Cuspidal systems for affine Khovanov-Lauda-Rouquier algebras
aa r X i v : . [ m a t h . R T ] D ec CUSPIDAL SYSTEMS FOR AFFINEKHOVANOV-LAUDA-ROUQUIER ALGEBRAS
ALEXANDER S. KLESHCHEVAbstract.
A cuspidal system for an affine Khovanov-Lauda-Rouquier al-gerba R α yields a theory of standard modules. This allows us to classifythe irreducible modules over R α up to the so-called imaginary modules.We make a conjecture on reductions modulo p of irreducible R α -modules,which generalizes James Conjecture. We also describe minuscule imaginarymodules, laying the groundwork for future study of imaginary Schur-Weylduality. We introduce colored imaginary tensor spaces and reduce a clas-sification of imaginary modules to one color. We study the characters ofcuspidal modules. We show that under the Khovanov-Lauda-Rouquier cat-egorification, cuspidal modules correspond to dual root vectors. Introduction
Khovanov-Lauda-Rouquier (KLR) algebras were defined in [ , , ]. Theirrepresentation theory is of interest for the theory of canonical bases, modularrepresentation theory, cluster theory, knot theory, and other areas of mathemat-ics. Let F be an arbitrary ground field. The KLR algebra R α = R α ( C , F ) is agraded unital associative F -algebra depending on a Lie type C and an element α of the non-negative part Q + of the corresponding root lattice.A natural approach to representation theory of R α is provided by a theory ofstandard modules. For KLR algebras of finite Lie type such a theory was firstdescribed in [ ], see also [ , , ]. Key features of this theory are as follows.There is a natural induction functor Ind α,β , which associates to an R α -module M and an R β -module N the R α + β -module M ◦ N := Ind α,β M ⊠ N for α, β ∈ Q + . We refer to this operation as the induction product . The functorInd α,β has an obvious right adjoint Res α,β .To every positive root β ∈ Φ + of the corresponding root system Φ, oneassociates a cuspidal module L β . We point out a remarkable property of cuspidalmodules which turns out to be key for building the theory of standard modules:the induction product powers L ◦ nβ are irreducible for all n >
0, see [ , Lemma6.6]. We make a special choice of a total order on Φ + , and let β > · · · > β N be the positive roots taken in this order. A root partition of α ∈ Q + is a tuple π = ( m , . . . , m N ) of nonnegative integers such that α = P Nn =1 m n β n . The setof root partitions of α is denoted by Π( α ). Mathematics Subject Classification. ALEXANDER S. KLESHCHEV
Given π = ( m , . . . , m N ) ∈ Π( α ) we define the corresponding standard mod-ule ∆( π ) as the induction product∆( π ) = L ◦ m β ◦ · · · ◦ L ◦ m N β N h sh ( π ) i , where h sh ( π ) i means that grading is shifted by an explicit integer sh ( π ). Thenthe head of ∆( π ) is proved to be irreducible, and, denoting this head by L ( π ),we get a complete irredundant system { L ( π ) | π ∈ Π( α ) } of irreducible R α -modules. Moreover, the decomposition matrix([∆( π ) : L ( σ )]) π,σ ∈ Π( α ) is unitriangular if we order its rows and columns according to the natural lexi-cographic order on root partitions.We now comment on the order on Φ + . In [ ], the so-called Lyndon orderis used, cf. [ ]. This is determined by a choice of a total order on the set I of simple roots. Once such a choice has been made, we have a lexicographicorder on the set h I i α of words of weight α . These words play the role ofweights in representation theory of R α . In particular each R α -module has itshighest word, and the highest word of an irreducible module determines theirreducible module uniquely up to an isomorphism. This leads to the naturalnotion of dominant words, namely the ones which occur as highest words in R α -modules (called good words in [ ]). The dominant words of cuspidal modulesare characterized among all dominant words by the property that they areLyndon words. It turns out that the dominant Lyndon words are in one-to-onecorrespondence with positive roots, and now we can compare positive roots bycomparing the corresponding dominant Lyndon words lexicographically. Thisgives a total order on Φ + called a Lyndon order. We point out that the cuspidalmodules themselves depend on the choice of a Lyndon order on Φ + .It is well-known that each Lyndon order is convex. However, there are ingeneral more convex orders on Φ + than Lyndon orders. Recently McNamara[ ] has found a remarkable generalization of the standard module theory whichworks for any convex order on Φ + . In this generalization the cuspidal modulesare defined via their restriction properties, which seems to be not quite asexplicit as the definition via highest words. However, all the other importantfeatures of the theory, including the simplicity of induction powers of cuspidalmodules, as well as the unitriangularity of decomposition matrices, remain thesame.In this paper, we begin to extend the results described above from finiteto affine root systems. To describe the results in more detail we need somenotation. Let the Lie type C be of arbitrary untwisted affine type . In particular,the simple roots are labeled by the elements of I = { , , . . . , l } . We have an(affine) root system Φ and the subset Φ + ⊂ Φ of positive roots . It is knownthat Φ + = Φ re+ ⊔ Φ im+ , where Φ re+ are the real roots , and Φ im+ = { nδ | n ∈ Z > } ,for the null-root δ , are the imaginary roots . USPIDAL SUSTEMS FOR AFFINE KLR ALGEBRAS 3
Following [ ], we define a convex preorder on Φ + as a preorder (cid:22) such thatthe following three conditions hold for all β, γ ∈ Φ + : β (cid:22) γ or γ (cid:22) β ; (1.1)if β (cid:22) γ and β + γ ∈ Φ + , then β (cid:22) β + γ (cid:22) γ ; (1.2) β (cid:22) γ and γ (cid:22) β if and only if β and γ are proportional . (1.3)Convex preorders are known to exist. It follows from (1.3) that β (cid:22) γ and γ (cid:22) β happens for β = γ if and only if both β and γ are imaginary. Moreover,it is easy to see that the set of real roots splits into two disjoint infinite setsΦ re ≻ := { β ∈ Φ re+ | β ≻ δ } and Φ re ≺ := { β ∈ Φ re+ | β ≺ δ } . (We write β ≺ γ if β (cid:22) γ but γ β ). In fact, one can label the real roots asΦ re+ = { ρ n | n ∈ Z =0 } so thatΦ re ≻ = { ρ ≻ ρ ≻ ρ ≻ . . . } and Φ re ≺ = {· · · ≻ ρ − ≻ ρ − ≻ ρ − } . (1.4)Root partitions are defined similarly to the case of finite root systems, ex-cept that now we need to take care of imaginary roots. We do this as fol-lows. Let α ∈ Q + . Define the set Π( α ) of root partitions of α to be theset of all pairs ( M, µ ), where M = ( m , m , . . . ; m ; . . . , m − , m − ) is a se-quence of nonnegative integers, and µ is an l -multipartition of m such that m δ + P n =0 m n ρ n = α . There is a natural partial order ‘ ≤ ’ on Π( α ), which isa version of McNamara’s bilexicographic order [ ], see (3.4).A cuspidal system (for a fixed convex preorder) is the following data:(Cus1) An irreducible R ρ -module L ρ assigned to every ρ ∈ Φ re+ , with the fol-lowing property: if β, γ ∈ Q + are non-zero elements such that ρ = β + γ and Res β,γ L ρ = 0, then β is a sum of (positive) roots less than ρ and γ is a sum of (positive) roots greater than ρ .(Cus2) An irreducible R nδ -module L ( µ ) assigned to every l -multipartition of n for every n ∈ Z ≥ , with the following property: if β, γ ∈ Q + \ Φ im+ are non-zero elements such that nδ = β + γ and Res β,γ L ( µ ) = 0, then β is a sum of real roots less than δ and γ is a sum of real roots greaterthan δ . It is required that L ( λ ) = L ( µ ) unless λ = µ .We call the irreducible modules L ρ from (Cus1) cuspidal modules , and theirreducible modules L ( µ ) from (Cus2) imaginary modules . It will be provedthat cuspidal systems exist for all convex preorders, and cuspidal modules (fora fixed preorder) are determined uniquely up to an isomorphism. However,it is clearly not the case for imaginary modules: they are defined up to apermutation of multipartitions µ of n . We give more comments on this afterthe Main Theorem.Now, given a root partition ( M, µ ) ∈ Π( α ) as above, we define the corre-sponding standard module ∆( M, µ ) := L ◦ m ρ ◦ L ◦ m ρ ◦ · · · ◦ L ( µ ) ◦ · · · ◦ L ◦ m − ρ − ◦ L m − ρ − h sh ( M, µ ) i , where sh ( M, µ ) is an explicit integer defined in (3.7).
ALEXANDER S. KLESHCHEV
Main Theorem.
For any convex preorder there exists a cuspidal system { L ρ | ρ ∈ Φ re+ } ∪ { L ( λ ) | λ ∈ P } . Moreover: (i) For every root partition ( M, µ ) , the standard module ∆( M, µ ) has ir-reducible head; denote this irreducible module L ( M, µ ) . (ii) { L ( M, µ ) | ( M, µ ) ∈ Π( α ) } is a complete and irredundant system ofirreducible R α -modules up to isomorphism. (iii) L ( M, µ ) ⊛ ≃ L ( M, µ ) . (iv) [∆( M, µ ) : L ( M, µ )] q = 1 , and [∆( M, µ ) : L ( N, ν )] q = 0 implies ( N, ν ) ≤ ( M, µ ) . (v) L ◦ nρ is irreducible for every ρ ∈ Φ re+ and every n ∈ Z > . This theorem, proved in Section 4, gives a ‘rough classification’ of irreducible R α -modules. The main problem is that we did not give a canonical definitionof individual imaginary modules L ( µ ). We just know that the amount of suchmodules for R nδ is equal to the number of l -multipartitions of n , and so wehave labeled them by such multipartitions in an arbitrary way. In fact, thereis a solution to this problem. It turns out that there is a beautiful rich theoryof imaginary representations of KLR algebras of affine type, which relies onthe so-called imaginary Schur-Weyl duality. This theory in particular allowsus to construct an equivalence between an appropriate category of imaginaryrepresentations of KLR algebras and the category of representations of theclassical Schur algebras. We will address these matters in the forthcomingwork [ ].In Section 5, we make some first steps in the study of imaginary representa-tions and describe explicitly the minuscule imaginary representations—the oneswhich correspond to the l -multipartitions of 1. We introduce colored imaginarytensor spaces and reduce a classification of imaginary modules to one color.Minuscule imaginary representations are also used in Sections 6.2 and 6.3 todescribe explicitly the cuspidal modules corresponding to the roots of the form nδ ± α i . In Section 6 we also explain how the characters of other cuspidal mod-ules can be computed by induction using the idea of minimal pairs which wassuggested in [ ]. In Section 4.7, we show that under the Khovanov-Lauda-Rouquier categorification, cuspidal modules correspond to dual root vectors ofa dual PBW basis.In conclusion, we would like to draw the reader’s attention to Conjecture 4.9,which asserts that reductions modulo p of irreducible modules over the KLRalgebras of affine type remain irreducible under an explicit assumption on thecharacteristic p . In type A (1) l (for level 1) this is equivalent to a block versionof the James Conjecture.Immediately after the first version of this paper has been posted, the paper[ ] has also been released on the arXiv. That paper suggestes a differentapproach to standard module theory for affine KLR algebras. Acknowledgements.
This paper has been influenced by the beautiful ideasof Peter McNamara [ ], who also drew my attention to the paper [ ] andsuggested a slightly more general version of the main result appearing hereafter the first version of this paper was released. I am also grateful to ArunRam and Jon Brundan for many useful conversations. USPIDAL SUSTEMS FOR AFFINE KLR ALGEBRAS 5 Preliminaries
Throughout the paper, F is a field of arbitrary characteristic p ≥
0. Denotethe ring of Laurent polynomials in the indeterminate q by A := Z [ q, q − ]. Weuse quantum integers [ n ] q := ( q n − q − n ) / ( q − q − ) ∈ A for n ∈ Z , and thequantum factorials [ n ] ! q := [1] q [2] q . . . [ n ] q . We have a bar-involution on A andon Q ( q ) ⊃ A with b q = q − .2.1. Lie theoretic notation.
Throughout the paper C = ( c ij ) i,j ∈ I is a Car-tan matrix of untwisted affine type , see [ , §
4, Table Aff 1]. We have I = { , , . . . , l } , where 0 is the affine vertex. Following [ , § h , Π , Π ∨ )be a realization of the Cartan matrix C , so we have simple roots { α i | i ∈ I } ,simple coroots { α ∨ i | i ∈ I } , and a bilinear form ( · , · ) on h ∗ such that c ij =2( α i , α j ) / ( α i , α i ) for all i, j ∈ I . We normalize ( · , · ) so that ( α i , α i ) = 2 if α i isa short simple root.The fundamental dominant weights { Λ i | i ∈ I } have the property that h Λ i , α ∨ j i = δ i,j , where h· , ·i is the natural pairing between h ∗ and h . We havethe integral weight lattice P = ⊕ i ∈ I Z · Λ i and the set of dominant weights P + = P i ∈ I Z ≥ · Λ i . For i ∈ I we define[ n ] i := [ n ] q ( αi,αi ) / , [ n ] ! i := [1] i [2] i . . . [ n ] i . Denote Q + := L i ∈ I Z ≥ α i . For α ∈ Q + , we write ht( α ) for the sum of itscoefficients when expanded in terms of the α i ’s.Let g ′ = g ( C ′ ) be the finite dimensional simple Lie algebra whose Cartanmatrix C ′ corresponds to the subset of vertices I ′ := I \ { } . The affine Liealgebra g = g ( C ) is then obtained from g ′ by a procedure described in [ ,Section 7]. We denote by W (resp. W ′ ) the corresponding affine Weyl group (resp. finite Weyl group ). It is a Coxeter group with standard generators { r i | i ∈ I } (resp. { r i | i ∈ I ′ } ), see [ , Proposition 3.13].Let Φ ′ and Φ be the root systems of g ′ and g respectively. Denote by Φ ′ + andΦ + the set of positive roots in Φ ′ and Φ, respectively, cf. [ , § δ the null-root. Let δ = a α + a α + · · · + a l α l . (2.1)By [ , Table Aff 1], we always have a = 1 . (2.2)We have δ − α = θ, (2.3)where θ is the highest root in the finite root system Φ ′ . Finally,Φ + = Φ im+ ⊔ Φ re+ , where Φ im+ = { nδ | n ∈ Z > } andΦ re+ = { β + nδ | β ∈ Φ ′ + , n ∈ Z ≥ } ⊔ {− β + nδ | β ∈ Φ ′ + , n ∈ Z > } . (2.4) ALEXANDER S. KLESHCHEV
Words.
