Existence of Kirillov-Reshetikhin crystals for near adjoint nodes in exceptional types
aa r X i v : . [ m a t h . QA ] O c t EXISTENCE OF KIRILLOV–RESHETIKHIN CRYSTALSFOR NEAR ADJOINT NODES IN EXCEPTIONAL TYPES
KATSUYUKI NAOI AND TRAVIS SCRIMSHAW
Abstract.
We prove that, in types E (1)6 , , , F (1)4 and E (2)6 , every Kirillov–Reshetikhinmodule associated with the node adjacent to the adjoint one (near adjoint node) hasa crystal pseudobase, by applying the criterion introduced by Kang et al. In order toapply the criterion, we need to prove some statements concerning values of a bilinearform. We achieve this by using the global bases of extremal weight modules. Introduction
Let g be an affine Kac–Moody Lie algebra, and denote by U ′ q ( g ) the associated quan-tum affine algebra without the degree operator. Kirillov–Reshetikhin (KR for short)modules are a distinguished family of finite-dimensional simple U ′ q ( g )-modules (see, forexample, [CP94]). In this article KR modules are denoted by W r,ℓ , where r is a node ofthe Dynkin diagram of g except the node 0 prescribed in [Kac90] and ℓ is a positive inte-ger. KR modules are known to have several good properties, such as their q -characterssatisfy the T ( Q , Y )-system relations, fermionic formulas for their graded characters,and so on (see [HKO +
99, Nak03, Her06, Her10], for example, and references therein).Another important (conjectural) property of a KR module is the existence of a crystalbase in the sense of Kashiwara, which was presented in [HKO +
99, HKO + W r,ℓ is multiplicity free as a U q ( g )-module, it is known to havea crystal pseudobase, where g is the subalgebra of g whose Dynkin diagram is obtainedfrom that of g by removing 0. In nonexceptional types, in which all W r,ℓ are multiplicityfree, this was shown by Okado and Schilling [OS08]. Recently this was also proved forall multiplicity free W r,ℓ of exceptional types by Biswal and the second author [BS20]in a similar fashion.On the other hand, if W r,ℓ is not multiplicity free, then the conjecture has been solvedin only a few cases so far. Kashiwara showed for all affine types that all fundamentalmodules W r, have crystal bases [Kas02], and in types G (1)2 and D (3)4 , the first authorverified the existence of a crystal pseudobase for all W r,ℓ [Nao18].We say a node r is near adjoint if the distance from 0 is precisely 2. The goal of thispaper is to show the conjecture for all KR modules associated with near adjoint nodesin exceptional types. This has already been done in [Nao18] for types G (1)2 and D (3)4 ,and our main theorem below covers all remaining types. Mathematics Subject Classification.
Key words and phrases. affine quantum group, Kirillov–Reshetikhin crystal, crystal pseudobase.
Theorem 1.
Assume that g is either of type E (1) n ( n = 6 , , , F (1)4 , or E (2)6 , and r is the near adjoint node. Then for every ℓ ∈ Z > , the KR module W r,ℓ has a crystalpseudobase. In particular, since a KR module W r,ℓ in type E (1)6 is multiplicity free if r is not thenear adjoint node, Theorem 1 solves the conjecture for all KR modules of this type.As with previous works [OS08, Nao18, BS20], Theorem 1 is proved by applying thecriterion for the existence of a crystal pseudobase introduced in [KKM + ℓ ). However, this appears to be quite dif-ficult to do in our cases. Hence we apply a more sophisticated method using the globalbasis of an extremal weight module introduced by Kashiwara [Kas94]. For example, itis previously known that a global basis is almost orthonormal [Nak04], and thereforethe required almost orthonormality of given vectors is deduced by connecting them witha global basis. The other conditions are also proved in a similar spirit.Besides the KR modules treated in this paper, there are several families of W r,ℓ forwhich the existence of crystal pseudobases remain open: r = 3 , E (1)7 , 3 ≤ r ≤ E (1)8 , and r = 3 in types F (1)4 and E (2)6 , where the labeling of nodes are given inFigure 1 in Subsection 3.1. We hope to study these in our future work.The paper is organized as follows. In Section 2, we recall the basic notions needed inthe proof of the main theorem. In Subsection 3.1, we reduce the main theorem to threestatements (C1)–(C3), and these are proved in Subsections 3.2–3.4. In Subsection 3.4,we use a certain relation (3.4.15) in W r,ℓ , whose proof is postponed to Appendix Asince, while straightforward, it is slightly lengthy and technical. Acknowledgments
The authors would like to thank Rekha Biswal for helpful discussions. This work ben-efited from computations using
SageMath [Sage19]. The first author was supportedby JSPS Grant-in-Aid for Young Scientists (B) No. 16K17563. The second author waspartially supported by the Australian Research Council DP170102648.
Index of notation
We provide for the reader’s convenience a brief index of the notation which is usedrepeatedly in this paper:Subsection 2.1: g , I , C = ( c ij ) i,j ∈ I , α i , h i , Λ i , δ , P , P + , Q , Q + , W , s i , I , ̟ i , P ∗ , d , P cl , q i , D , q s , U q ( g ), e i , f i , q h , U ′ q ( g ), U q ( n ± ), e ( n ) i , wt P , U q ( g J ), t i , ∆, wt P cl . XISTENCE OF KR CRYSTALS FOR NEAR ADJOINT NODES 3
Subsection 2.2: ˜ e i , ˜ f i , A , .Subsection 2.3: k u k .Subsection 2.4: V (Λ), v Λ , L (Λ), B (Λ), B (Λ), V (Λ) Z , B (Λ , − Λ ).Subsection 2.5: M a , ι a , ι , W r,ℓ , z r , L ( W r, ), w ℓ , ι k .Subsection 2.6: g , P , P +0 , V ( λ ).Subsection 3.1: W ℓ , I , J , R , R + , R + L , R , θ , θ J , e ( p ) r , E ( p ) r , c g , i , j , i [ k , k ], j [ k , k ], s r , Λ ∨ i , E p for p ∈ Z , wt, S ℓ .Subsection 3.3: ε i , E p for p ∈ Z , S ℓ , m ( p , . . . , p n : λ ).Subsection 3.4: a , m ( p , p ). 2. Preliminaries
Quantum affine algebra.
Let g be an affine Kac–Moody Lie algebra not oftype A (2)2 n over Q with index set I = { , , · · · , n } and Cartan matrix C = ( c ij ) i,j ∈ I .We assume that the index 0 coincides with the one prescribed in [Kac90] (we do notassume this for the other indices, and in fact later we use another labeling, see Figure 1in Subsection 3.1). Let α i and h i ( i ∈ I ) be the simple roots and simple corootsrespectively, Λ i ( i ∈ I ) the fundamental weights, δ the generator of null roots, P = L i Z Λ i ⊕ Z δ the weight lattice, P + = L i ∈ I Z ≥ Λ i ⊕ Z δ the set of dominant weights, Q = L i ∈ I Z α i the root lattice, Q + = P i ∈ I Z ≥ α i ⊆ Q , W the Weyl group withreflections s i ( i ∈ I ), and ( , ) a nondegenerate W -invariant bilinear form on P satisfying( α , α ) = 2. Set I = I \ { } , and ̟ i = Λ i − h K, Λ i i Λ for i ∈ I , where K ∈ P ∗ = Hom ( P, Z ) is the canonical central element. Let d ∈ P ∗ be the elementsatisfying h d, Λ i i = 0 ( i ∈ I ) and h d, δ i = 1. Set P cl = P/ Z δ , and let cl : P ։ P cl be thecanonical projection. For simplicity of notation, we will write α i , ̟ i for cl( α i ), cl( ̟ i )when there should be no confusion.Let q be an indeterminate. Set q i = q ( α i ,α i ) / ,[ m ] i = q mi − q − mi q i − q − i , [ n ] i ! = [ n ] i [ n − i · · · [1] i , and (cid:20) mn (cid:21) i = [ m ] i [ m − i · · · [ m − n + 1] i [ n ] i !for i ∈ I , m ∈ Z , n ∈ Z ≥ . Choose a positive integer D such that ( α i , α i ) / ∈ Z D − for all i ∈ I , and set q s = q /D . Let U q ( g ) be the quantum affine algebra, which isan associative Q ( q s )-algebra generated by e i , f i ( i ∈ I ), q h ( h ∈ D − P ∗ ) with certaindefining relations (see, for example, [Kas02]). Denote by U ′ q ( g ) the quantum affinealgebra without the degree operator, that is, the subalgebra of U q ( g ) generated by e i , f i ( i ∈ I ) and q h ( h ∈ D − P ∗ cl ). Let U q ( n + ) (resp. U q ( n − )) be the subalgebra generatedby e i (resp. f i ) ( i ∈ I ). For i ∈ I and n ∈ Z , set e ( n ) i = e ni / [ n ] i ! if n ≥
0, and e ( n ) i = 0otherwise. Define f ( n ) i analogously. We define a Q -grading U q ( g ) = L α ∈ Q U q ( g ) α by U q ( g ) α = { X ∈ U q ( g ) | q h Xq − h = q h h,α i X for h ∈ D − P ∗ } . If 0 = X ∈ U q ( g ) α , we write wt P ( X ) = α . For a proper subset J ⊂ I , denote by g J the corresponding simple Lie subalgebra, and by U q ( g J ) (resp. U q ( n + ,J ), U q ( n − ,J )) the Q ( q s )-subalgebra of U q ( g ) generated by e i , f i , q ± D − h i (resp. e i , f i ) with i ∈ J . K. NAOI AND T. SCRIMSHAW
Set t i = q ( α i ,α i ) h i / for i ∈ I , and denote by ∆ the coproduct of U q ( g ) defined by∆( q h ) = q h ⊗ q h , ∆( e ( m ) i ) = m X k =0 q k ( m − k ) i e ( k ) i ⊗ t − ki e ( m − k ) i , ∆( f ( m ) i ) = m X k =0 q k ( m − k ) i t m − ki f ( k ) i ⊗ f ( m − k ) i for h ∈ D − P ∗ , i ∈ I , m ∈ Z > .For a U q ( g )-module (resp. U ′ q ( g )-module) M and λ ∈ P (resp. λ ∈ P cl ), write M λ = { v ∈ M | q h v = q h h,λ i v for h ∈ D − P ∗ (resp. h ∈ D − P ∗ cl ) } , and if v ∈ M λ with v = 0, we write wt P ( v ) = λ (resp. wt P cl ( v ) = λ ). We will omit thesubscript P or P cl when no confusion is likely. We say a U q ( g )-module (or U ′ q ( g )-module) M is integrable if M = L λ M λ and the actions of e i and f i ( i ∈ I ) are locally nilpotent.Throughout the paper we will repeatedly use the following assertions. For i, j ∈ I such that i = j and r, s ∈ Z ≥ , it follows from the Serre relations that e ( r ) i e ( s ) j ∈ U q ( n + ) s ( α j − c ij α i ) e ( r + c ij s ) i if r + c ij s > ,e ( s ) j e ( r ) i ∈ e ( r + c ij s ) i U q ( n + ) s ( α j − c ij α i ) if r + c ij s > , (2.1.1)where U q ( n + ) α = U q ( n + ) ∩ U q ( g ) α . For i, j ∈ I such that c ij = c ji = − r, s, t ∈ Z ≥ ,we have e ( r ) i e ( s ) j e ( t ) i = r − s + t X m =0 (cid:20) r − s + tm (cid:21) i e ( t − m ) j e ( r + t ) i e ( s − t + m ) j if r + t ≥ s, (2.1.2)see [Lus93, Lemma 42.1.2]. Given a U q ( g )-module M , v ∈ M λ and r, s ∈ Z ≥ , we have e ( r ) i f ( s ) i v = min( r,s ) X k =0 (cid:20) r − s + h h i , λ i k (cid:21) i f ( s − k ) i e ( r − k ) i v, (2.1.3a) f ( r ) i e ( s ) i v = min( r,s ) X k =0 (cid:20) r − s − h h i , λ i k (cid:21) i e ( s − k ) i f ( r − k ) i v (2.1.3b)for i ∈ I , and e ( r ) i f ( s ) j = f ( s ) j e ( r ) i for i, j ∈ I such that i = j , see [ loc. cit. , Corollary 3.1.9].2.2. Crystal (pseudo)bases and global bases.
Let M be an integrable U q ( g )-module(or U ′ q ( g )-module). For i ∈ I , we have M = M λ ; h h i ,λ i≥ h h i ,λ i M n =0 f ( n ) i (ker e i ∩ M λ ) . Endomorphisms ˜ e i , ˜ f i ( i ∈ I ) on M called the Kashiwara operators are defined by˜ f i ( f ( n ) i u ) = f ( n +1) i u, ˜ e i ( f ( n ) i u ) = f ( n − i u XISTENCE OF KR CRYSTALS FOR NEAR ADJOINT NODES 5 for u ∈ ker e i ∩ M λ with 0 ≤ n ≤ h h i , λ i . These operators also satisfy that˜ e i ( e ( n ) i v ) = e ( n +1) i v, ˜ f i ( e ( n ) i v ) = e ( n − i v for v ∈ ker f i ∩ M µ with 0 ≤ n ≤ −h h i , µ i . Let A be the subring of Q ( q s ) consistingof rational functions without poles at q s = 0. A free A -submodule L of M is called a crystal lattice of M if M ∼ = Q ( q s ) ⊗ A L , L = L λ L λ where L λ = L ∩ M λ , and ˜ e i , ˜ f i ( i ∈ I ) preserve L . Definition 2.2.1 ([Kas91, KKM + . (1) A pair ( L, B ) is called a crystal base of M if(i) L is a crystal lattice of M , (ii) B is a Q -basis of L/q s L ,(iii) B = F λ B λ where B λ = B ∩ (cid:0) L λ /q s L λ ), (iv) ˜ e i B ⊆ B ∪ { } , ˜ f i B ⊆ B ∪ { } ,(v) for b, b ′ ∈ B and i ∈ I , ˜ f i b = b ′ if and only if ˜ e i b ′ = b .(2) ( L, B ) is called a crystal pseudobase of M if they satisfy the conditions (i), (iii)–(v),and (ii’) B = B ′ ⊔ ( − B ′ ) with B ′ a Q -basis of L/q s L .Recall that, if M and M are integrable U q ( g )-modules and ( L i , B i ) is a crystalbase of M i ( i = 1 , L ⊗ A L , B ⊗ B ) is a crystal base of M ⊗ M , where B ⊗ B = { b ⊗ b | b i ∈ B i } ⊆ ( L ⊗ A L ) /q s ( L ⊗ A L ).Let denote the automorphism of Q ( q s ) sending q s to q − s , and set A = { a | a ∈ A } .We also denote by the involutive Q -algebra automorphism of U q ( g ) defined by e i = e i , f i = f i , q h = q − h , a ( q s ) x = a ( q − s ) x for i ∈ I , h ∈ D − P ∗ , a ( q s ) ∈ Q ( q s ) and x ∈ U q ( g ). Let U q ( g ) Q be the Q [ q s , q − s ]-subalgebra of U q ( g ) generated by e ( n ) i , f ( n ) i , q h for i ∈ I , n ∈ Z > , h ∈ D − P ∗ . Definition 2.2.2 ([Kas91]) . (1) Let V be a vector space over Q ( q s ), L a free A -submodule, L ∞ a free A -submodule,and V Q a free Q [ q s , q − s ]-submodule. We say that ( L , L ∞ , V Q ) is balanced if each of L , L ∞ , and V Q generates V as a Q ( q s )-vector space, and the canonical map L ∩ L ∞ ∩ V Q → L /q s L is an isomorphism.(2) Let M be an integrable U q ( g )-module with a crystal base ( L, B ), be an involutionof M (called a bar involution ) satisfying xu = x u for x ∈ U q ( g ) and u ∈ M , and M Q a U q ( g ) Q -submodule of M such that M Q = M Q , u − u ∈ ( q s − M Q for u ∈ M Q . Assume that (
L, L, M Q ) is balanced, where L = { u | u ∈ L } . Then, letting G be theinverse of L ∩ L ∩ M Q ∼ → L/q s L , the set B = { G ( b ) | b ∈ B } forms a basis of M called a global basis of M (with respect to the bar involution ).Note that the global basis B is an A -basis of L . K. NAOI AND T. SCRIMSHAW
Polarization. A Q ( q s )-bilinear pairing ( , ) between U q ( g )-modules (resp. U ′ q ( g )-modules) M and N is said to be admissible if it satisfies( q h u, v ) = ( u, q h v ) , ( e ( m ) i u, v ) = ( u, q − m i t − mi f ( m ) i v ) , ( f ( m ) i u, v ) = ( u, q − m i t mi e ( m ) i v ) (2.3.1)for h ∈ D − P ∗ (resp. h ∈ D − P ∗ cl ), i ∈ I , m ∈ Z > , u ∈ M , v ∈ N . A bilinear form( , ) on M is called a prepolarization if it is symmetric and satisfies (2.3.1) for u, v ∈ M .A prepolarization is called a polarization if it is positive definite with respect to thefollowing total order on Q ( q s ): f > g if and only if f − g ∈ G n ∈ Z { q ns ( c + q s A ) | c ∈ Q > } , and f ≥ g if f = g or f > g . Throughout the paper, we use the notation k u k = ( u, u )for u ∈ M .2.4. Extremal weight modules.
