Demazure Character Formulas for Generalized Kac--Moody Algebras
aa r X i v : . [ m a t h . R T ] A p r Demazure Character Formulas for GeneralizedKac–Moody Algebras
Motohiro Ishii
Graduate School of Pure and Applied Sciences, University of Tsukuba,Tsukuba, Ibaraki 305-8571, Japan(e-mail: [email protected] ) Abstract
For a dominant integral weight λ , we introduce a family of U + q ( g )-submodules V w ( λ )of the irreducible highest weight U q ( g )-module V ( λ ) of highest weight λ for a general-ized Kac–Moody algebra g . We prove that the module V w ( λ ) is spanned by its globalbasis, and then give a character formula for V w ( λ ), which generalizes the Demazurecharacter formula for ordinary Kac–Moody algebras. For a (symmetrizable) Kac–Moody algebra g , the Demazure character formula describes the(formal) character of a
Demazure module , which is a U + ( g )-submodule generated by an extremalweight vector of an integrable highest weight U ( g )-module; this formula was proved by Kumar([ Kum ]) and Mathieu ([ M ]) independently by using geometric methods. Also, from an algebraicviewpoint, it is possible to formulate the notion of Demazure modules for an integrable highestweight U q ( g )-module. In fact, Littelmann ([ Li3 ]) gave a (conjectural) algebraic description ofthe Demazure character formula in terms of Kashiwara’s crystal bases for g of finite type. Soonafterward, this conjecture of Littelmann was proved by Kashiwara ([ Kas2 ]) generally for a Kac–Moody algebra. More precisely, Kashiwara showed that a Demazure module is spanned by itsglobal basis, and that there exists a basis at the crystal limit “ q = 0”, which is called a Demazurecrystal .In [
JKK, JKKS ], Kashiwara’s crystal basis theory was extended to the case of generalizedKac–Moody algebras. Also, Littelmann’s path model for representations of Kac–Moody algebraswas extended to the case of generalized Kac–Moody algebras by Joseph and Lamprou ([ JL ]).Therefore, it is natural to expect the existence of Demazure crystals for an irreducible highestweight module over a generalized Kac–Moody algebra. The purpose of this paper is to obtaina generalization of Demazure modules, their crystal bases, and a character formula for them forgeneralized Kac–Moody algebras.It is known that the representation theory of generalized Kac–Moody algebras are very similarto that of ordinary Kac–Moody algebras, and many results for Kac–Moody algebras are extended1o the case of generalized Kac–Moody algebras. However, there are some obstructions, whichcome from the existence of imaginary simple roots , for the study of the structure of an irreduciblehighest weight module over a generalized Kac–Moody algebra. Recall that a Demazure crystaldecomposes into a disjoint union of sl -strings of finite length in the case of ordinary Kac–Moody algebras. In contrast, in the case of generalized Kac–Moody algebras, sl -strings and“Heisenberg algebra”-strings corresponding to imaginary simple roots for an irreducible highestweight module are no longer of finite length. Hence, we cannot apply a method similar to thecase of ordinary Kac–Moody algebras to our setting. To overcome this difficulty, we introducea certain Kac–Moody algebra ˜ g defined from a given Borcherds–Cartan datum of a generalizedKac–Moody algebra g , and then we relate the representation theory of g to the one of ˜ g . Thisenables us to study the structure of an irreducible highest weight module over a generalizedKac–Moody algebra by comparing it with the corresponding one over an ordinary Kac–Moodyalgebra. In this way, under a certain condition, we can define Demazure modules for a generalizedKac–Moody algebra, and show that they are spanned by their global bases (Theorem 3). As aresult, we obtain a character formula for these Demazure modules for a generalized Kac–Moodyalgebra by introducing a “modified” Demazure operator for each imaginary simple root (Theorem4). Let us state our results more precisely. Let g be a generalized Kac–Moody algebra and ˜ g the associated Kac–Moody algebra (see § λ ∈ P + for g andthe corresponding dominant integral weight ˜ λ ∈ e P + for ˜ g (see § B ( λ ) and e B (˜ λ ) the sets of generalized Lakshmibai–Seshadri paths and ordinary Lakshmibai–Seshadri paths ,respectively. Then, there exists an embedding B ( λ ) ֒ → e B (˜ λ ) of path crystals (Proposition 2.5.1);note that this is not a morphism of crystals. By using this embedding, we have the followingdecomposition rules for U q ( g )-modules in the category O int (see [ JKK , § V ( λ )the irreducible highest weight U q ( g )-module of highest weight λ ∈ P + . Theorem 1
Let λ, µ ∈ P + be dominant integral weights for g . Then, we have an isomorphismof U q ( g ) -modules: V ( λ ) ⊗ V ( µ ) ∼ = M π ∈ B ( µ )˜ π : ˜ λ - dominant V (cid:0) λ + π (1) (cid:1) . Here, ˜ π ∈ e B (˜ µ ) denotes the image of π ∈ B ( µ ) under the embedding B ( µ ) ֒ → e B (˜ µ ) , and it is saidto be ˜ λ -dominant if ˜ π ( t ) + ˜ λ belongs to the dominant Weyl chamber of ˜ g for all t ∈ [0 , . Theorem 2
Let λ ∈ P + . For a subset S of the index set I of simple roots of g , we denote by g S the corresponding Levi subalgebra of g , by ˜ g e S the corresponding one for ˜ g (see § S ( µ ) the irreducible highest weight U q ( g S ) -module of highest weight µ . Then, V ( λ ) ∼ = M π ∈ B ( λ )˜ π : ˜ g e S - dominant V S (cid:0) π (1) (cid:1) as U q ( g S ) - modules . Here, ˜ π ∈ e B (˜ λ ) denotes the image of π ∈ B ( λ ) under the embedding B ( λ ) ֒ → e B (˜ λ ) , and it is saidto be ˜ g e S -dominant if ˜ π ( t ) belongs to the dominant Weyl chamber of ˜ g e S for all t ∈ [0 , . Also, we prove an analog of the
Parthasarathy–Ranga Rao–Varadarajan conjecture for generalizedKac–Moody algebras (Theorem 3.2.4), using the embedding of path models and the tensorproduct decomposition rule (Theorem 1).Now, let us describe our main results. For a dominant integral weight λ ∈ P + , we take andfix an element w in the monoid W (see Definition 2.3.1) satisfying the condition:there exists an expression w = v l r j l · · · v r j v , with j , . . . , j l ∈ I im and v , v , . . . , v l ∈W re (see § § α ∨ j s (cid:0) v s − r j s − · · · v r j v ( λ ) (cid:1) = 1 for all s = 1 , , . . . , l, where α ∨ j , j ∈ I, are the simple coroots of g .For such an element w ∈ W , we set V w ( λ ) := U + q ( g ) V ( λ ) wλ , where V ( λ ) µ ⊂ V ( λ ) is the weightspace of weight µ ∈ P . Theorem 3
For λ ∈ P + and w ∈ W satisfying the condition above, there exists a subset B w ( λ ) ⊂ B ( λ ) of the crystal basis B ( λ ) of V ( λ ) such that V w ( λ ) = M b ∈ B w ( λ ) C ( q ) G λ ( b ) , where { G λ ( b ) } b ∈ B ( λ ) denotes the global basis of V ( λ ) . For the element w ∈ W above, we can take a specific expression, a minimal dominant reducedexpression (see § w = w k r a k i k · · · w r a i w , with w , w , . . . , w k ∈ W re , a , . . . , a k ∈ Z > , and i , . . . , i k ∈ I im (all distinct). For each i ∈ I re , we define an operator D i on the group ring Z [ P ] := L µ ∈ P Z e µ of the weight lattice P of g by: D i ( e µ ) := e µ − e µ − (1+ α ∨ i ( µ )) α i − e − α i . For a reduced expression v = r j l · · · r j r j ∈ W re , with j , j , . . . , j l ∈ I re , we set D v := D j l · · · D j D j . Also, for each i ∈ I im and a ∈ Z ≥ , we define an operator D ( a ) i by D ( a ) i ( e µ ) := ( e µ if α ∨ i ( µ ) = 0 , P am =0 e µ − mα i otherwise . heorem 4 Let λ ∈ P + and w = w k r a k i k · · · w r a i w ∈ W be as above. Then, the (formal)character of the Demazure module V w ( λ ) is given as follows: ch V w ( λ ) = D w k D ( a k ) i k · · · D w D ( a ) i D w ( e λ ) . This paper is organized as follows. In Section 2, we recall some elementary facts aboutgeneralized Kac–Moody algebras and Joseph–Lamprou’s path model. Also, we review the con-struction of an embedding of Joseph–Lamprou’s path model into Littelmann’s path model. InSection 3, we give proofs of Theorems 1 and 2, and show an analog of the Parthasarathy–Ranga-Rao–Varadarajan conjecture for generalized Kac–Moody algebras. In Section 4, we introduceDemazure modules for generalized Kac–Moody algebras, and study the structure of them. Fi-nally in Section 5, we prove Theorems 3 and 4.
Acknowledgments
The author is grateful to Professor Satoshi Naito for reading the manuscript very carefully andfor valuable comments. He is also grateful to Professor Daisuke Sagaki for valuable comments.The author’s research was supported by the Japan Society for the Promotion of Science ResearchFellowships for Young Scientists.