Sequences of elements of I will be called words . The set of allwords is denoted h I i . If i = i . . . i d is a word, we denote | i | := α i + · · · + α i d ∈ Q + . For any α ∈ Q + we denote h I i α := { i ∈ h I i | | i | = α } . If α is of height d , then the symmetric group S d with simple permutations s , . . . , s d − acts on h I i α from the left by place permutations.Let i = i . . . i d and j = i d +1 . . . i d + f be two elements of h I i . Define the quantum shuffle product : i ◦ j := X q − e ( σ ) i σ (1) . . . i σ ( d + f ) ∈ A h I i , where the sum is over all σ ∈ S d + f such that σ − (1) < · · · < σ − ( d ) and σ − ( d + 1) < · · · < σ − ( d + f ), and e ( σ ) := P k ≤ d
Define the polynomials in the variables u, v { Q ij ( u, v ) ∈ F [ u, v ] | i, j ∈ I } as follows. For the case where the Cartan matrix C = A (1)1 , choose signs ε ij forall i, j ∈ I with c ij < ε ij ε ji = −
1. Then set: Q ij ( u, v ) := i = j ;1 if c ij = 0; ε ij ( u − c ij − v − c ji ) if c ij < i < j . (2.5)For type A (1)1 we define Q ij ( u, v ) := (cid:26) i = j ;( u − v )( v − u ) if i = j . (2.6)Fix α ∈ Q + of height d . The KLR-algebra R α is an associative graded unital F -algebra, given by the generators { i | i ∈ h I i α } ∪ { y , . . . , y d } ∪ { ψ , . . . , ψ d − } (2.7)and the following relations for all i , j ∈ h I i α and all admissible r, t :1 i j = δ i , j i , P i ∈h I i α i = 1; (2.8) y r i = 1 i y r ; y r y t = y t y r ; (2.9) ψ r i = 1 s r i ψ r ; (2.10)( y t ψ r − ψ r y s r ( t ) )1 i = i if i r = i r +1 and t = r + 1, − i if i r = i r +1 and t = r ,0 otherwise; (2.11) ψ r i = Q i r ,i r +1 ( y r , y r +1 )1 i (2.12) ψ r ψ t = ψ t ψ r ( | r − t | > ψ r +1 ψ r ψ r +1 − ψ r ψ r +1 ψ r )1 i = ( Q ir,ir +1 ( y r +2 ,y r +1 ) − Q ir,ir +1 ( y r ,y r +1 ) y r +2 − y r i if i r = i r +2 ,0 otherwise. (2.14) USPIDAL SUSTEMS FOR AFFINE KLR ALGEBRAS 7
The grading on R α is defined by setting:deg(1 i ) = 0 , deg( y r i ) = ( α i r , α i r ) , deg( ψ r i ) = − ( α i r , α i r +1 ) . It is pointed out in [ ] and [ , § F -algebra R α depends only on the Cartan matrix and α .Fix in addition a dominant weight Λ ∈ P + . The corresponding cyclotomicKLR algebra R Λ α is the quotient of R α by the following ideal: J Λ α := ( y h Λ ,α ∨ i i i | i = ( i , . . . , i d ) ∈ h I i α ) . (2.15)For each element w ∈ S d fix a reduced expression w = s r . . . s r m and set ψ w := ψ r . . . ψ r m . In general, ψ w depends on the choice of the reduced expression of w . Theorem 2.16. [ , Theorem 2.5] , [ , Theorem 3.7] The elements { ψ w y m . . . y m d d i | w ∈ S d , m , . . . , m d ∈ Z ≥ , i ∈ h I i α } form an F -basis of R α . There exists a homogeneous algebra anti-involution τ : R α −→ R α , i i , y r y r , ψ s ψ s (2.17)for all i ∈ h I i α , ≤ r ≤ d , and 1 ≤ s < d . If M = L d ∈ Z M d is a finitedimensional graded R α -module, then the graded dual M ⊛ is the graded R α -module such that ( M ⊛ ) n := Hom F ( M − n , F ), for all n ∈ Z , and the R α -actionis given by ( xf )( m ) = f ( τ ( x ) m ), for all f ∈ M ⊛ , m ∈ M, x ∈ R α .2.4. Basic representation theory of R α . For any ( Z -)graded F -algebra H , we denote by H -mod the abelian subcategory of all finite dimensional graded H -modules, with morphisms being degree-preserving module homomor-phisms, and [ H -mod] denotes the corresponding Grothendieck group. Then[ H -mod] is an A -module via q m [ M ] := [ M h m i ] , where M h m i denotes themodule obtained by shifting the grading up by m , i.e. M h m i n := M n − m . We denote by hom H ( M, N ) the space of morphism in H -mod. For n ∈ Z ,let Hom H ( M, N ) n := hom H ( M h n i , N ) denote the space of all homomorphismsthat are homogeneous of degree n . SetHom H ( M, N ) := M n ∈ Z Hom H ( M, N ) n . For graded H -modules M and N we write M ≃ N to mean that M and N areisomorphic as graded modules and M ∼ = N to mean that they are isomorphic as H -modules after we forget the gradings. For a finite dimensional graded vectorspace V = ⊕ n ∈ Z V n , its graded dimension is dim q V := P n ∈ Z (dim V n ) q n ∈ A .Given M, L ∈ H -mod with L irreducible, we write [ M : L ] q for the correspond-ing graded composition multiplicity , i.e. [ M : L ] q := P n ∈ Z a n q n , where a n isthe multiplicity of L h n i in a graded composition series of M .Going back to the algebras R α , every irreducible graded R α -module is finitedimensional [ , Proposition 2.12], and there are finitely many irreducible mod-ules in R α -mod up to isomorphism and grading shift [ , § ALEXANDER S. KLESHCHEV is a splitting field for R α [ , Corollary 3.19], so working with irreducible R α -modules we do not need to assume that F is algebraically closed. Finally, forevery irreducible module L , there is a unique choice of the grading shift so thatwe have L ⊛ ≃ L [ , Section 3.2]. When speaking of irreducible R α -moduleswe often assume by fiat that the shift has been chosen in this way.For i ∈ h I i α and M ∈ R α -mod, the i -weight space of M is M i := 1 i M. Wehave M = L i ∈h I i α M i . We say that i is a weight of M if M i = 0. Note fromthe relations that ψ r M i ⊂ M s r i . Define the (graded formal) character of M asfollows: ch q M := X i ∈h I i α (dim q M i ) i ∈ A h I i α . The character map ch q : R α -mod → A h I i α factors through to give an injective A -linear map ch q : [ R α -mod] → A h I i α , see [ , Theorem 3.17].2.5. Induction, coinduction, and duality.
Given α, β ∈ Q + , we set R α,β := R α ⊗ R β . Let M ⊠ N be the outer tensor product of the R α -module M andthe R β -module N . There is an injective homogeneous non-unital algebra ho-momorphism R α,β ֒ → R α + β , i ⊗ j ij , where ij is the concatenation of i and j . The image of the identity element of R α,β under this map is1 α,β := X i ∈h I i α , j ∈h I i β ij . Let Ind α + βα,β and Res α + βα,β be the induction and restriction functors:Ind α + βα,β := R α + β α,β ⊗ R α,β ? : R α,β -mod → R α + β -mod , Res α + βα,β := 1 α,β R α + β ⊗ R α + β ? : R α + β -mod → R α,β -mod . We often omit upper indices and write simply Ind α,β and Res α,β . These functorshave obvious generalizations to n ≥ γ ,...,γ n : R γ ,...,γ n -mod → R γ + ··· + γ n -mod , Res γ ,...,γ n : R γ + ··· + γ n -mod → R γ ,...,γ n -mod . The functor Ind γ ,...,γ n is left adjoint to Res γ ,...,γ n . If M a ∈ R γ a -Mod, for a = 1 , . . . , n , we define M ◦ · · · ◦ M n := Ind γ ,...,γ n M ⊠ · · · ⊠ M n . (2.18)In view of [ , Lemma 2.20], we havech q ( M ◦ · · · ◦ M n ) = ch q ( M ) ◦ · · · ◦ ch q ( M n ) . (2.19)The functors of induction and restriction have obvious parabolic analogues.Given a family ( α ab ) ≤ a ≤ n, ≤ b ≤ m of elements of Q + , set P na =1 α ab =: β b for all1 ≤ b ≤ m . Then we have functorsInd β ; ... ; β m α ,...,α n ; ... ; α m ,...,α nm and Res β ; ... ; β m α ,...,α n ; ... ; α m ,...,α nm The right adjoint to the functor Ind γ ,...,γ n is given by the coinduction:Coind γ ,...,γ n := Hom R γ ,...,γn (1 γ ,...,γ n R γ + ··· + γ n , ?)Induction and coinduction are related as follows: USPIDAL SUSTEMS FOR AFFINE KLR ALGEBRAS 9
Lemma 2.20. [ , Theorem 2.2] Let γ := ( γ , . . . , γ n ) ∈ Q n + , and V m ∈ R γ m -mod for m = 1 , . . . , n . Denote d ( γ ) = P ≤ m Let γ := ( γ , . . . , γ n ) ∈ Q n + , and V m ∈ R γ m -mod for m =1 , . . . , n . Denote d ( γ ) = P ≤ m Follows from Lemma 2.20 by uniqueness of adjoint functors as in theproof of [ , Corollary 3.7.4] (cid:3) Mackey Theorem. We state a slight generalization of the Mackey The-orem of Khovanov and Lauda [ , Proposition 2.18]. Given x ∈ S n and γ = ( γ , . . . , γ n ) ∈ Q n + , we denote xγ := ( γ x − (1) , . . . , γ x − ( n ) ) . Correspondingly, define the integer s ( x, γ ) := − X ≤ m Let γ = ( γ , . . . , γ n ) ∈ Q n + and β = ( β , . . . , β m ) ∈ Q m + with γ + · · · + γ n = β + · · · + β m =: α . Then for any M ∈ R γ -mod we have that Res β Ind γ M has filtration with factors of the form Ind β ; ... ; β m α ,...,α n ; ... ; α m ,...,α nm x ( α ) (cid:0) Res γ ; ... ; γ n α ,...,α m ; ... ; α n ,...,α nm M (cid:1) with α = ( α ab ) ≤ a ≤ n, ≤ b ≤ m running over all tuples of elements of Q + such that P mb =1 α ab = γ a for all ≤ a ≤ n and P na =1 α ab = β b for all ≤ b ≤ m , and x ( α ) is the permutation of mn which maps ( α , . . . , α m ; α , . . . , α m ; . . . ; α n , . . . , α nm ) to ( α , . . . , α n ; α , . . . , α n ; . . . ; α m , . . . , α nm ) . Proof. For m = n = 2 this follows from [ , Proposition 2.18]. The generalcase can be proved by the same argument or deduced from the case m = n = 2by induction. (cid:3) ALEXANDER S. KLESHCHEV Crystal operators. The theory of crystal operators has been developedin [ ], [ ] and [ ] following ideas of Grojnowski [ ], see also [ ]. We reviewnecessary facts for reader’s convenience.Let α ∈ Q + and i ∈ I . It is known that R nα i is a nil-Hecke algebra withunique (up to a degree shift) irreducible module, which we denote by L ( i n ).Moreover, dim q L ( i n ) = [ n ] ! i . We have functors e i : R α -mod → R α − α i -mod , M Res R α − αi,αi R α − αi ◦ Res α − α i ,α i M,f i : R α -mod → R α + α i -mod , M Ind α,α i M ⊠ L ( i ) . If L ∈ R α -mod is irreducible, we define˜ f i L := head( f i L ) , ˜ e i L := soc( e i L ) . A fundamental fact is that ˜ f i L is again irreducible and ˜ e i L is irreducible orzero. We refer to ˜ e i and ˜ f i as the crystal operators. These are operators on B ∪ { } , where B is the set of isomorphism classes of irreducible R α -modulesfor all α ∈ Q + . Define wt : B → P, [ L ] 7→ − α if L ∈ R α -mod. Theorem 2.24. [ ] The set B with the operators ˜ e i , ˜ f i and the function wt isthe crystal graph of the negative part U q ( n − ) of the quantized enveloping algebraof g . For any M ∈ R α -mod, we define ε i ( M ) := max { k ≥ | e ki ( M ) = 0 } . Then ε i ( M ) is also the length of the longest ‘ i -tail’ of weights of M , i.e. themaximum of k ≥ j d − k +1 = · · · = j d = i for some weight j =( j , . . . , j d ) of M . Define also ε ∗ i ( M ) := max { k ≥ | j = · · · = j k = i for a weight j = ( j , . . . , j d ) of M } to be the length of the longest ‘ i -head’ of weights of M . Proposition 2.25. [ , ] Let L be an irreducible R α -module, i ∈ I , and ε = ε i ( L ) . (i) ˜ e i ˜ f i L ∼ = L and if ˜ e i L = 0 then ˜ f i ˜ e i L ∼ = L ; (ii) ε = max { k ≥ | ˜ e ki ( L ) = 0 } ; (iii) Res α − εα i ,εα i L ∼ = ˜ e εi L ⊠ L ( i ε ) . Recall from (2.15) the cyclotomic ideal J Λ α . We have an obvious functor ofinflation infl Λ : R Λ α -mod → R α -mod and its left adjointpr Λ : R α -mod → R Λ α -mod , M M/J Λ α M. Lemma 2.26. [ , Proposition 2.4] Let L be an irreducible