mirkovic-v´ ilonen polytopes and khovanov-lauda-rouquier ...mathserver.neu.edu/~bwebster/mv.pdf ·...

Mirkovic-Vilonen polytopes andKhovanov-Lauda-Rouquier algebras

Peter Tingley BenWebsterDepartment of Mathematics and Statistics, Department of Mathematics,

Loyola University, Chicago Northeastern Universityptingley @ luc.edu b.webster @ neu.edu

Abstract. We describe how Mirkovic-Vilonen polytopes arise naturally fromthe categorification of Lie algebras using Khovanov-Lauda-Rouquier alge-bras. This gives an explicit description of the unique crystal isomorphismbetween simple representations of the KLR algebra and MV polytopes.

MV polytopes as defined from the geometry of the affine Grassmannianonly make sense for finite dimensional semi-simple Lie algebras, but ourconstruction actually gives a map from the infinity crystal to polytopes in allsymmetrizable Kac-Moody algebras. However, to make the map injectiveand have well-defined crystal operators on the image, we must in generaldecorate the polytopes with some extra information. We suggest that theresulting KLR polytopes are the general-type analogues of MV polytopes.

We give a combinatorial description of the resulting decorated polytopes inall affine cases, and show that this recovers the affine MV polytopes recentlydefined by Kamnitzer and Baumann and the first author in symmetric affinetypes. We also briefly discuss the situation beyond affine type.

Contents

Introduction 2Acknowledgements 41. Background 51.1. Crystals 51.2. Convex orders and charges 61.3. Pseudo-Weyl polytopes 91.4. Finite type MV polytopes 101.5. Rank 2 affine MV polytopes 111.6. Khovanov-Lauda-Rouquier algebras 142. KLR algebras and Lusztig data 172.1. Cuspidal decompositions 182.2. Lusztig data inKLR 222.3. Saito reflections onKLR 233. KLR polytopes and MV polytopes 263.1. KLR polytopes 263.2. The crystal corresponding to a face 293.3. Affine polytopes 35

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Mirkovic-Vilonen polytopes and Khovanov-Lauda-Rouquier algebras

3.4. An example 393.5. Proof of Theorem B and Theorem C 413.6. Beyond affine type 42References 44

Introduction

Let g be a complex semi-simple Lie algebra. In recent years, a number of para-metrizations of the crystal B(−∞) for U+

q (g) have been studied. In this paper, weconsider the relationship between two of these. In the first, the indexing set is theset MV of Mirkovic-Vilonen polytopes, as introduced by Anderson [And03] andstudied by Kamnitzer [Kam10, Kam07], building on Mirkovic and Vilonen’s work[MV07] on the geometry of the affine Grassmannian. In the second, the indexingset is the set KLR of simple gradable modules of Khovanov-Lauda-Rouquier (KLR)algebras, as developed by Lauda-Vazirani [LV11] and Kleshchev-Ram [KR]. Sinceboth of these sets index B(−∞), there is a unique crystal isomorphism between them,but this bijection has not previously been described directly.

Here we give a simple description of this bijection: There is a KLR algebra R(ν)attached to each positive sum ν =

∑aiαi of simple roots. For any two such weights

ν1, ν2, there is a natural inclusion R(ν1) ⊗ R(ν2) ↪→ R(ν1 + ν2). Let PL be the convexhull of the weights ν′ such that Resνν′,ν−ν′ L , 0.

Theorem A The map which sends a simple R(ν)-module L to PL is the unique crystalisomorphism betweenKLR andMV.

We feel Theorem A is interesting in its own right, but perhaps more important isthe fact thatKLR naturally indexes B(−∞) for any symmetrizable Kac-Moody algebra.Thus, one can try to use the map above to define Mirkovic-Vilonen polytopes outsideof finite type. However, one finds pairs of non-isomorphic simples with the samepolytopes (for example, this occurs in sl2 in the weight space 4δ), so the polytopesalone are not enough information to parametrize B(−∞).

As suggested by Dunlap [Dun10] and developed in [BKT], this problem can beovercome by decorating the edges of PL with extra information. In the current setting,the most natural data to associate to an edge is a semi-cuspidal representation ofa smaller KLR algebra (see Definition 2.2). In complete generality, there are manydifferent semi-cuspidal representations that can decorate a given edge, and we donot know a fully combinatorial description of the resulting object.

For edges parallel to real roots there is only one possible semi-cuspidal representa-tion, so it is safe to leave off the decoration. In particular, in finite type one can ignorethe decoration altogether, and the polytopes are described by Theorem A. The next

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Peter Tingley and Ben Webster

simplest types are the affine algebras. There the decoration does play an importantrole, but we nonetheless obtain a combinatorial object.

For now restrict to the case when g is affine of rank r + 1. Then g has only oneminimal imaginary root δ, and this has multiplicity r. It turns out that the semi-cuspidal representations that can be associated to a given edge of PL parallel to δ arenaturally indexed by an r-tuple of partitions (see Lemma 3.31). In fact, we can reducethe amount of information even further: as in [BKT], the (possibly degenerate) r-facesof PL parallel to δ are naturally indexed by the chamber coweights γ of an underlyingfinite type root system. Denote the face of PL corresponding to γ by PγL. We in factdecorate PL with just the data of a partition πγ for each chamber coweight γ (seeDefinition 3.32) in such a way that, for any edge E parallel to δ,

(1) E is a translate of

∑E⊂PγL

|πγ|

δ.The representation attached to such an edge E is determined in a natural way by{πγ : E ⊂ PγL}.