In this subsection, we recall some fundamental facts about generalized Kac–Moody algebras.For more details, we refer the reader to [ Bo , JKK , JKKS , JL , Kac , Kan ].Let I be a (finite or) countable index set. We call A = ( a ij ) i,j ∈ I a Borcherds–Cartan matrix ifthe following three conditions are satisfied: (1) a ii = 2 or a ii ∈ Z ≤ for each i ∈ I ; (2) a ij ∈ Z ≤ for all i, j ∈ I with i = j ; (3) a ij = 0 if and only if a ji = 0 for all i, j ∈ I with i = j .An index i ∈ I is said to be real if a ii = 2, and imaginary if a ii ≤
0. Denote by I re := { i ∈ I | a ii = 2 } the set of real indices, and by I im := { i ∈ I | a ii ≤ } = I \ I re the set of imaginaryindices. A Borcherds–Cartan matrix A is said to be symmetrizable if there exists a diagonalmatrix D = diag( d i ) i ∈ I , with d i ∈ Z > , such that DA is symmetric. Also, if a ii ∈ Z for all i ∈ I , then A is said to be even . Throughout this paper, we assume that the Borcherds–Cartanmatrix is symmetrizable and even.For a given Borcherds–Cartan matrix A = ( a ij ) i,j ∈ I , a Borcherds–Cartan datum is a quintuple (cid:0)
A, Π := { α i } i ∈ I , Π ∨ := { α ∨ i } i ∈ I , P, P ∨ (cid:1) , where Π and Π ∨ are the sets of simple roots and simplecoroots , respectively, P ∨ is a coweight lattice , and P := Hom Z ( P ∨ , Z ) is a weight lattice . We set4 := P ∨ ⊗ Z C , and call it the Cartan subalgebra . Let h ∗ := Hom C ( h , C ) denote the full dualspace of h . In this paper, we assume that Π ∨ ⊂ h and Π ⊂ h ∗ are both linearly independentover C . Let P + := { λ ∈ P | α ∨ i ( λ ) ≥ i ∈ I } be the set of dominant integral weights,and Q := L i ∈ I Z α i the root lattice ; we set Q + := P i ∈ I Z ≥ α i . Let g be the generalized Kac–Moody algebra associated with a Borcherds–Cartan datum ( A, Π, Π ∨ , P, P ∨ ). We have the rootspace decomposition g = L α ∈ h ∗ g α , where g α := { x ∈ g | ad( h )( x ) = h ( α ) x for all h ∈ h } , and h = g . Denote by ∆ := { α ∈ h ∗ | g α = { } , α = 0 } the set of roots, and by ∆ + the setof positive roots. Note that ∆ ⊂ Q ⊂ P and ∆ = ∆ + ⊔ ( − ∆ + ). Also, we define a Coxetergroup W re := h r i | i ∈ I re i group ⊂ GL( h ∗ ), where r i for i ∈ I re is a simple reflection, andset ∆ im := W re Π im , where Π im := { α i } i ∈ I im is the set of imaginary simple roots ; note that ∆ im ⊂ ∆ + . In addition, we set ∆ re := W re Π re and ∆ + re := ∆ re ∩ ∆ + , where Π re := { α i } i ∈ I re isthe set of real simple roots . As in the case of ordinary Kac–Moody algebras, it is easily checkedthat the coroot β ∨ := wα ∨ i of β = wα i ∈ ∆ + re ⊔ ∆ im is well-defined (see [ JL , § Definition 2.1.1
Let q be an indeterminate. The quantized universal enveloping algebra U q ( g ) associated with a Borcherds–Cartan datum ( A, Π, Π ∨ , P, P ∨ ) , with D = diag( d i ) i ∈ I as above, isa C ( q ) -algebra generated by the symbols e i , f i , i ∈ I , and q h , h ∈ P ∨ , subject to the followingrelations: · q = 1 , q h q h = q h + h for h , h ∈ P ∨ , · q h e i q − h = q h ( α i ) e i , q h f i q − h = q − h ( α i ) f i for h ∈ P ∨ and i ∈ I , · [ e i , f j ] = δ ij K i − K − i q i − q − i for i, j ∈ I , where we set q i := q d i and K i := q d i α ∨ i , · − a ij X r =0 ( − r (cid:20) − a ij r (cid:21) i e − a ij − ri e j e ri = 0 for i ∈ I re and j ∈ I , with i = j , · − a ij X r =0 ( − r (cid:20) − a ij r (cid:21) i f − a ij − ri f j f ri = 0 for i ∈ I re and j ∈ I , with i = j , · [ e i , e j ] = [ f i , f j ] = 0 if a ij = 0 .Here, we set [ n ] i := q ni − q − ni q i − q − i , [ n ] i ! := n Y k =1 [ k ] i , and (cid:20) mn (cid:21) i := [ m ] i ![ m − n ] i ![ n ] i ! for i ∈ I . If a ii <
0, then we set c i := − a ii ∈ Z > , and define { n } i := q cini − q − cini q cii − q − cii . If a ii = 0, we set { n } i := n . We define divided powers by e ( n ) i := e ni [ n ] i ! , f ( n ) i := f ni [ n ] i ! if i ∈ I re , and by e ( n ) i := e ni , f ( n ) i := f ni if i ∈ I im . Here we understand that e (0) i = f (0) i := 1, and e ( n ) i = f ( n ) i := 0 for5 <
0. Let U + q ( g ) and U − q ( g ) be the subalgebras of U q ( g ) generated by e i , i ∈ I , and f i , i ∈ I ,respectively.Let V ( λ ) be the irreducible highest weight U q ( g )-module of highest weight λ ∈ P + . Wedefine the Kashiwara operators ˜ e i , ˜ f i , i ∈ I , on V ( λ ) in the following way (see [ JKK , § V ( λ ) = L n ≥ f ( n ) i Ker( e i ) for each i ∈ I , and the weight space decomposition V ( λ ) = L µ ∈ P V ( λ ) µ , each weight vector v ∈ V ( λ ) µ , µ ∈ P , has the unique expression v = P n ≥ f ( n ) i v n such that (a) v n ∈ Ker( e i ) ∩ V ( λ ) µ + nα i , (b) if i ∈ I re and α ∨ i ( µ + nα i ) < n , then v n = 0, and (c)if i ∈ I im , n >
0, and α ∨ i ( µ + nα i ) = 0, then v n = 0. This expression for v is called the i -stringdecomposition . Then, we define ˜ f i v := P n ≥ f ( n +1) i v n and ˜ e i v := P n ≥ f ( n − i v n . Note that if i ∈ I im , then ˜ f i = f i .Let u λ ∈ V ( λ ) λ \{ } denote the highest weight vector, and B ( λ ) the crystal basis of V ( λ ) withthe crystal lattice L ( λ ). Here, L ( λ ) is a free module over the local ring { f ( q ) /g ( q ) | f ( q ) , g ( q ) ∈ C [ q ] , g (0) = 0 } generated by ˜ f i l · · · ˜ f i u λ , l ≥ i , . . . , i l ∈ I , and we have B ( λ ) = { ˜ f i l · · · ˜ f i u λ mod qL ( λ ) | l ≥ , i , . . . , i l ∈ I } \ { } . Let A := C [ q, q − ], and denote by V ( λ ) A the A -form of V ( λ ). Let ¯ : U q ( g ) → U q ( g ) be the C -algebra automorphism defined by q q − , q h q − h , e i e i , f i f i , for h ∈ P ∨ and i ∈ I . Also, we define a C -linear automorphism ¯ on V ( λ ) by Xu λ := Xu λ for X ∈ U q ( g ). Let { G λ ( b ) } b ∈ B ( λ ) denote the global basis of V ( λ ). We know from [ JKK , Theorem9.3] that the element G λ ( b ), b ∈ B ( λ ), is characterized by the following three conditions: (i) G λ ( b ) = G λ ( b ), (ii) G λ ( b ) ∈ V ( λ ) A ∩ L ( λ ), and (iii) G λ ( b ) ≡ b mod qL ( λ ). ˜ g We associate a Cartan matrix e A with a given Borcherds–Cartan matrix A = ( a ij ) i,j ∈ I as follows.Set ˜ I := { ( i, } i ∈ I re ⊔ { ( i, m ) } i ∈ I im ,m ∈ Z ≥ , and define a Cartan matrix e A := (˜ a ( i,m ) , ( j,n ) ) ( i,m ) , ( j,n ) ∈ ˜ I by ( ˜ a ( i,m ) , ( i,m ) := 2 for ( i, m ) ∈ ˜ I, ˜ a ( i,m ) , ( j,n ) := a ij for ( i, m ) , ( j, n ) ∈ ˜ I with ( i, m ) = ( j, n ) . Note that if A is symmetrizable, then so is e A (see [ I , Lemma 4.1.1]). Let us denote by (cid:0) e A, e Π := { ˜ α ( i,m ) } ( i,m ) ∈ ˜ I , e Π ∨ := { ˜ α ∨ ( i,m ) } ( i,m ) ∈ ˜ I , e P , e P ∨ (cid:1) the Cartan datum associated with e A , where e Π, e Π ∨ are the sets of simple roots and simple coroots, respectively, e P is a weight lattice, and e P ∨ isa coweight lattice. Let ˜ g be the associated Kac–Moody algebra, ˜ h := e P ∨ ⊗ Z C the Cartansubalgebra, and f W the Weyl group. Note that every permutation on the subset { ( i, m ) } m ∈ Z ≥ ⊂ ˜ I induces a (Dynkin) diagram automorphism of ˜ g for each i ∈ I im . We denote by S i thepermutation group on the subset { ( i, m ) } m ∈ Z ≥ ⊂ ˜ I for each i ∈ I im , and set Ω := Q i ∈ I im S i .6or notational simplicity, we write ( i , m ) := (( i s , m s )) ks =1 = (( i k , m k ) , . . . , ( i , m ) , ( i , m )) if i = ( i k , . . . , i , i ) ∈ I k and m = ( m k , . . . , m , m ) ∈ Z k for k ≥