For an arbitrary Λ ∈ P , let V (Λ) be the extremalweight module [Kas94] with generator v Λ , which is an integrable U q ( g )-module generatedby v Λ of weight Λ with certain defining relations. If Λ belongs to the W -orbit of adominant (resp. antidominant) weight, say Λ ◦ , then V (Λ) is a simple highest (resp.lowest) weight module with highest (resp. lowest) weight Λ ◦ . In [ loc. cit. ], it was shownfor any Λ ∈ P that V (Λ) has a crystal base (cid:0) L (Λ) , B (Λ) (cid:1) and (cid:0) L (Λ) , L (Λ) , V (Λ) Q (cid:1) is balanced, where the bar involution is defined by xv Λ = xv Λ for x ∈ U q ( g ), and V (Λ) Q = U q ( g ) Q v Λ . We denote by B (Λ) = { G ( b ) | b ∈ B (Λ) } ⊆ V (Λ)the associated global basis. Let U q ( g ) Z denote the Z [ q s , q − s ]-subalgebra of U q ( g ) gener-ated by e ( n ) i , f ( n ) i ( i ∈ I , n ∈ Z > ) and q h ( h ∈ D − P ∗ ), and set V (Λ) Z = U q ( g ) Z v Λ ⊆ V (Λ). The following proposition is due to [Kas91] for highest and lowest weight cases,and [Nak04] for level zero cases. Proposition 2.4.1.
Let Λ ∈ P . (1) There exists a polarization ( , ) on V (Λ) such that k v Λ k = 1 . (2) We have (cid:0) L (Λ) , L (Λ) (cid:1) ⊆ A , and (˜ e i u, v ) ≡ ( u, ˜ f i v ) mod q s A for u, v ∈ L (Λ) and i ∈ I . (3) B (Λ) is an almost orthonormal basis with respect to ( , ) , that is, ( v, v ′ ) ∈ δ vv ′ + q s A for v, v ′ ∈ B (Λ) . (4) We have L (Λ) = (cid:8) v ∈ V (Λ) (cid:12)(cid:12) k v k ∈ A (cid:9) , ± B (Λ) = (cid:8) v ∈ V (Λ) Z (cid:12)(cid:12) v = v, k v k ∈ q s A (cid:9) . Let Λ , Λ ∈ P + . By [Lus92] (see also [Kas94]), the triple (cid:0) L (Λ ) ⊗ A L ( − Λ ) , L (Λ ) ⊗ A L ( − Λ ) , V (Λ ) Q ⊗ Q [ q s ,q − s ] V ( − Λ ) Q (cid:1) in the tensor product V (Λ ) ⊗ V ( − Λ ) is balanced. Here the bar involution is definedby x ( v Λ ⊗ v − Λ ) = x ( v Λ ⊗ v − Λ ) for x ∈ U q ( g ) . XISTENCE OF KR CRYSTALS FOR NEAR ADJOINT NODES 7
Denote the associated global basis by B (Λ , − Λ ) = { G ( b ) | b ∈ B (Λ ) ⊗ B ( − Λ ) } ⊆ V (Λ ) ⊗ V ( − Λ ) . It is easily checked from the definition that v Λ ⊗ B ( − Λ ) ⊆ B (Λ , − Λ ) . (2.4.1)By the construction of the global basis of an extremal weight module in [Kas94,Subsection 8.2], the following lemma is obvious. Lemma 2.4.2.
Let Λ ∈ P , and suppose that Λ , Λ ∈ P + satisfy Λ − Λ = Λ . Thereexists a unique surjective U q ( g ) -module homomorphism Ψ from V (Λ ) ⊗ V ( − Λ ) to V (Λ) mapping v Λ ⊗ v − Λ to v Λ , and Ψ maps the subset { X ∈ B (Λ , − Λ ) | Ψ( X ) = 0 } bijectively to B (Λ) . Kirillov–Reshetikhin modules.
Given a U ′ q ( g )-module M , we define a U q ( g )-module M aff = Q ( q s )[ z, z − ] ⊗ M by letting e i and f i ( i ∈ I ) act by z δ i ⊗ e i and z − δ i ⊗ f i respectively, and q D − d on z k ⊗ M by the scalar multiplication by q ks . Set M a = M aff / ( z − a ) M aff for nonzero a ∈ Q ( q s ), which is again a U ′ q ( g )-module. Wedenote by ι a : M ∼ → M a the Q ( q s )-linear (not U ′ q ( g )-linear) isomorphism defined by ι a ( v ) = p a (1 ⊗ v ), where p a : M aff → M a is the projection. If no confusion is likely, wewill write ι for ι a sometimes.Let r ∈ I . In [Kas02], a U ′ q ( g )-module automorphism z r of weight δ is constructed onthe level-zero fundamental extremal weight module V ( ̟ r ), which preserves the globalbasis B ( ̟ r ). Set W r, = V ( ̟ r ) / ( z r − V ( ̟ r ) , which is a finite-dimensional simple integrable U ′ q ( g )-module called a fundamental mod-ule . Note that W r, ∼ = V ( ̟ r ). Let p : V ( ̟ r ) → W r, be the canonical projection, anddefine a bilinear form ( , ) on W r, by (cid:0) p ( u ) , p ( v ) (cid:1) = X k ∈ Z ( z kr u, v ) for u, v ∈ V ( ̟ r ) . (2.5.1)Since ( u, v ) = ( z r u, z r v ) holds for u, v ∈ V ( ̟ r ) by [Nak04, Lemma 4.7], this is a well-defined polarization on W r, . Let L ( W r, ) = p (cid:0) L ( ̟ r ) (cid:1) . It follows from Proposition 2.4.1that L ( W r, ) = { u ∈ W r, | k u k ∈ A } , and ( u, v ) ∈ A for any u, v ∈ L ( W r, ) . (2.5.2)Fix r ∈ I and ℓ ∈ Z > . Let w ∈ W r, denote a vector such that wt P cl ( w ) = ̟ r and k w k = 1. Hereafter we write ι k for ι q k ( k ∈ D − Z ). Set m = ( ( α r , α r ) / g : nontwisted affine type , g : twisted affine type . Let f W = W r, q m (1 − ℓ ) ⊗ W r, q m (3 − ℓ ) ⊗ · · · ⊗ W r, q m ( ℓ − ⊗ W r, q m ( ℓ − , and denote by w ℓ a vector of f W defined by w ℓ = ι m (1 − ℓ ) ( w ) ⊗ ι m (3 − ℓ ) ( w ) ⊗ · · · ⊗ ι m ( ℓ − ( w ) ⊗ ι m ( ℓ − ( w ) . K. NAOI AND T. SCRIMSHAW
The U ′ q ( g )-submodule W r,ℓ = U ′ q ( g ) w ℓ ⊆ f W is called the Kirillov–Reshetikhin module (KR module for short) associated with r, ℓ . Proposition 2.5.1.
Let r ∈ I , ℓ ∈ Z > . (1) W r,ℓ is a finite-dimensional simple integrable U ′ q ( g ) -module. (2) The weight space W r,ℓℓ̟ r is -dimensional and spanned by w ℓ . (3) The weight set { λ ∈ P cl | W r,ℓλ = 0 } coincides with the intersection of ℓ̟ r − P i ∈ I Z ≥ α i and the convex hull of the W -orbit of ℓ̟ r . (4) The vector w ℓ ∈ W r,ℓ satisfies e i w ℓ = 0 if i ∈ I and f i w ℓ = 0 if i ∈ I \ { r } . Proof.
The assertion (1) is proved in [OS08, Proposition 3.6]. The assertions (2) and(3) follow from [Kas02, Theorem 5.17], and (4) is proved from (3). (cid:3)
Next we shall recall how to define a prepolarization on W r,ℓ . There exists a unique U ′ q ( g )-module homomorphism R : W r, q m ( ℓ − ) ⊗ W r, q m ( ℓ − ⊗ · · · ⊗ W r, q m (1 − ℓ ) → W r, q m (1 − ℓ ) ⊗ · · · ⊗ W r, q m ( ℓ − ⊗ W r, q m ( ℓ − mapping ι m ( ℓ − ( w ) ⊗ · · · ⊗ ι m (1 − ℓ ) ( w ) to w ℓ , and its image is W r,ℓ (see [OS08]). Thefollowing lemma is proved straightforwardly. Lemma 2.5.2.
Assume that ℓ ∈ Z > , M k , N k (1 ≤ k ≤ ℓ ) are U ′ q ( g ) -modules, and ( , ) k : M k × N k → Q ( q s ) (1 ≤ k ≤ ℓ ) are admissible pairings. Then the Q ( q s ) -bilinearpairing ( , ) : ( M ⊗ · · · ⊗ M ℓ ) × ( N ⊗ · · · ⊗ N ℓ ) → Q ( q s ) defined by ( u ⊗ u ⊗ · · · ⊗ u ℓ , v ⊗ v ⊗ · · · ⊗ v ℓ ) = ( u , v ) ( u , v ) · · · ( u ℓ , v ℓ ) ℓ is admissible. The lemma gives an admissible pairing ( , ) between W r, q m ( ℓ − ⊗ · · · ⊗ W r, q m (1 − ℓ ) and W r, q m (1 − ℓ ) ⊗ · · · ⊗ W r, q m ( ℓ − ) , which defines a bilinear form ( , ) on W r,ℓ by (cid:0) R ( u ) , R ( v ) (cid:1) = (cid:0) u, R ( v ) (cid:1) for u, v ∈ W r, q m ( ℓ − ⊗ · · · ⊗ W r, q m (1 − ℓ ) . (2.5.3)By [KKM +
92, Proposition 3.4.3], ( , ) is a nondegenerate prepolarization on W r,ℓ , and k w ℓ k = 1 holds. We will use the following lemma later, whose proof is similar to thatof [Nao18, Lemma 3.6] Lemma 2.5.3.
Let r ∈ I and ℓ ∈ Z > , and set W = W r, q m ( ℓ − ⊗ W r,ℓ − q − m and W = W r, q m (1 − ℓ ) ⊗ W r,ℓ − q m . There are unique U ′ q ( g ) -module homomorphisms R : W → W r,ℓ and R : W r,ℓ → W satisfying R (cid:0) ι ( w ) ⊗ ι ( w ℓ − ) (cid:1) = w ℓ and R ( w ℓ ) = ι ( w ) ⊗ ι ( w ℓ − ) respectively, and for any u, v ∈ W we have (cid:0) R ( u ) , R ( v ) (cid:1) = (cid:0) u, R ◦ R ( v ) (cid:1) , where ( , ) is the admissible pairing between W and W obtained from Lemma 2.5.2. XISTENCE OF KR CRYSTALS FOR NEAR ADJOINT NODES 9
Criterion for the existence of a crystal pseudobase.
Following the previousworks [OS08, Nao18, BS20], we will prove Theorem 1 by applying a criterion for theexistence of a crystal pseudobase introduced in [KKM + g = g I for short. We identify the weight lattice P of g with the subgroup L i ∈ I Z ̟ i of P cl , and set P +0 = P i ∈ I Z ≥ ̟ i . For λ ∈ P +0 , denote by V ( λ ) the simpleintegrable U q ( g )-module with highest weight λ .Let A Z and K Z be the subalgebras of Q ( q s ) defined respectively by A Z = { f ( q s ) /g ( q s ) | f ( q s ) , g ( q s ) ∈ Z [ q s ] , g (0) = 1 } , K Z = A Z [ q − s ] . Let U ′ q ( g ) K Z denote the K Z -subalgebra of U ′ q ( g ) generated by e i , f i , q h ( i ∈ I, h ∈ D − P ∗ cl ). Proposition 2.6.1 ([KKM +
92, Propositions 2.6.1 and 2.6.2]) . Assume that M is afinite-dimensional integrable U ′ q ( g ) -module having a prepolarization ( , ) and a U ′ q ( g ) K Z -submodule M K Z such that ( M K Z , M K Z ) ⊆ K Z . We further assume that there exist weightvectors u k ∈ M K Z (1 ≤ k ≤ m ) satisfying the following conditions: (i) wt( u k ) ∈ P +0 for ≤ k ≤ m and M ∼ = L mk =1 V (cid:0) wt( u k ) (cid:1) as U q ( g ) -modules, (ii) ( u k , u l ) ∈ δ kl + q s A for ≤ k, l ≤ m , (iii) k e i u k k ∈ q − h h i , wt( u k ) i− i q s A for all i ∈ I and ≤ k ≤ m .Then ( , ) is a polarization, and the pair ( L, B ) with L = { u ∈ M | k u k ∈ A } and B = { b ∈ ( M K Z ∩ L ) / ( M K Z ∩ q s L ) | ( b, b ) = 1 } , where ( , ) is the Q -valued bilinear form on L/q s L induced by ( , ) , is a crystalpseudobase of M . From [KKM + U ′ q ( g ) K Z -submodule W r,ℓK Z = U ′ q ( g ) K Z w ℓ ⊆ W r,ℓ satisfies (cid:0) W r,ℓK Z , W r,ℓK Z (cid:1) ⊆ K Z . Hence if we show for M = W r,ℓ the existence of weight vectors u , . . . , u m satisfying (i)–(iii), Theorem 1 follows from Proposition 2.6.1. We will showthis in the next section with an explicit construction of the vectors u , . . . , u m .3. Proof of Theorem 1
Set of vectors.