Define a decorated affine pseudo-Weyl polytope to be a pair consisting of• a polytope P in the root lattice of gwith all edges parallel to roots, and• a choice of partition πγ for each chamber coweight γ of the underlying finite

type root system which satisfies condition (1) for each edge parallel to δ.Let PKLR be the set of such polytopes which arrive as PL for some L (which we callKLR-polytopes). As in finite type, we seek a combinatorial characterization of PKLR.

Notice that for every 2-face F of a decorated pseudo-Weyl polytope, the rootsparallel to F form a rank 2 sub-root system ∆F of either finite or affine type. If ∆F isof affine type, then F has two edges parallel to δ, which are of the form Eγ = F ∩ Pγand Eγ′ = F ∩ Pγ′ for unique chamber coweights γ, γ′. We would like to decoratethese imaginary edges with πγ and πγ′ , but this fails to satisfy (1) since Eγ and Eγ′

are too long. Instead, F is the Minkowski sum of the line segment(∑

γ:F⊂Pξ |πξ|

)δ

with a decorated pseudo-Weyl polytope F, obtained by shortening Eγ and Eγ′ anddecorating them with πγ and πγ′ . We will show that:

Theorem B For g an affine Lie algebra, the polytopes PL are precisely the decoratedpseudo-Weyl polytopes where every 2-dimensional face F satisfies

• If ∆F is a finite type root system, then F is an MV polytope for that root system(i.e. it satisfies the tropical Plucker relations from [Kam10]).• If ∆F is of affine type, then F is an MV polytope for that rank 2 affine algebra

(either sl2 or A(2)2 ) as defined in [BDKT].

The description of rank 2 affine MV polytopes in [BDKT] is combinatorial, so TheoremB gives a combinatorial characterization of KLR polytopes in all affine cases.

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In [BKT] analogues of MV polytopes were constructed in all symmetric affinetypes as decorated Harder-Narasimhan polytopes, and it was shown that these arecharacterized by their 2-faces. Thus Theorem B also allows us to understand therelationship between our decorated polytopes and those defined in [BKT]:

Theorem C Assume g is of affine type with symmetric Cartan matrix. Fix b ∈ B(−∞)and let L be the corresponding element of KLR. Our polytope PL and the decoratedHarder-Narasimhan polytope HNb from [BKT] have identical underlying polytopes.Furthermore, for each chamber coweight γ in the underlying finite type root system,the partition λγ decorating HNb as defined in [BKT, Sections 1.5 and 7.6] is thetranspose of our πγ.

It is natural to ask for an intrinsic characterization of the polytopes PL in the gen-eral Kac-Moody case. We do not even have a conjecture for a true combinatorialcharacterization, since the polytopes are decorated with various semi-cuspidal rep-resentations, which at the moment are not well-understood. Some difficulties thatcome up outside of affine type are discussed in Section 3.6. However, our con-struction does still satisfy the most basic properties one would expect, as we nowsummarize (see Corollaries 3.9 and 3.10 for precise statements).

Theorem D For g an arbitrary symmetrizable Kac-Moody algebra, the map fromKLR to polytopes with edges labeled by semi-cuspidal representations is injective.Furthermore, for each convex order on roots, the elements ofKLR are parameterizedby the tuples of representations of smaller KLR algebras decorating the edges alonga corresponding path through the polytope, generalizing the parameterization ofcrystals in finite type by Lusztig data.

As we were completing this paper, some independent work on similar problemsappeared: McNamara [McN] proved a version of Theorem D in finite type (amongstother theorems on the structure of these representations) and Kleshchev [Kle] gave ageneralization of this to affine type. While there was some overlap with the presentpaper, these other works were focused on a single convex order, rather than giving adescription of how different orders interact as we do in Theorems A, B and C.

Acknowledgements

We thank Arun Ram for first suggesting this connection to us, Dinakar Muthiahfor many interesting discussions, Monica Vazirani for pointing out the example ofSection 3.4, and Scott Carnahan for directing us to a useful reference on Borcherdsalgebras [Car]. P. T. was supported by NSF grants DMS-0902649, DMS-1162385 andDMS-1265555; B. W. was supported by the NSF under Grant DMS-1151473 and theNSA under Grant H98230-10-1-0199.

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1. Background

1.1. Crystals. Fix a symmetrizable Kac-Moody algebra and let Γ = (I,E) be its Dynkindiagram. We are interested in the crystal B(−∞) associated with U+(g). This is acombinatorial object arising from the theory of crystal bases for the correspondingquantum group (see e.g. [Kas95]). This section contains a brief explanation ofthe results we need, roughly following [Kas95] and [HK02], to which we refer thereader for details. We start with a combinatorial notion of crystal that includes manyexamples which do not arise from representations, but which is easy to characterizecombinatorially.

Definition 1.1 (see [Kas95, Section 7.2]) A combinatorial crystal is a set B along withfunctions wt: B → P (where P is the weight lattice), and, for each i ∈ I, εi, ϕi : B →Z ∪ {−∞} and ei, fi : B→ B t {∅}, such that

(i) ϕi(b) = εi(b) + 〈wt(b), α∨i 〉.(ii) ei increases ϕi by 1, decreases εi by 1 and increases wt by αi.

(iii) fib = b′ if and only if eib′ = b.(iv) If ϕi(b) = −∞, then eib = fib = ∅.

We often denote a combinatorial crystal simply by B, suppressing the other data.

Definition 1.2 A lowest weight combinatorial crystal is a combinatorial crystalwhich has a distinguished element b− (the lowest weight element) such that

(i) The lowest weight element b− can be reached from any b ∈ B by applying asequence of fi for various i ∈ I.

(ii) For all b ∈ B and all i ∈ I, ϕi(b) = max{n : f ni (b) , ∅}.

Notice that, for a lowest weight combinatorial crystal, the functions ϕi, εi and wtare determined by the fi and the weight wt(b−) of just the lowest weight element.