R α -module. Then pr Λ L = 0 if and only if ε ∗ i ( L ) ≤ h Λ , α ∨ i i for all i ∈ I . Extremal words and multiplicity one results. Let i ∈ I . Considerthe map θ ∗ i : h I i → h I i such that for j = ( j , . . . , j d ) ∈ h I i , we have θ ∗ i ( j ) = (cid:26) ( j , . . . , j d − ) if j d = i ;0 otherwise. (2.27)We extend θ ∗ i by linearity to a map θ ∗ i : A h I i → A h I i . USPIDAL SUSTEMS FOR AFFINE KLR ALGEBRAS 11 Let x be an element of A h I i . Define ε i ( x ) := max { k ≥ | ( θ ∗ i ) k ( x ) = 0 } . A word i a . . . i a b b ∈ h I i , with a , . . . , a b ∈ Z ≥ , is called extremal for x if a b = ε i b ( x ), a b − = ε i b − (( θ ∗ i b ) a b ( x )), . . . , a = ε i (( θ ∗ i ) a . . . ( θ ∗ i b ) a b ( x )). A weight i a . . . i a b b ∈ h I i α is called extremal for M ∈ R α -mod if it is an extremal wordfor ch q M ∈ A h I i , in other words, if a b = ε i b ( M ), a b − = ε i b − (˜ e a b i b M ), . . . , a = ε i (˜ e a i . . . ˜ e a b i b M ).The following useful result, which is a version of [ , Corollary 2.17], describesthe multiplicities of extremal weight spaces in irreducible modules. We denoteby 1 F the trivial module F over the trivial algebra R ≃ F . Lemma 2.28. Let L be an irreducible R α -module, and i = i a . . . i a b b ∈ h I i α bean extremal weight for L . Then dim q L i = [ a ] ! i . . . [ a b ] ! i b , and L ∼ = ˜ f a b i b ˜ f a b − i b − . . . ˜ f a i F . Moreover, i is not an extremal weight for any irreducible module L ′ = L .Proof. Follows easily from Proposition 2.25, cf. [ , Theorem 2.16]. (cid:3) Corollary 2.29. Let M ∈ R α -mod , and i = i a . . . i a b b ∈ h I i α be an extremalweight for M . Then we can write dim q M i = m [ a ] ! i . . . [ a b ] ! i b for some m ∈ A .Moreover, if L ∼ = ˜ f a b i b ˜ f a b − i b − . . . ˜ f a i F and L ⊛ ≃ L , then we have [ M : L ] q = m .Proof. Apply Lemma 2.28, cf. [ , Corollary 2.17]. (cid:3) Now we establish some useful ‘multiplicity-one results’. The first one showsthat in every irreducible module there is a weight space with a one dimensionalgraded component: Lemma 2.30. Let L be an irreducible R α -module, and i = i a . . . i a b b ∈ h I i α be an extremal weight for L . Set N := − b + P bm =1 a m ( α i m , α i m ) / . Then dim 1 i L N = dim 1 i L − N = 1 .Proof. This follows immediately from the equality dim q i L = [ a ] ! i . . . [ a b ] ! i b ,which comes from Lemma 2.28. (cid:3) The following result shows that any induction product of irreducible modulesalways has a multiplicity one composition factor. Proposition 2.31. Suppose that n ∈ Z > and for r = 1 , . . . , n , we have α ( r ) ∈ Q + , an irreducible R α ( r ) -module L ( r ) , and i ( r ) := i a ( r )1 . . . i a ( r ) k k ∈ h I i α ( r ) is anextremal weight for L ( r ) . Denote a m := P nr =1 a ( r ) m for all ≤ m ≤ k . Then j := i a . . . i a k k is an extremal weight for L (1) ◦ · · · ◦ L ( n ) , and the graded multiplicityof the ⊛ -self-dual irreducible module N ∼ = ˜ f a k i k ˜ f a k − i k − . . . ˜ f a i F in L (1) ◦ · · · ◦ L ( n ) is q m , where m := − P ≤ t
Proof. By Lemma 2.28, the multiplicity of i ( r ) in ch q L ( r ) is [ a ( r )1 ] ! i . . . [ a ( r ) k ] ! i k .By (2.19), we havech q ( L (1) ◦ · · · ◦ L ( n ) ) = ch q ( L (1) ) ◦ · · · ◦ ch q ( L ( n ) ) . It is easy to see that the weight j is an extremal weight for L (1) ◦ · · · ◦ L ( n ) ,and that j can be obtained only from the shuffle product i (1) ◦ · · · ◦ i ( n ) . Anelementary computation shows that j appears in i (1) ◦ · · · ◦ i ( n ) with multiplicity q m [ a ] ! i . . . [ a k ] ! i k . Now apply Corollary 2.29. (cid:3) Corollary 2.32. Let L be an irreducible R α -module and n ∈ Z > . Then thereis an irreducible R nα -module N which appears in L ◦ n with graded multiplicity q − ( α,α ) n ( n − / . In particular, the ungraded multiplicity of N is one.Proof. Apply Proposition 2.31 with L (1) = · · · = L ( n ) = L . (cid:3) Khovanov-Lauda-Rouquier categorification. We recall the Khovanov-Lauda-Rouquier categorification of the quantized enveloping algebra f obtainedin [ , , ]. We follow the presentation of [ , ]. Let f A ⊂ f be the A -formof the Lusztig’s quantum group f corresponding to the Cartan matrix C . This A -algebra is generated by the divided powers θ ( n ) i = θ ni / [ n ] ! i of the standardgenerators. The algebra f A has a Q + -grading f A = ⊕ α ∈ Q + ( f A ) α determinedby the condition that each θ i is in degree α i .There is a bilinear form ( · , · ) on f defined in [ , § § f ∗ A = (cid:8) y ∈ f (cid:12)(cid:12) ( x, y ) ∈ A for all x ∈ f A (cid:9) . Let ( θ ∗ i ) ( n ) be the map dual to the map f A → f A , x xθ ( n ) i . Finally, there is a coproduct r on f such that f is atwisted unital and counital bialgebra. Moreover, for all x, y, z ∈ f we have( xy, z ) = ( x ⊗ y, r ( z )) . (2.33)The field Q ( q ) possesses a unique automorphism called the bar involution such that q = q − . With respect to this involution, let b : f → f be theanti-linear algebra automorphism such that b ( θ i ) = θ i for all i ∈ I . Also let b ∗ : f → f be the adjoint anti-linear map to b with respect to Lusztig’s form,so ( x, b ∗ ( y )) = ( b ( x ) , y ) for all x, y ∈ f . The maps b and b ∗ preserve f A and f ∗ A , respectively.Let [ R -mod] = L α ∈ Q + [ R α -mod] denote the Grothendieck ring, which is an A -algebra via induction product and q n [ V ] = [ V h n i ]. Similarly the functors ofrestriction define a coproduct r on [ R -mod]. This product and coproduct make[ R -mod] into a twisted unital and counital bialgebra [ , Proposition 3.2].It [ , ] an explicit A -bialgebra isomorphisms γ ∗ : [ R -mod] ∼ → f ∗ A is con-structed; in fact [ ] establishes a dual isomorphism, see [ , Theorem 4.4] fordetails on this. Moreover, γ ∗ ([ V ⊛ ]) = b ∗ ( γ ∗ ([ V ])), and we have a commutativetriangle A h I i [ R -mod] f ∗ A ✲ γ ∗ ✑✑✑✸ ch q ◗◗◗❦ ι , (2.34) USPIDAL SUSTEMS FOR AFFINE KLR ALGEBRAS 13 where the map ι is defined as follows: ι ( x ) = X i =( i ,...,i d ) ∈h I i ( x, θ i . . . θ i d ) i ( x ∈ f ∗ A ) . Lemma 2.35. Let v ∗ be a dual canonical basis element of f , and i = i a . . . i a k k be an extremal word of ι ( v ∗ ) in the sence of Section 2.8. Then i appears in ι ( v ∗ ) with coefficient [ a ] ! i . . . [ a k ] ! i k .Proof. Apply induction on a + · · · + a k . The induction base is a + · · · + a k = 0,in which case v ∗ = 1 ∈ f ∗ A and ι (1) is the empty word. Recall the map θ ∗ i : A h I i → A h I i from (2.27). For all x ∈ f ∗ A we have ι (( θ ∗ i ) ( n ) ( x )) = ( θ ∗ i ) ( n ) ( ι ( x )),where in the right hand side ( θ ∗ i ) ( n ) = ( θ ∗ i ) n / [ n ] ! i . By [ , Proposition 5.3.1],( θ ∗ i k ) ( a ik ) ( v ∗ ) is again a dual canonical basis element, and by induction, the word i a . . . i a k − k − appears in ι (( θ ∗ i k ) ( a ik ) ( v ∗ )) with coefficient [ a ] ! i . . . [ a k − ] ! i k − . Theresult follows. (cid:3) Cuspidal systems and standard modules Convex preorders on Φ + . Recall the notion of a convex preorder onΦ + from (1.1)–(1.3). Convex preorders exist, see e.g. [ , Example 2.11(ii)]. Lemma 3.1. [ ] For any positive root β , the convex cones spanned by Φ + ( β ) := { γ ∈ Φ + | γ (cid:23) β } and Φ + \ Φ( β ) intersect only at the origin.Proof. The set { γ ∈ Φ + | γ (cid:23) β } is a terminal section for the preorder (cid:22) inthe sense of [ , Section 2.4]. By [ , Lemma 2.9], this set is biconvex, which isequivalent to the statement about the cones by [ , Remark 2.3]. (cid:3) Lemma 3.1 immediately implies the following properties:(Con1) Let ρ ∈ Φ re+ , m ∈ Z > , and mρ = P ba =1 γ a for some positive roots γ a . Assume that either γ a (cid:22) ρ for all a = 1 , . . . , b or γ a (cid:23) ρ for all a = 1 , . . . , b . Then b = m and γ a = ρ for all a = 1 , . . . , b .(Con2) Let β, κ be two positive roots, not both imaginary. If β + κ = P ba =1 γ a for some positive roots γ a (cid:22) β , then β (cid:23) κ .(Con3) Let ρ ∈ Φ im+ , and ρ = P ba =1 γ a for some positive roots γ a . If either γ a (cid:22) ρ for all a = 1 , . . . , b or γ a (cid:23) ρ for all a = 1 , . . . , b , then all γ a areimaginary.Indeed, for (Con1), we may assume that all γ a ≺ ρ , and apply the lemmawith β = ρ . For (Con2), taking into account (Con1), we may assume that all γ a ≺ β , and apply the lemma. For (Con3), we may assume that all γ a are realand apply the lemma with β = ρ .The Main Theorem from the introduction will be proved for an arbitraryconvex preorder, but later results which rely on the theory of imaginary repre-sentations, beginning from Section 5, require an additional assumption. Recallfrom (2.4) thatΦ re+ = { β + nδ | β ∈ Φ ′ + , n ∈ Z ≥ } ⊔ {− β + nδ | β ∈ Φ ′ + , n ∈ Z > } . A convex preorder (cid:22) will be called balanced ifΦ re ≻ = { β + nδ | β ∈ Φ ′ + , n ∈ Z ≥ } . (3.2) ALEXANDER S. KLESHCHEV Then of course we also have Φ re ≺ = {− β + nδ | β ∈ Φ ′ + , n ∈ Z > } . Balancedconvex preorders exist, see for example [ ]. For reader’s convenience we con-clude this section with a sketch of a construction of balanced preorders (it willnot be used in the paper). Let V be the R -span of Φ ′ . The affine group W actson V with affine transformations, see [ , Chapter 4] . This action induces asimply transitive action of W on the set of alcoves, which are the connectedcomponents of the complement of the affine hyperplanes H α,n = { v ∈ V | ( v, α ) = m } ( α ∈ Φ ′ + , m ∈ Z ) . Let C = { v ∈ V | < ( v, α ) < α ∈ Φ ′ + } be the fundamental alcove. A(possibly infinite) sequence of alcoves . . . , C , C , C = C , C − , C − , . . . is calledan alcove path if for each n there is a common wall for C n and C n +1 so that C n +1 is a reflection of C n in this wall. Let β n ∈ Φ ′ + and m n be defined from H β n ,m n := (cid:26) the common wall between C n − and C n if n > C n +1 and C n if n < simple if it crosses each affine hyperplaneexactly once. An alcove path is called balanced if m n ≥ n > m n < n < 0. For a balanced simple path set ρ n := (cid:26) β n + m n δ if n > − β n + m n δ if n < . Now we define a preorder on Φ + such that ρ ≻ ρ ≻ · · · ≻ nδ (cid:23) mδ ≻ · · · ≻ ρ − ≻ ρ − for all n, m ∈ Z > . The well-known geometric interpretation of thereduced expressions in terms of alcove paths [ , Section 4.4, 4.5] easily impliesthat this preorder is convex, and it is balanced by definition.3.2. Root partitions. Recall that I ′ = { , . . . , l } . We will consider the set P of l -multipartitions λ = ( λ ( i ) ) i ∈ I ′ , where each λ ( i ) = ( λ ( i )1 , λ ( i )2 , . . . ) is ausual partition. For all i ∈ I ′ , we denote | λ ( i ) | := λ ( i )1 + λ ( i )2 + . . . , and set | λ | := P i ∈ I ′ | λ ( i ) | . For m ∈ Z ≥ , denote P m := { λ ∈ P | | λ | = m } .We work with a fixed convex preorder (cid:22) on Φ + . Recall the notation (1.4).We will consider finitary sequences of non-negative integers of the form M = ( m , m , . . . ; m ; . . . , m − , m − ) . The set of all such sequences is denoted by Se . The left lexicographic order on Se is denoted ≤ l and the right lexicographic order on Se is denoted ≤ r . Wewill use the following bilexicographic partial order on Se : M ≤ N if and only if M ≤ l N and M ≥ r N. A root partition is a pair ( M, µ ) with M ∈ Se and µ ∈ P m . For a rootpartition ( M, µ ) we define M n := (cid:26) m n ρ n if n = 0 m δ if n = 0 , and set | M | = ( M , M , . . . ; M ; . . . , M − , M − ) . (3.3)This is a finitary sequence of elements of Q + . If P n ∈ Z M n = α we say that( M, µ ) is a root partition of α . In that case we have a parabolic subalgebra USPIDAL SUSTEMS FOR AFFINE KLR ALGEBRAS 15 R | M | ⊆ R α . Denote by Π( α ) the set of all root partitions of α . We will use thefollowing partial order on Π( α ):( M, µ ) ≤ ( N, ν ) if and only if M ≤ N and if M = N then µ = ν. (3.4)The positive subalgebra n + ⊂ g has a basis consisting of root vectors { E ρ , E nδ,i | ρ ∈ Φ re+ , n ∈ Z > , i ∈ I ′ } . For i ∈ I ′ , assign to a partition µ ( i ) = ( µ ( i )1 , µ ( i )2 , . . . ) a PBW monomial E µ ( i ) := E µ ( i )1 δ,i E µ ( i )2 δ,i . . . . Now, to a root partition ( M, µ ), we assign a PBW monomial E M,µ := E m ρ E m ρ . . . E µ (1) E µ (2) . . . E µ ( l ) . . . E m − ρ − E m − ρ − . Then { E M,µ | ( M, µ ) ∈ Π( α ) } is a basis of the weight space U ( n + ) α . Inparticular, | Π( α ) | = dim U ( n + ) α is the Kostant partition function of α . In viewof the isomorphism γ ∗ from (2.34), we conclude: Lemma 3.5. The number of irreducible R α -modules (up to isomorphism) is | Π( α ) | . Given a root partition ( M, µ ) and a ∈ Z , denote by ( M, µ ) ′ a the root partitionobtained from ( M, µ ) by ‘annihilating’ its a th component; to be more precise,( M, µ ) ′ a = ( M ′ , µ ′ ), where m ′ b = (cid:26) b = am b if b = a and µ ′ = (cid:26) ∅ if a = 0 µ otherwise. (3.6)Finally, sometimes we use a slightly different notation for the root partitions.For example, if ( M, µ ) is such that m = 2 , m = 1 , m − = 1 , and all other m a with a = 0 are zero, then we write ( M, µ ) = ( ρ , ρ , ρ , µ, ρ − ).3.3. Standard modules. We continue to work with a fixed convex preorder (cid:22) on Φ + and use the notation (1.4). Recall from the introduction the definitionof the corresponding cuspidal system. It consists of certain cuspidal modules L ρ for ρ ∈ Φ re+ and imaginary modules L ( µ ) for µ ∈ P satisfying the properties(Cus1) and (Cus2). For every α ∈ Q + and ( M, µ ) ∈ Π( α ), we define an integer sh ( M, µ ) := X n =0 ( ρ n , ρ n ) m n ( m n − / , (3.7)we define the R | M | -module L M,µ := L ◦ m ρ ⊠ L ◦ m ρ ⊠ · · · ⊠ L ( µ ) ⊠ · · · ⊠ L ◦ m − ρ − ⊠ L ◦ m − ρ − h sh ( M, µ ) i , (3.8)and we define the standard module ∆( M, µ ) := L ◦ m ρ ◦ L ◦ m ρ ◦ · · · ◦ L ( µ ) ◦ · · · ◦ L ◦ m − ρ − ◦ L ◦ m − ρ − h sh ( M, µ ) i . (3.9)Note that ∆( M, µ ) = Ind | M | L M,µ ∈ R α -mod. Lemma 3.10. Let ρ ∈ Φ re+ , L ρ be the corresponding cuspidal module, and n ∈ Z > . Then ( L ◦ nρ ) ⊛ ≃ L ◦ nρ h ( ρ, ρ ) n ( n − / i . In particular, the module L ◦ nρ h ( ρ, ρ ) n ( n − / i is ⊛ -self-dual.Proof. Recall that our standard choice of shifts of irreducible modules is so that L ⊛ ρ ≃ L ρ . Now the result follows from Lemma 2.21. (cid:3) ALEXANDER S. KLESHCHEV Lemma 3.11. We have L ⊛ M,µ ≃ L M,µ Proof. Follows from Lemma 3.10. (cid:3) Restrictions of standard modules. The proof of the following propo-sition is similar to [ , Lemma 3.3]. Proposition 3.12. Let ( M, µ ) , ( N, ν ) ∈ Π( α ) . Then: (i) Res | N | ∆( N, ν ) ≃ L N,ν . (ii) Res | M | ∆( N, ν ) = 0 implies M ≤ N .Proof. Let Res α | M | ∆( N, ν ) = 0. It suffices to prove that M ≥ l N or M ≤ r N implies that M = N and Res α | M | ∆( N, ν ) ∼ = L N,ν . We may assume that M ≥ l N ,the case M ≤ r N being similar. We apply induction on ht( α ) and consider threecases.Case 1: m a > a > 0. Pick the minimal such a , and let ( M ′ , µ ′ ) =( M, µ ) ′ a and ( N ′ , ν ′ ) = ( N, ν ) ′ a , see (3.6). By the Mackey Theorem 2.23,Res α | M | ∆( N, ν ) has filtration with factors of the formInd m a ρ a ; | M ′ | κ ,...,κ c ; γ V, where m a ρ a = κ + · · · + κ c , with κ , . . . , κ c ∈ Q + \ { } , and γ is a refinementof | M ′ | . Moreover, the module V is obtained by twisting and degree shifting asin (2.22) of a module obtained by restriction of L ⊠ n ρ ⊠ L ⊠ n ρ ⊠ · · · ⊠ L ( ν ) ⊠ · · · ⊠ L ⊠ n − ρ − ⊠ L ⊠ n − ρ − to a parabolic which has κ , . . . , κ c in the beginnings of the correspondingblocks. In particular, if V = 0, then for each b = 1 , . . . , c we have thatRes ρ k κ b ,ρ k − κ b L ρ k = 0 for some k = k ( b ) with n k = 0 or Res n δκ b ,n δ − κ b L ( ν ) = 0.If Res ρ k κ b ,ρ k − κ b L ρ k = 0, then by (Cus1), κ b is a sum of roots (cid:22) ρ k . Moreover,since M ≥ l N and n k = 0, we have that ρ k (cid:22) ρ a . Thus κ b is a sum of roots (cid:22) ρ a . On the other hand, if Res n δκ b ,n δ − κ b L ( ν ) = 0, then by (Cus2), either κ b is an imaginary root or it is a sum of real roots less than n δ . In either casewe conclude again that κ b is a sum of roots (cid:22) ρ a . Using (Con1), we can nowconclude that c = m a , and κ b = ρ a = ρ k ( b ) for all b = 1 , . . . , c . Hence n a ≥ m a .Since M ≥ l N , we conclude that n a = m a , andRes α | M | ∆( N, ν ) ∼ = L ◦ m a ρ a ⊠ Res α − m a ρ a | M ′ | ∆( N ′ , ν ′ ) . Now, since ht( α − m a ρ a ) < ht( α ), we can apply the inductive hypothesis.Case 2: m b = 0 for all b > 0, but m = 0. Since N ≤ l M , we also have that n b = 0 for all b > 0. Let ( M ′ , µ ′ ) = ( M, µ ) ′ a , ( N ′ , ν ′ ) = ( N, ν ) ′ a . By the MackeyTheorem 2.23, Res α | M | ∆( N, ν ) has filtration with factors of the formInd m δ ; | M ′ | κ ,...,κ c ; γ V, where m δ = κ + · · · + κ c , with κ , . . . , κ c ∈ Q + \ { } , and γ is a refinement of | M ′ | . Moreover, the module V is obtained by twisting and degree shifting of amodule obtained by parabolic restriction of the module L ( ν ) ⊠ · · · ⊠ L ⊠ n − ρ − ⊠ L ⊠ n − ρ − to a parabolic which has κ , . . . , κ c in the beginnings of the correspondingblocks. In particular, if V = 0, then either USPIDAL SUSTEMS FOR AFFINE KLR ALGEBRAS 17 (1) Res n δκ ,n δ − κ L ( ν ) = 0 and for b = 2 , . . . , c , there is k = k ( b ) < ρ k κ b ,ρ k − κ b L ρ k = 0, or(2) for b = 1 , . . . , c there is k = k ( b ) < ρ k κ b ,ρ k − κ b L ρ k = 0 . By (Cus1) and (Con3), only (1) is possible, and in that case, using also (Cus2),we must have c = 1 and κ = m δ . Since M ≥ l N , we conclude that n = m ,and Res α | M | ∆( N, ν ) ∼ = L ( ν ) ⊠ Res α − m δ | M ′ | ∆( N ′ , ν ) . Now, since ht( α − m δ ) < ht( α ), we can apply the inductive hypothesis.Case 3: m b = 0 for all b ≥ 0. This case is similar to Case 1. (cid:3) Rough classification of irreducible modules We continue to work with a fixed convex preorder (cid:22) on Φ + and use the no-tation (1.4). In this section we prove the Main Theorem from the introduction.4.1. Statement and the structure of the proof. We will prove the followingresult, which contains slightly more information than the Main Theorem: Theorem 4.1. For a given convex preorder, there exists a corresponding cus-pidal system { L ρ | ρ ∈ Φ re+ } ∪ { L ( λ ) | λ ∈ P } . Moreover: (i) For every root partition ( M, µ ) , the standard module ∆( M, µ ) has anirreducible head; denote this irreducible module L ( M, µ ) . (ii) { L ( M, µ ) | ( M, µ ) ∈ Π( α ) } is a complete and irredundant system ofirreducible R α -modules up to isomorphism. (iii) L ( M, µ ) ⊛ ≃ L ( M, µ ) . (iv) [∆( M, µ ) : L ( M, µ )] q = 1 , and [∆( M, µ ) : L ( N, ν )] q = 0 implies ( N, ν ) ≤ ( M, µ ) . (v) Res | M | L ( M, µ ) ≃ L M,µ and Res | N | L ( M, µ ) = 0 implies N ≤ M . (vi) L ◦ nρ is irreducible for all ρ ∈ Φ re+ and all n ∈ Z > . The rest of Section 4 is devoted to the proof of Theorem 4.1, which goes byinduction on ht( α ). To be more precise, we prove the following statements forall α ∈ Q + by induction on ht( α ):(1) For each ρ ∈ Φ re+ with ht( ρ ) ≤ ht( α ) there exists a unique up to isomor-phism irreducible R ρ -module L ρ which satisfies the property (Cus1).Moreover, L ρ then also satisfies the property (vi) of Theorem 4.1 ifht( nρ ) ≤ ht( α ).(2) For each n ∈ Z ≥ with ht( nδ ) ≤ ht( α ) there exist irreducible R nδ -modules { L ( µ ) | µ ∈ P n } which satisfy the property (Cus2).(3) The standard modules ∆( M, µ ) for all ( M, µ ) ∈ Π( α ), defined as in(3.9) using the modules from (1) and (2), satisfy the properties (i)–(v)of Theorem 4.1.The induction starts with ht( α ) = 0, and for ht( α ) = 1 the theorem is alsoclear since R α i is a polynomial algebra, which has only the trivial representation L α i . The inductive assumption will stay valid throughout Section 4. ALEXANDER S. KLESHCHEV Irreducible heads. In the following proposition, we exclude the caseswhere the standard module is either of the form L ◦ nρ for a real root ρ , or isimaginary of the form L ( λ ). The excluded cases will be dealt with in thisSections 4.3, 4.4 and 4.5. Proposition 4.2. Let ( M, µ ) ∈ Π( α ) , and suppose that there are integers a = b such that m a = 0 and m b = 0 . (i) ∆( M, µ ) has an irreducible head; denote this irreducible module L ( M, µ ) . (ii) If ( M, µ ) = ( N, ν ) , then L ( M, µ ) = L ( N, ν ) . (iii) L ( M, µ ) ⊛ ≃ L ( M, µ ) . (iv) [∆( M, µ ) : L ( M, µ )] q = 1 , and [∆( M, µ ) : L ( N, ν )] q = 0 implies ( N, ν ) ≤ ( M, µ ) . (v) Res | M | L ( M, µ ) ≃ L M,µ and Res | N | L ( M, µ ) = 0 implies N ≤ M .Proof. (i) and (v) If L is an irreducible quotient of ∆( M, µ ) = Ind | M | L M,µ ,then by adjointness of Ind | M | and Res | M | and the irreducibility of the R | M | -module L M,µ , which holds by the inductive assumption, we conclude that L M,µ is a submodule of Res | M | L . On the other hand, by Proposition 3.12(i) themultiplicity of L M,µ in Res | M | ∆( M, µ ) is one, so (i) follows. Note that we havealso proved the first statement in (v), while the second statement in (v) followsfrom Proposition 3.12(ii) and the exactness of the functor Res | M | .(iv) By (v), Res | N | L ( N, ν ) ∼ = L N,ν = 0. Therefore, if L ( N, ν ) is a compo-sition factor of ∆( M, µ ), then Res | N | ∆( M, µ ) = 0 by exactness of Res | N | . ByProposition 3.12, we then have N ≤ M and the first equality in (iv). If N < M ,then ( N, ν ) < ( M, µ ). If N = M , and ν = µ , then we get a contribution of L N,ν into Res | M | ∆( M, µ ), which contradicts (v).