaton: C n (1) case. We assume 1 r n 1. Let c =(c 1,c 2,...,c r ) be a sequence of ntegers such that [s/2] c 1 c 2 c r. For such c we defne a vector u m ( m r) nw s (r) nductvely by u m = e (2c m) r m e(2c m) 2 e (2c m) 1 e (c m) u m 1, where u s the vector n () of Proposton 3.5. Set u(c) =u r.theweghtofu(c) s gven by r λ(c) = 2(c j c j+1 )ϖ r j, j=

14 EXISTENCE OF KR CRYSTALS 199 wherewehavesetc = s/2,c r+1 =,andϖ should be understood as. λ(c) represents all ω n (4.1) when c runs over all possble sequences. In ths subsecton, besdes (4.2) we also use [ ] l m defned by (4.2) wth q replaced by q = q 2. (Recall that we have renormalzed the nner product (, )onp n such a way that (ɛ,ɛ j )= δ j.) We are to calculate the values of (u(c),u(c)) and (e j u(c),e j u(c)). Snce the calculaton goes parallel to the case of D n (1) treated n [27], we only gve here ntermedate results as a lemma. We wrte u 2 for (u, u). Lemma 4.3. (1) u m 2 = q c m(s c m ) [ s c m ] u m 1 2, (2) e j u(c) =f j>r, (3) e j u(c) 2 = q 2β j f j u(c) 2 + q βj 1 [β j ] u(c) 2 f 1 j r, whereβ j = h j,λ(c) =2(c r+1 j c r j ), (4) f j u(c) 2 = 1 m r m r j+1 From ths lemma we have [ ] q c m(s c m ) s c m q c r j+1(s 1 c r j+1 ) [ ] s 1 c r j+1 q 2c r j 1 [2c r j ]. Proposton 4.4. (1) (u(c),u(c)) = r [ m=1 qc m(s c m ) s c m ], (2) q 2s 2cr j 1 [2s 2c r j ] (e j u(c),e j u(c)) = r m=1 q(c m δ m,r j+1 )(s c m )[ s δm,r j+1 ] c m δ m,r j+1 f 1 j r, f r<j n. Note that q = q for,n, q n = q 2,andq s = q under the renormalzaton. Snce h j,λ(c) = β j =2(c r j c r+1 j ), we have (u(c),u(c)) 1+q s A and (e j u(c),e j u(c)) q s q 2(1+ h j,λ(c) ) j A for j I. By Proposton 3.7 ths proves Theorem Calculaton of prepolarzaton: A (2) 2n,D(2) n+1 cases. We assume 1 r n for A (2) 2n and 1 r n 1forD(2) n+1. Let c =(c 1,c 2,...,c r ) be a sequence of ntegers such that s c 1 c 2 c r. For such c we defne a vector u m ( m r) nw s (r) nductvely by u m = e (c m) r m e (c m) 1 e (c m) u m 1, where u s the vector n () of Proposton 3.5. Set u(c) =u r.theweghtofu(c) s gven by r λ(c) = (c j c j+1 )ϖ r j, j=

15 2 MASATO OKADO AND ANNE SCHILLING wherewehavesetc = s, c r+1 =,andϖ should be understood as. λ(c) represents all ω n (4.1) when c runs over all possble sequences. In ths subsecton, besdes (4.2) we also use [ ] l m defned by (4.2) wth q replaced by q = q 1/2. As n the prevous subsecton, we only gve here ntermedate results as a lemma. As before we wrte u 2 for (u, u). Lemma 4.5. (1) u m 2 = q c m(2s c m )[ 2s c m ] u m 1 2, (2) e j u(c) =f j>r, (3) e j u(c) 2 = q 2β j f j u(c) 2 + q βj 1 [β j ] u(c) 2 f 1 j r, whereβ j = h j,λ(c) = c r+1 j c r j, (4) r [ ] 2s 2δ f j u(c) 2 (1) = q cr j 1 [c r j ] + m=1 r m=1 q c m(2s 2δ (1) c m ) c m q (c m+δ (1) δ (2) )(2s δ (1) +δ (2) c m ) [2s c r j +1] 2, where δ (1) = δ m,r j+1,δ (2) = δ m,r j. From ths lemma we have [ 2s 2δ (1) ] c m δ (1) δ (2) Proposton 4.6. (1) (u(c),u(c)) = r [ m=1 qc m(2s c m ) 2s c m ], (2) { q 2β j f j u(c) 2 + q βj 1 [β j ] u(c) 2 f 1 j r, (e j u(c),e j u(c)) = f r<j