It will be convenient for us to consider a slightly stronger notion, which is lesscommon in the literature:

Definition 1.3 A bicrystal is a set B with 2 different crystal structures whose weightfunctions agree. We will always use the convention of placing a star superscript onall data for the second crystal structure, so e∗i , f ∗i , ϕ

∗

i , etc.

We will consider one very important example of a bicrystal: the universal lowestweight crystal B(−∞) along with the usual crystal operators and Kashiwara’s ∗-crystal operators, which are the conjugates e∗i = ∗ei∗, f ∗i = ∗ fi∗ of the usual operatorsby Kashiwara’s involution ∗ : B(−∞) → B(−∞) (see [Kas93, 2.1.1]). The involution ∗is a crystal limit of the bar involution of U+

q (g), but it also has a simple combinatorialdefinition in each of the models of interest to us.

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The following is a rewording of [KS97, Proposition 3.2.3] designed to make theroles of the usual crystal operators and the ∗-crystal operators more symmetric:

Proposition 1.4 Fix a bicrystal B. Assume (B, ei, fi) and (B, e∗i , f ∗i ) are both lowestweight combinatorial crystals with same the same lowest weight element b−, wherethe other data is determined by setting wt(b−) = 0. Assume further that, for alli , j ∈ I and all b ∈ B,

(i) ei(b), e∗i (b) , 0.(ii) e∗i e j(b) = e je∗i (b),

(iii) For all b ∈ B, ϕi(b) + ϕ∗i (b) − 〈wt(b), α∨i 〉 ≥ 0(iv) If ϕi(b) + ϕ∗i (b) − 〈wt(b), α∨i 〉 = 0 then ei(b) = e∗i (b),(v) If ϕi(b) + ϕ∗i (b) − 〈wt(b), α∨i 〉 ≥ 1 then ϕ∗i (ei(b)) = ϕ∗i (b) and ϕi(e∗i (b)) = ϕi(b).

(vi) If ϕi(b) + ϕ∗i (b) − 〈wt(b), α∨i 〉 ≥ 2 then eie∗i (b) = e∗i ei(b).

then (B, ei, fi) ' (B, e∗i , f ∗i ) ' B(−∞), and e∗i = ∗ei∗, f ∗i = ∗ fi∗, where ∗ is Kashiwara’sinvolution. Furthermore, these conditions are always satisfied by B(−∞) along withits operators ei, fi, e∗i , f ∗i .

Proof. We simply explain how [KS97, Proposition 3.2.3] implies our statement, refer-ring the reader there for specialized notation. Define the map

B→ B ⊗ Bi

b 7→ ( f ∗i )ϕ∗

i (b)(b) ⊗ eϕ∗i (b)i bi.

Then one can check that our conditions imply all the conditions from [KS97, Propo-sition 3.2.3], so that result implies the crystal structure on B defined by ei, fi is iso-morphic to B(−∞). The remaining statements then follow from [KS97, Theorem3.2.2]. �

We will also make use of Saito’s crystal reflections from [Sai94].

Definition 1.5 Fix b ∈ B(−∞) with ϕ∗i (b) = 0. The Saito reflection of b is σib =

(e∗i )εi(b) fϕi(b)

i b. There is also a dual notion of Saito reflection defined by σ∗i (b) := ∗(σi(∗b))which is defined for those b such that ϕi(b) = 0

The operation σi does in fact reflect the weight of b by si, as the name suggests.

1.2. Convex orders and charges. Fix a symmetrizable Kac-Moody algebra g withroot system ∆ and Cartan subalgebra h. Let ∆min

+ be the set of positive roots α suchthat xα is not a root for any 0 < x < 1 (this is all positive roots in finite type).

Definition 1.6 A convex (pre)order is a (pre)order � on ∆min+ such that, given S,S′ ⊂

∆min+ with S ∪ S′ = ∆irr

+ and α � α′ for all α ∈ S, α′ ∈ S′, the convex cones spanR≥0S and

spanR≥0S′ intersect only at the origin.

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Notice that any preorder on ∆min+ extends to a preorder on all positive roots, where

proportional roots are equivalent.

Definition 1.7 A charge is a linear function c : h∗ → C such that c(αi) , 0 for eachsimple root αi and such that all c(αi)’s (and thus the images of all positive roots) liein the upper half-plane.

Every charge defines a preorder>c on ∆min+ be settingα ≥c β if and only if arg(c(α)) ≥

arg(c(β)), where arg is the argument of the complex number (taking a branch cut oflog which does not intersect the upper half plane), and this is clearly convex. If cis generic, >c is a total order. We will need the following notion of “reflection” forconvex orders and charges.

Definition 1.8 Fix a convex order � such that αi is lowest (reps. greatest). Define anew convex order �si by

β � γ⇔ siβ �si siγ β, γ , αi

and αi greatest (resp. lowest) for �si .Similarly, for a charge c such that argαi is lowest (resp. greatest) amongst positive

roots, define a new charge csi by csi(ν) = c(si(ν)). This will not always send ∆+ to theupper half plane, but it will send it to some half plane, and we can then rotate tomake that the upper half plane (in the end we only care about the order on roots, sothe precise rotation does not matter).

It should be clear from the definitions that the reflections for charges and convexorders are compatible in the sense that, for all charges c such that αi is greatest orlowest, (>c)si and >csi coincide.

Definition 1.9 Fix a pair (α,�), where α is a positive root and � is a convex orderon ∆min

+ . A charge c is said to be (α,�) compatible if, for all β ∈ ∆+ such thatα− β ∈ spanZ≥0

{αi}, we have α ≺ β if and only if α <c β and α � β if and only if α >c β.