0, and call it an ordered index if m r = 1 for i r ∈ I re , and if m x s = s for all s = 1 , , . . . , t , where { x , x , . . . , x t } = { ≤ x ≤ k | i x = i } , with 1 ≤ x < x < · · · < x t ≤ k , for each i ∈ I im . We set I := S ∞ k =1 I k , e I := S ∞ k =1 ˜ I k ,and denote by e I ord the set of all ordered indices. Since the m for which ( i , m ) ∈ e I ord is determineduniquely by i , we have a bijection I → e I ord , i ( i , m ) . Note that Ω acts on e I diagonally. Example 2.2.1 If I = { , , } , I re = { } and I im = { , } , then ˜ I = { (1 , } ⊔ { ( , m ) , ( , m ) } m ∈ Z ≥ . An ordered index ( i , m ) ∈ e I ord corresponding to i = ( , , , , , , , , , , , ) ∈I is as follows: ( i , m ) = (cid:0) ( , , (1 , , ( , , ( , , (1 , , ( , , ( , , (1 , , ( , , ( , , (1 , , ( , (cid:1) . We also define the subset e I gen ⊂ e I of generic indices in e I by e I gen := { ( i , m ) = (( i s , m s )) ks =1 ∈ e I | m s = m t for s = t with i s = i t ∈ I im } ; note that e I ord ⊂ e I gen ⊂ e I , and that e I gen = Ω e I ord . Inparticular, e I gen is stable under the action of Ω. W In this subsection, we give a brief review of the monoid W and its properties. For more details,we refer the reader to [ I , § r i ∈ GL( h ∗ ) , i ∈ I im , by r i ( µ ) := µ − α ∨ i ( µ ) α i for µ ∈ h ∗ , then the inverse of r i is given by r − i ( µ ) = µ + 11 − a ii α ∨ i ( µ ) α i for µ ∈ h ∗ . Note that this r i has an infinite order in GL( h ∗ ). Definition 2.3.1 ([ I , Definition 2 . . . Let W denote the monoid generated by the symbols ˜ r i , i ∈ I , subject to the following relations:(1) ˜ r i = 1 for all i ∈ I re ;(2) if i, j ∈ I re , i = j, and the order of r i r j ∈ GL( h ∗ ) is m ∈ { , , , } , then we have (˜ r i ˜ r j ) m = (˜ r j ˜ r i ) m = 1 ;(3) if i ∈ I im , then for all j ∈ I \ { i } such that a ij = 0 , we have ˜ r i ˜ r j = ˜ r j ˜ r i . Each element w ∈ W can be written as a product w = ˜ r i ˜ r i · · · ˜ r i k of generators ˜ r i , i ∈ I . Ifthe number k is minimal among all the expressions for w of the form above, then k is called the length of w and the expression ˜ r i ˜ r i · · · ˜ r i k is called a reduced expression . In this case, we write ℓ ( w ) = k . Since the r i ∈ GL( h ∗ ), i ∈ I , satisfy the conditions (1) , (2) and (3) in Definition 2.3.1,we have the following (well-defined) homomorphism of monoids:7 −→ GL( h ∗ ) , ˜ r i r i , for i ∈ I. For simplicity, we write r i for ˜ r i in W . Remark that W re and h r i | i ∈ I re i monoid ⊂ W areisomorphic as groups, where h r i | i ∈ I re i monoid denotes the submonoid of W generated by r i , i ∈ I re ; note that this submonoid is in fact a group by Definition 2.3.1 (1). Hence we may (anddo) regard the group W re as a submonoid of W . For β = wα i ∈ ∆ + re ⊔ ∆ im , i ∈ I , w ∈ W re , theelement r β := wr i w − is well-defined (see [ I , § Definition 2.3.2 ([ I , Definition 2 . . . For w ∈ W and β ∈ ∆ + re ⊔ ∆ im , we write w → r β w if ℓ ( r β w ) > ℓ ( w ) . Also, we define a partial order ≤ on W as follows: w ≤ w ′ in W if there exist w , w , . . . , w l ∈ W such that w = w → w → · · · → w l = w ′ . Note that if v ≤ w in W and w = r i l · · · r i r i is a reduced expression, then there exists a reducedexpression v = r i xp · · · r i x r i x , l ≥ x p > · · · > x > x ≥
1, by the
Exchange Property of W (see[ I , § i, j ∈ I im with a ij = 0,then r i and r i r j are not comparable even though r i is a subword of r i r j and each of them is a(unique) reduced expression. Definition 2.3.3 ([ I , Definition 2 . . . Let w = w k r i k · · · w r i w ∈ W , with i , . . . , i k ∈ I im and w , w , . . . , w k ∈ W re , be a reduced expression, namely, ℓ ( w ) = ℓ ( w k ) + 1 + · · · + ℓ ( w ) + 1 + ℓ ( w ) . We call this expression a dominant reduced expression if it satisfies r i s w s − r i s − · · · w r i w ( P + ) ⊂ P + for all s = 1 , , . . . , k . Note that every w ∈ W has at least one dominant reduced expression. Indeed, if we choose anexpression of the form above in such a way that the sequence (cid:0) ℓ ( w ) , ℓ ( w ) , . . . , ℓ ( w k ) (cid:1) is minimalin lexicographic order among all the reduced expressions of the form above, then it is a dominantreduced expression (see [ I , § In this subsection, following [ JL ], we review Joseph–Lamprou’s path model for generalized Kac–Moody algebras. For more details, we refer the reader to [ JL , I ].Let h R denote a real form of h , and h ∗ R its full dual space. Let P be the set of all piecewise-linear continuous maps π : [0 , −→ h ∗ R such that π (0) = 0 and π (1) ∈ P , where we set[0 ,
1] := { t ∈ R | ≤ t ≤ } . Also, we set H πi ( t ) := α ∨ i (cid:0) π ( t ) (cid:1) for t ∈ [0 , m πi := min { H πi ( t ) | H πi ( t ) ∈ Z , t ∈ [0 , } . 8ow we define the root operators e i , f i : P −→ P ⊔ { } for i ∈ I . First, we set f i + ( π ) :=max { t ∈ [0 , | H πi ( t ) = m πi } . If f i + ( π ) <
1, then we can define f i − ( π ) := min { t ∈ [ f + i ( π ) , | H πi ( t ) = m πi + 1 } . In this case, we set( f i π )( t ) := π ( t ) t ∈ [0 , f i + ( π )] ,π (cid:0) f i + ( π ) (cid:1) + r i (cid:0) π ( t ) − π (cid:0) f i + ( π ) (cid:1)(cid:1) t ∈ [ f i + ( π ) , f i − ( π )] π ( t ) − α i t ∈ [ f i − ( π ) , . Otherwise (i.e., if f i + ( π ) = 1), we set f i π := .Next, we define the operator e i for i ∈ I re . Set e i + ( π ) := min { t ∈ [0 , | H πi ( t ) = m πi } . If e i + ( π ) >
0, then we can define e i − ( π ) := max { t ∈ [0 , e i + ( π )] | H πi ( t ) = m πi + 1 } . In this case, weset ( e i π )( t ) := π ( t ) t ∈ [0 , e i − ( π )] ,π (cid:0) e i − ( π ) (cid:1) + r i (cid:0) π ( t ) − π (cid:0) e i − ( π ) (cid:1)(cid:1) t ∈ [ e i − ( π ) , e i + ( π )] π ( t ) + α i t ∈ [ e i + ( π ) , . Otherwise (i.e., if e i + ( π ) = 0), we set e i π := .Finally, we define the operator e i for i ∈ I im . Set e i − ( π ) := f i + ( π ). If e i − ( π ) < t ∈ [ e i − ( π ) ,
1] such that H πi ( t ) ≥ m πi + 1 − a ii , then we can define e i + ( π ) := min { t ∈ [ e i − ( π ) , | H πi ( t ) = m πi + 1 − a ii } . We set e i π := if e i − ( π ) = 1, or e i − ( π ) < H πi ( t ) 1] such that H πi ( t ) ≥ m πi + 1 − a ii and H πi ( s ) ≤ m πi − a ii for some s ∈ [ e i + ( π ) , e i π )( t ) := π ( t ) t ∈ [0 , e i − ( π )] ,π (cid:0) e i − ( π ) (cid:1) + r − i (cid:0) π ( t ) − π (cid:0) e i − ( π ) (cid:1)(cid:1) t ∈ [ e i − ( π ) , e i + ( π )] π ( t ) + α i t ∈ [ e i + ( π ) , . Let λ ∈ P + . We write µ ≥ ν for µ, ν ∈ W λ := { wλ ∈ P | w ∈ W} if there exists a sequenceof elements µ =: λ , λ , . . . , λ s − , λ s := ν in W λ and positive roots β , . . . , β s ∈ ∆ + re ⊔ ∆ im suchthat λ i − = r β i λ i and β ∨ i ( λ i ) > i = 1 , . . . , s . This relation ≥ on W λ defines a partial order.For µ, ν ∈ W λ and β ∈ ∆ + re ⊔ ∆ im , we write µ β ←− ν if µ = r β ν , β ∨ ( ν ) > 0, and µ covers ν bythis partial order. Note that the direction of the arrow β ←− defined above is opposite to that in[ JL , § Definition 2.4.1 ([ JL , § . . . For a rational number a ∈ (0 , and µ, ν ∈ W λ with µ ≥ ν ,an a -chain for ( µ, ν ) is a sequence µ =: ν β ←− ν β ←− · · · β s ←− ν s := ν of elements in W λ such thatfor each i = 1 , , . . . , s, aβ ∨ i ( ν i ) ∈ Z > if β i ∈ ∆ + re , and aβ ∨ i ( ν i ) = 1 if β i ∈ ∆ im . Definition 2.4.2 ([ JL , § . . . Let λ := ( λ > λ > · · · > λ s ) be a sequence of elements in W λ , and a := (0 = a < a < · · · < a s = 1) a sequence of rational numbers. Then, the pair := ( λ ; a ) is called a generalized Lakshmibai–Seshadri path (GLS path for short) of shape λ ifit satisfies the following conditions (called the chain condition): (i) there exists an a i -chain for ( λ i , λ i +1 ) for each i = 1 , , . . . , s − ; (ii) there exists a -chain