n, where β j and f j u(c) 2 are gven n the prevous lemma. Note that q = q for,n, q n = q 2 for A (2) 2n,=q1/2 for D (2) n+1,andq s = q 1/2 under the renormalzaton. Snce h j,λ(c) = β j = c r j c r+1 j, we have (u(c),u(c)) 1+q s A and (e j u(c),e j u(c)) q s q 2(1+ h j,λ(c) ) j A for j I. By Proposton 3.7 ths proves Theorem Combnatoral crystal B r,s of type D (1) n,b (1) n,a (2) 2n 1 In ths secton we revew the combnatoral crystal Br,s of [3, 32] of type D n (1), B n (1),andA (2) 2n 1 and prove some prelmnary results that wll be needed n Secton 6 to establsh the equvalence of B r,s and B r,s Type D n, B n,andc n crystals. Crystals assocated wth a U q (g)-module when g s a smple Le algebra of nonexceptonal type, were studed by Kashwara and Nakashma [18]. Here we revew the combnatoral structure n terms of tableaux of the crystals of type X n = D n, B n,andc n snce these are the fnte subalgebras relevant to the KR crystals of type D n (1), B n (1),andA (2) 2n 1. For g = D n (1),B n (1),orA (2) 2n 1,anyg domnant weght ω wthout a spn component can be expressed as ω = c ϖ for nonnegatve ntegers c and the sum

16 EXISTENCE OF KR CRYSTALS 21 D (1) n n n-2 n-1 n n n n n-1 n-2 n B (1) n n-1 n n n-1 n n A (2) 2n n-1 n n-1 n n Table 2. KR crystal B 1,1 runs over all =1, 2,...,n not a spn node. As explaned earler we represent ω by the partton that has exactly c columns of heght. FortypeD n,thscanbe extended by assocatng a column of heght n 1wthϖ n 1 + ϖ n and a column of heght n wth 2ϖ n. For type B n one may assocate a column of heght n wth 2ϖ n.conversely,fω s a partton, we wrte c (ω) for the number of columns of ω of heght. From now on we dentfy parttons and domnant weghts n ths way. The crystal graph B(ϖ 1 ) of the vector representaton for type D n, B n,andc n s gven n Table 2 by removng the arrows n the crystal B 1,1 of type D (1) n, B (1) n, and A (2) 2n 1, respectvely. The crystal B(ϖ l)forl not a spn node can be realzed as the connected component of B(ϖ 1 ) l contanng the element l (l 1) 1, where we use the ant-kashwara conventon for tensor products. Smlarly, the crystal B(ω) labeled by a domnant weght ω = ϖ l1 + + ϖ lk wth l 1 l 2 l k not contanng spn nodes can be realzed as the connected component n B(ϖ l1 ) B(ϖ lk ) contanng the element u ϖl1 u ϖlk,whereu ϖ s the hghest weght element n B(ϖ ). As shown n [18], the elements of B(ω) canbe labeled by tableaux of shape ω n the alphabet {1, 2,...,n,n,...,1} for types D n and C n and the alphabet {1, 2,...,n,, n,...,1} for type B n. For the explct rules of type D n, B n,andc n tableaux we refer the reader to [18]; see also [11] Defnton of B r,s. Let g be of type D n (1), B n (1),orA (2) 2n 1 wth the underlyng fnte Le algebra g of type X n = D n,b n,orc n, respectvely. The combnatoral crystal Br,s s defned as follows. As an X n -crystal, Br,s decomposes nto the followng rreducble components (5.1) Br,s = B(ω), ω for 1 r n not a spn node. Here B(ω) sthex n -crystal of hghest weght ω and the sum runs over all domnant weghts ω that can be obtaned from sϖ r by