For any fixed α there are only finitely many positive roots βwith α−β ∈ spanZ≥0{αi},

so it follows easily from the definition of convex order that, for any pair (α,�), thereexists a (α,�)-compatible charge. Not all convex orders arise from charges, but inmany instances the existence of (α,�)-compatible charges will allow us to restrict toconvex orders that do.

The following is well known; however, since we have used a slightly unusualdefinition of convex order, we include a proof for completeness.

Proposition 1.10 Assume g is of finite type. There is a bijection between convexorders on ∆+ and expressions i = i1 · · · iN for the longest word w0, which is given bysending i to the order αi1 � si1αi2 � si1si2αi3 � · · · � si1 · · · siN−1αiN .

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Proof. First, fix a reduced expression. It is well known that

{αi1 , si1αi2 , si1si2αi3 , · · · , si1 · · · siN−1αiN}

is an enumerate of the positive roots, so we have defined a total ordering on positiveroots. Any pair S,S′ as in Definition 1.6 for this order is of the form

S = {αi1 , si1αi2 , . . . , si1 · · · sir−1αir}, and S′ = {si1 · · · sirαir+1 , . . . , si1 · · · siN−1αiN}.

The convex cones for these sets are separated by the hyperplane defined by sir · · · si1ρ∨,

so are clearly disjoint. This proves that all the orders coming from reduced expres-sions in this way are convex.

Now fix a convex order �. There is a unique greatest root, which must be asimple root αi1 , since otherwise it would be in the span of the other positive roots,contradicting convexity. The convex order �si1 as defined above also has a greatestroot αi2 . Define i3 in the same way using �s1s2 and continue as many times as there arepositive roots. The list αi1 , si1αi2 , si1si2αi3 , . . . , si1 · · · siN−1αiN is a complete, irredundantlist of positive roots. This implies that i is a reduced expression for w0. Furthermore,if we apply the procedure in the statement to create an order on positive roots fromthis expression, we clearly end up with our original convex order. �

Of course, if g is of infinite type, applying the technique of this proof will resultnot in a reduced word for the longest element (which does not exist), but an infinitereduced word i1, i2, i3, . . . in I as well as a dual sequence . . . , i−3, i−2, i−1 constructedfrom looking at lowest elements. The corresponding lists of roots

αi1 � si1αi2 � si1si2αi3 � · · · and · · · � si−1si−2αi−3 � si−1αi−2 � αi−1

are totally ordered, but don’t contain every root. In the affine case, only δ will belacking, but for hyperbolic algebras, we can miss many real roots, or even simpleroots. In general, the first list will contain all real roots larger than any imaginaryroot, and the second all such roots smaller than any imaginary root.

Definition 1.11 Fix a convex order �. For each b ∈ B(−∞) and each real root αsuch that α � β for all imaginary roots β, we define an integer a�α(b). This is doneinductively by setting aαi(b) = ϕi(b) if αi lowest for �, and

aα(b) = a�

sisiα

(σ∗i ( fϕi(b)i b))

for all other roots. The collection of such aα(b) is called the crystal-theoretic realLusztig data for b with respect to �.

Similarly, define the dual crystal theoretic Lusztig data a∗α for all real roots smallerthan all imaginaries by switching starred and unstarred operators. For finite typegroups, this is just the Lusztig data for the opposite order, but in infinite types it isnew information.

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Remark 1.12 In finite type, Theorem A, along with results in [Kam10, Kam07], showsthat the aαi(b) are the exponents in Lusztig’s PBW basis element corresponding to bfor an appropriate reduced expression of w0. This explains the terminology.

1.3. Pseudo-Weyl polytopes. We use notation as in the previous subsection.

Definition 1.13 A pseudo-Weyl polytope is a convex polytope P in h∗ with all edgesparallel to roots.

Definition 1.14 For a pseudo-Weyl polytope P, let µ0(P) be the vertex of P such that〈µ0(P), ρ∨〉 is lowest, and µ0(P) the vertex where this is highest (these are vertices asfor all roots 〈α, ρ∨〉 , 0).

Lemma 1.15 Fix a pseudo-Weyl polytope P and a convex total order � on ∆min+ . There

is a unique path P� through the 1-skeleton of P from µ0(P) to µ0(P) which passesthrough at most one edge parallel to each root, and these appear in decreasing orderaccording to �.

Proof. Let {β1, β2, . . . , βr} ∈ ∆min+ be the minimal roots that are parallel to edges in P,

ordered by β1 � β2 � · · · � βr. Since � is convex, for each 1 ≤ k ≤ r − 1, one canfind φk ∈ h such that 〈βr, φk〉 > 0 for r ≤ k, and 〈βr, φk〉 < 0 for r > k. Let φ0 = ρ∨

and φr = −ρ∨. Construct a path φt in coweight space for t ranging from 0 to r by, fort = k + q for 0 ≤ q < 1 letting φt = (1 − q)φk + qφk+1. As t varies from 0 to r, the locusin the polytope where φt takes on its highest value is generically a vertex of P, butoccasionally defines an edge. The set of edges that come up is the required path. �

Definition 1.16 Fix a pseudo-Weyl polytope P and a convex order �. For eachα ∈ ∆min

+ , define a�α(P) to be the unique non-negative number such that the edge inP� parallel to α is a translate of a�α(P)α. We call the collection {a�α(P)} the geometricLusztig data of P with respect to �.

Lemma 1.17 Let P be a pseudo-Weyl polytope and e an edge of P. Then there existsa charge c such that >c is a total order and e ⊂ P>c . In particular, a pseudo-Weylpolytope P is uniquely determined by its Lusztig data with respect to all convexorders >c coming from charges.