(ii) If L ( M, µ ) ∼ = L ( N, ν ), then we deduce from (iv) that ( M, µ ) ≤ ( N, ν ) and( N, ν ) ≤ ( M, µ ), whence ( M, µ ) = ( N, ν ).(iii) follows from (v) and Lemma 3.11. (cid:3) Imaginary modules. In this subsection we assume that α = nδ for some n ∈ Z ≥ . Then Proposition 4.2, yields | Π( α ) | − | P n | (pairwise non-isomorphic)irreducible modules, namely the modules L ( M, µ ) corresponding to the rootpartitions ( M, µ ) such that m a = 0 for some a = 0. Let us label the re-maining | P n | irreducible R nδ -modules by the elements of P n in some way, cf.Lemma 3.5. So we get irreducible R nδ -modules { L ( µ ) | µ ∈ P n } , and then { L ( M, µ ) | ( M, µ ) ∈ Π( α ) } is a complete and irredundant system of irreducible R α -modules up to isomorphism. Our next goal is Lemma 4.3 which proves thatthe modules { L ( µ ) | µ ∈ P n } are imaginary in the sense of (Cus2).We need some terminology. Let ( M, µ ) be a root partition. We say thata real root ρ a (resp. an imaginary root m δ ) appears in the support of M if m a > m > κ be the largest root appearing in the support of M , and β (cid:23) κ . Note that if β is real then L β ◦ ∆( M, µ ) is, up to a degree shift,a standard module again. If β = nδ is imaginary, ν ∈ P n , and κ is real, then L ( ν ) ◦ ∆( M, µ ) is again a standard module. Lemma 4.3. Let λ ∈ P n . Suppose that β, γ ∈ Q + \ Φ im+ are non-zero elementssuch that nδ = β + γ and Res β,γ L ( λ ) = 0 . Then β is a sum of real roots lessthan δ and γ is a sum of real roots greater than δ . USPIDAL SUSTEMS FOR AFFINE KLR ALGEBRAS 19 Proof. We prove that β is a sum of real roots less than δ , the proof that γ is a sum of real roots greater than δ being similar. Let L ( M, µ ) ⊠ L ( N, ν )be an irreducible submodule of Res β,γ L ( λ ) = 0, so that ( M, µ ) ∈ Π( β ) and( N, ν ) ∈ Π( γ ). Note that ht( β ) , ht( γ ) < ht( α ), so the modules L ( M, µ ) , L ( N, ν )are defined by induction.Let χ be the largest root appearing in the support of M . If χ ≤ δ , then,since β is not an imaginary root, we conclude that β is a sum of real roots lessthan δ . So we may assume that χ ∈ Φ re ≻ . Moreover, Res βχ,β − χ L ( M, µ ) = 0, andhence Res χ,γ + β − χ L ( λ ) = 0. So we may assume from the beginning that β ∈ Φ re ≻ and L ( M, µ ) ≃ L β . Moreover, we may assume that β is the largest possiblereal root for which Res β,γ L ( λ ) = 0.Now, let κ be the largest root appearing in the support of N . If κ is a realroot, we have the cuspidal module L κ . If κ is imaginary, then let us denote by L κ the module L ( ν ). Then we have a non-zero map L β ⊠ L κ ⊠ V → Res β,κ,γ − κ L ( λ ),for some non-zero R γ − κ -module V . By adjunction, this yields a non-zero map f : (Ind β + κβ,κ L β ⊠ L κ ) ⊠ V → Res β + κ,γ − κ L ( λ )If κ = γ note that β = γ , since it has been assumed that β, γ Φ im+ . Now weconclude that β ≺ γ , for otherwise L ( λ ) is a quotient of the standard module L β ◦ L γ , which contradicts the definition of the imaginary module L ( λ ). Now,since nδ = β + κ , we have by (Con3) that β ≺ nδ ≺ γ , in particular β ≺ nδ (cid:22) δ as desired.Next, let κ = γ , and pick a composition factor L ( M ′ , µ ′ ) of Ind β + κβ,κ L β ⊠ L κ ,which is not in the kernel of f . By the assumption on the maximality of β ,every root κ ′ in the support of M ′ satisfies κ ′ (cid:22) β . Thus β + κ is a sum of roots (cid:22) β . Now (Con2) implies that κ (cid:22) β , and so by adjointness, L ( λ ) is a quotientof the standard module L β ◦ ∆( N, ν ), which is a contradiction. (cid:3) We now establish a useful property of imaginary modules: Lemma 4.4. Let µ ∈ P r and ν ∈ P s with r + s = n . Then all compositionfactors of L ( µ ) ◦ L ( ν ) are of the form L ( κ ) for κ ∈ P n .Proof. Let L ( K, κ ) be a composition factor of L ( µ ) ◦ L ( ν ). We need to prove that k a = 0 for all a = 0, i.e. L ( K, κ ) = L ( κ ). If this is not the case, there is a > k a = 0. Pick the smallest such a , and set ( K ′ , κ ′ ) := ( K, κ ) ′ a , see (3.6). ByProposition 4.2(v), we have that Res | K | L ( K, κ ) = 0, so Res | K | ( L ( µ ) ◦ L ( ν )) = 0.We apply the Mackey Theorem to conclude that the last module has a filtrationwith factors of the form Ind k a ρ a ; | K ′ | λ ,λ ; γ V, where k a ρ a = λ + λ , γ is a refinement of | K ′ | , andRes λ ,rδ − λ L ( µ ) = 0 = Res λ ,sδ − λ L ( ν ) . By the inductive assumption, we know that L ( µ ) and L ( ν ) satisfy (Cus2), i.e. λ and λ are either imaginary roots or a sum of the roots of the form ρ b with b < 0. In either case, λ and λ are sums of the roots less than ρ a , and then sois k a ρ a . This contradicts (Con1). (cid:3) ALEXANDER S. KLESHCHEV Cuspidal modules. Throughout this subsection we assume that α = ρ n ∈ Φ re+ for some n = 0. Let ( M, µ ) ∈ Π( α ) be a root partition of α . Thereis a trivial root partition, denoted ( α ), and defined as ( α ) = ( M, ∅ ), where m n = 1, and m a = 0 for all a = n . Proposition 4.2 yields | Π( α ) | − R α -modules, namely the ones which correspond to the non-trivial root partitions ( M, µ ). We define the cuspidal module L α to be the miss-ing irreducible R α -module, cf. Lemma 3.5. Then, of course, we have that { L ( M, µ ) | ( M, µ ) ∈ Π( α ) } is a complete and irredundant system of irreducible R α -modules up to isomorphism. We now prove that L α satisfies the property(Cus1) and is uniquely determined by it. To be more precise: Lemma 4.5. If β, γ ∈ Q + are non-zero elements such that α = β + γ and Res β,γ L α = 0 , then β is a sum of roots less than α and γ is a sum of rootsgreater than α . Moreover, this property characterizes L α among the irreducible R α -modules uniquely up to isomorphism and degree shift.Proof. We prove that β is a sum of roots less than α , the proof that γ is a sumof roots greater than α being similar. Let L ( M, µ ) ⊠ L ( N, ν ) be an irreduciblesubmodule of Res β,γ L α , so that ( M, µ ) ∈ Π( β ) and ( N, ν ) ∈ Π( γ ). Let χ be thelargest root appearing in the support of M . Then Res χ,β − χ L ( M, µ ) = 0, andhence Res χ,γ + β − χ L α = 0. If we can prove that χ is a sum of roots less than α , then by (Con1), (Con3), χ is a root less than α , whence, by the maximalityof χ , we have that β is a sum of roots less than α . So we may assume fromthe beginning that β is a root and L ( M, µ ) = L β (if β is imaginary, L β isinterpreted as L ( µ )). Moreover, we may assume that β is the largest possibleroot for which Res β,γ L α = 0.Now, let κ be the largest root appearing in the support of N . If κ is a realroot, we have the cuspidal module L κ . If κ is imaginary, then we interpret L κ as L ( ν ). Then we have a non-zero map L β ⊠ L κ ⊠ V → Res β,κ,γ − κ L α , for some 0 = V ∈ R γ − κ -mod. By adjunction, this yields a non-zero map f : (Ind β,κ L β ⊠ L κ ) ⊠ V → Res β + κ,γ − κ L α . If κ = γ , then we must have β ≺ γ , for otherwise L α is a quotient of thestandard module L β ◦ L γ , which contradicts the definition of the cuspidal module L α . Now, since α = β + κ , we have by (Con1) that β ≺ α ≺ γ , in particular β ≺ α as desired.Next, let κ = γ , and pick a composition factor L ( M ′ , µ ′ ) of Ind β + κβ,κ L β ⊠ L κ ,which is not in the kernel of f . By the assumption on the maximality of β ,every root κ ′ in the support of M ′ satisfies κ ′ (cid:22) β . Thus β + κ is a sum of roots (cid:22) β . If β and κ are not both imaginary, then (Con2) implies that κ (cid:22) β , andso by adjointness, L α is a quotient of the standard module L β ◦ ∆( N, ν ), whichis a contradiction.If β and κ are both imaginary, then ∆( N, ν ) = L ( ν ) ◦ ∆( N ′ , ∅ ) for N ′ suchthat a maximal root appearing in the support of N ′ is of the form ρ a with a < L α is a quotient of L ( µ ) ◦ L ( ν ) ◦ L ( N ′ , ∅ ).It now follows from Lemma 4.4 that L α is a quotient of the standard module USPIDAL SUSTEMS FOR AFFINE KLR ALGEBRAS 21 of the form L ( λ ) ◦ L ( N ′ , ∅ ) for some composition factor L ( λ ) of L ( µ ) ◦ L ( ν ), sowe get a contradiction again, since L α is cuspidal.The second statement of the lemma is clear since, in view of Proposition 4.2(v)and (Con1), the irreducible modules L ( M, µ ), corresponding to non-trivial rootpartitions ( M, µ ) ∈ Π( α ), do not satisfy the property (Cus1). (cid:3) Powers of cuspidal modules. Assume finally that α = nρ for some ρ ∈ Φ re+ and n ∈ Z > . Lemma 4.6. The induced module L ◦ nρ is irreducible for all n ∈ Z > .Proof. In view of Proposition 4.2, we have the irreducible modules L ( M, µ ) forall root partitions ( M, µ ) ∈ Π( α ), except for ( N, ν ) = ( ρ n ) for which ∆( N, ν ) = L ◦ nρ . By (Con1), we have that N ≤ M for all ( M, µ ) ∈ Π( α ), and if M = N ,then ( M, µ ) = ( N, ν ). By Proposition 4.2(v), we conclude that L ◦ nρ has only onecomposition factor L appearing with certain multiplicity c ( q ) ∈ A , and suchthat L = L ( M, µ ) for all ( M, µ ) ∈ Π( α ) \ { ( N, ν ) } . Finally, by Corollary 2.32,we conclude that L ◦ nρ ∼ = L . (cid:3) The proof of Theorem 4.1 is now complete.4.6. Reduction modulo p . In this section we work with two fields: F ofcharacteristic p > K of characteristic 0. We use the corresponding indicesto distinguish between the two situations. Given an irreducible R α ( K )-module L K for a root partition π ∈ Π( α ) we can pick a (graded) R α ( Z )-invariant lattice L Z as follows: pick a homogeneous weight vector v ∈ L K and set L Z := R α ( Z ) v .The lattice L Z can be used to reduce modulo p :¯ L := L Z ⊗ Z F. In general, the R α ( F )-module ¯ L depends on the choice of the lattice L Z .However, we have ch q ¯ L = ch q L K , so by linear independence of charactersof irreducible R α ( F )-modules, composition multiplicities of irreducible R α ( F )-modules in ¯ L are well-defined. In particular, we have well-defined decompositionnumbers d π,σ := [ ¯ L ( π ) : L F ( σ )] q ( π, σ ∈ Π( α )) , which depend only on the characteristic p of F , since prime fields are splittingfields for irreducible modules over KLR algebras. Lemma 4.7. Let L K be an irreducible R α ( K ) -module and let i = i a . . . i a b b bean extremal weight for L K . Let N be the irreducible ⊛ -selfdual R α ( F ) -moduledefined by N := ˜ f a k i k . . . ˜ f a i F . Then [ ¯ L : N ] q = 1 .Proof. Reduction modulo p preserves formal characters, so the result followsfrom Corollary 2.29. (cid:3) Proposition 4.8. Let ( M, µ ) , ( N, ν ) ∈ Π( α ) . Then d ( M,µ ) , ( N,ν ) = 0 implies N ≤ M . In particular, reduction modulo p of any cuspidal module is an irre-ducible cuspidal module again: ¯ L ρ ≃ L ρ,F .Proof. By Theorem 4.1(v), which holds over any field, we conclude that