17 22 MASATO OKADO AND ANNE SCHILLING the removal of vertcal domnoes, where ϖ are the fundamental weghts of X n as defned n secton 5.1. The addtonal operators ẽ and f are defned as f = σ f 1 σ, (5.2) ẽ = σ ẽ 1 σ, where σ s the crystal analogue of the automorphsm of the Dynkn dagram that nterchanges nodes and 1. The nvoluton σ s defned n Defnton Defnton of σ. To defne σ we frst need the noton of ± dagrams. A ± dagram P of shape Λ/λ s a sequence of parttons λ µ Λ such that Λ/µ and µ/λ are horzontal strps. We depct ths ± dagram by the skew tableau of shape Λ/λ n whch the cells of µ/λ are flled wth the symbol + and those of Λ/µ are flled wth the symbol. Wrte Λ = outer(p )andλ = nner(p ) for the outer and nner shapes of the ± dagram P. For type A (2) 2n 1 and r = n, the nner shape λ s not allowed to be of heght n. When drawng parttons or tableaux, we use the French conventon where the parts are drawn n ncreasng order from top to bottom. There s a bjecton Φ : P b from ± dagrams P of shape Λ/λ to the set of X n 1 -hghest weght vectors b of X n 1 -weght λ n B Xn (Λ). Here X n 1 s the subalgebra whose Dynkn dagram s obtaned from that of X n by removng node 1. There s a natural homomorphsm of the weght lattces π : P (X n ) P (X n 1 ), )=α X n 1 1 and π(ϖ X n )=ϖ X n 1 1, and the partton λ s dentfed where π(α X n wth the X n 1 weghts under π. We dentfy the Kashwara operators f X n 1 wth under the embeddng. Explctly, the bjecton Φ s constructed as follows. Defne a strng of operators f X n f a := f a1 fa2 f al such that Φ(P )= f a u,whereu s the hghest weght vector n B Xn (Λ), where f s the Kashwara crystal operator correspondng to f. Start wth a = (). Scan the columns of P from rght to left. For each column of P for whch a + can be added, append (1, 2,...,h)to a,whereh s the heght of the added +. Next scan P from left to rght and for each column that contans a n P,appendto a the strng (1, 2,...,n,n 2,n 3,...,h)fortypeD n, (1, 2,...,n 1,n,n,n 1,...,h)fortypeB n,and(1, 2,...,n 1,n,n 1,...,h) for type C n,whereh s the heght of the n P. Note that for type C n the strngs (1, 2,...,h)and(1, 2,...,n 1,n,n 1,...,h) are the same for h = n, whchs why empty columns of heght n are excluded for ± dagrams of type A (2) 2n 1. By constructon the automorphsm σ commutes wth f and ẽ for =2, 3,...,n. Hence t suffces to defne σ on X n 1 hghest weght elements. Because of the bjecton Φ between ± dagrams and X n 1 -hghest weght elements, t suffces to defne the map on ± dagrams. Let P be a ± dagram of shape Λ/λ. Let c = c (λ) be the number of columns of heght n λ for all 1 <rwth c = s λ 1. If r 1 (mod 2), then n P, above each column of λ of heght, theremustbea+ora. Interchange the number of such + and symbols. If r (mod 2), then n P, above each column of λ of heght, ether there are no sgns or a par. Suppose there are p pars above the columns of heght. Change ths to (c p ) pars. The result s S(P ), whch has the same nner shape λ as P but a possbly dfferent outer shape.