Proof. Since e is an edge of P, there is a functional φ ∈ h such that

e = {p ∈ P : 〈p, φ〉 is greatest}.

If e is parallel to the root β, this means 〈β, φ〉 = 0, and φ may be chosen so that〈β′, φ〉 , 0 for all other β′ which are parallel to edges of P. For any linear functionf : h→ R such that f (∆+) ⊂ R+, define a charge c f by

c f (p) = φ(p) + f (p)i.

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For generic f , the charge c f satisfies the required conditions. �

The following should be thought of as a general-type analogue of the fact that, infinite type, each reduced expression for w0 can be obtained from any other reducedexpression by a finite number of braid moves. In fact, this statement can be gener-alized to include all convex orders, not just those coming from charges, but we onlyneed the simpler version.

Lemma 1.18 Let P be a Pseudo-Weyl polytope and c, c′ two generic charges. Thenthere is a sequence of generic charges c0, c1, . . . ck such that P>c0 = P>c , P>ck = P>c′ , and,for all k ≤ j < k, P>cj and P>cj+1 differ by moving around a single 2-face of P in the twopossible directions.

Proof. Let ∆res be the set of root directions that appear as edges in P. For 0 ≤ t ≤ 1,let ct = (1 − t)c + tc′. Clearly this is a charge. We can deform c, c′ slightly, withoutchanging the order of any of the roots in ∆res, such that

• For all but finitely many t, ct induces a total order on ∆res.• For those t where ct does not induce a total order, there is exactly one argument

0 < at < π such that more then one root in ∆res has argument at. Furthermore,the span of the roots with argument at is 2 dimensional.

Denote the values of t where ct does not induce a total order by s1, . . . sk−1. Fix t1, . . . , tk

with0 = t0 < s1 < t1 < s2 . . . < tk−1 < sk−1 < tk = 1.

Then c j = ct j is the required sequence. �

1.4. Finite type MV polytopes. In finite type, Anderson [And03] and Kamnitzer[Kam10, Kam07] developed a realization of B(−∞) where the underlying set consistsof Mirkovic-Vilonen (MV) polytopes. These are certain polytopes in weight space.Here we will need are the certain characterization theorems, which we now discuss.

Proposition 1.19 Assume g finite type. There is a unique map b→ Pb from B(−∞) topseudo-Weyl polytopes such that

(i) wt(b) = µ0(Pb) − µ0(Pb).(ii) If� is a convex order with minimal root αi, then for all β , αi, a�β (Pei(b)) = a�β (Pb),

and a�αi(Pei(b)) = a�αi

(Pb) + 1.(iii) If � is a convex order with minimal root αi and ϕi(Pb) = 0, then for all β , αi,

a�β (Pb) = a�si

si(β)(Pσib) and aαi(Pσib) = 0.

This map is the unique bicrystal isomorphism between B(−∞) and the set of MVpolytopes. �

Proof. The first step is to show that there is at most one map b→ Pb satisfying the con-ditions. To see this we proceed by induction. Consider the reverse-lexicographical

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order on collections of integers a = (aβk)1≤k≤N. Assume a is minimal such that, forsome convex order

β1 � β2 � · · · · · · � βN

and for two maps b → Pb and b → P′b satisfying the conditions, a≺βk(Pb) = ak for all k,

but a≺βk(P′b) , ak for some k. If aN , 0 we can reduce to a smaller such example using

condition (ii). Otherwise, as long as some ak , 0, we can reduce to a smaller suchexample using (iii). Clearly the map is unique if all ak = 0, so this proves uniqueness.

So, it remains to show that b → MVb does satisfy both conditions. But this isimmediate from [Sai94, Proposition 3.4.7] and the fact that the integers a�βk

(MVb)agree with the exposnents in the Lusztig’s PBW monomial corresponding to b, whichis shown in [Kam10, Theorem 7.2]. �

The following is immediate from 1.19

Corollary 1.20 In finite type, the geometric Lusztig data for an MV polytope willalways agree with the crystal theoretic Lusztig data for the corresponding elementof B(−∞) (see Definition 1.11).

We also need the following standard facts about MV polytopes:

Theorem 1.21 ([Kam10, Theorem D]) The MV polytopes are exactly those pseudo-Weyl polytopes such that all 2-faces are MV polytopes for the corresponding rank 2root system. �

Theorem 1.22 ([Kam10, 4.2]) An MV polytope is uniquely determined by its Lusztigdata with respect to any one convex order on positive roots. �

1.5. Rank 2 affine MV polytopes. We briefly review the MV polytopes associatedto the affine root systems sl2 and A(2)

2 in [BDKT], and recall a characterization of theresulting polytopes developed in [MT].

The sl2 and A(2)2 root systems correspond to the affine Dynkin diagrams

• •

0 1 ,sl2 : • •

0 1 .A(2)

2 :

The corresponding symmetrized Cartan matrices are

sl2 : N =

(2 −2−2 2

), A(2)

2 : N =

(2 −4−4 8

).

Denote the simple roots by α0, α1, where in the case of A(2)2 the short root is α0. Define

δ = α0 + α1 for sl2 and δ = 2α0 + α1 for A(2)2 .

The dual Cartan subalgebra h∗ of g is a three dimensional vector space containingα0, α1. This has a standard non-degenerate bilinear form (·, ·) such that (αi, α j) = Ni, j.

11


Notice that (α0, δ) = (α1, δ) = 0. Fix fundamental coweights ω0, ω1 which satisfy(αi, ω j) = δi, j, where we are identifying coweight space with weight space using (·, ·).