for ( λ s , λ ) . In this paper, we think of the pair π = ( λ ; a ) as a path belonging to P by π ( t ) := P j − i =1 ( a i − a i − ) λ i + ( t − a j − ) λ j for a j − ≤ t ≤ a j and j = 1 , , . . . , s. We denote by B ( λ ) the set of all GLSpaths of shape λ .Now, we define a crystal structure on B ( λ ). Let π ∈ B ( λ ). We set wt( π ) := π (1) ∈ P . Foreach i ∈ I re , we set ε i ( π ) := − m πi , ϕ i ( π ) := α ∨ i (cid:0) wt( π ) (cid:1) − m πi = H πi (1) − m πi . Then, we have ε i ( π ) = max { n ∈ Z ≥ | e ni π ∈ P } , and ϕ i ( π ) = max { n ∈ Z ≥ | f ni π ∈ P } . For each i ∈ I im , weset ε i ( π ) := 0 , ϕ i ( π ) := α ∨ i (cid:0) wt( π ) (cid:1) . By the definitions, we have ϕ i ( π ) = ε i ( π ) + α ∨ i (cid:0) wt( π ) (cid:1) for all i ∈ I . Next, we define the Kashiwara operators on B ( λ ). We use the root operators e i , i ∈ I re , and f i , i ∈ I, on P as Kashiwara operators. For e i , i ∈ I im , we use the “cutoff” of the rootoperators e i , i ∈ I im , on P (cid:0) ⊃ B ( λ ) (cid:1) , that is, if e i π / ∈ B ( λ ), then we set e i π := in B ( λ ) even if e i π = in P . Thus, B ( λ ) is endowed with a crystal structure. From [ JL , Proposition 6.3.5], wehave B ( λ ) = F π λ \ { } , where F is the monoid generated by the Kashiwara operators f i , i ∈ I . In this subsection, we give a brief review of the construction of an embedding of Joseph–Lamprou’s path model for a given generalized Kac–Moody algebra g into Littelmann’s pathmodel for an associated Kac–Moody algebra ˜ g . For more details, we refer the reader to [ I , § A, Π, Π ∨ , P, P ∨ ) be a Borcherds–Cartan datum, and ( e A, e Π, e Π ∨ , e P , e P ∨ ) be an associatedCartan datum as in § µ ∈ W P + = W re P + , we take (and fix) an element ˜ µ ∈ e P such that ˜ α ∨ ( i,m ) (˜ µ ) = α ∨ i ( µ ) for all ( i, m ) ∈ ˜ I. (1)Let e B (˜ µ ) be the set of all (G)LS paths of shape ˜ µ for ˜ g , and e F the monoid generated by the Kashi-wara operators f ( i,m ) , ( i, m ) ∈ ˜ I . Following the notation of § F i = f i k · · · f i f i ∈ F for i = ( i k , . . . , i , i ) ∈ I , and F ( i , m ) = f ( i k ,m k ) · · · f ( i ,m ) f ( i ,m ) ∈ e F for ( i , m ) = (( i s , m s )) ks =1 ∈ e I . Proposition 2.5.1 ([ I , Proposition 4 . . . For a dominant integral weight λ ∈ P + , the map e : B ( λ ) −→ e B (˜ λ ) , π = F i π λ ˜ π := F ( i , m ) π ˜ λ , is well-defined and injective, where the m for which ( i , m ) ∈ e I ord is determined uniquely by i ∈ I . B ( λ ) denotes the crystal basis of the irreducible highest weight U q ( g )-module V ( λ ) of highest weight λ ∈ P + . Since we know from [ I , Theorem 6.1.1] that B ( λ ) ∼ = B ( λ ) ascrystals, we obtain the following embedding by Proposition 2.5.1: e : B ( λ ) ֒ → e B (˜ λ ) , b = e F i u λ ˜ b := e F ( i , m ) ˜ u ˜ λ , for ( i , m ) ∈ e I ord , (2)where e B (˜ λ ) denotes the crystal basis of the irreducible highest weight U q (˜ g )-module e V (˜ λ ) ofhighest weight ˜ λ ∈ e P + , e F i := ˜ f i k · · · ˜ f i ˜ f i , e F ( i , m ) := ˜ f ( i k ,m k ) · · · ˜ f ( i ,m ) ˜ f ( i ,m ) are monomials ofthe Kashiwara operators, and u λ ∈ V ( λ ), ˜ u ˜ λ ∈ e V (˜ λ ) are the highest weight vectors. U q ( g ) -modules in the category O int In this subsection, we show the decomposition rules for U q ( g )-modules stated in Theorems 1 and2. For this purpose, we recall the following decomposition rules for path crystals. Theorem 3.1.1 ([ I , Theorem 7 . . . Let λ, µ ∈ P + . Then, we have an isomorphism of crys-tals: B ( λ ) ⊗ B ( µ ) ∼ = G π ∈ B ( µ )˜ π : ˜ λ - dominant B (cid:0) λ + π (1) (cid:1) . Here, ˜ π ∈ e B (˜ µ ) denotes the image of π ∈ B ( µ ) under the embedding B ( µ ) ֒ → e B (˜ µ ) , and it is saidto be ˜ λ -dominant if ˜ π ( t ) + ˜ λ belongs to the dominant Weyl chamber of ˜ g for all t ∈ [0 , . Let S ⊂ I be a subset. We set S re := S ∩ I re and S im := S ∩ I im . Also, we set e S := { ( i, } i ∈ S re ⊔ { ( i, m ) } i ∈ S im ,m ∈ Z ≥ . Let us denote by g S (resp., ˜ g e S ) the Levi subalgebra of g corresponding to S (resp., the Levi subalgebra of ˜ g corresponding to e S ), and denote by B S ( λ )the set of all GLS paths of shape λ for g S . Theorem 3.1.2 ([ I , Theorem 7 . . . Let λ ∈ P + . Then, we have an isomorphism of g S -crystals: B ( λ ) ∼ = G π ∈ B ( λ )˜ π : ˜ g e S - dominant B S (cid:0) π (1) (cid:1) . Here, ˜ π ∈ e B (˜ λ ) denotes the image of π ∈ B ( λ ) under the embedding B ( λ ) ֒ → e B (˜ λ ) , and it is saidto be ˜ g e S -dominant if ˜ π ( t ) belongs to the dominant Weyl chamber of ˜ g e S for all t ∈ [0 , . From [ JKK , Theorems 3.7 and 7.1], we know the existence and uniqueness of the crystalbasis of U q ( g )-modules in the category O int , and the complete reducibility for U q ( g )-modulesin the category O int (see [ JKK , Definition 3.1] for the definition of the category O int ). Since11 ( λ ) ⊗ V ( µ ) , λ, µ ∈ P + , belongs to O int for g , and V ( λ ) belongs to O int for g S (as a U q ( g S )-module), Theorems 1 and 2 follow immediately from Theorems 3.1.1 and 3.1.2. In this subsection, we prove an analog of the Parthasarathy–Ranga Rao–Varadarajan conjecturefor generalized Kac–Moody algebras.In what follows, we denote by [ µ ] the unique element in W re µ ∩ P + for µ ∈ W P + . By anargument similar to the one for [ Li1 , Proposition 7.1], we can show the following lemma. Lemma 3.2.1 Let λ , µ ∈ P + , and π := ( µ , . . . , µ l ; a , a , . . . , a l ) ∈ B ( µ ) . If λ + π ( a p ) ∈ P + forall ≤ p ≤ l − , then there exists a λ -dominant path π ′ ∈ B ( µ ) such that λ + π ′ (1) = [ λ + π (1)] . Let r ( i,m ) , ( i, m ) ∈ ˜ I , denote the simple reflection of the Weyl group f W of ˜ g , f W re ⊂ f W thesubgroup of f W generated by r ( i, , i ∈ I re , and f W gen := { R ( i , m ) ∈ f W | ( i , m ) ∈ e I gen } , where R ( i , m ) := r ( i l ,m l ) · · · r ( i ,m ) for ( i , m ) = (( i s , m s )) ls =1 ∈ e I . Note that f W gen is not closed undermultiplication, and that f W re is isomorphic as a group to the submonoid W re of W ; we write thisisomorphism as e : W re ∼ = −−→ f W re , w = r i l · · · r i ˜ w := r ( i l , · · · r ( i , , for i , . . . , i l ∈ I re . (3)Also, we can easily check that the map f W gen → GL( h ∗ ) , R ( i , m ) R i , is well-defined, where R i := r i l · · · r i for i = ( i l , . . . , i ) ∈ I . Lemma 3.2.2 Let λ ∈ P + . If F ( i , m ) π ˜ λ , ( i , m ) ∈ e I gen , is not in e B (˜ λ ) , then F ( i , ¯m ) π ˜ λ is not in e B (˜ λ ) , where the element ( i , ¯m ) ∈ e I ord is determined uniquely by i ∈ I . Proof. Since ( i , m ) ∈ e I gen , there exists a permutation ω ∈ Ω on e I that sends ( i , m ) to ( i , ¯m )(see § e B (˜ λ ), induced by the diagramautomorphism of ˜ g corresponding to the ω ∈ Ω, that sends F ( i , m ) π ˜ λ to F ( i , ¯m ) π ˜ λ (see [ NS , Lemma3.1.1]). Therefore, we see that F ( i , ¯m ) π ˜ λ = . (cid:3) For a path η = F ( i , m ) π ˜ λ ∈ e B (˜ λ ) with ( i , m ) ∈ e I gen , we set ¯ η := F ( i , ¯m ) π ˜ λ , with ( i , ¯m ) ∈ e I ord .From the proof of Lemma 3.2.2, ¯ η is independent of the choice of an expression for η of the form F ( i , m ) π ˜ λ with ( i , m ) ∈ e I gen . The following lemma is obvious. Lemma 3.2.3 Let η ∈ e B (˜ λ ) , and write η (1) as ˜ λ − P c ( i,m ) ˜ α ( i,m ) , c ( i,m ) ∈ Z ≥ . If c ( i,m ) ∈ { , } for all i ∈ I im and m ∈ Z ≥ , then there exists a path π ∈ B ( λ ) such that ˜ π = ¯ η in e B (˜ λ ) . i = ( i k , i k − , . . . , i ) ∈ I , we set i [ s ] := ( i s , i s − , . . . , i ) , ≤ s ≤ k . Let