anycomposition factor of ¯ L ρ is isomorphic to L ρ,F up to a degree shift. Now useLemma 4.7. (cid:3) ALEXANDER S. KLESHCHEV We complete this section with a version of the James conjecture for anyaffine type. In the case where the Cartan matrix C = A (1) l and Λ = Λ , this isequivalent to a block version of the classical James Conjecture, cf. [ , Section10.4]. The bound on p is inspired by [ , (3.4)] and [ , (3.11)]. Conjecture 4.9. Let α ∈ Q + , and L K be an irreducible R α ( K ) -module whichfactors through R Λ α ( K ) . Then reduction modulo p of L K is irreducible provided p > (Λ , α ) − ( α, α ) / . In view of Lemma 2.26, if α = P i ∈ I m i α i , then every R α -module certainlyfactors through R Λ α for Λ = P i ∈ I m i Λ i , although usually a much smaller Λcould be used.4.7. Cuspidal modules and dual PBW bases. Recall the Q + -graded A -algebras f ∗ A and f A and Q ( q )-algebras f ∗ and f . Suppose that we are givenelements { E ∗ ρ ∈ ( f ∗ A ) ρ | ρ ∈ Φ re+ } ∪ { E ∗ λ ∈ ( f A ) | λ | δ | λ ∈ P } . (4.10)Recalling the notation (1.4), for a root partition ( M, µ ) we then define thecorresponding dual PBW monomial E ∗ M,µ := ( E ∗ ρ ) m ( E ∗ ρ ) m . . . E ∗ µ . . . ( E ∗ ρ − ) m − ( E ∗ ρ − ) m − ∈ f ∗ A . We say that (4.10) is a dual PBW family if the following properties are satisfied:(i) (‘convexity’) if β ≻ γ are positive roots then E ∗ γ E ∗ β − q − ( β,γ ) E ∗ β E ∗ γ isan A -linear combination of elements E ∗ M,µ with ( M, µ ) < ( β, γ ); hereif β = nδ is imaginary, then E ∗ β is interpreted as E ∗ µ and ( β, γ ) isinterpreted as ( µ, γ ) ∈ Π( β + γ ) for an arbitrary µ ∈ P n , and similarlyfor γ (both β and γ cannot be imaginary since then β γ );(ii) (‘basis’) { E ∗ M,µ | ( M, µ ) ∈ Π( α ) } is an A -basis of ( f ∗ A ) α for all α ∈ Q + ;(iii) (‘orthogonality’)( E ∗ M,µ , E ∗ N,ν ) = δ M,N ( E ∗ µ , E ∗ ν ) Y n ∈ Z =0 (( E ∗ ρ n ) m n , ( E ∗ ρ n ) m n );(iv) (‘bar-triangularity’) b ∗ ( E ∗ M,µ ) = E ∗ M,µ + an A -linear combination ofPBW monomials E ∗ N,ν for ( N, ν ) < ( M, µ ).The following result shows in particular that the elements E ∗ ρ of the dualPBW family are determined uniquely up to signs (for a fixed preorder (cid:22) ): Lemma 4.11. Assume that (4.10) is a dual PBW family. (i) The elements of (4.10) are b ∗ -invariant. (ii) Suppose that we are given another family { ′ E ∗ ρ ∈ ( f ∗ A ) ρ | ρ ∈ Φ re+ } ∪{ ′ E ∗ λ ∈ ( f A ) | λ | δ | λ ∈ P } of b ∗ -invariant elements which satisfies thebasis and orthogonality properties. Then E ∗ ρ = ± ′ E ∗ ρ for all ρ ∈ Φ re+ ,and for any µ ∈ P n , we have that E ∗ µ is an A -linear combination ofelements ′ E ∗ ν with ν ∈ P n .Proof. (i) The convexity of (cid:22) implies that for ρ ∈ Φ re+ the trivial root partition( ρ ) ∈ Π( ρ ) is a minimal element of Π( ρ ) and for µ ∈ P n the trivial root USPIDAL SUSTEMS FOR AFFINE KLR ALGEBRAS 23 partition ( µ ) ∈ Π( nδ ) is a minimal element of Π( nδ ). So the bar-triangularityproperty (iv) implies that the elements of a dual PBW family are b ∗ -invariant.Part (ii) has two statements, one for E ∗ ρ with ρ ∈ Φ re+ and another for E ∗ µ with µ ∈ P n . Let α := ρ in the first statement and α := nδ in the second. Weprove (ii) by induction on ht( α ), the induction base being clear. For the firststatement, by the basis property of dual PBW families, we can write ′ E ∗ ρ = cE ∗ ρ + X ( M,µ ) ∈ Π( ρ ) \{ ( ρ ) } c M,µ E ∗ M,µ ( c, c M,µ ∈ A ) . (4.12)Fix for a moment a non-trivial root partition ( M, µ ) ∈ Π( ρ ). By the or-thogonality property of dual PBW families and non-degeneracy of the form( · , · ), there is a Q ( q )-linear combination X M,µ of elements E ∗ M,ν with ν ∈ P | µ | such that ( E ∗ π , X M,µ ) = δ π, ( M,µ ) for all π ∈ Π( ρ ). So pairing the right handside of (4.12) with X M,µ yields c M,µ . On the other hand, by the inductive as-sumption, each E ∗ M,ν is a linear combination of elements of the form ′ E ∗ M,λ . Sousing the orthogonality property for the primed family in (ii), we must have( ′ E ∗ ρ , X M,µ ) = 0 for all non-trivial root partitions ( M, ν ) ∈ Π( ρ ). So c M,µ = 0.Thus ′ E ∗ ρ = cE ∗ ρ . Furthermore, the elements ′ E ∗ ρ and E ∗ ρ belong to the algebra f ∗ A and are parts of its A -bases, whence ′ E ∗ ρ = ± q n E ∗ ρ . Since both ′ E ∗ ρ and E ∗ ρ are b ∗ -invariant, we conclude that n = 0.Now, we prove the second statement in (ii). We can write E ∗ µ as ′ E ∗ µ = X λ ∈ P n c λ E ∗ λ + X ( N,ν ) ∈ Π( nδ ) with | ν | We now show that under the Khovanov-Lauda-Rouquier categorification (seeSection 2.9), cuspidal systems yield dual PBW families. Proposition 4.13. The following set of elements in f ∗ A { E ∗ ρ := γ ∗ ([ L ρ ]) | ρ ∈ Φ re+ } ∪ { E ∗ µ := γ ∗ ([ L ( µ )]) | λ ∈ P } (4.14) is a dual PBW family. Moreover, { E ∗ ρ | ρ ∈ Φ re+ } is a subset of Lusztig’s dualcanonical basis.Proof. (i) Under the categorification map γ ∗ , the graded duality ⊛ correspondsto b ∗ , so γ ∗ ([ L ]) is b ∗ -invariant for any ⊛ -self-dual R α -module L . Moreover,under γ ∗ , the induction product corresponds to the product in f ∗ A , so the con-vexity condition (i) follows from Theorem 4.1(iv) and Lemma 2.21. Now, notethat E ∗ M,µ = γ ∗ ([∆( M, µ )]), so the conditions (ii) and (iv) follow from The-orem 4.1(iv) again. It remains to establish the orthogonality property (iii).Under γ ∗ , the coproduct r corresponds to the map on the Grothendieck groupinduces by Res. So using (2.33), we get( E ∗ M,µ , E ∗ N,ν ) = (cid:0) ( E ∗ ρ ) m ⊗ · · · ⊗ E ∗ µ ⊗ · · · ⊗ ( E ∗ ρ − ) m − , γ ∗ ([Res | M | ∆( N, ν )]) (cid:1) . ALEXANDER S. KLESHCHEV By Proposition 3.12, Res | M | ∆( N, ν ) = 0 unless M = N , and for M = N wehave Res | M | ∆( N, ν ) = L ◦ m ρ ⊠ · · · ⊠ L ( ν ) ⊠ · · · ⊠ L ◦ m − ρ − . Since the form ( · , · ) issymmetric, the orthogonality follows from the preceding remarks.(ii) For symmetric Cartan matrices we can deduce that each E ∗ ρ is a dualcanonical basis element using Proposition 4.8 and the main result of [ ]. Ingeneral, we can argue as follows. It is known that the elements of the dualcanonical basis parametrized by the real roots ρ coincide with the correspond-ing elements of the dual PBW basis, see [ , Proposition 8.2]. By [ , Propo-sition 40.2.4], the dual PBW basis (with an arbitrary choice of a b ∗ -invariant A -basis of the ‘imaginary part’ P , cf. [ , Section 40.2.3] and [ , ]) satisfiesthe properties of Lemma 4.11(ii). So the elements E ∗ ρ of our dual PBW familybelong to the dual canonical basis up to signs. In view of the commutativityof the triangle (2.34), it now suffices to find for an arbitrary element v ∗ of thedual canonical basis just one word i ∈ h I i such that the coefficient of i in ι ( v ∗ )evaluated at q = 1 is positive. But this follows from Lemma 2.35. (cid:3) Remark 4.15. For certain special convex preorders, which we refer to as Beckpreorders , (dual) PBW families have been constructed in [ , ]. Fix a Beckpreorder and denote by { ′ E ∗ ρ ∈ ( f ∗ A ) ρ | ρ ∈ Φ re+ } ∪ { ′ E ∗ λ ∈ ( f A ) | λ | δ | λ ∈ P } the corresponding dual PBW family. By Lemma 4.11(ii), ′ E ∗ ρ = ± E ∗ ρ for all ρ ∈ Φ re+ . In fact, ′ E ∗ ρ = E ∗ ρ for all ρ ∈ Φ re+ by Proposition 4.13 since the realdual root elements of Beck-Chari-Pressley basis are known to belong to dualcanonical basis.5. Minuscule representations and imaginary tensor spaces In this section we study the ‘smallest’ imaginary representations, namelythe imaginary representations of R δ . Then we consider induction powers ofthese minuscule representations, which turn out to play a role of tensor spaces.Denote e := ht( δ ) . Throughout the section we assume that our convex preorder (cid:22) is balanced,see (3.2). In particular, this implies that α i ≻ nδ ≻ α for all n ∈ Z > and i ∈ I ′ = { , . . . , l } . So for any imaginary irreducible representation L of R nδ ,we conclude using (Cus2) that Res α i ,nδ − α i L = 0 for all i ∈ I ′ , i.e. all weights i = ( i , . . . , i d ) of L have the property that i = 0.5.1. Minuscule representations. Note that | P | = l , so there are exactly l imaginary irreducible representations of R δ . We call these representations mi-nuscule . The following lemma shows that a description of minuscule imaginarymodules is equivalent to a description of the irreducible R Λ δ -modules. Lemma 5.1. Let L be an irreducible R δ -module. The following are equivalent: (i) L is minuscule imaginary; (ii) L factors through to the cyclotomic quotient R Λ δ ; (iii) we have i = 0 for any weight i = ( i , . . . , i e ) of L .Proof. By (2.2), there is exactly one 0 among the entries i , . . . , i e of an arbitraryword i ∈ h I i δ . Now (ii) and (iii) are equivalent by Lemma 2.26. The implication(i) = ⇒ (iii) follows from the remarks in the beginning of Section 5. Finally, USPIDAL SUSTEMS FOR AFFINE KLR ALGEBRAS 25 let L ( M, µ ) be an irreducible R δ -module, which is not imaginary, i.e. thereis a = 0 with m a = 0. Then, since P a ∈ Z M a = δ , we conclude that there is a > m a = 0. Let a be the smallest positive integer with m a = 0. Then ρ a ∈ Φ ′ + , in particular, j = 0 for all weights j = ( j , . . . ) of L ( ρ a ). In viewof Theorem 4.1(v), we have L M,µ ⊆ Res | M | L ( M, µ ). In particular, there is aweight i = ( i , . . . ) of L ( M, µ ) with i = 0. (cid:3) We always consider R Λ α -modules as R α -modules via infl Λ . Lemma 5.2. Let β ∈ Φ ′ + . The cuspidal module L δ − β factors through R Λ δ − β and it is the only irreducible R Λ δ − β -module.Proof. Let ( M, µ ) ∈ Π( δ − β ). In view of Lemma 2.26, it suffices to prove thatif ( M, µ ) is non-trivial in the sense of Section 4.4 then i = 0 for some weight i = ( i , . . . ) of L ( M, µ ). But if ( M, µ ) is non-trivial, then there is a > m a = 0. Take the smallest such a . Then ρ a ∈ Φ ′ + , so j = 0 for all weights j = ( j , . . . ) of L ( ρ a ). By Theorem 4.1(v), we have L M,µ ⊆ Res | M | L ( M, µ ). Inparticular, there is a weight i = ( i , . . . ) of L ( M, µ ) with i = 0. (cid:3) Corollary 5.3. The minuscule imaginary modules are exactly { L δ,i := ˜ f i L δ − α i | i ∈ I ′ } . Moreover, e j L δ,i = 0 for all j ∈ I \ { i } . Thus, for each i ∈ I ′ , the minusculeimaginary module L δ,i can be characterized uniquely up to isomorphism as theirreducible R Λ δ -module such that i e = i for all weights i = ( i , . . . , i e ) of L δ,i .Proof. If L and L ′ are two minuscule imaginary modules, with e i L = 0 and e i L ′ = 0, then by Lemmas 5.1 and 5.2, we have that ˜ e i L ∼ = ˜ e i L ′ , whence L ∼ = L ′ by Proposition 2.25(i). It follows by a counting argument that for eachminuscule imaginary module L there exists exactly one i with e i L = 0, andthen, by Lemma 