The set of positive roots for sl2 is

(2) {α0, α0 + δ, α0 + 2δ, . . .} t {α1, α1 + δ, α1 + 2δ, . . .} t {δ, 2δ, 3δ . . .},

where the first two families consist of real roots and the third family of imaginaryroots. The set of positive roots for A(2)

2 is

(3) ∆+re = {α0 +kδ, α1 +2kδ, α0 +α1 +kδ, 2α0 +(2k+1)δ | k ≥ 0} and ∆+

im = {kδ | k ≥ 1},

where ∆+re consists of real roots and ∆+

im of imaginary roots. We draw these as

...... ...... ...

α0

α0 + δ

α0 + 2δ

α0 + 3δ

α1

α1 + δ

α1 + 2δ

α1 + 3δkδ

sl2

...

α0

2α0 + δ

2α0 + 3δ

α1

α1 + 2δ

kδ

A(2)2

Definition 1.23 Label the positive real roots by rk, rk for k ∈ Z>0 by:

• For sl2: rk = α1 + (k − 1)δ and rk = α0 + (k − 1)δ.• For A(2)

2 :

rk =

{α1 + (k − 1)δ if k is odd,α0 + α1 + k−2

2 δ if k is even,rk =

{α0 + k−1

2 δ if k is odd,2α0 + (k − 1)δ if k is even.

There are exactly two convex orders on ∆min+ : the order �+

r1 �+ r2 �+ · · · �+ δ �+ · · · �+ r2�+ r1,

and the reverse of this order, which we denote by �−.

Definition 1.24 A rank 2 affine decorated pseudo-Weyl polytope is a pseudo-Weylpolytope along with a choice of two partitions aδ = (λ1 ≥ λ2 ≥ · · · ) and aδ = (λ1 ≥

λ2 ≥ · · · ) such that µ∞ − µ∞ = |aδ|δ and µ∞ − µ∞

= |aδ|δ. Here |aδ| = λ1 + λ2 + · · · and|aδ| = λ1 + λ2 + · · · .

Definition 1.25 The right Lusztig data of a decorated pseudo-Weyl polytope P is therefinement a = (aα)α∈∆+

of the Lusztig data from Section 1.3 with respect to �+ (whichrecords the lengths of the edges parallel to each root up one side of P), where, for

12


α , δ, aα = a�+α (P), and aδ is as in Definition 1.24. Similarly the left Lusztig data is

a = (aα)α∈∆+where, for α , δ, aα = a�−α (P), and aδ is as in Definition 1.24.

••

•

•

•

•

•

•

•

•

•

•

•

•

••

•

•

•

•

α1

α1 + δ

α1 + 2δ

δ

α0 + 2δ

α0α1

α1 + δ

α1 + 3δ

δ

α0 + 3δ

α0 + 2δ

α0 + δ

α0 µ0µ1

µ2

µ3 = µ4 = · · · = µ∞

µ3 = µ4 = · · · = µ∞

µ1 = µ2

µ1

µ2

µ3

µ∞

= · · · = µ5 = µ4

µ∞ = · · · = µ5= µ4

µ3= µ2

µ1

µ3= µ2

µ0

α1α0

Figure 1. An sl2 MV polytope. The partitions labeling the verticaledges are indicated by including extra vertices on the vertical edges,such that the edge is cut into the pieces indicated by the partition. Herethe Lusztig data is

a1 = 2, a2 = 1, a3 = 1, λ = (9, 2, 1, 1), a3 = 1, a1 = 1,a1 = 1, a2 = 2, a3 = 1, a4 = 1, λ = (2, 1, 1), a4

= 1, a2= 1, a1

= 5,and all other ak, ak, ak, a

k are 0.

In [BDKT], the first author and collaborators combinatorially define a setMV ofdecorated pseudo-Weyl polytopes, which they call rank 2 affine MV polytopes. Wewill not need the details of this construction, but will instead use the following resultfrom [MT]. Assume g is of type. Define `0 and `1 by δ = `0α0 + `1α1 (so `0 = `1 = 1 forsl2, and `0 = 2, `1 = 1 for A(2)

2 .

Theorem 1.26 [MT, Theorem 3.10] There is a unique map b→ Pb from B(−∞) to typeg decorated pseudo-Weyl polytopes (considered up to translation) such that, for allb ∈ B(−∞), the following hold.

13


(i) wt(b) = µ0(Pb) − µ0(Pb).(ii.1) aα0(Pe0b) = aα0(Pb) + 1, and for all other root directions aα(Pe0b) = aα(Pb);(ii.2) aα1(Pe1b) = aα1(Pb) + 1, and for all other root directions aα(Pe1b) = aα(Pb);(ii.3) aα1(Pe∗1b) = aα1(Pb) + 1, and for all other root directions aα(Pe∗1b) = aα(Pb);(ii.4) aα0(Pe∗0b) = aα0(Pb) + 1, and for all other root directions aα(Pe∗0b) = aα(Pb).Let σ0, σ1 denote Saito’s reflections.(iii.1) If aα0(Pb) = 0, then for all α , α0, aα(Pb) = as0(α)(Pσ0(b)) and aα0(Pσ0(b)) = 0;(iii.2) if aα1(Pb) = 0, then for all α , α1, aα(Pb) = as1(α)(Pσ1(b)) and aα1(Pσ1(b)) = 0;(iii.3) if aα0(Pb) = 0, then for all α , α0, aα(Pb) = as0(α)(Pσ∗0(b)) and aα0(Pσ∗0(b)) = 0;(iii.4) if aα1(Pb) = 0, then for all α , α1, aα(Pb) = as1(α)(Pσ∗1(b)) and aα1(Pσ∗1(b)) = 0.

(iv) If aβ(Pb) = 0 for all real roots β and aδ(Pb) = λ , 0 then:

aα1(Pb) = `1λ1; aδ(Pb) = λ\λ1; aα0(Pb) = `0λ1;

aβ(Pb) = 0 for all other β ∈ ∆+.