18 EXISTENCE OF KR CRYSTALS 23 Defnton 5.1. Let b B r,s and ẽ a := ẽ a1 ẽ a2 ẽ al be such that ẽ a (b) sax n 1 hghest weght crystal element. Defne f a := f al fal 1 f a1.then (5.3) σ(b) := f a Φ S Φ 1 ẽ a (b). It was shown n [3] that B r,s s regular Propertes of B r,s. For the proof of unqueness we wll requre the acton of ẽ 1 on X n 2 hghest weght elements, where X n 2 s the Dynkn dagram obtaned by removng nodes 1 and 2 from X n. As we have seen n secton 5.3, the X n 1 -hghest weght elements n the branchng X n X n 1 can be descrbed by ± dagrams. Smlarly the X n 2 -hghest weght elements n the branchng X n 1 X n 2 can be descrbed by ± dagrams. Hence each X n 2 -hghest weght vector s unquely determned by a par of ± dagrams (P, p) such that nner(p )=outer(p). The dagram P specfes the X n 1 -component B Xn 1 (nner(p )) n B Xn (outer(p )), and p specfes the X n 2 component nsde B Xn 1 (nner(p )). Let Υ denote the map (P, p) b from a par of ± dagrams to a X n 2 hghest weght vector. To descrbe the acton of ẽ 1 on an X n 2 hghest weght element or by Υ equvalently on (P, p) perform the followng algorthm: (1) Successvely run through all + n p from left to rght and, f possble, par t wth the leftmost yet unpared + n P weakly to the left of t. (2) Successvely run through all n p from left to rght and, f possble, par t wth the rghtmost yet unpared n P weakly to the left. (3) Successvely run through all yet unpared + n p from left to rght and, f possble, par t wth the leftmost yet unpared n p. Lemma 5.1 ([3, Lemma 5.1]). If there s an unpared + n p, ẽ 1 moves the rghtmost unpared + n p to P. Otherwse, f there s an unpared n P, ẽ 1 moves the leftmost unpared n P to p. Otherwse,ẽ 1 annhlates (P, p). In ths paper, we wll only requre the case of Lemma 5.1 when a from P moves to p. Schematcally, f a from a par n P moves to p, then the followng happens: or , where the blue mnus s the mnus n P that s beng moved and the red mnus s the new mnus n p. Smlarly, schematcally f a not part of a par n P moves to p, then or For any b B r,s, let nner(b) be the nner shape of the ± dagram correspondng to the X n 1 hghest weght element n the component of b. Furthermore, recall that B r,s s regular, so that n partcular ẽ and ẽ 1 commute. We can now state the lemma needed n the next secton. Lemma 5.2. Let b B r,s be an X n 2 hghest weght vector correspondng under Υ to the tuple of ± dagrams (P, p) where nner(p) =outer(p). Assume that ε (b),ε 1 (b) >. Then nner(b) s strctly contaned n nner(ẽ (b)), nner(ẽ 1 (b)), and nner(ẽ ẽ 1 (b)).

19 24 MASATO OKADO AND ANNE SCHILLING Proof. By assumpton p does not contan any and ẽ 1 s defned. Hence ẽ 1 moves a n P to p. Ths mples that the nner shape of b s strctly contaned n the nner shape of ẽ 1 (b). The nvoluton σ does not change the nner shape of b (only the outer shape). By the same arguments as before, the nner shape of b s strctly contaned n the nner shape of ẽ 1 σ(b). Snce σ does not change the nner shape, ths s stll true for ẽ (b) =σẽ 1 σ(b). Now let us consder ẽ ẽ 1 (b). For the change n nner shape we only need to consder ẽ 1 σẽ 1 (b), snce the last σ does not change the nner shape. By the same arguments as before, ẽ 1 moves a from P to p and σ does not change the nner shape. The next ẽ 1 wll move another n σẽ 1 (b) top. Hence p wll have grown by two, so that the nner shape of ẽ 1 σẽ 1 (b) s ncreased by two boxes. 