5.2, we must have ˜ e i L ∼ = L δ − α i and L ∼ = ˜ f i L δ − α i . (cid:3) For each i ∈ I ′ , we refer to the minuscule module L δ,i described in Corol-lary 5.3 as the minuscule module of color i . Let µ ( i ) := ( ∅ , . . . , ∅ , (1) , ∅ , . . . , ∅ ) ∈ P ( i ∈ I ′ ) (5.4)be the l -multipartition of 1 with the partition (1) in the i th component. Weassociate to it the minuscule module L δ,i : L ( µ ( i )) := L δ,i ( i ∈ I ′ ) . (5.5) Lemma 5.6. Let i ∈ I ′ . Then ε i ( L δ,i ) = 1 .Proof. Otherwise e i ( L δ,i ) = 0, whence Λ − δ + 2 α i is a weight of V (Λ ), whichis a contradiction. (cid:3) Remark 5.7. The minuscule modules are defined over Z . To be more precise,for each i ∈ I ′ , there exists an R δ ( Z )-module L δ,i, Z which is free finite rankover Z and such that L δ,i, Z ⊗ F is the minuscule imaginary module L δ,i,F over R δ ( F ) for any ground field F . To construct L δ,i, Z , recall that a prime fieldis a splitting field for R α . Now, start with the minuscule module L δ,i, Q over Q , pick any weight vector v and consider the lattice L δ,i, Q := R δ ( Z ) v . Then ALEXANDER S. KLESHCHEV L δ,i, Z ⊗ Z Q ∼ = L δ,i, Q . To see that L δ,i, Z ⊗ Z F is the minuscule module L δ,i,F over any filed F , it suffices to prove that L δ,i, Z ⊗ Z F is irreducible. If L ( M, µ )is a composition factor of L δ,i, Z ⊗ Z F with m a = 0 for some a = 0, then weget a contradiction with the definition of an imaginary module. So, taking intoaccount the character information, all composition factors of L δ,i, Z ⊗ Z F areof the form L δ,i,F . Now, in fact we must have L δ,i, Z ⊗ Z F ≃ L δ,i,F using themultiplicity one result from Lemma 4.7.5.2. Imaginary tensor spaces. The imaginary tensor space of color i is the R nδ -module M n,i := L ◦ nδ,i . In this definition we allow n to be zero, in which case M ,i is interpreted as thetrivial module over the trivial algebra R . Lemma 5.8. M ⊛ n ≃ M n . Proof. This comes from Lemma 2.21 using ( δ, δ ) = 0. (cid:3) A composition factor of M n,i is called an imaginary module of color i . Weremark that by Lemma 4.4 such composition factor is indeed an imaginarymodule in the sense of (Cus2). Another application of Lemma 4.4 now gives: Lemma 5.9. All composition factors of M n , ◦ · · · ◦ M n l ,l are imaginary. We next observe that if an irreducible R nδ -module L (with n > 0) is imagi-nary of color i ∈ I ′ , then L cannot be imaginary of color j ∈ I ′ , i.e. the coloris well defined. Indeed, if L is imaginary of color i , then by (2.19) we have that ε i ( L ) > ε j ( L ) = 0 for any j = i . Lemma 5.10. Let i ∈ I ′ and n , . . . , n a ∈ Z > . Set n := n + · · · + n a . Thenall composition factors of Res n δ,...,n a δ M n,i are of the form L ⊠ · · · ⊠ L a where L , . . . , L a are imaginary of color i .Proof. By the Mackey Theorem, Res n δ,...,n a δ M n,i has filtration with factors ofthe form Ind n δ ; ... ; n a δν ,...,ν n ; ... ; ν a ,...,ν na V, where P nm =1 ν mb = n b δ for all b = 1 , . . . , a , P ab =1 ν mb = δ for all m = 1 , . . . , n ,and V is obtained by an appropriate twisting of the module(Res ν ,...,ν a L δ,i ) ⊠ · · · ⊠ (Res ν n ,...,ν na L δ,i ) . If ν m = 0 and ν m = δ for some m , then by Lemma 4.3, we have that ν m is asum of real roots less than δ , which leads to a contradiction with P nm =1 ν m = n δ . So we deduce that ν m = δ for n different values of m , and ν m = 0 forall other values of m . Then L ⊠ L ⊠ · · · ⊠ L a is a composition factor of M n ,i ⊠ Res n δ,...,n a δ M n − n ,i , and the lemma follows by induction. (cid:3) Corollary 5.11. Let i ∈ I ′ and n , . . . , n a ∈ Z ≥ . Set n := n + · · · + n a . If L is an imaginary irreducible R nδ -module of color i , then all composition factorsof Res n δ,...,n a δ L are of the form L ⊠ · · · ⊠ L a where L , . . . , L a are imaginaryof color i . USPIDAL SUSTEMS FOR AFFINE KLR ALGEBRAS 27 Proof. Follows from Lemma 5.10, since by definition L is a composition factorof M n,i . (cid:3) Reduction to one color. The goal of this section is to prove: Theorem 5.12. Let n ∈ Z ≥ , and suppose that for each i ∈ I ′ , we have anirredundant family { L i ( λ ) | λ ⊢ n } of irreducible imaginary R nδ -modules ofcolor i . For a multipartition λ = ( λ (1) , . . . , λ ( l ) ) ∈ P n , define L ( λ ) := L ( λ (1) ) ◦ · · · ◦ L l ( λ ( l ) ) . Then { L ( λ ) | λ ∈ P n } is a complete and irredundant system of imaginaryirreducible R nδ -modules. We prove the theorem by induction on n . The induction base is clear.Throughout this section we work under the induction hypothesis. Lemma 5.13. Let λ, µ ∈ P n with λ ( i ) ⊢ n i for i = 1 , . . . , l . If the irreducible R n δ,...,n l δ -module L ( λ (1) ) ⊠ · · · ⊠ L l ( λ ( l ) ) appears as a composition factor in Res n δ,...,n l δ L ( µ ) , (5.14) then λ = µ , and the multiplicity of this composition factor is one.Proof. Let µ ( i ) ⊢ m i for i = 1 , . . . , l . By the Mackey Theorem, the module in(5.14) has filtration with factors of the formInd n δ ; ... ; n l δν ,...,ν l ; ... ; ν l ,...,ν ll V, (5.15)where P li =1 ν ij = n j δ for all j ∈ I ′ , P lj =1 ν ij = m i δ for all i ∈ I ′ , and V isobtained by an appropriate twisting of the module(Res ν ,...,ν l L ( µ (1) )) ⊠ · · · ⊠ (Res ν l ,...,ν ll L l ( µ ( l ) )) . (5.16)Assume that the module in (5.15) is non-zero.Since each L i ( µ ( i ) ) is imaginary and Res ν i ,...,ν il L i ( µ ( i ) ) = 0, it follows byLemma 4.3 that either ν i = n i δ for some n i, ∈ Z ≥ , or ν i a sum of real rootsless than m i δ . Since P li =1 ν i = n δ , we conclude that the second option is im-possible. Next, we claim that also each ν i = n i δ for some n i ∈ Z ≥ . Indeed,since Res ν i ,...,ν il L i ( µ ( i ) ) = 0, we have that Res ν i + ν i ,m i δ − ν i − ν i L i ( µ ( i ) ) = 0.By Lemma 4.3, either ν i + ν i is an imaginary root, or it is a sum of real rootsless than m i δ . Since we already know that the ν i, are imaginary roots (orzero), the equality P li =1 ν i = n δ implies that ν i = n i δ for some n i ∈ Z ≥ .Continuing this way, we establish that all ν ij are of the form n ij δ .Now, by Corollary 5.11, all composition factors of Res ν i ,...,ν il L i ( µ ( i ) ) are ofthe form L i ( µ ( i ) ⊠ · · · ⊠ L i ( µ ( il ) ). Then the module in (5.14) has filtrationwith factors of the form (cid:0) L ( µ (11) ) ◦ · · · ◦ L l ( µ ( l ) (cid:1) ⊠ · · · ⊠ (cid:0) L ( µ (1 l ) ) ◦ · · · ◦ L l ( µ ( ll ) ) (cid:1) . By the inductive hypothesis, each L ( µ (1 j ) ) ◦ · · · ◦ L l ( µ ( lj ) ) is irreducible, and L ( µ (1 j ) ) ◦ · · · ◦ L l ( µ ( lj ) ) ∼ = L j ( λ ( j ) )if and only if µ ( jj ) = λ ( j ) and µ ( ij ) = ∅ for all i = j . Thus ν jj = n j δ , ν ij = 0for all i = j . We conclude that m j = n j and µ ( j ) = λ ( j ) for all j . (cid:3) ALEXANDER S. KLESHCHEV Corollary 5.17. The module L ( λ ) has simple head; denote it by L λ . Themultiplicity of L λ in L ( λ ) is one.Proof. If an irreducible module L is in the head of L ( λ ), then by the adjunctionof Ind and Res, we have that L ( λ (1) ) ⊠ · · · ⊠ L l ( λ ( l ) ) ⊆ Res n δ,...,n l δ L . Now theresult follows from Lemma 5.13 with λ = µ . (cid:3) Corollary 5.18. If λ = µ , then L λ = L µ .Proof. Assume that L λ ∼ = L µ . Then L µ is a quotient of L ( λ ). By the adjunctionof Ind and Res, we have that L ( λ (1) ) ⊠ · · · ⊠ L l ( λ ( l ) ) ⊆ Res n δ,...,n l δ L µ . Inparticular, L ( λ (1) ) ⊠ · · · ⊠ L l ( λ ( l ) ) is a composition factor of Res n δ,...,n l δ L ( µ ).Now, by Lemma 5.13, we have λ = µ . (cid:3) Now we can finish the proof of Theorem 5.12. By counting using Theorem 4.1,Lemma 5.9, and Corollary 5.18, we see that { L λ | λ ∈ P n } is a complete andirredundant set of irreducible imaginary R nδ -modules. It remains to prove that L ( µ ) is irreducible, i.e. L ( µ ) = L µ , for each µ . If L ( µ ) is not irreducible, let L λ = L µ be an irreducible submodule in the socle of L ( µ ), see Corollary 5.17.Then there is a nonzero homomorphism L ( λ ) → L ( µ ), whence by the adjunctionof Ind and Res, we have that L ( λ (1) ) ⊠ · · · ⊠ L l ( λ ( l ) ) ⊆ Res n δ,...,n l δ L ( µ ). Now,by Lemma 5.13, we have λ = µ . Theorem 5.12 is proved.5.4. Homogeneous modules. In the remainder of Section 5 we describe theminuscule imaginary modules more explicitly for symmetric (affine) Cartanmatrices. This is done using the theory of homogeneous representations devel-oped in [ ], which we review next. Throughout this subsection we assumethat the Cartan matrix C is symmetric. As usual, we work with an arbitraryfixed α ∈ Q + of height d . A graded R α -module is called homogeneous if it isconcentrated in one degree.Let i ∈ h I i α . We call s r ∈ S d an admissible transposition for i if c i r ,i r +1 =0. The weight graph G α is the graph with the set of vertices h I i α , and with i , j ∈ h I i α connected by an edge if and only if j = s r i for some admissibletransposition s r for i .Recall from Section 2.1 the Weyl group W = h r i | i ∈ I i . Let C be aconnected component of G α , and i = ( i , . . . , i d ) ∈ C . We set w C := r i d . . . r i ∈ W. Clearly the element w C depends only on C and not on i ∈ C . An element w ∈ W is called fully commutative if any reduced expression for w can be obtainedfrom any other by using only the Coxeter relations that involve commutinggenerators, see e.g. [ ]. For an integral weight Λ ∈ P , an element w ∈ W iscalled Λ -minuscule if there is a reduced expression w = r i l . . . r i such that h r i k − . . . r i Λ , α ∨ i k i = 1 (1 ≤ k ≤ l ) , cf. [ , Section 2]. By [ , Proposition 2.1], if w is Λ-minuscule for some Λ ∈ P ,then w is fully commutative.A connected component C of G α is called homogeneous (resp. strongly ho-mogeneous ) if for some (equivalently every) i = ( i , . . . , i d ) ∈ C , we have that USPIDAL SUSTEMS FOR AFFINE KLR ALGEBRAS 29 r i d . . . r i is a reduced expression for a fully commutative (resp. minuscule ) el-ement w C ∈ W , cf. [ , Sections 3.2, Definition 3.5, Proposition 3.7]. In thatcase, there is an obvious one-to-one correspondence between the elements i ∈ C and the reduced expressions of w C . Lemma 5.19. [ , Lemma 3.3] A connected component C of G α is homo-geneous if and only if for some i = ( i , . . . , i d ) ∈ C the following conditionholds: if i r = i s for some r < s then there exist t, u such that r < t < u < s and c i r ,i t = c i r ,i u = − . (5.20)The main theorem on homogeneous representations is: Theorem 5.21. [ , Theorems 3.6, 3.10, (3.3)](i) Let C be a homogeneous connected component of G α . Let L ( C ) be thevector space concentrated in degree with basis { v i | i ∈ C } labeled bythe elements of C . The formulas j v i = δ i , j v i ( j ∈ h I i α , i ∈ C ) ,y r v i = 0 (1 ≤ r ≤ d, i ∈ C ) ,ψ r v i = (cid:26) v s r i if s r i ∈ C , otherwise; (1 ≤ r < d, i ∈ C ) define an action of R α on L ( C ) , under which L ( C ) is a homogeneousirreducible R α -module. (ii) L ( C ) = L ( C ′ ) if C = C ′ , and every homogeneous irreducible R α -module, up to a degree shift, is isomorphic to one of the modules L ( C ) . (iii) If β, γ ∈ Q + with α = β + γ , then Res β,γ L ( C ) is either zero or irre-ducible. Minuscule representations in simply laced types. Throughout thissubsection we assume that the Cartan matrix C is symmetric. Lemma 5.22. Let i ∈ I ′ . Then we can write Λ − δ + α i = w ( i )Λ for a unique Λ -minuscule element w ( i ) ∈ W .Proof. Let θ be the highest root in the finite root system Φ ′ . Pick a (unique)minimal length element u of the finite Weyl group W ′ with uθ = α i . Now, take w ( i ) = ur . Note that w ( i )(Λ ) = ur (Λ ) = u (Λ − α ) = u (Λ − α − θ + θ ) = u (Λ − δ + θ )= Λ − δ + u ( θ ) = Λ − δ + α i . Since the α -string through β has length 0 or 1 for any distinct roots α, β ∈ Φ ′ ,we deduce that u is θ -minuscule, and the lemma follows. (cid:3) By the theory described in Section 5.4, the minuscule element w ( i ) con-structed in Lemma 5.22 is of the form w C ( i ) for some strongly homogeneouscomponent C ( i ) of G δ − α i . Lemma 5.23. Let i ∈ I ′ , d := e − δ − α i ) and j = ( j , . . . , j d ) ∈ C ( i ) .Then: (i) j = 0 ; ALEXANDER S. KLESHCHEV (ii) j d is connected to i in the Dynkin diagram, i.e. c j d ,i < ; (iii) if j b = i for some b , then there are at least three indices b , b , b suchthat b < b < b < b ≤ d such that c i,b = c i,b = c i,b = − .Proof. (i) is clear from the construction of w ( i ) which always has r as the lastsimple reflection in its reduced decomposition.(ii) Let w ( i ) = r j d . . . r j be a reduced decomposition. By definition of aminuscule element, we conclude that h Λ − δ + α i , α ∨ j d i < 0, so h α i , α ∨ j d i < j b = i , then, using the definition of a minuscule element and theequality w ( i )Λ = r j d . . . r j Λ = Λ − δ + α i , we see that h r j b +1 . . . r j d (Λ − δ + α i ) , α ∨ i i = h r j b r j b − . . . r j Λ , α ∨ j b i = − . This implies (iii), since h Λ − δ + α i , α ∨ i i = 2. (cid:3) Corollary 5.24. Let i ∈ I ′ . Then the cuspidal module L δ − α i is the homoge-neous module L ( C ( i )) .Proof. By Lemmas 5.23(i) and 2.26, the module L ( C ( i )) factors through H Λ δ − α i .So L ( C ( i )) ∼ = L δ − α i by Lemma 5.2. (cid:3) Proposition 5.25. Let i ∈ I ′ . The set of concatenations C i := { j i | j ∈ C ( i ) } is a homogeneous component of G δ , and the corresponding homogeneous R δ -module L ( C i ) is isomorphic to the minuscule imaginary module L δ,i .Proof. By Lemmas 5.19 and 5.23(ii),(iii), we have that C i is a homogeneousconnected component of G δ . By Lemmas 5.23(i) and 2.26, the correspondinghomogeneous representation L ( C i ) factors through to R Λ δ , and so it must beone of the minuscule representations L δ, , . . . , L δ,l , see Corollary 5.3. Finally,by the second statement in Corollary 5.3, we must have L ( C i ) ∼ = L δ,i . (cid:3) Example 5.26. Let C = A (1) l and i ∈ I ′ . Then L δ,i is the homogeneous irre-ducible R δ -module with characterch q L δ,i = 0 (cid:0) (12 . . . i − ◦ ( l, l − , . . . , i + 1) (cid:1) i. For example, L δ, and L δ,l are 1-dimensional with charactersch q L δ, = (0 , l, l − , . . . , , ch q L δ,l = (01 . . . l ) , while for l ≥ 3, the module L δ,l − is ( l − q L δ,l − = l − X r =0 (0 , , . . . , r, l, r + 1 , . . . , l − . More on cuspidal modules In this section we first work again with an arbitrary convex preorder (cid:22) , andthen in subsections 6.2 and 6.3 we assume that the preorder is balanced. USPIDAL SUSTEMS FOR AFFINE KLR ALGEBRAS 31 Minimal pairs. Let ρ ∈ Φ re+ . A pair of positive roots ( β, γ ) is called a minimal pair for ρ if(i) β + γ = ρ and β ≻ γ ;(ii) for any other pair ( β ′ , γ ′ ) satisfying (i) we have β ′ ≻ β or γ ′ ≺ γ .In view of convexity, ( β, γ ) is a minimal pair for ρ if and only if ( β, γ ) is aminimal element of Π( ρ ) \ { ( ρ ) } . A minimal pair ( β, γ ) is called real if both β and γ are real roots. Lemma 6.1. Let ρ ∈ Φ re+ and ( β, γ ) be a minimal pair for ρ . If L is a compo-sition factor of the standard module ∆( β, γ ) = L ( β ) ◦ L ( γ ) , then L ∼ = L ( β, γ ) or L ∼ = L ρ .Proof. Use the minimality of ( β, γ ) in Π( ρ ) \ { ( ρ ) } and Theorem 4.1(iv). (cid:3) Remark 6.2. Let ( β, γ ) be a real minimal pair for ρ ∈ Φ re+ . Denote p β,γ := max { n ∈ Z ≥ | β − nγ ∈ Φ + } . The argument as in the proof of [ , Theorem 4.2] shows that in the Grothendieckgroup we have[ L γ ◦ L δ ] − q − ( β,γ ) [ L β ◦ L γ ] = q − p β,γ (1 − q p β,γ − ( β,γ )) )[ L ρ ] . (6.3)So one can compute the character of the cuspidal module L ρ by induction onht( ρ ), provided ρ possesses a real minimal pair, cf. Lemma 6.9 below. Remark 6.4. By Lemma 6.1, we can write in the Grothendieck group[ L β ◦ L γ ] = [ L ( β, γ )] + m ( q )[ L ρ ] . Now, by Lemma 2.21, we also have[ L γ ◦ L β ] = q − ( β,γ ) [ L ( β, γ )] + q − ( β,γ ) m ( q − )[ L ρ ] . So (6.3) implies q − ( β,γ ) ( m ( q − ) − m ( q )) = q − p β,γ (1 − q p β,γ − ( β,γ )) ) , whence m ( q ) − m ( q − ) = q p β,γ − ( β,γ ) − q ( β,γ ) − p β,γ . Now, assume that the Cartan matric C is symmetric. Then by the main resultof [ ], we have that m ( q ) ∈ q Z [ q ], and so the last equality implies m ( q ) = q p β,γ − ( β,γ ) , (6.5)i.e. there is a short exact sequence0 −→ L ρ h p β,γ − ( β, γ ) i −→ L β ◦ L γ −→ L ( β, γ ) −→ . (6.6)Note that for symmetric C we always have p β,γ = 0 and p β,γ − ( β, γ ) = 1.We conjecture that this also holds in non-simply laced affine types (a similarresult for all finite types is established in [ , Theorem 4.7]): Conjecture 6.7. For non-symmetric C , let ρ ∈ Φ re+ , and ( β, γ ) be a real mini-mal pair for ρ . Then there still is a short exact sequence of the form (6.6). Example 6.8. Let n ∈ Z > and i ∈ I ′ . Assume that the preorder is balanced.(i) If ρ = nδ + α i , then ( α i + ( n − δ, δ ) is a minimal pair for ρ .(ii) If n > ρ = nδ − α i , then ( δ, ( n − δ − α i ) is a minimal pair for ρ . ALEXANDER S. KLESHCHEV Lemma 6.9. Assume that the preorder is balanced. Let ρ be a non-simplepositive root. Then there exists a real minimal pair for ρ , unless ρ is of theform nδ ± α i .Proof. If ρ ∈ Φ re ≻ is not of the form nδ + α i , then we can always write ρ as asum of two roots in Φ re ≻ , and so there exists a real minimal pair for ρ .If ρ ∈ Φ re ≺ is not of the form nδ − α i and n ≥ 2, then we can write ρ as a sumof two roots in Φ re ≺ , and so again there exists a real minimal pair for ρ . Finally,in the special case where ρ is a non-simple root of the form δ − α for α ∈ Φ ′ + ,by an argument of [ , Lemma 2.1] we can write ρ as a sum of two real roots,which implies the result. (cid:3) In view of the lemma, the cuspidal modules corresponding to the roots ofthe form nδ ± α i play a special role. In Sections 6.2 and 6.3 we will investigatethem in detail.6.2. Cuspidal modules L nδ + α i . We continue to assume (until the end ofthe paper) that the convex preorder (cid:22) is balanced. We will now use a slightlydifferent notation for the root partitions. For example, if ( M, µ ) is such that m = 2 , m = 1 , m = 1 , m − = 1 , all other m a = 0, and µ = µ ( i ) as in (5.4),then we write ( M, µ ) = ( ρ , ρ , ρ , δ ( i ) , ρ − ).Fix i ∈ I ′ . In this section we consider the cuspidal modules correspondingto the real roots of the form nδ + α i for i ∈ I ′ . Fix also an extremal weight i = i a . . . i a k k (6.10)of the minuscule imaginary module L δ,i , see Section 2.8. Recall from Corol-lary 5.3 and Lemma 5.6 that i k = i and a k = 1. We will use the concatenations i n ∈ h I i nδ , i n i ∈ h I i nδ + α i and also the special weight i { n } := i na . . . i na k − k − i n +1 ∈ h I i nδ + α i . Proposition 6.11. Let i ∈ I ′ , n ∈ Z > , α = nδ + α i , and β = ( n − δ + α i .Then: (i) The standard module ∆( β, δ ( i ) ) = L β ◦ L δ,i has composition series oflength two with head L ( β, δ ( i ) ) and socle L α h ( α i , α i ) / i . (ii) We have ch q L α = 1 q i − q − i (cid:0) (ch q L β ) ◦ (ch q L δ,i ) − (ch q L δ,i ) ◦ (ch q L β ) (cid:1) . (iii) We have ch q L α = 1 q i − q − i n X m =0 ( − m (ch q L δ,i ) ◦ m ◦ i ◦ (ch q L δ,i ) ◦ ( n − m ) . (iv) The weight i { n } is an extremal weight of L α .Proof. We apply induction on n . Consider the induced modules W := L β ◦ L δ,i and W := L δ,i ◦ L β . When evaluated at q = 1, the formal characters of thesetwo modules are the same. It follows from the linear independence of ungradedformal characters of irreducible R α -modules that W and W have the samecomposition factors, but possibly with different degree shifts. We also know that USPIDAL SUSTEMS FOR AFFINE KLR ALGEBRAS 33 the graded multiplicity of L ( β, δ ( i ) ) in W = ∆( β, δ ( i ) ) is 1. By Lemma 2.21,we have that W ⊛ ≃ W , so the graded multiplicity of L ( β, δ ( i ) ) in W is also 1.In view of Lemma 6.1 and Example 6.8(i), in the Grothendieck group [ R α -mod]we now have [ W i ] = [ L ( β, δ ( i ) )] + c i [ L ρ ] ( i = 1 , c i ∈ A such that b c = c .To compute c and c , we look at the multiplicity of the weight i { n } in W .By induction, i { n − } is extremal in L β . Let N be a ⊛ -selfdual irreducible R α -module such that N ∼ = ˜ f n +1 i ˜ f na k − . . . ˜ f na i F . By Proposition 2.31, i { n } is anextremal weight for W . An elementary computation using Proposition 2.31also shows that N appears in W with graded multiplicity q i . So we must have N ≃ L α , and c = q i . We have proved (i) and (iv). Part (ii) easily follows from(i), and (ii) implies (iii) by induction on n . (cid:3) Cuspidal modules L nδ − α i . Fix i ∈ I ′ . In this section we consider thecuspidal modules corresponding to the real roots of the form nδ − α i for i ∈ I ′ .Recall that we have i k = i and a k = 1 for the extremal weight i of L δ,i pickedin (6.10). So in view of Corollary 5.3 and Lemma 5.6, the weight j = i a . . . i a k − k − is an extremal weight of L δ − α i . We will use the notation i [ n ] := i n . . . i ne − i n − e ∈ h I i nδ − α i . Proposition 6.12. Let i ∈ I ′ , n ∈ Z > , and α = nδ − α i , β = ( n − δ − α i .Then: (i) The standard module ∆( δ ( i ) , β ) = L δ,i ◦ L β has composition series oflength two with head L ( δ ( i ) , β ) and socle L α h ( α i , α i ) / i . (ii) We have ch q L α = 1 q i − q − i (cid:0) (ch q L δ,i ) ◦ (ch q L β ) − (ch q L β ) ◦ (ch q L δ,i ) (cid:1) . 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