The image of this map is exactlyMV as defined in [BDKT]. �

Remark 1.27 This theorem implies that, for any rank-2 affine MV polytope, any bi-convex order �, and any real root α, the crystal theoretic Lusztig data aα agrees withthe Lusztig data aα for the corresponding MV polytope for all real roots α. In fact, itfollows from Corollary 3.13 below that this holds for all affine algebras, regardless ofthe rank.

1.6. Khovanov-Lauda-Rouquier algebras. In this section, we recall the basic factsabout the Khovanov-Lauda-Rouquier algebras (sometimes called quiver Hecke al-gebras) attached to the Lie algebra g as defined for Kac-Moody algebras in [KL09,Rou], and extended to the case of Borcherds algebras in [KOP].

This is an algebra R built out of generic string diagrams, i.e. immersed 1-dimensional submanifolds of R2 whose boundary lies on the lines y = 0 and y = 1,where each string (i.e each immersed copy of the interval) projects homeomorphi-cally to [0, 1] under the projection to the y-axis (so in particular there are no closedloops). These are assumed to be generic in the sense that

• no points lie on 3 or more components• no components intersect non-transversely.

Each string is labeled with a simple root of the corresponding Kac-Moody algebra,and each string is allowed to carry dots at any point where it does not intersect another(but with only finitely many dots in each diagram). All diagrams are considered upto isotopy preserving all these conditions.

Define a product on the space of k-linear combinations of these diagrams, wherethe product ab of two diagrams is formed by stacking a on top of b, shrinking verticallyby a factor of 2, and smoothing kinks; if the labels of the line y = 0 for a and y = 1 forb cannot be isotoped to match, the product is 0.

14


This product gives the space of k-linear combinations of these diagrams the struc-ture of an algebra, which has the following generators, which depend on a sequencei = (i1, . . . , in) of nodes of the Dynkin diagram:

• The idempotent ei which is straight lines labeled with (i1, . . . , in).• The element yi

k which is just straight lines with a dot on the kth strand.• The element ψi

k which is a crossing of the i and i + 1st strand.

i1 i2 in

· · ·

ei

i1 i j in

· · ·· · ·

yik

i1 i j i j+1 in

· · ·· · ·

ψik

In order to arrive at the KLR algebra R, we must impose the relations shown inFigure 2. All of these relations are local in nature, that is, if we recognize a smallpiece of a diagram which looks like the LHS of a relation, we can replace it with theRHS, leaving the rest unchanged. The relations depend on a choice of a polynomialQi j(u, v) ∈ k[u, v] for each pair i , j. Let C = (ci j) be the Cartan matrix of g and di becoprime integers so that d jci j = dic ji. We assume each polynomial is homogeneous ofdegree 〈αi, α j〉 = −2d jci j = −2dic ji when u is given degree 2di and v degree 2d j. We willalways assume that the leading order of Qi j in u is −c ji, and that Qi j(u, v) = Q ji(v,u).

In [LV11, 1.1.4], Lauda and Vazirani define an automorphism σ : R→ R which upto sign reflects the diagrams through the vertical axis. We let Mσ denote the twist ofan R-module by this automorphism.

While some other aspects of the representation theory of R are quite sensitive tothe choice of k and Qi j (for example, the dimensions of simple modules), none ofthe theorems we prove will depend on it; the reader is free to imagine that we havechosen their favorite field and worked with it throughout.

Since the diagrams allowed in R never change the sum of the simple roots labelingthe strands, it breaks up as a direct sum of algebras R � ⊕ν∈Q+R(ν), where Q+ is thepositive part of the root lattice, and for ν =

∑aiαi, R(ν) is the part of the algebra

with exactly ai strings colored with each simple root αi. In particular, for any simpleR-module L, there is a unique ν such that R(ν) · L = L. We call this the weight of L.We let Li denote the unique 1-dimensional simple of R(αi).

For any two positive elements of the root lattice µ, ν, then is an inclusion R(µ) ⊗R(ν) ↪→ R(µ + ν) given by horizontal juxtaposition. We use

Resµ+νµ,ν (−) = ResR(µ+ν)

R(µ)⊗R(ν)(−) Indµ+νµ,ν (−) = R(µ + ν) ⊗R(µ)⊗R(ν) −

to denote the functors of restriction and extension of scalars along this map.

15


i j

=

i j

unless i = j

i i

=

i i

+

i i

i i

=

i i

+

i i

i i

= 0 and

i j

=

ji

Qi j(y1, y2)

ki j

=

ki j

unless i = k , j

ii j

=

ii j

+

ii j

Qi j(y3, y2) −Qi j(y1, y2)y3 − y1

Figure 2. The relations of the KLR algebra. These relations are insen-sitive to labeling of the plane.

Definition 1.28 Fix representations L of R(µ) and L′ of R(ν); then we have an inducedmodule

L ◦ L′ := Indµ+νµ,ν (L � L′).

See [KL09, §2.6] for a more extensive discussion of this functor.

16


We will only ever consider modules over R(ν) which are finite dimensional and onwhich all the yi

k’s act nilpotently; for simple modules, this is equivalent to Lauda andVazirani’s condition that their modules are gradable. The following result of Laudaand Vazirani is crucial to us:

Proposition 1.29 ([LV11, Section 5.1]) The setKLR of isomorphism classes of gradablesimple modules over R carry a crystal structure with operators defined by

eiL = cosoc(L ◦Li) fiL = cosoc(HomR(ν−αi)⊗R(αi)(R(ν − αi) ⊗Li,Resνν−αi,αiL)),

e∗i L = cosoc(Li ◦ L) f ∗i L = cosoc(HomR(αi)⊗R(ν−αi)(Li ⊗ R(ν − αi),Resναi,ν−αiL)),

and this bicrystal is isomorphic to B(−∞). The map (−)σ : KLR → KLR is intertwinedwith the Kashiwara involution of B(−∞).