In the rest of this paper, assume that g is either of type E (1) n ( n = 6 , , F (1)4 or E (2)6 and the nodes of the Dynkin diagram is labeled as in Figure 1.We have q i = q / ( g : F (1)4 , i = 3 , , q i = q ( g : E (2)6 , i = 3 , , q i = q (otherwise) . From now on, for i ∈ I such that q i = q we write [ m ] for [ m ] i , [ n ]! for [ n ] i !, and (cid:20) mn (cid:21) for (cid:20) mn (cid:21) i . Note that in all types r = 2 is the unique near adjoint node. In the sequel, wewill consider W ,ℓ only and, hence, write W ℓ for W ,ℓ . (cid:16) E (1)6 (cid:17) • ◦ ◦ ◦ • tttt ❏❏❏❏ • ◦ (cid:16) E (1)7 (cid:17) • ◦ ◦ • • tttt ❏❏❏❏ • ◦ ◦ (cid:16) E (1)8 (cid:17) • ◦ ◦ • • • • tttt ❏❏❏❏ • ◦ (cid:16) F (1)4 (cid:17) ◦ ◦ • / / • ◦ (cid:16) E (2)6 (cid:17) ◦ ◦ • o o • ◦ Figure 1.
Dynkin diagrams of types E (1)6 , , , F (1)4 , and E (2)6 ( • : nodesbelonging to J ) g θ θ J E (1)6 α + α + α + α + α α + α + α = − ̟ + ̟ + ̟ = − ̟ + ̟ + ̟ − ̟ − ̟ E (1)7 α + 2 α + α + 2 α + 2 α + α α + 2 α + α + α = − ̟ + ̟ = − ̟ + ̟ − ̟ E (1)8 α + 2 α + 3 α + 4 α + 2 α + 3 α + 2 α α + 2 α + 2 α + 2 α + α + α = − ̟ + ̟ = − ̟ + ̟ − ̟ F (1)4 α + 2 α + 2 α = − ̟ + 2 ̟ α + 2 α = − ̟ + 2 ̟ − ̟ E (2)6 α + α + α = − ̟ + ̟ α + α = − ̟ + ̟ − ̟ Table 1.
Explicit forms of θ and θ J Let us prepare several notation. Define two subsets I and J of I by I = I \ { } ,and J = { , , } ( g : E (1)6 ) , { , , , } ( g : E (1)7 ) , { , , , , , } ( g : E (1)8 ) , { , } ( g : F (1)4 , E (2)6 ) . Let R ⊆ Q denote the root system of g , and R + = R ∩ Q + the set of positive roots.For a subset L ⊂ I denote by R L the root subsystem of R generated by the simpleroots corresponding to the elements of L , and let R + L = R L ∩ R + . We write R = R I .Let θ be the highest short root of R if g is of type E (2)6 , and the highest root of R otherwise. Define θ J ∈ R J similarly (see Table 1). XISTENCE OF KR CRYSTALS FOR NEAR ADJOINT NODES 11
For i ∈ I and k ∈ Z , set E ( k ) i = ( e (2 k ) i if g is of type F (1)4 and α i is short ,e ( k ) i otherwise . For p ∈ Z and a sequence r = ( r k r k − · · · r ) of elements of I (in this paper we alwaysread such sequences from right to left), we use the abbreviations e ( p ) r = e ( p ) r k e ( p ) r k − · · · e ( p ) r and E ( p ) r = E ( p ) r k · · · E ( p ) r . (3.1.1)Set c g = ( g : F (1)4 ) , , and choose a sequence i = ( i L i L − · · · i i ) of elements of I satisfying i = 2 , s i L · · · s i ( α ) = θ , and h h i k , s i k − · · · s i ( α ) i = − c g for 1 ≤ k ≤ L. (3.1.2)Similarly, choose a sequence j = ( j L ′ j L ′ − · · · j j ) of elements of J satisfying j = 2 , s j L ′ · · · s j ( α ) = θ J , and h h j k , s j k − · · · s j ( α ) i = − c g for 1 ≤ k ≤ L ′ . In the rest of this paper, we fix i = ( i L · · · i ) and j = ( j L ′ · · · j ) satisfying theseconditions. For 0 ≤ k ≤ k ≤ L , denote by i [ k , k ] the subsequence ( i k i k − · · · i k )of i , and let i [ k , k ] be the empty set if k < k . We define j [ k , k ] similarly. For asequence r = ( r ℓ r ℓ − · · · r ) of elements of I , set s r = s r ℓ · · · s r ∈ W , and let s r be theidentity element of W if r is the empty set. Let h α ∈ P ∗ ( α ∈ R ) denote the coroots,and Λ ∨ i ∈ P ∗ ⊗ Z Q ( i ∈ I ) elements satisfying h Λ ∨ i , α j i = δ ij for i, j ∈ I . Lemma 3.1.1. (1)
Neither of the subsequences i [ L, and j [ L ′ , contains . (2) We have h h i , θ i = 0 for all i ∈ J . (3) For any p ∈ Z ≥ , we have wt P (cid:0) E ( p ) i [ k, (cid:1) = ps i [ k, ( α ) (0 ≤ k ≤ L ) and wt P (cid:0) E ( p ) j [ k, (cid:1) = ps j [ k, ( α ) (0 ≤ k ≤ L ′ ) . In particular, wt P (cid:0) E ( p ) i (cid:1) = pθ and wt P (cid:0) E ( p ) j (cid:1) = pθ J hold. (4) Both s i and s j are reduced expressions. (5) If α ∈ R + satisfies s − i [ L, ( α ) ∈ − R + (resp. s − j [ L ′ , ( α ) ∈ − R + ), then we have h h α , θ i > (resp. h h α , θ J i > ). (6) For any p ∈ Z ≥ , E ( p ) i (resp. E ( p ) j ) does not depend on the choice of i (resp. j ).Proof. The assertion (1) is obvious since h Λ ∨ , θ i = h Λ ∨ , θ J i = 1 (see Table 1), and (2)is checked directly. The assertion (3) is easily seen from the conditions on i and j . Wewill show the assertion (4) for s i (the proof for s j is similar). By the condition on i , wehave for any 0 ≤ k ≤ L that h s i [ L,k +1] ( h i k ) , θ i = h h i k , s i [ k, ( α ) i > . Since h h i , θ i ≥ i ∈ I and s i [ L,k +1] ( α i k ) ∈ R , this implies that s i [ L,k +1] ( α i k )is a positive root for any k , which implies that s i is reduced. Let us show the assertion (5) for s i [ L, (the proof for s j [ L ′ , is similar). There exists 1 ≤ k ≤ L such that α = s i [ L,k +1] ( α i k ), and we have h h α , θ i = h s i [ L,k +1] ( h i k ) , θ i = h h i k , s i [ k, ( α ) i > , as required. Finally, let us show the assertion (6) for E ( p ) i (the proof for E ( p ) j is similar).If g is either of type F (1)4 or E (2)6 , i = (43) is the unique choice. Hence we may assumethat g is of type E (1) n ( n = 6 , , i ′ = ( i ′ L , . . . , i ′ ) is another choice.Since P Lk =0 α i k = P L k =0 α i ′ k = θ , we have L = L . Let r be the smallest number suchthat i r = i ′ r , and let s be the smallest number such that r < s and i r = i ′ s . Then since h h i r , r − X k =0 α i k i = − h h i ′ s , s − X k =0 α i ′ k i and i ′ k = i r for r ≤ k < s , we have h h i r , α i ′ k i = 0 for r ≤ k < s . Hence setting i ′′ = ( i ′ L · · · i ′ s +1 i ′ s − · · · i ′ r i r · · · i ) , we have E ( p ) i ′ = E ( p ) i ′′ . By repeating this argument we can show that E ( p ) i ′ = E ( p ) i , andhence the assertion (6) is proved. (cid:3) For p = ( p , p , . . . , p ) ∈ Z , we write E p = e ( p )0 e ( p )1 e ( p )2 E ( p ) j E ( p ) i e ( p )10 ∈ U q ( n + ) , and define a map wt : Z → P bywt( p )= ( p − p − p − p + 2 p − p ) ̟ + ( − p + 2 p − p ) ̟ + ( p − p ) γ + ( p − p ) γ , where we set γ = ̟ + cl( θ ) ∈ P +0 and γ = ̟ + γ + cl( θ J ) ∈ P +0 . (3.1.3)For ℓ ∈ Z > , define a finite subset S ℓ ⊆ Z ≥ by S ℓ = (cid:8) ( p , . . . , p ) ∈ Z ≥ (cid:12)(cid:12) p ≤ p ≤ p ≤ p ≤ p , p + p + p − p ≤ p ≤ p + ℓ } . Note that if p ∈ S ℓ , then wt P cl ( E p w ℓ ) = wt( p ) + ℓ̟ ∈ P +0 . As stated in the final partof the previous section, Theorem 1 is proved once we show the following. Proposition 3.1.2.
For any ℓ ∈ Z > , the vectors { E p w ℓ | p ∈ S ℓ } ⊆ W ℓ satisfy thefollowing conditions: (C1) W ℓ ∼ = L p ∈ S ℓ V (cid:0) wt( p ) + ℓ̟ (cid:1) as U q ( g ) -modules, (C2) ( E p w ℓ , E p ′ w ℓ ) ∈ δ p , p ′ + q s A for p , p ′ ∈ S ℓ , (C3) k e i E p w ℓ k ∈ q − h h i , wt( p ) i− ℓδ i − i q s A for i ∈ I and p ∈ S ℓ . XISTENCE OF KR CRYSTALS FOR NEAR ADJOINT NODES 13
Proof of (C1) in Proposition 3.1.2.
By [Nak03, Her06, DFK08], the multi-plicities of a KR module are known to coincide with the cardinalities of highest weightrigged configurations . In our cases, explicit formulas for the number of them have beenobtained using the Kleber algorithm [Kle98], and hence we have the following.
Proposition 3.2.1 ([Scr20, Section 9]) . Let ℓ ∈ Z > . Define a subset T ℓ ⊆ Z ≥ by T ℓ = { r = ( r , r , . . . , r ) ∈ Z ≥ | r + r + r + r ≤ ℓ, r + 2 r ≤ r } , and a map wt T : Z ≥ → P by wt T ( r ) = ( r − r − r ) ̟ + ( − r − r − r − r + r ) ̟ + r γ + r γ , where γ , γ are given in (3.1.3) . Then we have W ℓ ∼ = M r ∈ T ℓ V (cid:0) wt T ( r ) + ℓ̟ (cid:1) ⊕ (1+ r − r − r ) as U q ( g ) -modules. Now (C1) is easily deduced from Proposition 3.2.1. Indeed, the map φ : Z → Z defined by φ ( p , . . . , p ) ( p , p − p − p , p − p , p − p , p − p )sends S ℓ to T ℓ , wt T ◦ φ = wt holds, and for any r ∈ T ℓ , φ − ( r ) ∩ S ℓ = { r + k (1 , , , , , | r ≤ k ≤ r + r − r − r } , where r = ( r + r + r + r + r , r + r + r , r + r , r , , r ) , and hence Proposition 3.2.1 is equivalent to (C1).3.3. Proof of (C2) in Proposition 3.1.2.
In this and next subsections, we need toconsider prepolarizations on several types of modules (extremal weight modules, KRmodules, or tensor products of them) simultaneously. Therefore, when we would like toindicate what prepolarization we are considering, we will occasionally write ( , ) M and k k M for ( , ) and k k on a module M .We begin with the following lemma. Lemma 3.3.1.
Let M be a U ′ q ( g ) -module with a prepolarization ( , ) , and u ∈ M λ forsome λ ∈ P cl . Assume that f u = e u = f u = 0 . Then for any p , p ′ ∈ Z ≥ with p = p ′ , ( E p u, E p ′ u ) = 0 holds.Proof. Set p = ( p , . . . , p ) and p ′ = ( p ′ , . . . , p ′ ). We may assume that p ≥ p ′ . By theadmissibility, we have ( E p u, E p ′ u ) = q c ( E p − p ε u, f ( p )0 E p ′ u ) , where c is a certain integer and ε i = (0 , . . . , , | {z } i , , . . . ,
0) (1 ≤ i ≤
6) is the standardbasis of Z . Since e ( a )1 e ( b )0 u = 0 if a > b by (2.1.1), it follows from (2.1.3) that f ( p )0 E p ′ u = δ p p ′ q c ′ E p ′ − p ε u with c ′ ∈ Z , and hence we may (and do) assume that p = p ′ = 0. If we furtherassume that p = p ′ , then p = p ′ implies wt P cl ( E p u ) = wt P cl ( E p ′ u ), which forces( E p u, E p ′ u ) = 0.Hence we may assume that p > p ′ . In this case, we have( E p u, E p ′ u ) = q c ′′ ( E p − p ε u, f ( p )1 E p ′ u ) (3.3.1)with c ′′ ∈ Z , and by applying (2.1.1) and (2.1.3), it is easily proved that f ( p )1 E p ′ u ∈ e U q ( g ) u. Since f E p − p ε u = 0, (3.3.1) implies ( E p u, E p ′ u ) = 0, and the assertion is proved. (cid:3) Since the vector w ℓ ∈ W ℓ satisfies the assumption of the lemma, ( E p w ℓ , E p ′ w ℓ ) = 0follows if p = p ′ . In order to verify (C2) in Proposition 3.1.2, it remains to show k E p w ℓ k ∈ q s A for p ∈ S ℓ . Lemma 3.3.2.
For any p = ( p , . . . , p ) ∈ Z ≥ such that p − p + p ≤ ℓ , we have k E p w ℓ k ∈ (1 + qA ) k E p − p ε w ℓ k .Proof. We have k E p w ℓ k = q p (3 ℓ − p + p − p ) ( E p − p ε w ℓ , f ( p )0 E p w ℓ ) . (3.3.2)Since f E p − p ε w ℓ = 0 holds, it follows from (2.1.3) that(3.3.2) = q p (3 ℓ − p + p − p ) (cid:20) ℓ − p + p p (cid:21) k E p − p ε w ℓ k ∈ (1 + qA ) k E p − p ε w ℓ k . The lemma is proved. (cid:3)
In the sequel, we regard Z as a subgroup of Z via Z ∋ p ֒ → ( p , ∈ Z . Hence for p = ( p , . . . , p ) ∈ Z , we have E p = e ( p )1 e ( p )2 E ( p ) j E ( p ) i e ( p )10 . For ℓ ∈ Z > , set S ℓ = S ℓ ∩ Z = { ( p , . . . , p ) | p ≤ p ≤ p ≤ p , p + p + p − p ≤ p ≤ p + ℓ } . By the lemma, the proof of the assertion k E p w ℓ k ∈ q s A for p ∈ S ℓ is reduced to thecase p ∈ S ℓ . An idea for the proof of this assertion is to use the almost orthonormalityof B ( ℓ̟ ), the global basis of the extremal weight module V ( ℓ̟ ). To do this we needto show that E p v ℓ̟ ∈ ± B ( ℓ̟ ) ∪ { } for p ∈ Z ≥ . For this purpose, we prepare severallemmas. Lemma 3.3.3.
Let Λ ∈ P and i ∈ I , and assume that u ∈ ± B (Λ) . (1) If f ( n ) i u ∈ ± B (Λ) ∪ { } for all n > , then we have e ( n ) i u ∈ ± B (Λ) ∪ { } for all n > . (2) In particular, if f i u = 0 then e ( n ) i u ∈ ± B (Λ) ∪ { } for all n > . XISTENCE OF KR CRYSTALS FOR NEAR ADJOINT NODES 15
Proof.