6. Equvalence of B r,s and B r,s of type D (1) n, B (1) n, and A (2) 2n 1 In ths secton all crystals are of type D n (1), B n (1),orA (2) 2n 1 wth correspondng classcal subalgebra of type X n = D n,b n,c n, respectvely. Let B and B be regular crystals of type D n (1), B n (1),orA (2) 2n 1 wth ndex set I = {, 1, 2,...,n}. WesaythatB B s an somorphsm of J-crystals f B and B agree as sets and all arrows colored J are the same. Proposton 6.1. Suppose that there exst two somorphsms: Ψ : B r,s B as an somorphsm of {1, 2,...,n}-crystals, Ψ 1 : B r,s B as an somorphsm of {, 2,...,n}-crystals. Then Ψ (b) =Ψ 1 (b) for all b B r,s and hence there exsts an I-crystal somorphsm Ψ: B r,s B. Remark 6.1. Note that Ψ and Ψ 1 preserve weghts, that s, wt (b) =wt(ψ (b)) = wt (Ψ 1 (b)) for all b B r,s. Ths s due to the fact that f all but one coeffcent m j are known for a weght Λ = n j= m jλ j, then the mssng m j s also determned by the level condton. Proof. If Ψ (b) =Ψ 1 (b) forab n a gven X n 1 -component C, thenψ (b )=Ψ 1 (b ) for all b C snce ẽ Ψ (b ) = Ψ (ẽ b )andẽ Ψ 1 (b ) = Ψ 1 (ẽ b )for J = {2, 3,...,n}. Hence t suffces to prove Ψ (b) =Ψ 1 (b) for only one element b n each X n 1 -component C. We are gong to establsh the theorem for b correspondng to the pars of ± dagrams (P, p) where nner(p) = outer(p). Note that ths s an X n 2 -hghest weght vector, but not necessarly an X n 1 -hghest weght vector. We proceed by nducton on nner(b) by contanment. Frst suppose that both ε (b),ε 1 (b) >. By Lemma 5.2, the nner shape of ẽ ẽ 1 b,ẽ b,andẽ 1 b s bgger than the nner shape of b, so that by nducton hypothess Ψ (ẽ ẽ 1 b)=ψ 1 (ẽ ẽ 1 b), Ψ (ẽ b)=ψ 1 (ẽ b), and Ψ (ẽ 1 b)=ψ 1 (ẽ 1 b). Therefore we obtan ẽ ẽ 1 Ψ (b) =ẽ Ψ (ẽ 1 b)=ẽ Ψ 1 (ẽ 1 b)=ψ 1 (ẽ ẽ 1 b)=ψ (ẽ ẽ 1 b) =ẽ 1 Ψ (ẽ b)=ẽ 1 Ψ 1 (ẽ b)=ẽ 1 ẽ Ψ 1 (b). Ths mples that Ψ (b) =Ψ 1 (b). Next we need to consder the cases when ε (b) =orε 1 (b) =, whch comprses the base case of the nducton. Let us frst treat the case ε 1 (b) =. Recall that

20 EXISTENCE OF KR CRYSTALS 25 nner(p) = outer(p) sothatp contans only empty columns. Hence t follows from the descrpton of the acton of ẽ 1 of Lemma 5.1, that ε 1 (b) = f and only f P conssts only of empty columns or columns contanng +. Clam. Ψ (b) =Ψ 1 (b) for all b correspondng to the par of ± dagrams (P, p) wherep contans only empty columns and columns wth +, and nner(p) = outer(p). The clam s proved by nducton on k, whch s defned to be the number of empty columns n P of heght strctly smaller than r. For k = the clam s true by weght consderatons. Now assume the clam s true for all k <kand we wll establsh the clam for k. Suppose that Ψ 1 (b) =Ψ ( b) where b b. By weght consderatons b must correspond to a par of ± dagrams ( P,p), where P has the same columns contanng + as P, but some of the empty columns of P of heght h strctly smaller than r could be replaced by columns of heght h + 2 contanng. Denote by k + the number of columns of P contanng +. Then m := ε (b) =k + + k, snce under σ all empty columns n P become columns wth ± and columns contanng + become columns wth. By Lemma 5.1, then ẽ 1 acts on (S(P ),p)as often as there are mnus sgns n S(P ), whch s k + + k. Setˆb =ẽ a 1 b, wherea>s the number of columns n P contanng. If(ˆP,ˆp) denotes the tuple of ± dagrams assocated to ˆb, then compared to ( P,p)all from the pars n P moved to p. Note that ˆP has only k a<kempty columns of heght less than r, sothatby nducton hypothess Ψ (ˆb) =Ψ 1 (ˆb). Hence (6.1) Ψ 1 (b) =Ψ ( b) =Ψ ( f a 1 ˆb) = f a 1 Ψ (ˆb) = f a 1 Ψ 1 (ˆb). Note that Hence but ε (ˆb) =ε ( b) =m a<m. ẽ m Ψ 1 (b) =Ψ 1 (ẽ m b), ẽ m f a 1 Ψ 1 (ˆb) = f a 1 Ψ 1 (ẽ m ˆb) = whch contradcts (6.1). Ths mples that we must have b = b provng the clam. The case ε (b) = can be proven n a smlar fashon to the case ε 1 (b) =. Usng the explct acton of S on P and Lemma 5.1, t follows that ε (b) =fand only f P conssts only of columns contanng or pars. Clam. Ψ (b) =Ψ 1 (b) for all b correspondng to the par of ± dagrams (P, p)wherep contans only columns wth and columns wth pars, and nner(p) = outer(p). By nducton on the number of pars n P, ths clam can be proven smlarly as before (usng the fact that S changes columns wth nto columns wth + and columns wth pars nto empty columns). Proof of Theorem 1.2. Both crystals B r,s and B r,s have the same classcal decomposton (5.1) as X n crystals wth ndex set {1, 2,...,n} and {, 2, 3,...,n} by

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