λ, µ ∈ P + and R i , R j ∈ W , i = ( i k , . . . , i , i ), j = ( j l , . . . , j , j ) ∈ I , be such that α ∨ i s ( R i [ s − λ ) =1 (cid:0) resp ., α ∨ j t ( R j [ t − µ ) = 1 (cid:1) if i s ∈ I im (resp ., j t ∈ I im ). We take sequences m = ( m k , . . . , m , m )and n = ( n l , . . . , n , n ) of positive integers such that ( i , m ) , ( j , n ) ∈ e I gen , and such that m s = n t if i s = j t in I im . Also, we set ν := R i λ + R j µ ∈ P and ¯ ν := R ( i , m ) ˜ λ + R ( j , n ) ˜ µ ∈ e P ; note that¯ ν = ˜ ν in general, where ˜ ν ∈ e P is defined by equation (1) in § ν ∈ W P + . Withthis notation, we have the following. Theorem 3.2.4 If ¯ ν ∈ e P + , then V ( ν ) appears in the irreducible decomposition of V ( λ ) ⊗ V ( µ ) . Proof. If we take M ∈ Z ≥ such that ( i, M ) does not appear in ( i , m ) and ( j , n ) for any i ∈ I , then the equality α ∨ i ( ν ) = ˜ α ∨ ( i,M ) (¯ ν ) holds for all i ∈ I by the definition of ¯ ν . Since˜ α ∨ ( i,M ) (¯ ν ) ≥ 0, we deduce that ν is a dominant integral weight. By Theorem 1, it suffices toshow that there exists π ∈ B ( µ ) such that ˜ π ∈ e B (˜ µ ) is ˜ λ -dominant and λ + π (1) = ν . Set η := (cid:0) R − i , m ) R ( j , n ) ˜ µ ; 0 , (cid:1) ∈ e B (˜ µ ), with R ( i , m ) , R ( j , n ) ∈ f W gen . By Lemma 3.2.1, there existsa ˜ λ -dominant path η ∈ e B (˜ µ ) such that ˜ λ + η (1) = [˜ λ + η (1)] = ¯ ν . If we write η (1) as˜ µ − P c ( i,m ) ˜ α ( i,m ) , then c ( i,m ) ∈ { , } for all i ∈ I im and m ∈ Z ≥ by the definition of ¯ ν .Therefore, by Lemma 3.2.3, there exists π ∈ B ( µ ) such that ˜ π = ¯ η . From this, we conclude that˜ π ∈ e B (˜ µ ) is ˜ λ -dominant and λ + π (1) = ν . (cid:3) Corollary 3.2.5 Let λ, µ ∈ P + , w , w ∈ W re , and set ν = w λ + w µ . If ν ∈ P + , then V ( ν ) appears in the irreducible decomposition of V ( λ ) ⊗ V ( µ ) . Remark 3.2.6 For general elements w , w in the monoid W , the statement of the corollaryabove is false. To see this, we take λ, µ ∈ P + and i ∈ I im such that α ∨ i ( λ ) = 0 and α ∨ i ( µ ) = 1 .Then, ν := λ + r i µ = λ + µ − α i is a dominant integral weight for g . However, we have (cid:0) B ( λ ) ⊗ B ( µ ) (cid:1) ν = (cid:8) π λ ⊗ f i π µ (cid:9) , and e i ( π λ ⊗ f i π µ ) = π λ ⊗ e i f i π µ = π λ ⊗ π µ = . Therefore, there isno highest weight vector of weight ν , and V ( ν ) cannot be an irreducible component of V ( λ ) ⊗ V ( µ ) .As for ˜ g , we have ¯ ν := ˜ λ + r ( i, ˜ µ , ˜ α ∨ ( i, (¯ ν ) = ˜ α ∨ ( i, (˜ λ ) + ˜ α ∨ ( i, ( r ( i, ˜ µ ) = α ∨ i ( λ ) − α ∨ i ( µ ) = − ,and hence we see that ¯ ν is not a dominant integral weight for ˜ g . Let λ ∈ P + and w = v l r j l · · · v r j v ∈ W , with j , . . . , j l ∈ I im , v , . . . , v l ∈ W re , be such that α ∨ j s (cid:0) v s − r j s − · · · v r j v ( λ ) (cid:1) = 1 for all s = 1 , , . . . , l. (4)13ote that the condition (4) for w and λ is independent of the choice of an expression for w sincethis condition is preserved under the relations (1) , (2), and (3) of Definition 2.3.1. For such anelement w ∈ W , we set V w ( λ ) := U + q ( g ) V ( λ ) wλ , where V ( λ ) µ ⊂ V ( λ ) denotes the weight spaceof weight µ ∈ P . We call V w ( λ ) the Demazure (sub)module of V ( λ ) of lowest weight wλ . Since V w ( λ ) depends only on the weight wλ ∈ P , we may assume that the element w = R i , with i = ( i k , . . . , i , i ), satisfies the following condition (recall the notation i [ t ] = ( i t , . . . , i , i ) , t =1 , . . . , k ): α ∨ i t ( R i [ t − λ ) > t = 1 , , . . . , k. (5)We will show that the Demazure module V w ( λ ) is generated by a single weight vector. Namely,we will show that dim V ( λ ) wλ = 1; note that the action of W on the set of weights of V ( λ ) doesnot necessarily preserve the weight multiplicities. Before doing this, we recall the action of W re on B ( λ ) (see [ JL , § π ∈ B ( λ ) and v = r i k · · · r i r i ∈ W re , with i , i , . . . , i k ∈ I re , define S v π := x α ∨ ik ( R i [ k − wt( π )) i k · · · x α ∨ i ( R i [1] wt( π )) i x α ∨ i (wt( π )) i π, (6)where x ai := f ai if a ≥ , x ai := e − ai if a < 0, and i = ( i k , . . . , i , i ). It is obvious thatwt( S v π ) = v (cid:0) wt( π ) (cid:1) . Lemma 4.1.1 If w ∈ W and λ ∈ P + satisfy condition (4), then we have dim V ( λ ) wλ = 1 . Proof. Let w = v l r j l · · · v r j v , with v , . . . , v l ∈ W re , j , . . . , j l ∈ I im , be a reduced expressionsatisfying condition (4). It suffices to show that B ( λ ) wλ = 1. As remarked above, we mayassume further that this expression for w satisfies condition (5).If we set π := S v l f j l · · · S v f j S v π λ , then the x ’s in the expression (6) for S v s , s = 0 , , . . . , l, are all f by condition (5), and hence π = π wλ by condition (4). Also, by using the fact [ I , Lemma4.1.5], we can deduce that the element ˜ π ∈ e B (˜ λ ) (see Proposition 2.5.1) can be written as ˜ π = S ˜ v l f ( j l ,m l ) · · · S ˜ v f ( j ,m ) S ˜ v π ˜ λ = π ˜ w ˜ λ , where (( j s , m s )) ls =1 ∈ e I ord , ˜ w := ˜ v l r ( j l ,m l ) · · · ˜ v r ( j ,m ) ˜ v ∈ f W , and each ˜ v s ∈ f W re corresponds to v s ∈ W re via the isomorphism (3) in § B ( λ ) wλ under the embedding of Proposition 2.5.1 is contained in e B (˜ λ ) ˜ w ˜ λ ; note thatif π and π in B ( λ ) satisfy wt( π ) = wt( π ), then wt(˜ π ) = wt(˜ π ). Since e B (˜ λ ) ˜ w ˜ λ = { π ˜ w ˜ λ } , weconclude that B ( λ ) wλ = { π wλ } , and hence B ( λ ) wλ = 1. (cid:3) Remark 4.1.2 Since the element ˜ w ∈ f W in the proof of Lemma 4.1.1 is a minimal cosetrepresentative of a coset in f W / f W ˜ λ , where f W ˜ λ ⊂ f W denotes the isotropy subgroup for ˜ λ ∈ e P + (see [ Hu , § . 10 and § . ), by the uniqueness of minimal coset representative, ˜ w is independentof the choice of an expression for w satisfying conditions (4) and (5) for λ ∈ P + . .2 Minimal dominant reduced expressions In this subsection, we introduce a specific expression for w ∈ W , which satisfies conditions (4)and (5) for λ ∈ P + , in order to state Theorem 4 (see § Lemma 4.2.1 Let w ∈ W satisfy conditions (4) and (5) for λ ∈ P + . Then, w has an expression w = w k r a k i k · · · w r a i w , with w , . . . , w k ∈ W re and a , . . . , a k ∈ Z ≥ , where i , . . . , i k ∈ I im areall distinct. Moreover, if s ∈ { ≤ t ≤ k | a t > } , then a i s ,i s = 0 , i.e., the ( i s , i s ) -entry of theBorcherds–Cartan matrix A is zero. Proof. Let w = v l r j l · · · v r j v , with v , . . . , v l ∈ W re , j , . . . , j l ∈ I im , be a reduced expressionsatisfying conditions (4) and (5) for λ ∈ P + . If j s = j t with s > t , then we have α ∨ j s (cid:0) v s − r j s − · · · v t r j t ( v t − r j t − · · · v r j v ( λ )) (cid:1) = α ∨ j t (cid:0) v t − r j t − · · · v r j v ( λ ) (cid:1) by condition (4). Also, by condition (5), r j s commutes with v s − , . . . , v t +1 , v t and r j s − , . . . , r j t +1 , r j t since α ∨ j s is an anti-dominant integral coweight by [ JL , Lemma 2.1.11]. In particular, we obtain a j s ,j s = 0. (cid:3) We fix an expression w = w k r a k i k · · · w r a i w given in Lemma 4.2.1 for which the sequence (cid:0) ℓ ( w ) , ℓ ( w ) , . . . , ℓ ( w k ) (cid:1) is minimal in lexicographic order among all such expressions of w .Then, this is a dominant reduced expression (see Definition 2.3.3). We call this expression a minimal dominant reduced expression (with respect to λ ∈ P + ).Here we collect some fundamental properties of minimal dominant reduced expression w = w k r a k i k · · · w r a i w ; these properties follow directly from the definition and by induction on k and ℓ ( w ). For each 0 ≤ s ≤ k , let w s = r s,ℓ s · · · r s, r s, , with r s,p := r α s,p , α s,p ∈ Π re , be a (fixed)reduced expression, where ℓ s := ℓ ( w s ). Lemma 4.2.2 With the notation above, the following statements hold.