Let us prove the assertion (1) (note that (2) is just a special case). Since u ∈± B (Λ), it follows from Proposition 2.4.1 (4) that e ( n ) i u is bar-invariant and e ( n ) i u ∈ V (Λ) Z for any n >
0. Hence, again by the same proposition, it suffices to show that k e ( n ) i u k ∈ q s A for n > e ( n ) i u = 0. Set L = { v ∈ V (Λ) | k v k ∈ q s A } ⊆ L (Λ) . Let λ ∈ P be the weight of u , and set λ i = h h i , λ i ∈ Z . Write u = N X k =max(0 , − λ i ) f ( k ) i u k , where u k ∈ ker e i ∩ V (Λ) λ + kα i . Here we set N = max { k ∈ Z ≥ | u k = 0 } . By Proposition 2.4.1 (2), it follows for every u k that k f ( m ) i u k k = k ˜ f mi u k k ∈ (1 + q s A ) k u k k if 0 ≤ m ≤ k + λ i . (3.3.3)We shall show that u k ∈ q k ( λ i + k ) i L for every k by the descending induction. For0 ≤ n ≤ N + λ i , we have0 = f ( n ) i u = N X k =max(0 , − λ i ) (cid:20) k + nk (cid:21) i f ( k + n ) i u k ∈ ± B (Λ) ⊆ L (3.3.4)by the assumption. Since f ( k + N + λ i ) i u k = 0 for k < N , (3.3.4) with n = N + λ i implies (cid:20) N + λ i N (cid:21) i f (2 N + λ i ) i u N ∈ L . Hence we have u N ∈ q N ( N + λ i ) i L by (3.3.3), and theinduction begins. Next let k be an integer such that max(0 , − λ i ) ≤ k < N . By (3.3.4)with n = k + λ i , we have N X k = k (cid:20) k + k + λ i k (cid:21) i f ( k + k + λ i ) i u k ∈ L . (3.3.5)It is easily checked from the admissibility that f ( k + k + λ i ) i u k ’s are pairwise orthogonalwith respect to the polarization, and then it follows from (3.3.5) that f (2 k + λ i ) i u k ∈ q k ( k + λ i ) L , since the induction hypothesis implies for k > k that (cid:20) k + k + λ i k (cid:21) i f ( k + k + λ i ) i u k ∈ q k ( k − k ) i L ⊆ q s L (Λ) . Hence u k ∈ q k ( k + λ i ) i L holds by (3.3.3), as required.Now assume that 0 < n ≤ N . It follows from (2.1.3) that e ( n ) i u = N X k =max(0 , − λ i ) (cid:20) k + n + λ i n (cid:21) i f ( k − n ) i u k , (3.3.6)and since we have (cid:20) k + n + λ i n (cid:21) i f ( k − n ) i u k ( ∈ q ( k − n )( k + λ i ) i L ( k ≥ n ) , = 0 (otherwise) by the above argument, (3.3.6) and the pairwise orthogonality of f ( l ) i u k ’s imply e ( n ) i u ∈ L . Since e ( n ) i u = 0 for n > N , this completes the proof. (cid:3) Lemma 3.3.4.
Let p = ( p , p , p , p ) ∈ Z ≥ . In V ( − ℓ Λ ) , we have the following: (1) For any ≤ k ≤ L , we have f i k E ( p ) i [ k − , e ( p )1 e ( p )0 v − ℓ Λ = e i k E ( p ) i [ k, e ( p )1 e ( p )0 v − ℓ Λ = 0 . (2) For any i ∈ I such that h h i , θ i = 0 , we have f i E ( p ) i e ( p )1 e ( p )0 v − ℓ Λ = 0 . (3) For any ≤ k ≤ L ′ , we have f j k E ( p ) j [ k − , E ( p ) i e ( p )1 e ( p )0 v − ℓ Λ = e j k E ( p ) j [ k, E ( p ) i e ( p )1 e ( p )0 v − ℓ Λ = 0 . (4) For any i ∈ J such that h h i , θ J i = 0 , we have f i E ( p ) j E ( p ) i e ( p )1 e ( p )0 = 0 .Proof. Set v = e ( p )1 e ( p )0 v − ℓ Λ and Λ = wt P ( v ) = − ℓ Λ + p α + p α .(1) We have s − i [ k − , wt P ( f i k E ( p ) i [ k − , v ) = Λ + p α − s − i [ k − , ( α i k ) , and since s − i [ k − , ( α i k ) is a positive root in R I \{ , } by Lemma 3.1.1 (1) and (4), theright-hand side does not belong to − ℓ Λ + Q + . Hence f i k E ( p ) i [ k − , v = 0 holds. Since h h i k , wt( E ( p ) i [ k − , v ) i = − c g p , we also have e ( c g p +1) i k E ( p ) i [ k − , v = 0, and the proof of (1)is complete.(2) We have s − i [ L, wt( f i E ( p ) i v ) = Λ + p α − s − i [ L, ( α i ) , (3.3.7)and s − i [ L, ( α i ) ∈ R +1 by Lemma 3.1.1 (5). Moreover, we have s − i [ L, ( α i ) = α since α i = θ , and hence the right-hand side of (3.3.7) does not belong to − ℓ Λ + Q + , whichimplies (2).(3) Set W = U q ( g J ) E ( p ) i v . The assertion (2), together with Lemma 3.1.1 (2), impliesthat W λ = 0 unless λ ∈ Λ + p θ + Q + . Using this, the assertion (3) is proved by asimilar argument to that of (1). Finally the proof of the assertion (4) is similar to thatof (2). (cid:3) Lemma 3.3.5.
Let ℓ ∈ Z > . (1) For any ( p , . . . , p ) ∈ Z ≥ , the vector e ( p )2 E ( p ) j E ( p ) i e ( p )1 e ( p )0 v − ℓ Λ in V ( − ℓ Λ ) belongs to ± B ( − ℓ Λ ) ∪ { } . (2) For any p = ( p , . . . , p ) ∈ Z ≥ , E p v − ℓ Λ ∈ V ( − ℓ Λ ) belongs to ± B ( − ℓ Λ ) ∪ { } .Proof. Obviously, f v − ℓ Λ = f e ( p )0 v − ℓ Λ = f e ( p )1 e ( p )0 v − ℓ Λ = 0 XISTENCE OF KR CRYSTALS FOR NEAR ADJOINT NODES 17 holds. Then the assertion (1) is proved by applying Lemma 3.3.3 (2) repeatedly usingLemma 3.3.4. For any n >
0, it is easily seen using (2.1.3) that f ( n )1 e ( p )2 E ( p ) j E ( p ) i e ( p )10 v − ℓ Λ = e ( p )2 E ( p ) j E ( p ) i e ( p − n )1 e ( p )0 v − ℓ Λ , which belongs to ± B ( − ℓ Λ ) ∪ { } by (1). Hence it follows from Lemma 3.3.3 that E p v − ℓ Λ ∈ ± B ( − ℓ Λ ) ∪ { } . The assertion (2) is proved. (cid:3) Now we prove the following.
Proposition 3.3.6.
Let ℓ ∈ Z > . For any p ∈ Z ≥ , the vector E p v ℓ̟ ∈ V ( ℓ̟ ) belongs to ± B ( ℓ̟ ) ∪ { } .Proof. By Lemma 3.3.5 (2) and (2.4.1), we have v ℓ Λ ⊗ E p v − ℓ Λ ∈ ± B ( ℓ Λ , − ℓ Λ ) ∪ { } , and then Lemma 2.4.2 implies that E p v ℓ̟ ∈ ± B ( ℓ̟ ) ∪ { } as required, since ̟ =Λ − . (cid:3) Next we will show that k E p v ℓ̟ k V ( ℓ̟ ) = k E p w ⊗ ℓ k W ) ⊗ ℓ for p ∈ Z ≥ . Before doingthat we prepare a lemma, which is also used in the next subsection. Lemma 3.3.7.
Let M , . . . , M n be integrable U ′ q ( g ) -modules, λ = ( λ , . . . , λ n ) an n -tuple of elements of P cl , and u k ∈ (cid:0) M k ) λ k (1 ≤ k ≤ n ) . Assume that each u k satisfies e i u k = 0 for i ∈ I . Then for any p ∈ Z ≥ , the vector E p ( u ⊗ · · · ⊗ u n ) ∈ M ⊗ · · · ⊗ M n can be written in the form E p ( u ⊗ · · · ⊗ u n ) = X p ,..., p n ∈ Z ≥ ; p + ··· + p n = p q m ( p ,..., p n : λ ) E p u ⊗ · · · ⊗ E p n u n , (3.3.8) where m ( p , . . . , p n : λ ) ∈ D − Z are certain numbers depending only on p , . . . , p n and λ .Proof. By the definition of the coproduct, E p ( u ⊗ · · · ⊗ u n ) is a sum of vectors of theform q m n O k =1 e ( s k )1 e ( r k )2 e ( h kL ′ ) j L ′ · · · e ( h k ) j e ( g kL ) i L · · · e ( g k ) i e ( b k )1 e ( a k )0 u k . (3.3.9)Since e ( b k )1 e ( a k )0 u k = 0 if b k > a k by (2.1.1) and P k a k = P k b k = p , the vector (3.3.9)becomes 0 unless a k = b k for all k .Take a sufficiently large positive integer ℓ . For any k , there is a U q ( n + )-modulehomomorphism from V ( − ℓ Λ ) to M k mapping v − ℓ Λ to u k , which follows from thewell-known fact that V ( − ℓ Λ ) is generated by v − ℓ Λ as a U q ( n + )-module with relations e ℓ +10 v − ℓ Λ = 0 and e i v − ℓ Λ = 0 ( i ∈ I ) . Then since P k g kt = 2 p if g is of type F (1)4 and t = 0 and P k g kt = p otherwise, we seefrom Lemma 3.3.4 (1) that the vector (3.3.9) becomes 0 unless c g g k = g k = · · · = g kL for all k . By a similar argument using Lemma 3.3.4 (3), we also see that the vector (3.3.9)with c g g k = g k = · · · = g kL becomes 0 unless c g h k = h k = · · · = h kL ′ for all k . Theproof is complete. (cid:3) Proposition 3.3.8.
Let ℓ ∈ Z > and p ∈ Z ≥ . (1) We have k E p v ℓ̟ k V ( ℓ̟ ) = k E p w ⊗ ℓ k W ) ⊗ ℓ . (2) If E p w ⊗ ℓ = 0 , we have k E p w ⊗ ℓ k ∈ q s A .Proof. (1) First we show the following: k E p v ℓ̟ k V ( ℓ̟ ) = k E p v ⊗ ℓ̟ k V ( ̟ ) ⊗ ℓ for p ∈ Z ≥ . (3.3.10)By [Nak04], there exists an injective U q ( g )-module homomorphism Φ from V ( ℓ̟ )to V ( ̟ ) ⊗ ℓ mapping v ℓ̟ to v ⊗ ℓ̟ . Although Φ does not preserve the values of thepolarizations in general, the relations between ( , ) V ( ℓ̟ ) and ( , ) V ( ̟ ) ⊗ ℓ are explicitlydescribed in [ loc. cit. ], which we recall here. Define a Q ( q s )[ t ± ]-valued bilinear form(( , )) t on V ( ̟ ) by (( u, v )) t = X k ∈ Z t m ( z − m u, v ) V ( ̟ ) , where z is the automorphism on V ( ̟ ) in Subsection 2.5. Define a Q ( q s )[ t ± , . . . , t ± ℓ ]-valued bilinear form (( , )) on V ( ̟ ) ⊗ ℓ by ℓ O k =1 u k , ℓ O k =1 v k !! = ℓ Y k =1 (( u k , v k )) t k . Then by [Nak04, Proposition 4.10], it holds for u, v ∈ V ( ℓ̟ ) that( u, v ) = 1 ℓ ! ((Φ( u ) , Φ( v ))) Y k = m (1 − t k t − m ) , (3.3.11)where [ f ] denotes the constant term in f .For p , p ′ ∈ Z ≥ such that p = p ′ , we have ( E p w , E p ′ w ) W = 0 by Lemma 3.3.1.Then by (2.5.1), this, together with the weight consideration, implies( z − m E p v ̟ , E p ′ v ̟ ) V ( ̟ ) = 0 unless p = p ′ and m = 0 . Hence in particular, it follows that(( E p v ̟ , E p ′ v ̟ )) t = ( E p v ̟ , E p ′ v ̟ ) V ( ̟ ) for p , p ′ ∈ Z ≥ , (3.3.12)which implies (( E p v ⊗ ℓ̟ , E p ′ v ⊗ ℓ̟ )) = ( E p v ⊗ ℓ̟ , E p ′ v ⊗ ℓ̟ ) V ( ̟ ) ⊗ ℓ by Lemma 3.3.7. Now Equa-tion (3.3.11) implies for p ∈ Z ≥ that k E p v ℓ̟ k V ( ℓ̟ ) = 1 ℓ ! (( E p v ⊗ ℓ̟ , E p v ⊗ ℓ̟ )) Y k = m (1 − t k t − m ) = k E p v ⊗ ℓ̟ k V ( ̟ ) ⊗ ℓ · ℓ ! Y k = m (1 − t k t − m ) = k E p v ⊗ ℓ̟ k V ( ̟ ) ⊗ ℓ , and the claim (3.3.10) is proved. XISTENCE OF KR CRYSTALS FOR NEAR ADJOINT NODES 19
In order to verify the assertion (1), by (3.3.10) it suffices to show k E p v ⊗ ℓ̟ k V ( ̟ ) ⊗ ℓ = k E p w ⊗ ℓ k W ) ⊗ ℓ for p ∈ Z ≥ . We see from (2.5.1) that( p ( u ) , p ( v )) = (( u, v )) t (cid:12)(cid:12) t =1 for u, v ∈ V ( ̟ ) . Hence by (3.3.12), we have( E p w , E p ′ w ) W = ( E p v ̟ , E p ′ v ̟ ) V ( ̟ ) for p , p ′ ∈ Z ≥ , and then k E p v ⊗ ℓ̟ k V ( ̟ ) ⊗ ℓ = k E p w ⊗ ℓ k W ) ⊗ ℓ follows by Lemma 3.3.7. The assertion (1)is proved.(2) Since ( , ) ( W ) ⊗ ℓ is positive definite, v ∈ ( W ) ⊗ ℓ satisfies k v k = 0 if and only if v =0. Hence the assertion (2) follows from (1), Proposition 3.3.6 and Proposition 2.4.1 (3). (cid:3) Proposition 3.3.9.
Let ℓ ∈ Z > . If p ∈ S ℓ , then E p w ⊗ ℓ = 0 , and hence k E p w ⊗ ℓ k ∈ q s A follows from Proposition 3.3.8.Proof. Let us prove the assertion by the induction on ℓ . First assume that ℓ = 1. Inthis case, we have S = { , ε , ε + ε , (2 , , , , } . If p ∈ S \ { ε + ε } , E p v ̟ = 0 is checked from the following elementary fact: for anintegrable U ′ q ( g )-module M , λ ∈ P cl and i ∈ I ,if u ∈ M λ \ { } , then e ( k ) i u = 0 for 0 ≤ k ≤ −h h i , λ i . (3.3.13)On the other hand, by Proposition 2.5.1 we have k e e e w k = q − ( e e w , f e e e w ) = q − ( e e w , e e e f w )= k e e f w k = q − ( e f w , e e f f w )= k e f f w k = q ( f f w , f e f f w ) = q [2] k f f w k = q [2] . Hence we have e e e w = 0, and then E ε + ε w = 0 is proved by applying (3.3.13).Thus the case ℓ = 1 is proved.Assume ℓ >
1. By Lemma 3.3.7, E p w ⊗ ℓ can be written in the form E p ( w ⊗ w ⊗ ( ℓ − ) = X p + p = p q m ( p , p : ̟ , ( ℓ − ̟ ) E p w ⊗ E p w ⊗ ( ℓ − , and for the vectors { E p w | p ∈ Z ≥ such that E p w = 0 } are linearly independentby Lemma 3.3.1, it is enough to show the existence of p satisfying E p w = 0 and E p − p w ⊗ ( ℓ − = 0 . (3.3.14)If p < p + ℓ , then p = satisfies (3.3.14) by the induction hypothesis since p ∈ S ℓ − .Assume that p = p + ℓ , and set k = max { ≤ k ≤ | p k = 0 } . If k = 2,set p = (2 , , . . . , | {z } k , , . . . , E p w = 0 follows from (3.3.13), and it is easilychecked that p − p ∈ S ℓ − . Therefore (3.3.14) holds. Finally if k = 2, p = (1 , , , , (cid:3) The following lemma connects values of the prepolarizations on ( W ) ⊗ ℓ and W ℓ . Lemma 3.3.10.
Let ℓ ∈ Z > , and X, Y ∈ U ′ q ( g ) . Suppose that the images of X, Y under the ℓ -iterated coproduct ∆ ( ℓ ) : U ′ q ( g ) → U ′ q ( g ) ⊗ ℓ are written in the forms ∆ ( ℓ ) ( X ) = N X k =1 f k ( q s ) X k, ⊗ X k, ⊗ · · · ⊗ X k,ℓ , and ∆ ( ℓ ) ( Y ) = N X m =1 g m ( q s ) Y m, ⊗ Y m, ⊗ · · · ⊗ Y m,ℓ respectively, where N , N ∈ Z ≥ , f k , g k ∈ Q ( q s ) , and X k,j , Y m,j ∈ U ′ q ( g ) are vectorshomogeneous with respect to the Q -grading. We further assume that, for any ≤ k ≤ N and ≤ m ≤ N ,if ℓ Y j =1 ( X k,j w , Y m,j w ) W = 0 , then wt P ( X k,j ) = wt P ( Y m,j ) for all ≤ j ≤ ℓ. (3.3.15) Then we have ( Xw ⊗ ℓ , Y w ⊗ ℓ ) ( W ) ⊗ ℓ = ( Xw ℓ , Y w ℓ ) W ℓ .Proof. By (2.5.3), we have( Xw ℓ , Y w ℓ ) W ℓ = (cid:16) X (cid:0) ι ℓ − ( w ) ⊗ · · · ⊗ ι − ℓ ( w ) (cid:1) , Y (cid:0) ι − ℓ ( w ) ⊗ · · · ⊗ ι ℓ − ( w ) (cid:1)(cid:17) . (3.3.16)For an arbitrary homogeneous vector Z ∈ U ′ q ( g ) β and k ∈ Z , we have Zι k ( w ) = q k h d,β i ι k ( Zw ) . Hence setting wt P ( X k,j ) = β k,j and wt P ( Y m,j ) = γ m,j , it follows that X (cid:0) ι ℓ − ( w ) ⊗ · · · ⊗ ι − ℓ ( w ) (cid:1) = X k f k ( q s ) q P j ( ℓ +1 − j ) h d,β k,j i ι ( X k, w ) ⊗ · · · ⊗ ι ( X k,ℓ w ) , and Y (cid:0) ι − ℓ ( w ) ⊗ · · · ⊗ ι ℓ − ( w ) (cid:1) = X m g m ( q s ) q P j (2 j − ℓ − h d,γ m,j i ι ( Y m, w ) ⊗ · · · ⊗ ι ( Y m,ℓ w ) . Then we have(3.3.16) = X k,m f k ( q s ) g m ( q s ) q P j ( ℓ +1 − j ) h d,β k,j − γ m,j i Y j ( X k,j w , Y m,j w ) W = X k,m f k ( q s ) g m ( q s ) Y j ( X k,j w , Y m,j w ) W = ( Xw ⊗ ℓ , Y w ⊗ ℓ ) ( W ) ⊗ ℓ by the assumption, and the assertion is proved. (cid:3) Now the following proposition, together with Proposition 3.3.9, completes the proofof (C2) in Proposition 3.1.2.
Proposition 3.3.11.
Let ℓ ∈ Z > . For any p ∈ Z ≥ , we have k E p w ⊗ ℓ k W ) ⊗ ℓ = k E p w ℓ k W ℓ . XISTENCE OF KR CRYSTALS FOR NEAR ADJOINT NODES 21
Proof.
It suffices to show that X = Y = E p satisfy the assumptions of Lemma 3.3.10.The vector ∆ ( ℓ ) ( E p ) can be written in the form P k q m k s q H k E k ⊗ · · · ⊗ q H kℓ E kℓ , where m k ∈ Z , E kj are some products of e ( m ) i ’s and H kj ∈ D − P ∗ cl . By Lemma 3.3.7, q H k E k w ⊗ · · · ⊗ q H kℓ E kℓ w = 0 unless E kj = E p j (1 ≤ j ≤ ℓ ) for some p j ∈ Z ≥ ,and then Q j ( q H kj E kj w , q H mj E mj w ) = 0 implies E kj = E mj for all j by Lemma 3.3.1.Hence (3.3.15) is obviously satisfied, and the proof is complete. (cid:3) Proof of (C3) in Proposition 3.1.2.
First we show the case i = 1. The proof issimilar to [Nao18, proof of Eq. (3.3) with i = 1]. We reproduce it here for the reader’sconvenience. Lemma 3.4.1.
For any p ∈ S ℓ , we have k e E p w ℓ k ∈ q − h h , wt( p ) i A. Proof.
Set p = h h , wt( p ) i = p − p − p − p + 2 p − p ≥ . We have k e E p w ℓ k = q − p − ( E p w ℓ , f e E p w ℓ ) = q − p − (cid:16) [ − p ] k E p w ℓ k + ( E p w ℓ , e f E p w ℓ ) (cid:17) ≡ q − p k f E p w ℓ k mod q − p A, where we have used the fact k E p w ℓ k ∈ q s A by (C2) (which we have already proved).Hence it suffices to show that k f E p w ℓ k ∈ A . Set r = 3 ℓ − p + p . It is easily checkedthat f ( k )0 f E p − p ε w ℓ = 0 for k >
1, and hence we have k f E p w ℓ k = q p ( r − p − ( f E p − p ε w ℓ , f ( p )0 e ( p )0 f E p − p ε w ℓ )= q p ( r − p − (cid:18)(cid:20) r − p (cid:21) k f E p − p ε w ℓ k + (cid:20) r − p − (cid:21) ( f E p − p ε w ℓ , e f f E p − p ε w ℓ ) (cid:19) ∈ k f E p − p ε w ℓ k A + q r − p ) k f f E p − p ε w ℓ k A. (3.4.1)It follows that k f E p − p ε w ℓ k = q p + p − ( E p − p ε w ℓ , e f E p − p ε w ℓ )= q p + p − (cid:16) [ p + p ] k E p − p ε w ℓ k + ( E p − p ε w ℓ , f e E p − p ε w ℓ ) (cid:17) = q p + p − (cid:16) [ p + p ] k E p − p ε w ℓ k + q p + p +1 [ p + 1] k E p + ε − p ε w ℓ k (cid:17) ∈ A. Moreover, it is easily checked that f f E p − p ε w ℓ = [3 ℓ − p + 1] E p − ε − p ε , and hence it also follows that q r − p ) k f f E p − p ε w ℓ k = q r − p ) [3 ℓ − p + 1] k E p − ε − p ε w ℓ k ∈ q p − p ) A ⊆ A. Hence k f E p w ℓ k ∈ A follows from (3.4.1), and the proof is complete. (cid:3) When we show (C3) for i ∈ I \ { } , as we did in the proof of (C2), we may assumethat p ∈ S ℓ (cid:0) = S ℓ ∩ Z ≥ (cid:1) by the following lemma. Lemma 3.4.2.
For any p = ( p , . . . , p ) ∈ Z ≥ such that p − p + p ≤ ℓ and i ∈ I \ { } , we have k e i E p w ℓ k ∈ (1 + qA ) k e i E p − p ε w ℓ k .Proof. Since e i E p w ℓ = e ( p )0 e i E p − p ε w ℓ , the same proof for Lemma 3.3.2 holds here. (cid:3) Our next goal is to give estimates for the values k e i E p w ⊗ ℓ k ( i ∈ I \ { } ). For thispurpose, let us prepare some lemmas. The proof of the following lemma is almost thesame with that of Lemma 3.3.3, with L replaced by L (Λ). Lemma 3.4.3.
Let Λ ∈ P and i ∈ I , and assume that u ∈ V (Λ) is a weight vector. If f ( n ) i u ∈ L (Λ) for all n ∈ Z ≥ , then e ( n ) i u ∈ L (Λ) for all n > . Lemma 3.4.4.
Let Λ , λ ∈ P , i ∈ I , and u ∈ V (Λ) λ , and assume that u ∈ q a L (Λ) , and f i u ∈ q b L (Λ) for some a, b ∈ D − Z . Set r i = ( α i , α i ) / . (1) We have e i u ∈ q min( a,b − r i h h i ,λ i ) L (Λ) . (2) Further assume that h h i , λ i ≤ and f (2) i u = 0 . Then we have e ( n ) i u ∈ q min( a,b − r i ( h h i ,λ i + n − L (Λ) for any n > . Proof.
Set λ i = h h i , λ i ∈ Z , and write u = N X k =max(0 , − λ i ) f ( k ) i u k , where u k ∈ ker e i ∩ V (Λ) λ + kα i . We have f i u = N X k =max(0 , − λ i ) [ k + 1] i f ( k +1) i u k ∈ q b L (Λ) , and since f ( k +1) i u k ’s are pairwise orthogonal with respect to ( , ), it follows from Propo-sition 2.4.1 (4) that [ k + 1] i f ( k +1) i u k ∈ q b L (Λ) for every k . Then since f ( k +1) i u k = 0 for k ≥ max(0 , − λ i + 1) such that u k = 0, we have u k ∈ q b + r i k L (Λ) for max(0 , − λ i + 1) ≤ k ≤ N (3.4.2)by Proposition 2.4.1 (2). We have e i u = N X k =max(1 , − λ i ) [ k + λ i + 1] i f ( k − i u k , and hence if λ i ≥
0, (3.4.2) implies e i u ∈ q b − r i λ i L (Λ) and the assertion (1) holds. When λ i <
0, we need to show further that f ( − λ i − i u − λ i ∈ q min( a,b − r i λ i ) L (Λ) . (3.4.3)Similarly as above, we see that u ∈ q a L (Λ) implies u k ∈ q a L (Λ) for all k , and hence(3.4.3) follows. The proof of (1) is complete. Under the assumption of (2), we may XISTENCE OF KR CRYSTALS FOR NEAR ADJOINT NODES 23 put N = − λ i + 1 and we have e ( n ) i u = f ( − λ i − n ) i u − λ i + [ n + 1] i f ( − λ i +1 − n ) i u − λ i +1 , whichbelongs to q min( a,b − r i ( λ i + n − L (Λ). Hence (2) is also proved. (cid:3) Lemma 3.4.5.
Assume that the sequence i satisfies the following condition: there exists ≤ m ≤ L such that i m , i m +1 , . . . , i L are pairwise distinct, c i m < , and c i k i k +1 = − for m ≤ k ≤ L − . Let ℓ ∈ Z > and ( p , p , p , p ) ∈ Z ≥ , and set v k = f i k f i k +1 · · · f i L E ( p ) i e ( p )1 e ( p )0 v − ℓ Λ ∈ V ( − ℓ Λ ) for m ≤ k ≤ L. (1) We have v k = e ( c g p − i [ L,k ] E ( p ) i [ k − , e ( p )1 e ( p )0 v − ℓ Λ for m ≤ k ≤ L. (2) We have v k ∈ ± B ( − ℓ Λ ) ∪ { } for m ≤ k ≤ L . (3) If g is not of type E (2)6 , we have E ( p ) i [ k − ,m ] e ( p )2 v k ∈ ± B ( − ℓ Λ ) ∪ { } for m ≤ k ≤ L. (3.4.4) On the other hand if g is of type E (2)6 , we have E ( p ) i [ k − ,m ] e ( p )2 v k ∈ q min(0 ,p − p − L ( − ℓ Λ ) for m ≤ k ≤ L. Proof.
The assertion (1) is easily proved using (2.1.3) and Lemma 3.3.4 (1).(2) Set v = e ( p )1 e ( p )0 v − ℓ Λ and Λ = wt P ( v ) = − ℓ Λ + p α + p α , and fix m ≤ k ≤ L .By (the proof of) Lemma 3.3.5, we have e ( c g p − i k E ( p ) i [ k − , v ∈ ± B ( − ℓ Λ ) ∪ { } . (3.4.5)For each k < k ′ ≤ L , we have s − i [ k ′ − , wt P ( f i k ′ e ( c g p − i [ k ′ − ,k ] E ( p ) i [ k − , v ) = Λ + p α − s − i [ k ′ − , ( α i k + · · · + α i k ′ )= Λ + p α − s − i [ L, ( α i k + · · · + α i k ′− ) (3.4.6)by the assumption on i . We have h h i L , θ i > i , andthen it is easily checked that h h i r , θ i = 0 for m ≤ r ≤ L − − ℓ Λ + Q + by Lemma 3.1.1 (5), whichimplies f i k ′ e ( c g p − i [ k ′ − ,k ] E ( p ) i [ k − , v = 0 for all k ′ . Now the assertion (2) follows from (3.4.5)by applying Lemma 3.3.3 (2) repeatedly.(3) First assume that g is not of type E (2)6 . We shall prove the assertion by theinduction on k . In the case k = m , since v m ∈ ± B ( − ℓ Λ ) ∪ { } by (2) it suffices to showthat f v m = 0, and as above, this is done by checking s − i [ L, wt P ( f v m ) / ∈ − ℓ Λ + Q + .Hence the induction begins. Assume that k > m . It follows from Lemma 3.3.4 (2) that f v k = f i k · · · f i L f E ( p ) i v = 0 , If g is not of type E (1)6 , this condition, together with the condition (3.1.2) on i , uniquely determinesthe sequence ( i L , i L − , . . . , i m ) (see Figure 1 and Table 1). In type E (1)6 , on the other hand, there aretwo possibilities; (5 ,
3) or (6 , and f i k ′ E ( p ) i [ k ′ − ,m ] e ( p )2 v k = f i k · · · f i L E ( p ) i [ k ′ − ,m ] e ( p )2 f i k ′ E ( p ) i v = 0 for any m ≤ k ′ ≤ k − . Hence we have E ( p ) i [ k − ,m ] e ( p )2 v k ∈ ± B ( − ℓ Λ ) ∪ { } . Since f ( p ) i k − E ( p ) i [ k − ,m ] e ( p )2 v k = δ p E ( p ) i [ k − ,m ] e ( p )2 v k − for p ∈ Z > , (3.4.4) is now proved from the induction hypothesis and Lemma 3.3.3.Next assume that g is of type E (2)6 . In this case i = (432), L = 2, m = 1 and v k = e ( p − e ( p − δ k )3 e ( p )2 e ( p )1 e ( p )0 v − ℓ Λ ( k = 1 , . We have f ( p )2 v = ( [ p − p + 1] e ( p − e ( p )1 e ( p )0 v − ℓ Λ ∈ q − p + p L ( − ℓ Λ ) ( p = 1) , p ∈ Z > )(note that v = 0 if p > p ), and hence it follows from Lemma 3.4.4 (2) that e ( p )2 v ∈ q min(0 ,p − p − L ( − ℓ Λ ). On the other hand, since f v = 0 we have e ( p )2 v ∈± B ( − ℓ Λ ) ∪ { } , and then e ( p )3 e ( p )2 v ∈ q min(0 ,p − p − L ( − ℓ Λ ) also follows since f ( p )3 e ( p )2 v = δ p e ( p )2 v for p ∈ Z > . The proof is complete. (cid:3) Lemma 3.4.6.