(a) α ∨ s,p ( r s,p − · · · r s, r s, r a s − i s − · · · w r a i w λ ) = 1 for all ≤ s ≤ k − and ≤ p ≤ ℓ s .(b) α ∨ s,p ( α s,p − ) = − for all ≤ s ≤ k − and < p ≤ ℓ s .(c) α ∨ s,p ( α t,q ) = 0 and α ∨ s,p ( α i t ) = 0 if ( t, q ) < ( s, p − in lexicographic order.(d) α ∨ i s ( α s − , ) = − and α ∨ i s ( α t,q ) = 0 if ( t, q ) < ( s − , in lexicographic order.(e) α ∨ , ( λ ) = 1 and α ∨ s,p ( λ ) = 0 for ≤ s ≤ k − and ≤ p ≤ ℓ s such that ( s, p ) = (0 , .(f ) If s = 0 or w = 1 , then α ∨ i s ( λ ) = 0 for ≤ s ≤ k − .(g) If w = 1 , then α ∨ i ( λ ) = 1 . .3 Description of Demazure modules This subsection is devoted to the proof of Proposition 4.3.1 below. Before doing this, we fixsome notation. Let g re ⊂ g be the Levi subalgebra corresponding to I re ; note that g re isan ordinary Kac–Moody algebra. For a (fixed) reduced expression v = r j l · · · r j r j ∈ W re ,with j l , . . . , j , j ∈ I re , we set F m v := f m l j l · · · f m j f m j and e F m v := ˜ f m l j l · · · ˜ f m j ˜ f m j . Also, for i = ( i l , . . . , i , i ), set F mi := f m l i l · · · f m i f m i and F max i := f max i l · · · f max i , where f max j u , j ∈ I re , u ∈ V ( λ ), denotes the element f pj u = 0, p ≥ 0, such that f p +1 j u =0. By convention, we write F v := F m v if m = (1 , . . . , , Proposition 4.3.1 Let w ∈ W satisfy conditions (4) and (5) for λ ∈ P + , and fix a minimaldominant reduced expression w = w k r a k i k · · · w r a i w . Then, with the notation above, we have(1) V w ( λ ) = P m , ǫ C ( q ) F m k w k f ǫ k i k · · · F m w f ǫ i F m w u λ , where the summation is over all m = ( m s ) ks =0 ∈ Q ks =0 Z ℓ ( w s ) ≥ and all ǫ = ( ǫ t ) kt =1 , ≤ ǫ t ≤ a t , ≤ t ≤ k .(2) V v ( λ ) ⊂ V w ( λ ) if v ≤ w in W . Part (2) of Proposition 4.3.1 follows from part (1), and part (1) for k = 1 is established byLemmas 4.3.2 and 4.3.3 below; part (1) for k ≥ k .Now, suppose that k = 1 for the expression of w above, and write w = R i r ai R j , with a ∈ Z ≥ , i ∈ I im , i = ( i s , . . . , i , i ) , j = ( j t , . . . , j , j ), such that R i , R j ∈ W re are reduced expres-sions. In this case, we have V w ( λ ) = U + q ( g ) u wλ , where u wλ := F max i f ai F j u λ = F max i f ai F max j u λ ∈ V ( λ ) wλ \ { } by Lemma 4.2.2. Set J := I \ { i } , and denote by g J ⊂ g the Levi subalge-bra corresponding to J . If we write (cid:0) e i U + q ( g J ) (cid:1) m := (cid:0) e i U + q ( g J ) (cid:1) · · · (cid:0) e i U + q ( g J ) (cid:1) ( m times), then U + q ( g ) = L ∞ m =0 U + q ( g J ) (cid:0) e i U + q ( g J ) (cid:1) m and we have the following by weight consideration: V w ( λ ) = ∞ M m =0 U + q ( g J ) (cid:0) e i U + q ( g J ) (cid:1) m u wλ = a M m =0 U + q ( g J ) (cid:0) e i U + q ( g J ) (cid:1) m u wλ , (7)where the number a in the right-hand side of (7) is the one appearing in the expression w = R i r ai R j . By Lemma 4.2.2, we have f mi F j u λ ∈ V ( λ ) r mi R j λ \ { } and F max i f mi F j u λ ∈ V ( λ ) R i r mi R j λ \{ } , 0 ≤ m ≤ a . Note that r mi R j λ ∈ P + for all m ≥ R i r mi R j is a (minimal) dominantreduced expression. Therefore, each f mi F j u λ for 1 ≤ m ≤ a is a U q ( g re )-highest weight vector. Lemma 4.3.2 For each ≤ m ≤ a, we have U + q ( g J ) (cid:0) e i U + q ( g J ) (cid:1) a − m u wλ = X n ∈ Z s ≥ C ( q ) F ni f mi F j u λ . roof. We proceed by descending induction on m . If m = a , then we have e j U + q ( g J ) u wλ = { } for all j ∈ J \ I re by weight consideration. Hence it follows that U + q ( g J ) u wλ = U + q ( g re ) u wλ . Also, U + q ( g re ) u wλ is identical to P n C ( q ) F ni f ai F j u λ by [ Kas2 , Corollary 3.2.2]. Thus, Lemma 4.3.2follows in this case.Suppose that 1 ≤ m ≤ a − 1. By the induction hypothesis, we have U + q ( g J ) (cid:0) e i U + q ( g J ) (cid:1) a − m u wλ = U + q ( g J ) e i X n C ( q ) F ni f m +1 i F j u λ , which is identical to U + q ( g J ) P C ( q ) F ni e i f m +1 i F j u λ since i does not appear in i and each F ni inthe summation commutes with e i . Since a ii = 0 by Lemma 4.2.1, it follows that e i f m +1 i F j u λ = { m + 1 } i | {z } = m +1( ≥ h α ∨ i ( λ − α j − · · · − α j t ) | {z } =1 − a ii m | {z } =0 i i f mi F j u λ by [ JKK , Lemma 2.5]. Therefore, we obtain U + q ( g J ) X C ( q ) F ni e i f m +1 i F j u λ = U + q ( g J ) X C ( q ) F ni f mi F j u λ . (8)Also, we see that the right-hand side of (8) is identical to U + q ( g re ) P C ( q ) F ni f mi F j u λ by weightconsideration, which is identical to P C ( q ) F ni f mi F j u λ by [ Kas2 , Corollary 3.2.2]. This provesLemma 4.3.2. (cid:3) Lemma 4.3.3 We have U + q ( g J ) (cid:0) e i U + q ( g J ) (cid:1) a u wλ = P n , m C ( q ) F ni F mj u λ . Proof. If we set m = 0 in the proof of Lemma 4.3.2, then the same argument shows that U + q ( g J ) (cid:0) e i U + q ( g J ) (cid:1) a u wλ = U + q ( g re ) X C ( q ) F ni F mj u λ . (9)Now, it is easily seen that the right-hand side of (9) is identical to P C ( q ) F ni F mj u λ by directcalculation. This proves Lemma 4.3.3. (cid:3) Ψ λ,w and Φ λ,w Let ˜ g re ⊂ ˜ g denote the Levi subalgebra corresponding to ˜ I re := { ( i, } i ∈ I re . Since g re (see § g re are ordinary Kac–Moody algebras with the same Cartan datum, there exists anembedding of C ( q )-algebras:Ψ re : U q ( g re ) ֒ → U q (˜ g re ) , e i e ( i, , f i f ( i, , q α ∨ i q ˜ α ∨ ( i, . (10)According to the minimal dominant reduced expression w = w k r a k i k · · · w r a i w , the element ˜ w ∈ f W (see Remark 4.1.2) is expressed as ˜ w = ˜ w k R ( i k , a k ) · · · ˜ w R ( i , a ) ˜ w , with a s = ( a s , . . . , , , ≤ ≤ k , where each ˜ w s ∈ f W re corresponds to w s ∈ W re via the isomorphism (3) in § i, m ) = (( i, m l ) , . . . , ( i, m )) if m = ( m l , . . . , m ); recall the notation R ( i , m ) = r ( i l ,m l ) · · · r ( i ,m ) .Note that this expression for ˜ w has properties similar to those in Lemma 4.2.2 for ˜ λ ∈ e P + . Let e V ˜ w (˜ λ ) denote the Demazure submodule, corresponding to ˜ w , of the irreducible highest weight U q (˜ g )-module e V (˜ λ ) of highest weight ˜ λ . We can think of e V ˜ w (˜ λ ) as a U + q ( g re )-module via Ψ re (see(10)).In this subsection, we construct a U + q ( g re )-linear maps Ψ λ,w : V w ( λ ) ֒ → e V ˜ w (˜ λ ) and Φ λ,w : e V ˜ w (˜ λ ) ։ V w ( λ ) such that Φ λ,w ◦ Ψ λ,w = id V w ( λ ) . For this purpose, we first study the U + q ( g re )-module structure of V w ( λ ) and e V ˜ w (˜ λ ). As in § F m ˜ v , e F m ˜ v and F m ( i , n ) for ˜ v ∈ f W and( i , n ) ∈ e I . Set F m ˜ w k ,..., ˜ w s := F m k ˜ w k · · · F m s ˜ w s ∈ U − q (˜ g ), m = ( m t ) kt = s ∈ Q kt = s Z ℓ ( ˜ w t ) ≥ , and ˜ u n := F ( i s , n s ) F ˜ w s − · · · F ( i , n ) F ˜ w ˜ u ˜ λ ∈ e V (˜ λ ), n = ( n t ) st =1 , n t ⊂ { , , . . . , a t } . Here we understand that˜ u n = ˜ u ˜ λ if s = 0. Note that if n t > 1, then a i t ,i t = 0 by Lemma 4.2.1, and hence all the f ( i t ,m ) , m ≥ 1, commute with each other. Therefore, the element F ( i t , n t ) is independent of thechoice of an ordering of the elements in n t . Also, set F m w k ,...,w s := F m k w k · · · F m s w s ∈ U − q ( g ) and u n := f n s i s F w s − · · · f n i F w u λ ∈ V ( λ ); note that u n depends only on the cardinalities n t ,1 ≤ t ≤ s . Hence we write u ǫ := u n for ǫ = ( ǫ t ) st =1 if ǫ t = n t , 1 ≤ t ≤ s . If n t = ∅ for 1 ≤ t ≤ s , then u n (resp., ˜ u n ) is a highest weight vector for U q ( g re ) (resp., U q (˜ g re )) ofweight r n s i s w s − · · · r n i w ( λ ) ∈ P (resp., R ( i s , n s ) ˜ w s − · · · R ( i , n ) ˜ w (˜ λ ) ∈ e P ) by Lemma 4.2.2.Moreover, in this case, we have α ∨ i (cid:0) r n s i s w s − · · · r n i w ( λ ) (cid:1) = ˜ α ∨ ( i, (cid:0) R ( i s , n s ) ˜ w s − · · · R ( i , n ) ˜ w (˜ λ ) (cid:1) for all i ∈ I re , and hence an isomorphism U q ( g re ) u n ∼ = −→ U q (˜ g re )˜ u n of U q ( g re )-modules via Ψ re (see(10)). Lemma 4.4.1 As U + q ( g re ) -modules, V w ( λ ) and e V ˜ w (˜ λ ) decompose as follows:(1) V w ( λ ) = L ks =0 L ǫ (cid:0)P m C ( q ) F m w k ,...,w s u ǫ (cid:1) , where the summation is over all m = ( m t ) kt = s ∈ Q kt = s Z ℓ ( w t ) ≥ and all ǫ = ( ǫ t ) st =1 , with ≤ ǫ t ≤ a t , ≤ t ≤ s, for each ≤ s ≤ k .