Let ℓ ∈ Z > and p = ( p , . . . , p ) ∈ Z ≥ . (1) We have e E p v − ℓ Λ ∈ q min(0 , − p + p ) L ( − ℓ Λ ) . (2) If g is not of type E (2)6 and i ∈ I \ { , } , we have e i E p v − ℓ Λ ∈ q min(0 , −h h i , wt( p ) i ) i L ( − ℓ Λ ) . (3) If g is of type E (2)6 , we have e E p v − ℓ Λ ∈ q , − p + p ) − δ p ,p L ( − ℓ Λ ) , and e E p v − ℓ Λ ∈ q , − p + p ) − δ p ,p L ( − ℓ Λ ) . Proof. (1) It suffices to show that f E p v − ℓ Λ ∈ q − p + p L ( − ℓ Λ ) by Lemmas 3.3.5 and3.4.4. Since f E ( p ) j E ( p ) i E ( p )10 v − ℓ Λ = 0 by Lemma 3.3.4 (4), it follows from the weightconsideration that E p v − ℓ Λ = 0 if p > p , and hence we may assume that p ≤ p . Bya direct calculation, we have f E p v − ℓ Λ = [ p − p + 1] E p − ε v − ℓ Λ , which belongs to q − p + p L ( − ℓ Λ ), as required.(2) It suffices to show that f i E p v − ℓ Λ ∈ L ( − ℓ Λ ). The proof is divided into threecases. First assume that h h i , θ i = h h i , θ J i = 0. In this case Lemma 3.3.4 implies f i E p v − ℓ Λ = 0, and hence the assertion holds. Next assume that h h i , θ J i >
0. ByLemma 3.1.1 (6), we may assume that the sequence j is chosen so that j L ′ = i . Foreach n ∈ Z ≥ , set v n = e ( c g p − i E ( p ) j [ L ′ − , E ( p ) i e ( p − n )1 e ( p )0 v − ℓ Λ . XISTENCE OF KR CRYSTALS FOR NEAR ADJOINT NODES 25
We easily see using Lemma 3.3.4 (3) that f i E p v − ℓ Λ = e ( p )1 e ( p )2 v , and f ( n )1 e ( p )2 v = e ( p )2 v n for any n ∈ Z ≥ . (3.4.7)Hence by Lemma 3.4.3, it suffices to show that e ( p )2 v n ∈ ± B ( − ℓ Λ ) ∪ { } ⊆ L ( − ℓ Λ )for any n . We have v n ∈ ± B ( − ℓ Λ ) ∪ { } by (the proof of) Lemma 3.3.5. Since α + α i is a positive root (see Table 1), we have s − j [ L ′ , wt P ( f v n ) = wt P ( E ( p ) i e ( p − n )1 e ( p )0 v − ℓ Λ ) + p α − s − j [ L ′ , ( α + α i ) / ∈ wt P ( E ( p ) i e ( p − n )1 e ( p )0 v − ℓ Λ ) + Q + , and the same argument as in the proof of Lemma 3.3.4 (3) shows that this implies f v n = 0. Hence e ( p )2 v n ∈ ± B ( − ℓ Λ ) ∪ { } holds, as required. Finally assume that h h i , θ i >
0. We may assume that the sequence i is chosen so that i L = i , and theassumption of Lemma 3.4.5 is satisfied. Let m be as in the assumption. Further, wemay also assume that the sequence j is chosen so that j k = i m + k − for 1 ≤ k ≤ L − m .For each n ∈ Z ≥ , set u n = f i E ( p ) i e ( p − n )1 e ( p )0 v − ℓ Λ = e ( c g p − i E ( p ) i [ L − , e ( p − n )1 e ( p )0 v − ℓ Λ . As above it is enough to show for any n that e ( p )2 E ( p ) j u n ∈ ± B ( − ℓ Λ ) ∪ { } . (3.4.8)It follows from Lemma 3.4.5 (3) that E ( p ) j [ L − m, u n ∈ ± B ( − ℓ Λ ) ∪ { } . We easily seefrom Figure 1 and Table 1 that { j ∈ J | c ij = 0 } = { i L − } , and { ≤ k ≤ L ′ | j k = i L − } = 1 . Then, since j L − m = i L − , we have c ij k = 0 for L − m < k ≤ L ′ , and hence we have f j k E ( p ) j [ k − , u n = f i f j k E ( p ) j [ k − , E ( p ) i e ( p − n )1 e ( p )0 v − ℓ Λ = 0 for all L − m < k ≤ L ′ by Lemma 3.3.4. Similarly, f E ( p ) j u n = 0 is proved. Now (3.4.8) is shown usingLemma 3.3.3, and the proof of (2) is complete.(3) We shall prove f E p v − ℓ Λ = e ( p )1 e ( p )2 e ( p − e ( p )2 e ( p )432 e ( p )10 v − ℓ Λ ∈ q min(0 ,p − p − L ( − ℓ Λ ) , which implies the former assertion by Lemma 3.4.4, and for this it is enough to showfor any n ∈ Z ≥ that f ( n )1 e ( p )2 e ( p − e ( p )2 e ( p )432 e ( p )10 v − ℓ Λ = e ( p )2 e ( p − e ( p )2 e ( p )432 e ( p − n )1 e ( p )0 v − ℓ Λ ∈ q min(0 ,p − p − L ( − ℓ Λ ) (3.4.9)by Lemma 3.4.3. We have e ( p − e ( p )2 e ( p )432 e ( p − n )1 e ( p )0 v − ℓ Λ ∈ ± B ( − ℓ Λ ) ∪ { } ⊆ L ( − ℓ Λ ) , and since f ( p )2 e ( p − e ( p )2 e ( p )432 e ( p − n )1 e ( p )0 v − ℓ Λ = ( [ p − n − p + 1] e ( p − e ( p )432 e ( p − n )1 e ( p )0 v − ℓ Λ ∈ q − p + p + n L ( − ℓ Λ ) ( p = 1) , p ∈ Z > )(note that the left-hand side is 0 if p > p − n ), (3.4.9) follows from Lemma 3.4.4, asrequired. The latter assertion is proved in a similar manner using Lemma 3.4.5. (cid:3) Now we obtain the following estimates for k e i E p w ⊗ ℓ k . Proposition 3.4.7.
Let ℓ ∈ Z > and p = ( p , . . . , p ) ∈ Z ≥ . (1) We have k e E p w ⊗ ℓ k ∈ q , − p + p ) A. (2) If i ∈ I \ { , } , we have k e i E p w ⊗ ℓ k ∈ q , −h h i , wt( p ) i ) − i A. Proof.
By (2.4.1), Lemma 2.4.2, [Nak04, Theorem 1 (2)] and the definition of L ( W ),we have p ⊗ ℓ ◦ Φ ◦ Ψ (cid:0) v ℓ Λ ⊗ L ( − ℓ Λ ) (cid:1) ⊆ L ( W ) ⊗ ℓ , where Ψ : V ( ℓ Λ ) ⊗ V ( − ℓ Λ ) → V ( ℓ̟ ) is the homomorphism given in the lemma,Φ : V ( ℓ̟ ) ֒ → V ( ̟ ) ⊗ ℓ is the one satisfying Φ( v ℓ̟ ) = v ⊗ ℓ̟ , and p : V ( ̟ ) ։ W is thecanonical projection. The assertions follow from this and Lemma 3.4.6. (cid:3) Let M , . . . , M n and u k ∈ ( M k ) λ k (1 ≤ k ≤ n ) be as in Lemma 3.3.7. We see thatthe vector e i E p ( u ⊗ · · · ⊗ u n ) for i ∈ I and p ∈ Z ≥ can be written in the form e i E p ( u ⊗ · · · ⊗ u p )= n X k =1 X p ,..., p n ∈ Z ≥ ; p + ··· + p n = p q m ( p ,..., p n : λ ,i,k ) E p u ⊗ · · · ⊗ e i E p k u k ⊗ · · · ⊗ E p n u n (3.4.10)with some m ( p , . . . , p n : λ , i, k ) ∈ D − Z . Now the following lemma, together withProposition 3.4.7 (2), completes the proof of (C3) for i ∈ I \ { , } . Lemma 3.4.8.
Let i ∈ I \ { , } . (1) If p , p ′ ∈ Z ≥ satisfy ( e i E p w , E p ′ w ) = 0 , then we have wt P ( e i E p ) = wt P ( E p ′ ) . (2) For any p ∈ Z ≥ and ℓ ∈ Z > , we have k e i E p w ⊗ ℓ k W ) ⊗ ℓ = k e i E p w ℓ k W ℓ . Proof.
Since ( e i E p w , E p ′ w ) = 0 implies wt P ( e i E p ) ∈ wt P ( E p ′ )+ Z δ , in order to prove(1) it is enough to show that ( e i E p w , E p ′ w ) = 0 if p = p . Since e j , f j ( j = 0 , e i , this follows from the same argument as in the proof of Lemma 3.3.1.Then we see from Lemma 3.3.1 and (3.4.10) that X = Y = e i E p satisfy the assumptionsof Lemma 3.3.10, and hence the assertion (2) is proved. (cid:3) It remains to prove (C3) for i = 2 and p ∈ S ℓ , which is more involved. We will provethe following stronger statement, and the proof will occupy the rest of this paper. XISTENCE OF KR CRYSTALS FOR NEAR ADJOINT NODES 27
Proposition 3.4.9.
Let ℓ ∈ Z > . For any p = ( p , p , p , p , p ) ∈ Z ≥ , we have k e E p w ℓ k ∈ q , − p + p ,p − p − ℓ ) − A. (3.4.11) Lemma 3.4.10.
Let ℓ ∈ Z > . If p ∈ Z ≥ satisfies ( W ) ⊗ ℓ ∋ E p w ⊗ ℓ = 0 , then p ≤ ℓ and p j ≤ min(2 ℓ, p ) for j ∈ { , , } .Proof. By Lemma 3.3.7, it is enough to show the assertion for ℓ = 1. In this case, since h h , ̟ i = − f w = 0, p ≤ h h , wt( e ( p )10 w ) i = − p + 1 and f (2)2 e ( p )10 w = 0 , we have e ( p +1)2 e ( p )10 w = 0, which implies p ≤ p . We easily see using Lemma 3.3.4 (2)that V ( ℓ Λ ) ⊗ V ( − ℓ Λ ) ∋ f (2)2 E ( p ) i e ( p )10 ( v Λ ⊗ v − ) = 0 , and then the existence of the map p ◦ Ψ : V (Λ ) ⊗ V ( − ) → W implies that f (2)2 E ( p ) i e ( p )10 w = 0. Hence p ≤ p is proved by the weight consideration. Simi-larly p ≤ p is proved from Lemma 3.3.4 (4). Finally we have to show that p j ≤ j ∈ { , , } even if p = 3. Similarly as above, these are deduced from the fact that f e (3)10 w = 0, and this fact follows since w is an extremal weight vector (see [Kas02,Theorem 5.17]). The proof is complete. (cid:3) In the sequel, we use the symbol a = (1 , , , , ∈ Z ≥ . The difficulty in the case i = 2 is that the statements of Lemma 3.4.8 for i = 2 do nothold in general. Instead, we have the following. Lemma 3.4.11.
Let ℓ ∈ Z > , and assume that either w = w ⊗ ℓ ∈ ( W ) ⊗ ℓ or w = w ℓ ∈ W ℓ . For any p , p ′ ∈ Z ≥ , we have ( e E p w, E p ′ w ) = 0 unless p ′ = p − a or p ′ = p + ε . Proof.
By the weight consideration, it is enough to show that ( e E p w, E p ′ w ) = 0 holdsif p < p ′ or p − > p ′ . If p < p ′ , the proof is similar to that of Lemma 3.3.1.Assume that p − > p ′ . It follows from (2.1.2) that( e E p w, E p ′ w ) = (cid:0) ( e ( p − e e − [ p − e ( p )1 e ) E p − p ε w, E p ′ w ) . (3.4.12)As in Lemma 3.3.1, it can be proved using p − > p ′ that( e ( p − e e E p − p ε w, E p ′ w ) = 0 = ( e ( p )1 e E p − p ε w, E p ′ w ) , and hence the right-hand side of (3.4.12) is zero. The proof is complete. (cid:3) We shall prove Proposition 3.4.9 by the induction on ℓ . By Proposition 3.4.7 (1) with ℓ = 1, we have k e E p w k ∈ q , − p + p ) A ⊆ q , − p + p ,p − p − − A for any p ∈ Z ≥ , and hence the induction begins. Throughout the rest of this section,fix ℓ ∈ Z > and assume that (3.4.11) holds for this ℓ . Our goal is to prove (3.4.11) with ℓ replaced by ℓ + 1, that is, k e E p w ℓ +1 k ∈ q , − p + p ,p − p − ℓ − − A for any p ∈ Z ≥ . (3.4.13)From now on, we write m ( p , p ) = m ( p , p ; ̟ , ℓ̟ ) for p , p ∈ Z ≥ for short (the right-hand side is defined in Lemma 3.3.7). For any p ∈ Z ≥ we have E p w ⊗ ( ℓ +1)1 = X p , p ∈ Z ≥ ; p + p = p q m ( p , p ) E p w ⊗ E p w ⊗ ℓ . (3.4.14) Lemma 3.4.12.
For p , p ∈ Z ≥ with p k = ( p k , . . . , p k ) , we have m ( p , p ) = − X j =1 p j p j + ( p + p + p ) p + p ( − p + p + p + p ) + ℓ (3 p − p − p − p ) . Proof.
Given weight vectors u , u of some U ′ q ( g )-modules, it follows for i ∈ I and p ∈ Z ≥ that e ( p ) i ( u ⊗ u ) = X p ,p ∈ Z ≥ ; p + p = p q − p ( h h i , wt( u ) i + p ) i e ( p ) i u ⊗ e ( p ) i u . In particular, if e ( p +1) i u = 0, e ( p +1) i u = 0 and h h i , wt( u ) i = − p , it follows that e ( p + p ) i ( u ⊗ u ) = e ( p ) i u ⊗ e ( p ) i u . Using these equalities, the assertion is obtainedstraightforwardly by calculating the coefficient of E p w ⊗ E p w ⊗ ℓ in E p + p w ⊗ ( ℓ +1)1 . (cid:3) Lemma 3.4.13.