(2) e V ˜ w (˜ λ ) = L ks =0 L n (cid:0)P m C ( q ) F m ˜ w k ,..., ˜ w s ˜ u n (cid:1) , where the summation is over all m = ( m t ) kt = s ∈ Q kt = s Z ℓ ( ˜ w t ) ≥ and all n = ( n t ) st =1 , with ∅ 6 = n t ⊂ { , , . . . , a t } , ≤ t ≤ s, for each ≤ s ≤ k . Proof. We give a proof only for (1); the proof for (2) is similar. By Lemma 4.2.2 and Proposition4.3.1 (1), it is easily seen that { u ǫ } ǫ , where the ǫ runs as in Lemma 4.4.1 (1), forms a completeset of representatives of linearly independent U q ( g re )-highest weight vectors in V w ( λ ). Therefore,the summation P ks =0 P ǫ (cid:0)P m C ( q ) F m w k ,...,w s u ǫ (cid:1) is a direct sum with respect to s and ǫ . Now, theequality in (1) follows from Proposition 4.3.1 (1). Also, from the arguments in § P m C ( q ) F m w k ,...,w s u ǫ is stable under the action of U + q ( g re ). Thus, theexpression for V w ( λ ) in Lemma 4.4.1 (1) is a decomposition as U + q ( g re )-modules. (cid:3) 18y using the descriptions of V w ( λ ) and e V ˜ w (˜ λ ) in Lemma 4.4.1, define Φ λ,w : e V ˜ w (˜ λ ) ։ V w ( λ ) by F m ˜ w k ,..., ˜ w s ˜ u n F m w k ,...,w s u n , and Ψ λ,w : V w ( λ ) ֒ → e V ˜ w (˜ λ ) by F m w k ,...,w s u ǫ = F m w k ,...,w s u n F m ˜ w k ,..., ˜ w s ˜ u n for n = ( n t ) st =1 , n t = { , , . . . , ǫ t } , 1 ≤ ǫ t ≤ a t , 1 ≤ t ≤ s , with ǫ = ( ǫ t ) st =1 . Since U q (˜ g re )˜ u n ∼ = U q ( g re ) u n as U q ( g re )-modules, we deduce that P m C ( q ) F m ˜ w k ,..., ˜ w s ˜ u n ∼ = −→ P m C ( q ) F m w k ,...,w s u n as U + q ( g re )-modules via the map Φ λ,w for all n as in Lemma 4.4.1 (2). Thus, Φ λ,w is well-defined,surjective, and U + q ( g re )-linear. Also, we can verify that Ψ λ,w is well-defined, injective, and U + q ( g re )-linear. Note that Ψ λ,w is also surjective (and hence an isomorphism) if and only if a = · · · = a k = 1. By the definitions, we have Φ λ,w ◦ Ψ λ,w = id V w ( λ ) . In this subsection, we recall some fundamental properties of the global basis of e V ˜ w (˜ λ ). Also, westudy certain symmetries of e V ˜ w (˜ λ ), which come from diagram automorphisms of ˜ g .Let e B (˜ λ ) and { e G ˜ λ ( b ′ ) } b ′ ∈ e B (˜ λ ) denote the crystal basis and the global basis of e V (˜ λ ) with thecrystal lattice e L (˜ λ ), and let e V (˜ λ ) A ⊂ e V (˜ λ ) denote the A -form of e V (˜ λ ). If we set e B ˜ w (˜ λ ) := { e F m k ˜ w k e F ( i k , n k ) · · · e F m ˜ w e F ( i , n ) e F m ˜ w ˜ u ˜ λ mod q e L (˜ λ ) | ( m s ) ks =0 ∈ k Y s =0 Z ℓ ( ˜ w s ) ≥ , n t ⊂ { , , . . . , a t } , ≤ t ≤ k } \ { } ⊂ e B (˜ λ ) , then we have e V ˜ w (˜ λ ) = L b ′ ∈ e B ˜ w (˜ λ ) C ( q ) e G ˜ λ ( b ′ ) by [ Kas2 , Proposition 3.2.3]. Let B w ( λ ) ⊂ B ( λ )denote the inverse image of e B ˜ w (˜ λ ) under the embedding B ( λ ) ֒ → e B (˜ λ ) of (2) in § e B w (˜ λ ) ⊂ e B (˜ λ ) the image of B w ( λ ) under this embedding. Then, these subsets can be written as B w ( λ ) = { e F m k w k ˜ f ǫ k i k · · · e F m w ˜ f ǫ i e F m w u λ mod qL ( λ ) | ( m s ) ks =0 ∈ k Y s =0 Z ℓ ( w s ) ≥ , ≤ ǫ t ≤ a t , ≤ t ≤ k } \ { } , and e B w (˜ λ ) = { e F m k ˜ w k e F ( i k , n k ) · · · e F m ˜ w e F ( i , n ) e F m ˜ w ˜ u ˜ λ mod q e L (˜ λ ) | ( m s ) ks =0 ∈ k Y s =0 Z ℓ ( ˜ w s ) ≥ , n t = { , , . . . , ǫ t } , ≤ ǫ t ≤ a t , ≤ t ≤ k } \ { } . In § 5, we will show that the subset B w ( λ ) ⊂ B ( λ ) satisfies the condition in Theorem 3.Let us set ˜ J s := { ( i s , m ) ∈ ˜ I | ≤ m ≤ a s } for 1 ≤ s ≤ k , with each i s ∈ I im and a s appearing in the expression w = w k r a k i k · · · w r a i w . Also, we set Ω λ,w := Q ks =1 S ( ˜ J s ) ⊂ Ω, where S ( ˜ J s ) denote the permutation group on ˜ J s (see § ω ∈ Ω λ,w on ˜ I isexpressed as: ω ( j, n ) = ( j, n ) if ( j, n ) / ∈ S ks =1 ˜ J s , and ω ( i s , m ) = ( i s , m ′ ) for some 1 ≤ m ′ ≤ a s if( i s , m ) ∈ ˜ J s . From this observation, we can show that e V ˜ w (˜ λ ) and e B ˜ w (˜ λ ) is stable under the actionof Ω λ,w . By using the description of Lemma 4.4.1 for e V ˜ w (˜ λ ), each ω ∈ Ω λ,w sends the element19 m ˜ w k ,..., ˜ w s ˜ u n , with n = ( n s , . . . , n ), to the element of the form F m ˜ w k ,..., ˜ w s ˜ u n ′ , with n ′ = ( n ′ s , . . . , n ′ )such that n t = n ′ t for 1 ≤ t ≤ s . Note that this expression for the action of Ω λ,w on e V ˜ w (˜ λ )shows that Φ λ,w ◦ ω = Φ λ,w for all ω ∈ Ω λ,w . Also, for each n ′ = ( n ′ s , . . . , n ′ ) with n ′ t = n t for 1 ≤ t ≤ s , we can find ω ∈ Ω λ,w such that ω ( F m ˜ w k ,..., ˜ w s ˜ u n ) = F m ˜ w k ,..., ˜ w s ˜ u n ′ . In particular, ifwe take n ′ = ( n ′ s , . . . , n ′ ) such that n ′ t = { , , . . . , n t } for 1 ≤ t ≤ s , then the corresponding ω ∈ Ω λ,w (not necessarily unique) sends F m ˜ w k ,..., ˜ w s ˜ u n into the image of the map Ψ λ,w . By the sameargument, for each b ′ ∈ e B ˜ w (˜ λ ), we can find ω ∈ Ω λ,w such that ω ( b ′ ) lies in e B w (˜ λ ). Namely, wehave e B ˜ w (˜ λ ) = Ω λ,w e B w (˜ λ ). Note that e G ˜ λ ◦ ω = ω ◦ e G ˜ λ for all ω ∈ Ω λ,w (see [ S , Lemma 3.4]). This subsection is devoted to the proof of Proposition 4.6.1 below. For b ′ ∈ e B ˜ w (˜ λ ), we set G ( b ′ ) := Φ λ,w (cid:0) e G ˜ λ ( b ′ ) (cid:1) ∈ V w ( λ ). Proposition 4.6.1 The element G ( b ′ ) ∈ V w ( λ ) , b ′ ∈ e B ˜ w (˜ λ ) , is a global basis element. Proof. To complete the proof, in view of the characterization of a global basis element (see § G ( b ′ ) = G ( b ′ ), (ii) G ( b ′ ) ∈ V ( λ ) A ∩ L ( λ ), and (iii) G ( b ′ ) mod qL ( λ ) ∈ B ( λ ) \{ } . Note that we know from § ′ e G ˜ λ ( b ′ ) = e G ˜ λ ( b ′ ), (ii) ′ e G ˜ λ ( b ′ ) ∈ e V ˜ w (˜ λ ) ∩ e V (˜ λ ) A ∩ e L (˜ λ ),and (iii) ′ e G ˜ λ ( b ′ ) ≡ b ′ mod q e L (˜ λ ). Then, the equality (i) follows from (i) ′ . Obviously, we haveΦ λ,w (cid:0) e V ˜ w (˜ λ ) ∩ e V (˜ λ ) A (cid:1) ⊂ V ( λ ) A . Therefore, by (ii) ′ , it suffices to verify that Φ λ,w (cid:0) e V ˜ w (˜ λ ) ∩ e L (˜ λ ) (cid:1) ⊂ L ( λ ) to show (ii). For this purpose, the following is enough: Claim 1 If e F ( i , m ) is of the form e F m k ˜ w k e F ( i k , n k ) · · · e F m ˜ w e F ( i k , n ) e F m ˜ w , with ( m s ) ks =0 ∈ Q ks =0 Z ℓ ( ˜ w s ) ≥ , n t ⊂{ , , . . . , a t } , ≤ t ≤ k , then we have Φ λ,w ( e F ( i , m ) ˜ u ˜ λ ) = e F i u λ . Proof of Claim 1. Set v = e F ( i , m ) ˜ u ˜ λ . We proceed by induction on the length l of ( i , m ) =(( i s , m s )) ls =1 . We assume that Φ λ,w ( v ) = e F i u λ , and show that Φ λ,w ( ˜ f ( i,m ) v ) = ˜ f i e F i u λ for all( i, m ) ∈ ˜ I such that ˜ f ( i,m ) e F ( i , m ) is also of