Let p , p ∈ Z ≥ , and assume that E p w = 0 and E p w ⊗ ℓ = 0 . Then m ( p , p ) ≥ holds.Proof. Let p = p + p . By (3.4.14) and Lemma 3.3.1, it follows that k E p w ⊗ ( ℓ +1)1 k = X p ′ + p ′ = p q m ( p ′ , p ′ ) k E p ′ w k k E p ′ w ⊗ ℓ k . Then Proposition 3.3.8 (2) implies that, if E p ′ w and E p ′ w ⊗ ℓ are both nonzero, then m ( p ′ , p ′ ) ≥
0. Hence the assertion is proved. (cid:3)
For p ∈ Z ≥ , we have k e E p w ℓ +1 k = (cid:16) e E p (cid:0) ι ℓ ( w ) ⊗ ι − ( w ℓ ) (cid:1) , e E p (cid:0) ι − ℓ ( w ) ⊗ ι ( w ℓ ) (cid:1)(cid:17) by Lemma 2.5.3, and e E p ( ι ± ℓ ( w ) ⊗ ι ∓ ( w ℓ ))= X p , p ∈ Z ≥ ; p + p = p q m ( p , p ) ± ℓp ∓ p (cid:16) ι ± ℓ ( E p w ) ⊗ ι ∓ ( e E p w ℓ )+ q −h h , wt( E p w ℓ ) i ι ± ℓ ( e E p w ) ⊗ ι ∓ ( E p w ℓ ) (cid:17) . Set x ( p ) = −h h , wt( E p w ℓ ) i = p − p + p − ℓ for p ∈ Z ≥ . It follows from Lemma 3.4.11 that k e E p w ℓ +1 k = Z + Z + Z + Z , where Z = X q m ( p , p ) k E p w k · k e E p w ℓ k ,Z = [2] q ℓ +1 X q m ( p , p )+ m ( p − a , p + a )+ x ( p ) ( e E p w , E p − a w )( E p w ℓ , e E p + a w ℓ ) ,Z = 2 X q m ( p , p )+ m ( p + ε , p − ε )+ x ( p ) ( e E p w , E p + ε w )( E p w ℓ , e E p − ε w ℓ ) ,Z = X q m ( p , p )+2 x ( p ) k e E p w k · k E p w ℓ k . Here all the sums are over the set { p , p ∈ Z ≥ | p + p = p } . Now it suffices to showthat Z + Z + Z + Z belongs to the subset of Q ( q s ) in (3.4.13).First we shall show that Z does. For k ∈ Z , write[ k ] + = ( [ k ] ( k > k ≤ . Lemma 3.4.14.
Let p ∈ Z ≥ , and set k = p − p − ℓ + 1 . (1) The vector ( f E p − [ k ] + E p − ε ) v ℓ̟ ∈ V ( ℓ̟ ) belongs to ± B ( ℓ̟ ) ∪ { } . (2) We have ( f E p − [ k ] + E p − ε ) w ⊗ ℓ ∈ L ( W ) ⊗ ℓ .Proof. (1) By Lemma 2.4.2, it is enough to show that ( f E p − [ k ] + E p − ε )( v ℓ Λ ⊗ v − ℓ Λ )belongs to ± B ( ℓ Λ , − ℓ Λ ) ∪ { } . The bar-invariance is obvious, and it is easily checkedthat( f E p − [ k ] + E p − ε )( v ℓ Λ ⊗ v − ℓ Λ )= f v ℓ Λ ⊗ E p v − ℓ Λ + ( q ℓ [ p − p + 1] − [ k ] + ) v ℓ Λ ⊗ E p − ε v − ℓ Λ . We have f v ℓ Λ ∈ B ( ℓ Λ ), E p v − ℓ Λ ∈ ± B ( − ℓ Λ ) ∪ { } by Lemma 3.3.5, and( q ℓ [ p − p + 1] − [ k ] + ) v ℓ Λ ⊗ E p − ε v − ℓ Λ ∈ qL ( ℓ Λ ) ⊗ L ( − ℓ Λ )since p − p + 1 < E p − ε v − ℓ Λ = 0 (see the proof of Lemma 3.4.6 (1)). Hencewe have ( f E p − [ k ] + E p − ε )( v ℓ Λ ⊗ v − ℓ Λ ) ∈ ± B ( ℓ Λ , − ℓ Λ ) ∪ { } , as required. Theassertion (2) follows from (1) since the map p ⊗ ℓ ◦ Φ : V ( ℓ̟ ) → ( W ) ⊗ ℓ sends L ( ℓ̟ )to L ( W ) ⊗ ℓ . (cid:3) We need the following relation in W ℓ : there exists a certain element c ℓ ∈ ± q s A such that e E p w ℓ = c ℓ E p − a + ε f w ℓ + [ p − p + 1] E p + ε w ℓ (3.4.15)for p ∈ Z ≥ . It is a rather straightforward computation, but we will give a proof inAppendix A (Proposition A.1) as it is somewhat lengthy and technical. Lemma 3.4.15.
Let p ∈ Z ≥ . (1) We have ( e E p w ℓ , E p − a w ℓ ) ∈ q min(0 ,p − p − ℓ ) A. (2) When ℓ = 1 , the following stronger statement holds: ( e E p w , E p − a w ) ∈ q max(0 ,p − p − A. Proof. (1) By (3.4.15) and Lemma 3.3.1, we have( e E p w ℓ , E p − a w ℓ ) = c ℓ ( E p − a + ε f w ℓ , E p − a w ℓ ) . (3.4.16)It is easily checked that X = E p − a + ε f and Y = E p − a satisfy the assumptions ofLemma 3.3.10, and hence we have(3.4.16) = c ℓ ( E p − a + ε f w ⊗ ℓ , E p − a w ⊗ ℓ ) ( W ) ⊗ ℓ . (3.4.17)A calculation using Lemma 3.3.4 shows that E p − a + ε f ( v ℓ Λ ⊗ v − ℓ Λ ) = ( f E p − a + ε + [ − p + p + ℓ + 1] E p − a )( v ℓ Λ ⊗ v − ℓ Λ ) , and then the existence of the map V ( ℓ Λ ) ⊗ V ( − ℓ Λ ) → ( W ) ⊗ ℓ implies that(3.4.17) = c ℓ ( f E p − a + ε w ⊗ ℓ + [ − p + p + ℓ + 1] E p − a w ⊗ ℓ , E p − a w ⊗ ℓ ) ( W ) ⊗ ℓ . (3.4.18)By Lemma 3.4.14 (2), we have f E p − a + ε w ⊗ ℓ + [ − p + p + ℓ + 1] E p − a w ⊗ ℓ ≡ [ − p + p + ℓ + 1] + E p − a w ⊗ ℓ mod L ( W ) ⊗ ℓ , and hence it follows from Proposition 3.3.8 and (2.5.2) that( f E p − a + ε w ⊗ ℓ + [ − p + p + ℓ + 1] E p − a w ⊗ ℓ , E p − a w ⊗ ℓ ) ∈ q min(0 ,p − p − ℓ ) A. Now the assertion (1) is proved since c ℓ ∈ ± q s A .(2) We may assume that E p − a w = 0, and hence that p ≤ p − p − p ≥
2. First assume that p − p = 2.By (3.4.16) and (3.4.18), it suffices to show that( f E p − a + ε w , E p − a w ) ∈ qA, (3.4.19)and we may assume that the two vectors are both nonzero. Since the two vectors f E p − a + ε ( v Λ ⊗ v − ) and v Λ ⊗ E p − a v − both belong to ± B (Λ , − ) and are ob-viously linearly independent, we see from Lemma 2.4.2 that f E p − a + ε v ̟ and E p − a v ̟ both belong to ± B ( ̟ ) and are linearly independent. Moreover since their P -weightsare the same, ( f E p − a + ε v ̟ , z k E p − a v ̟ ) = 0 if k = 0. Hence (3.4.19) follows fromProposition 2.4.1 (3) and (2.5.1). XISTENCE OF KR CRYSTALS FOR NEAR ADJOINT NODES 31
It remains to show the assertion in the case p − p = 3, that is, p = 3 and p = 0.By the admissibility, we have( e E p w , E p − a w ) = q p +1 ( E p w , f E p − a w ) . Since E p w and f E p − a w both belong to L ( W ) and E p − a w = 0 implies p ≥
1, thisbelongs to q A . The proof is complete. (cid:3) Now we show the following proposition, which assures that Z belongs to the setin (3.4.13). Proposition 3.4.16.
Let p , p ∈ Z ≥ , and set p = p + p . Then we have q m ( p , p )+ m ( p − a , p + a )+ x ( p ) ( e E p w , E p − a w )( E p w ℓ ,e E p + a w ℓ ) ∈ q min(0 ,p − p − ℓ )+ p − p − A, where p = ( p , . . . , p ) and x ( p ) = −h h , wt( E p w ℓ ) i .Proof. Set p i = ( p i , . . . , p i ) ( i = 1 , m ( p , p ) + x ( p ) = m ( p − a , p + a ) + p − p − . (3.4.20)We may assume that E p − a w = 0 and E p + a w ℓ = 0. By the induction hypothesis, itfollows from Proposition 2.6.1 that the prepolarization ( , ) W ℓ is positive definite, andhence E p + a w ℓ = 0 implies E p + a w ⊗ ℓ = 0 by Proposition 3.3.11. Then it follows fromLemmas 3.4.13 and 3.4.15 that q m ( p , p )+ m ( p − a , p + a )+ x ( p ) ( e E p w , E p − a w )( E p w ℓ , e E p + a w ℓ ) ∈ q m ( p − a , p + a )+ p − p − · q max(0 ,p − p − · q min(0 ,p − p − ℓ +1) A ⊆ q min(0 ,p − p − ℓ )+ p − p − A. The assertion is proved. (cid:3)
Next we shall show that Z belongs to the set in (3.4.13). Lemma 3.4.17.
Assume that p ∈ Z ≥ satisfies E p w ⊗ ℓ = 0 . (1) If p > p + ℓ , then E p + ε w ⊗ ℓ = 0 . (2) If p > p , then either E p − ε w ⊗ ℓ = 0 or E p + ε w ⊗ ℓ = 0 holds.Proof. (1) First consider the case ℓ = 1. By (the proof of) Lemma 3.4.14 (1) andLemma 2.4.2, the vector ( f E p + ε − [ p − p − E p ) w is either 0, or not proportionalto E p w . In both cases we have f E p + ε w = 0, and hence the assertion (1) is provedfor ℓ = 1.Assume that ℓ >
1. Obviously, E p w ⊗ ℓ = 0 implies E p w ⊗ · · · ⊗ E p ℓ w = 0 for some p , . . . , p ℓ ∈ Z ≥ such that p = p + · · · + p ℓ . The assumption implies that there existssome k such that p k − p k >
1, and then E p k + ε w = 0 holds by the argument for ℓ = 1.Since the nonzero vectors of the form E p ′ w ⊗ · · · ⊗ E p ′ ℓ w are linearly independent byLemma 3.3.1, this implies that E p + ε w ⊗ ℓ is nonzero. The assertion is proved. (2) First assume that ℓ = 1. If p = 0, E p − ε w = 0 obviously holds, and hence wemay assume p ≥
1. That E p w = 0 implies p ≤ p = 2 and p = 1. If h h , wt( E p − ε w ) i = p − p − p − ≤ − , then E p + ε w = 0 follows, and hence we may assume that p > p + p . If p = 0,since (2.1.2) implies e E ( p ) i e ( p )10 w = 0, we have e E p − ε w = E p w + e (2)2 E p − ε w = E p w = 0 , which implies E p − ε w = 0. It is also checked similarly that E p − ε w = 0 holds if p = 0. The remaining case is p = (3 , , , ,
1) only, and in this case E (3 , , , , w = 0is proved from (3.3.13) and f E (3 , , , , w = E (0 , , , , e (2)1 e (3)0 w = 0 . The proof for ℓ = 1 is complete. Then the same argument used in the proof of (1) alsoworks here, and (2) for general ℓ is proved. (cid:3) Lemma 3.4.18.
Let p , p ∈ Z ≥ be such that E p w = 0 and E p w ⊗ ℓ = 0 . (1) If p > p + 1 , then m ( p , p ) ≥ − p + p + ℓ . (2) If p > p , then we have m ( p , p ) ≥ − p + p .Proof. (1) By Lemma 3.4.17 (1), we have E p + ε w = 0, and hence m ( p + ε , p ) ≥ m ( p , p ) = m ( p + ε , p ) − p + p + ℓ by Lemma 3.4.12, the assertion (1) follows.(2) By Lemma 3.4.17 (2), we have either E p − ε w ⊗ ℓ = 0 or E p + ε w ⊗ ℓ = 0, andhence either m ( p , p − ε ) ≥ m ( p , p + ε ) ≥ m ( p , p ) = m ( p , p − ε ) − p + p and m ( p , p ) = m ( p , p + ε ) + p , in both cases m ( p , p ) ≥ − p + p holds, and the proof is complete. (cid:3) Now the following proposition implies that Z belongs to the set in (3.4.13). Proposition 3.4.19.
Assume that p , p ∈ Z ≥ satisfy E p w = 0 and E p w ℓ = 0 .Setting p = p + p , we have q m ( p , p ) k E p w k · k e E p w ℓ k ∈ q , − p + p ,p − p − ℓ − − A. (3.4.21) Proof.
Set N = min(0 , − p + p , p − p − ℓ −
1) and N = min(0 , − p + p , p − p − ℓ ) . Since k E p w k ∈ q s A by Proposition 3.3.8 and k e E p w ℓ k ∈ q N − A by (3.4.11)with p replaced by p (which we are assuming to hold), it suffices to show that m ( p , p ) + N ≥ N. (3.4.22)If N = 0, this follows from Lemma 3.4.13. Moreover if N = − p + p <
0, this holdssince m ( p , p ) + ( − p + p ) ≥ ( − p + p ) + ( − p + p ) = − p + p by Lemma 3.4.18 (2). Finally assume that N = p − p − ℓ . If p ≤ p + 1,then (3.4.22) holds since N ≥ N + ( p − p −
1) = p − p − ℓ − . On the other hand if p > p + 1, (3.4.22) follows from Lemma 3.4.18 (1). The proofis complete. (cid:3) Finally, we shall show that Z + Z belongs to the set in (3.4.13), which completesthe proof of Proposition 3.4.9. By a similar calculation that we did for k e E p w ℓ +1 k ,we have k e E p w ⊗ ( ℓ +1)1 k = W + W + W + W , where W = X q m ( p , p ) k E p w k · k e E p w ⊗ ℓ k ,W = 2 X q m ( p , p )+ m ( p − a , p + a )+ x ( p ) ( e E p w , E p − a w )( E p w ⊗ ℓ , e E p + a w ⊗ ℓ ) ,W = 2 X q m ( p , p )+ m ( p + ε , p − ε )+ x ( p ) ( e E p w , E p + ε w )( E p w ⊗ ℓ , e E p − ε w ⊗ ℓ ) ,W = X q m ( p , p )+2 x ( p ) k e E p w k · k E p w ⊗ ℓ k . We have W = Z by Proposition 3.3.11. Moreover, the equality( E p w ⊗ ℓ , e E p − ε w ⊗ ℓ ) ( W ) ⊗ ℓ = ( E p w ℓ , e E p − ε w ℓ ) W ℓ is proved for any p by checking X = E p and Y = e E p − ε satisfy the assumptionsof Lemma 3.3.10, and hence W = Z follows. On the other hand, the left-hand side k e E p w ⊗ ( ℓ +1)1 k belongs to q , − p + p ) by Proposition 3.4.7 (1). Hence in order toshow that Z + Z (= W + W ) belongs to the set in (3.4.13), it is enough to prove thatboth W and W do. The assertion for W is deduced from the following lemma. Lemma 3.4.20.
For any p , p ∈ Z ≥ , we have q m ( p , p ) k E p w k · k e E p w ⊗ ℓ k ∈ q , − p + p ) A, where we set p = p + p .Proof. We may assume that E p w = 0 and E p w ⊗ ℓ = 0. We have k E p w k ∈ q s A by Proposition 3.3.8, and k e E p w ⊗ ℓ k ∈ q , − p + p ) A by Proposition 3.4.7 (1). If − p + p ≥
0, the assertion follows from Lemma 3.4.13. Otherwise we have m ( p , p ) ≥− p + p by Lemma 3.4.18 (2), and hence the assertion is proved. (cid:3) The assertion for W is easily proved from the following lemma and (3.4.20). Lemma 3.4.21.
For any p ∈ Z ≥ , we have ( e E p w ⊗ ℓ , E p − a w ⊗ ℓ ) ∈ A. (3.4.23) Proof.
We proceed by the induction on ℓ . The assertion for the base case of ℓ = 1follows from Lemma 3.4.15 (2). Assume (3.4.23) for a fixed ℓ and any p . Our task is to prove this with ℓ replaced by ℓ + 1. We have( e E p w ⊗ ( ℓ +1)1 , E p − a w ⊗ ( ℓ +1)1 )= X p + p = p q m ( p , p ) (cid:16) q m ( p − a , p )+ x ( p ) ( e E p w , E p − a w ) k E p w ⊗ ℓ k + q m ( p , p − a ) k E p w k ( e E p w ⊗ ℓ , E p − a w ⊗ ℓ ) (cid:17) . By the induction hypothesis and Lemma 3.4.13, q m ( p , p )+ m ( p , p − a ) k E p w k ( e E p w ⊗ ℓ , E p − a w ⊗ ℓ ) ∈ A holds. On the other hand, E p w ⊗ ℓ = 0 implies p ≥ p by Lemma 3.4.10. Since m ( p , p ) + x ( p ) = m ( p − a , p ) + p − p by Lemma 3.4.12, it also follows from the induction hypothesis that q m ( p , p )+ m ( p − a , p )+ x ( p ) ( e E p w , E p − a w ) k E p w ⊗ ℓ k ∈ A. The proof is complete. (cid:3)
Appendix
A.The goal of this appendix is to show the following.
Proposition A.1.
Let ℓ ∈ Z > . There exists an element c ℓ ∈ ± q s A such that e E p w ℓ = c ℓ E p − a + ε f w ℓ + [ p − p + 1] E p + ε w ℓ for any p ∈ Z ≥ . A fundamental tool for the proof is the braid group action on U q ( g ) introduced byLusztig. For i ∈ I , let T i = T ′′ i, be the algebra automorphism of U q ( g ) in [Lus93,Chapter 37]. For a sequence i p · · · i of elements of I , write T i p ··· i = T i p · · · T i . Here wecollect the properties of T i ; for the proofs, see [Lus90, Lus93]. Lemma A.2. (a)
For i ∈ I and α ∈ Q , we have T i U q ( g ) α = U q ( g ) s i ( α ) . (b) For i, j ∈ I and p ∈ Z > , we have T i ( e ( p ) j ) = − c ij p X k =0 ( − q i ) − k e ( − c ij p − k ) i e ( p ) j e ( k ) i . (c) For i, j ∈ I , we have T i T j · · · | {z } c ij c ji +2 = T j T i · · · | {z } c ij c ji +2 . (d) If i p · · · i is a reduced word, then T i p ··· i ( e i ) ∈ U q ( n + ) . Moreover, if we furtherassume that s i p · · · s i ( α i ) = α j for some j ∈ I , then we have T i p ··· i ( e i ) = e j . (e) Let i, j ∈ I be such that c ij = c ji = − and p ∈ Z > . Then we have e i e ( p ) j = e ( p − j T i ( e j ) + q − p e ( p ) j e i and T i ( e j ) e j = qe j T i ( e j ) . XISTENCE OF KR CRYSTALS FOR NEAR ADJOINT NODES 35 (f)
Let M be an integrable U q ( g ) -module, and i ∈ I . There is a Q ( q s ) -linear auto-morphism e T i (denoted by T ′′ i, in [Lus93] ) satisfying e T i ( Xm ) = T i ( X ) e T i ( m ) for X ∈ U q ( g ) and m ∈ M . Moreover if m ∈ M λ for λ ∈ D − P and f i m = 0 , wehave e T i ( e ( p ) i m ) = ( − p q p ( − λ i − p +1) i e ( − λ i − p ) i m for p ∈ Z ≥ , where we set λ i = h h i , λ i . Lemma A.3.
The word ji = ( j L ′ · · · j i L · · · i ) is reduced.Proof. For any 0 ≤ k ≤ L , we have h s j s i [ L,k +1] ( h i k ) , θ i = h h i k , s i [ k, ( α ) i > , which implies s j s i [ L,k +1] ( α i k ) ∈ R +1 . This, together with Lemma 3.1.1 (4), implies theassertion. (cid:3) In the sequel, we write i = i [ L,
1] and j = j [ L ′ ,
1] for short.
Lemma A.4.
Let M be an integrable U q ( g ) -module, v ∈ M \ { } , and p ∈ Z > . (1) If e i v = 0 ( i ∈ { } ⊔ J \ { } ) and e e v = 0 , then T j ( e ) v = 0 . (2) If e i v = 0 ( i ∈ { } ⊔ J \ { } ) , then T j ( e ( p )2 ) v = E ( p )1 j v = ( − q ) p T j ( e ( p )1 ) v . (3) If e i v = 0 ( i ∈ J \ { } ) , then T j ( e ( p )2 ) v = E ( p ) j v . (4) We have T i ( e ) = e T i ( e ) − q − T i ( e ) e . (5) If e i v = 0 ( i ∈ I ) , then T i ( e ) v = E i v . (6) We have e T i ( e ( p )2 ) = T i ( e ( p − ) T i ( e ) + q − p T i ( e ( p )2 ) e . (7) If e v = 0 , then e T j ( e ( p )2 ) v = T j ( e ( p − ) T j ( e ) v . (8) If e i v = 0 ( i ∈ I ) , then T i ( e ) e ( p )1 v = e ( p − T i ( e ) v . (9) If e i v = 0 ( i ∈ I ) , then T i ( e ) e ( p )0 v = e ( p − T i ( e ) v . (10) We have T j ( e ) e ( p )0 = e ( p − T j ( e ) + q − p e ( p )0 T j ( e ) . (11) We have T j ( e ) e ( p )1 = q p e ( p )1 T j ( e ) . (12) We have e T j ( e ) T i ( e ) = T j ( e ) T i ( e ) e . (13) If e i v = 0 ( i ∈ I \ { , } ) , then T i ( e ( p )2 ) v = E ( p ) i v = a p T ji ( e ( p )2 ) v with somenonzero a ∈ Q ( q s ) .Proof. Let us prepare some notation. For a subset L ⊆ I and Λ ∈ − P + , denote by V L (Λ) the U q ( g L )-submodule of V (Λ) generated by v Λ , which is isomorphic to the simplelowest weight U q ( g L )-module whose lowest weight is the restriction of Λ on P i ∈ L D − h i .Let us prove the assertion (1). Set J ′ = { } ⊔ J , and ℓ = max { m ∈ Z ≥ | e ( m )2 v = 0 } .By the well-known fact for the defining relations (see the proof of Lemma 3.3.7), thereis a U q ( n + ,J ′ )-module homomorphism from V J ′ ( − ℓ Λ ) to M mapping v − ℓ Λ to v . Hencewe may assume that v = v − ℓ Λ , and then the assertion (1) is proved as follows: ByLemma A.2 (b) and (f), T j ( e ) v = T j ( e e − q − e e ) v = e T j (cid:0) ( e e − q − e e ) v (cid:1) = 0 . Next we shall prove the assertion (2). As above, we may assume that v = v − ℓ Λ forsome ℓ ∈ Z > . The first equality is proved using Lemma A.2 (f) as follows: T j ( e ( p )2 ) v = e T j ( e ( p )2 v ) = E ( p )1 j v. By Lemma A.2 (b), we have T j ( e ( p )1 ) v = X k ( − q ) − k T j ( e ( p − k )2 e ( p )1 e ( k )2 ) v = X k ( − q ) − k T j ( e ( p − k )2 ) e ( p )1 T j ( e ( k )2 ) v, and since f T j ( e ( k )2 ) v = 0, e ( p )1 T j ( e ( k )2 ) v = 0 holds unless k = p . Now the secondequality is proved similarly as above. The proofs of the assertions (3)–(5) are similar.The assertion (6) is proved as follows: By Lemma A.2 (b) and (e), we have e T i ( e ( p )2 ) = T i ( e e ( p )2 ) = T i ( e ( p − T ( e ) + q − p e ( p )2 e )= T i ( e ( p − ) T i ( e ) + q − p T i ( e ( p )2 ) e . The assertions (7)–(10) are proved similarly.The assertion (11) is proved as follows: By Lemma A.2 (e), we have T j ( e ) e ( p )1 = T j (cid:16) T ( e ) e ( p )1 (cid:17) = q p T j (cid:16) e ( p )1 T ( e ) (cid:17) = q p e ( p )1 T j ( e ) . The assertion (12) is proved as follows: Since s s ( α ) = α , from Lemma A.2 (d),it follows that e T j ( e ) T i ( e ) = T j ( e e ) T i ( e ) = T j ( e ) e T i ( e )= T j ( e ) T i ( e e ) = T j ( e ) T i ( e ) e . Finally let us show the assertion (13). As above, setting ℓ = max { m ∈ Z ≥ | e ( m )2 v =0 } , we may assume that v = v − ℓ Λ , and the first equality is proved similarly. To provethe other one, note first that wt P (cid:0) T ji ( e ( p )2 ) (cid:1) = pθ , anddim V I ( − ℓ Λ ) − ℓ Λ + pθ = ( ≤ p ≤ ℓ ) , p > ℓ ) , (A.1)which is proved by taking the classical limit and applying the Poincar´e–Birkhoff–Witttheorem. Moreover, since T ji ( e ( p )2 ) v = e T ji ( e ( p )2 e T − ji ( v )) and h h , wt P e T − ji ( v ) i = h h θ , − ℓ Λ i = − ℓ, we have T ji ( e ( p )2 ) v = 0 if and only if 0 ≤ p ≤ ℓ . Hence for each 1 ≤ p ≤ ℓ there issome nonzero a p ∈ Q ( q s ) such that a p T ji ( e ( p )2 ) v = E ( p ) i v , and T ji ( e ( p )2 ) v = E ( p ) i v = 0if p > ℓ . It remains to prove that a p = a p , which we show by the induction on p . Thecase p = 1 is trivial. Assume that p >
1. By Lemma 3.3.4 and weight considerations,we see that e i E ( p − i v = 0 for i ∈ I \ { , } , and hence it follows from the inductionhypothesis that T ji ( e ( p )2 ) v = a − p +11 [ p ] − T ji ( e ) E ( p − i v = a − p [ p ] − E (1) i E ( p − i v XISTENCE OF KR CRYSTALS FOR NEAR ADJOINT NODES 37 (note that E ( p ) i = (cid:16) E (1) i (cid:17) ( p ) by our convention (3.1.1)). Hence it suffices to show that E (1) i E ( p − i v = [ p ] E ( p ) i v . It is proved by a direct calculation that f ( c g p ) i · · · f ( c g p ) i L E (1) i E ( p − i v = e e ( p − v = [ p ] e ( p )2 v = [ p ] f ( c g p ) i · · · f ( c g p ) i L E ( p ) i v, which implies E (1) i E ( p − i v = [ p ] E ( p ) i v by (A.1). The proof of (13) is complete. (cid:3) Lemma A.5.
For any ℓ ∈ Z > and ( p , p , p ) ∈ Z ≥ , we have e E ( p ) j E ( p ) i e ( p )10 w ℓ = E ( p − j E ( p − i e ( p − E a w ℓ . Proof. If p < p , the left-hand side is 0 by (2.1.1), and so is the right-hand side since e i e E a w ℓ ∈ W ℓℓ̟ + α i = 0 . Hence we may assume that p ≥ p . Set w = e ( p )10 w ℓ , and w ′ = E ( p ) i e ( p )10 w ℓ . We have e i w ′ = 0 for i ∈ { } ⊔ J \ { } and e i E (1) j w ′ = 0 for i ∈ J by Lemma 3.3.4 and (2.1.2), and therefore we have the following; e E ( p ) j E ( p ) i e ( p )10 w ℓ = e E ( p ) j w ′ (3) = e T j ( e ( p )2 ) w ′ (7) = T j ( e ( p − ) T j ( e ) w ′ (2) (3) = E ( p − j e E j w ′ , where a number over an equality indicates which assertion of Lemma A.4 is used there.Since e i w = 0 for i ∈ I \ { } and e e w = 0, we have the following; e E j w ′ (2) = − qT j ( e ) E ( p ) i w (13) = − qa p T j ( e ) T ji ( e ( p )2 ) w (6) = − qa p T j (cid:16) T i ( e ( p − ) T i ( e ) + q − p T i ( e ( p )2 ) e (cid:17) w (1) (4) = − qa p T ji ( e ( p − ) T j ( e ) T ji ( e ) w (2) (13) = E ( p − i e E (1) j E (1) i w. Finally, we have e E (1) j E (1) i w = e E (1) j E (1) i e ( p )10 w ℓ (2) (13) = − qT j ( e ) T i ( e ) e ( p )10 w ℓ (8) (9) = − qT j ( e ) e ( p − T i ( e ) w ℓ (11) = − q p e ( p − T j ( e ) e ( p − T i ( e ) w ℓ (10) = − q p e ( p − (cid:16) e ( p − T j ( e ) + q − p +1 e ( p − T j ( e ) (cid:17) T i ( e ) w ℓ (12) = − qe ( p − T j ( e ) T i ( e ) w ℓ (2) (5) = e ( p − E a w ℓ . The assertion is proved. (cid:3)
Proof of Proposition A.1.
By (2.1.2) and Lemma A.5, we have e E p w ℓ = (cid:16) e ( p − e ( p +1)2 e E ( p ) j E ( p ) i e ( p )10 + [ p − p + 1] E p + ε (cid:17) w ℓ = (cid:0) E p − a + ε E a + [ p − p + 1] E p + ε (cid:1) w ℓ . Hence it suffices to show that E a w ℓ = c ℓ f w ℓ holds for some c ℓ ∈ ± q s A . We seefrom Proposition 3.1.2 (C1) that dim W ℓℓ̟ − α = 1, and hence we have E a w ℓ = c ℓ f w ℓ for some c ℓ ∈ Q ( q s ). Now c ℓ ∈ ± q s A follows since both k E a w ℓ k and k f w ℓ k belongto 1 + q s A . The proof is complete. (cid:3) References [BS20] R. Biswal and T. Scrimshaw. Existence of Kirillov–Reshetikhin crystals for multiplicity freenodes.
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Institute of Engineering, Tokyo University of Agriculture and Technology,2-24-16 Naka-cho, Koganei-shi, Tokyo 184-8588, JAPAN
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