the form given as in Claim 1.Case 1: i ∈ I im .Since ( i, m ) does not appear in ( i , m ), we have v ∈ Ker( e ( i,m ) ) and hence ˜ f ( i,m ) v = f ( i,m ) v . Hencewe deduce that Φ λ,w ( ˜ f ( i,m ) v ) = Φ λ,w ( f ( i,m ) v ) = f i Φ λ,w ( v ). Also, we have ˜ f i Φ λ,w ( v ) = f i Φ λ,w ( v )since ˜ f i = f i if i ∈ I im (see § i ∈ I re .Let v = P n ≥ f ( n )( i, v n , with v n ∈ Ker( e ( i, ), be the ( i, § e V ˜ w (˜ λ ) is stable under the action of e ( i, , we can show, by induction on n , that v n , f ( n )( i, v n ∈ e V ˜ w (˜ λ )20or all n ≥ 0. Hence the equality Φ λ,w ( v ) = P n ≥ f ( n ) i Φ λ,w ( v n ) holds. Also, we have Φ λ,w ( v n ) ∈ Ker( e i ) since e i Φ λ,w ( v n ) = Φ λ,w ( e ( i, v n ) = 0. Therefore, this expression for Φ λ,w ( v ) gives the i -string decomposition. Thus we have ˜ f i Φ λ,w ( v ) = P n ≥ f ( n +1) i Φ λ,w ( v n ), which is also identicalto the image of the element ˜ f ( i, v = P n ≥ f ( n +1)( i, v n under the map Φ λ,w . This proves Claim 1. (cid:4) Now, let us show (iii). Note that Claim 1 above shows that Φ λ,w induces the map Φ λ,w : e B ˜ w (˜ λ ) → B ( λ ). Moreover, it is easily seen that the image of this map is B w ( λ ), and that theequality Φ λ,w ◦ e = id B w ( λ ) holds, where e denotes (the restriction of) the map of (2) in § b ∈ e B w (˜ λ ), which is the imageof an element b ∈ B w ( λ ) under the map e , then we have G (˜ b ) = Φ λ,w (cid:0) e G ˜ λ (˜ b ) (cid:1) ≡ Φ λ,w (˜ b ) ≡ b mod qL ( λ ), where the second equality is due to (iii) ′ . For a general element b ′ ∈ e B ˜ w (˜ λ ), thereexists a diagram automorphism ω ∈ Ω λ,w such that ω − ( b ′ ) = ˜ b ∈ e B w (˜ λ ), where ˜ b is as above.Since e G ˜ λ ◦ ω = ω ◦ e G ˜ λ and Φ λ,w ◦ ω = Φ λ,w (see § G ( b ′ ) = Φ λ,w (cid:0) e G ˜ λ ( ω (˜ b )) (cid:1) =Φ λ,w (cid:0) ω (cid:0) e G ˜ λ (˜ b ) (cid:1)(cid:1) = Φ λ,w (cid:0) e G ˜ λ (˜ b ) (cid:1) = G (˜ b ). Therefore, we deduce that G ( b ′ ) ≡ b mod qL ( λ ). Thiscompletes the proof of Proposition 4.6.1. (cid:3) In the same way as above, we can also show that the map Ψ λ,w has properties similar tothose for Φ λ,w , and hence induces the map Ψ λ,w : B w ( λ ) → e B ˜ w (˜ λ ). In fact, this is identical tothe map e . Corollary 4.6.2 With the notation above, the following hold.(1) The maps Φ λ,w and Ψ λ,w induce maps Φ λ,w : e B ˜ w (˜ λ ) ։ B w ( λ ) and Ψ λ,w : B w ( λ ) ֒ → e B ˜ w (˜ λ ) such that Φ λ,w ◦ Ψ λ,w = id B w ( λ ) .(2) The equalities G λ ◦ Φ λ,w = Φ λ,w ◦ e G ˜ λ and e G ˜ λ ◦ Ψ λ,w = Ψ λ,w ◦ G λ hold.(3) Let b ∈ B w ( λ ) , ˜ b = Ψ λ,w ( b ) , and ω ∈ Ω λ,w . Then, the equality G (cid:0) ω (˜ b ) (cid:1) = G λ ( b ) holds. Proof of Theorem 3. If we write the element Ψ λ,w ( v ) ∈ e V ˜ w (˜ λ ) for v ∈ V w ( λ ) as Ψ λ,w ( v ) = P c ω (˜ b ) e G ˜ λ (cid:0) ω (˜ b ) (cid:1) , with c ω (˜ b ) ∈ C ( q ), then we have v = Φ λ,w (Ψ λ,w ( v )) = P c ω (˜ b ) G λ ( b ) by Corollary4.6.2 (3). Therefore, we conclude that V w ( λ ) = L b ∈ B w ( λ ) C ( q ) G λ ( b ) . This proves Theorem 3. (cid:3) .2 Proof of Theorem 4 Before starting the proof of Theorem 4, we fix some notation. Let B w ( λ ), e B ˜ w (˜ λ ), and e B w (˜ λ )denote the subsets of path crystals corresponding to B w ( λ ), e B ˜ w (˜ λ ), and e B w (˜ λ ) via the isomor-phisms B ( λ ) ∼ = B ( λ ) and e B (˜ λ ) ∼ = e B (˜ λ ) of crystals, respectively. Define subsets B m of B w ( λ ) for m = 1 , 2, by B := (cid:8) F m k − w k − f ǫ k − i k − · · · F m w f ǫ i F m w π λ ∈ B w ( λ ) (cid:12)(cid:12) m s ∈ Z ℓ ( w s ) ≥ , ≤ ǫ s ≤ a s for each s (cid:9) , B := { f ǫi k π ∈ B w ( λ ) | π ∈ B , ≤ ǫ ≤ a k } , and set e B m := Ψ λ,w ( B m ) ⊂ e B w (˜ λ ), m = 1 , Proof of Theorem 4. If w ∈ W re , then ch B w ( λ ) = D w ( e λ ) by [ Kas2 , Proposition 3.3.5], andthe assertion of Theorem 4 follows in this case. Let w = w k r a k i k · · · w r a i w be a minimal domi-nant reduced expression. We assume that the equality ch B = D w k − D ( a k − ) i k − · · · D w D ( a ) i D w ( e λ )holds, and show that ch B w ( λ ) = D w k D ( a k ) i k (ch B ).First, we show that ch B = D ( a k ) i k (ch B ). Note that f ǫi k π ∈ B is not for ǫ ≥ α ∨ i k (cid:0) π (1) (cid:1) > JL , Lemma 4.1.6 (1)]). Therefore, we deduce thatch B = X π ∈ B α ∨ ik ( π (1)) > a k X ǫ =0 e f ǫik π (1) + X π ∈ B α ∨ ik ( π (1))=0 e π (1) , (11)and that the right-hand side of (11) is identical to D ( a k ) i k (ch B ) by the definition of D ( a k ) i k .Now, we show that ch B w ( λ ) = D w k (ch B ). By [ Kas2 , Proposition 3.3.5], we have ch e B ˜ w (˜ λ ) = D ˜ w ( e ˜ λ ) = D ˜ w k D ˜ w − k ˜ w ( e ˜ λ ). Then, we can deduce that ch e B w (˜ λ ) = D ˜ w k (ch e B ) since e B w (˜ λ ) = { F m ˜ w k η | η ∈ e B , m ∈ Z ℓ ( ˜ w k ) ≥ } ⊂ e B ˜ w (˜ λ ), and e B has the string property (see [ Jo , Lemma in § B w ( λ ) = ch Φ λ,w (cid:0)e B w (˜ λ ) (cid:1) = D w k (ch Φ λ,w ( e B )) = D w k (ch B ).Consequently, we obtain ch B w ( λ ) = D w k D ( a k ) i k (ch B ). This proves Theorem 4 by inductionon k . (cid:3) References [ BB ] A. Bj¨orner and F. Brenti, Combinatorics of Coxeter Groups, Grad. Texts in Math., vol. ,Springer, New York, 2005.[ Bo ] R. E. Borcherds, Generalized Kac–Moody algebras, J. Algebra (1988), 501-512.[ HK ] J. Hong and S.-J. Kang, Introduction to Quantum Groups and Crystal Bases, Grad. Studiesin Math. vol. , Amer. Math. Soc., Providence, RI, 2002. Hu ] J. E. Humphreys, Reflection Groups and Coxeter Groups, Camb. Stud. in Adv. Math., ,Cambridge Univ. Press, Cambridge, 1990.[ I ] M. Ishii, Path model for representations of generalized Kac–Moody algebras, J. Algebra (2013), 277-300.[ JKK ] K. Jeong, S.-J. Kang, and M. Kashiwara, Crystal bases for quantum generalized Kac–Moodyalgebras, Proc. Lond. Math. Soc. (2005), 395-438.[ JKKS ] K. Jeong, S.-J. Kang, M. Kashiwara, and D.-U. Shin, Abstract crystals for quantum general-ized Kac–Moody algebras, Int. Math. Res. Not., 2007 Art. ID rnm001, 19 pages.[ Jo ] A. Joseph, A decomposition theorem for Demazure crystals, J. Algebra (2003) 562-578.[ JL ] A. Joseph and P. Lamprou, A Littelmann path model for crystals of generalized Kac–Moodyalgebras, Adv. Math. (2009), 2019-2058.[ Kac ] V. G. Kac, Infinite Dimensional Lie Algebras, 3rd ed., Cambridge Univ. Press, Cambridge,1990.[ Kan ] S.-J. Kang, Quantum deformations of generalized Kac–Moody algebras and their modules, J.Algebra (1995), 1041-1066.[ Kas1 ] M. Kashiwara, On crystal bases of the q -analogue of universal enveloping algebras, Duke Math.J. (1991), 465-509.[ Kas2 ] M. Kashiwara, The crystal base and Littelmann’s refined Demazure character formula, DukeMath. J. (1993), 839-858.[ Kum ] S. Kumar, Demazure character formula in arbitrary Kac–Moody setting, Invent. Math. (1987), no. 2, 395-423.[ Li1 ] P. Littelmann, A Littlewood–Richardson rule for symmetrizable Kac–Moody algebras, Invent.Math. (1994), 329-346.[ Li2 ] P. Littelmann, Paths and root operators in representation theory, Ann. of Math. (1995),499-525.[ Li3 ] P. Littelmann, Crystal graphs and Young tableaux, J. Algebra (1995), no. 1, 65-87.[ M ] O. Mathieu, Formules de Demazure–Weyl, et g´en´eralisation du th´eor`eme de Borel–Weil–Bott,C. R. Acad. Sci. Paris S´er. I Math. (1986), no. 9, 391-394.[ NS ] S. Naito and D. Sagaki, Lakshmibai–Seshadri paths fixed by a diagram automorphism, J.Algebra (2001), 395-412.[ S ] D. Sagaki, Crystal bases, path models, and a twining character formula for Demazure modules,Publ. Res. Inst. Math. Sci. (2002), 245-264.(2002), 245-264.