Remark 1.30 Our conventions are dual to those of [LV11], since we consider B(−∞)rather than B(∞).

Remark 1.31 In [LV11], the operator fi was actually defined as a socle, not a cosocle;however, as noted by Khovanov and Lauda [KL09, §3.2], all simple modules over theKLR algebra are self-dual, and HomR(ν−αi)⊗R(αi)(R(ν − αi) ⊗Li,Resνν−αi,αi

−) commuteswith duality, so its socle and cosocle when applied to a simple module are isomorphic.

It is shown in [KL09, 2.5] that, for all ν,

(4)

ψσ n∏

k=1

(yik)

rk

ei | wt(i) = ν, r1, . . . , rn ≥ 0, σ ∈ Sn

is a basis for R(ν), where ψσ is an arbitrary diagram which permutes its strands asthe permutation σ with no double crossings.

Remark 1.32 One can consider the “character”

ch(M) =∑

i

dimq(eiM) · w[i]

as an element of F , the free C[q, q−1]-module generated by words in the nodes of theDynkin diagram. As shown in [KL09, 2.20], it follows from (4) that

ch(M1 ◦M2) = ch(M1)ch(M2)

where the product on the right is the usual shuffle product.

2. KLR algebras and Lusztig data

Kleshchev and Ram’s work [KR] studying simple representations of KLR algebrasin terms of Lyndon word combinatorics allows one to construct a Lusztig datum foreach KLR module with respect to any convex order which arises from a lexicographicorder on Lyndon words. We now extend this to obtain a Lusztig datum for any convex

17


Figure 3. Examples of c-cuspidal, c-semi-cuspidal, and non-c-semi-cuspidal paths.

order. In general, we can no longer use the same type of combinatorics on words thatthey develop, and instead our main tool is the notion of a cuspidal representationwith respect to a “charge”.

2.1. Cuspidal decompositions. Let i = i1 · · · in be a word in the nodes of the Dynkindiagram and let αi =

∑nk=1 αik . Fix a charge c, and consider the preorder < on positive

elements of the root lattice induced by taking arguments with respect to this charge,as in Section 1.2.

Definition 2.1 The top of a word i is the maximal element which appears as the sumof a proper left prefix of the word; that is

top(i) = max1≤ j<n

αi1···i j .

We call a word in the simple roots c-cuspidal if top(i) < αi and c-semi-cuspidal iftop(i) ≤ αi

Geometrically, we can visualize our word as a path in the weight lattice, and thenpicture its image in the complex plane under c. A word is c-cuspidal if this pathstays strictly clockwise of the line from the beginning to the end of the word andc-semi-cuspidal if stays weakly clockwise of this line, as shown in Figure 3.

Definition 2.2 The top of a module over R is the maximum among the tops of alli such that eiM , 0. We call a simple module over R(ν) cuspidal if top(L) < ν, andsemi-cuspidal if top(L) ≤ ν.

Obviously, a simple representation is (semi-)cuspidal if and only if all words whichappear in its character are (semi-)cuspidal.

Theorem 2.3 Fix a charge c. If L1, . . . ,Lh are semi-cuspidal representations withwt(L1) > · · · > wt(Lh), then L1 ◦ · · · ◦ Lh has a unique simple quotient. Furthermore,

18


every gradable simple appears this way for a unique sequence of semi-cuspidalrepresentations.

Remark 2.4 Theorem 2.3 holds even if c is not generic.

In order to prove Theorem 2.3, we introduce a more general compatibility conditionon representations:

Definition 2.5 We call an h-tuple (L1, . . . ,Lh) unmixing if

Resν1+···+νhν1,...,νh

(L1 ◦ · · · ◦ Lh) = L1 � · · · � Lh.

Lemma 2.6 If (L1, . . . ,Lh) is an unmixing h-tuple, then L1 ◦ · · · ◦Lh has a unique simplequotient, which throughout we’ll denote A(L1, . . . ,Lh).

Proof. Let e denote the idempotent in R(ν) projecting to Resν1+···+νhν1,...,νh

(−). Then L1◦· · ·◦Lh isgenerated by any non-zero vector in the image of e; thus, a submodule M ⊂ L1◦· · ·◦Lh

is proper if and only if it is killed by e. It follows that the sum of any two propersubmodules is still killed by e, and thus again proper. There is thus a unique maximalproper submodule of L1 ◦ · · · ◦ Lh, which is to say this module has a simple cosocle.This establishes the theorem. �

Lemma 2.7 If L1, . . . ,Lh are semi-cuspidal representations with wt(L1) > · · · > wt(Lh),then the h-tuple (L1, . . . ,Lh) is unmixing.

Proof. Let i = i1i2 · · · ih be a word such that ei(L1◦· · ·◦Lh) , 0 and wt(i1) = ν1, . . . ,wt(ih) =

νh. Fix words j1, . . . , jh such that ejkLk , 0, and so that i is a shuffle of these words.For 1 ≤ k, g ≤ h, let ig

k be the subword of jg which appear as letters in ik, and letνg

k = wt(jgk ).

For each g, νg1 is the sum of the roots in a prefix of a word in the character of Lg.

Thus by semi-cuspidality, ν11 ≤ ν1 and, for g > 1, we have νg

1 ≤ νk < ν1. Clearlyν1

1 + · · · + νh1 = ν1, so this is only possible if ν1

1 = ν1, and so j1 = i1. Applying thisargument inductively, we see that jk = ik for all k. This immediately implies that(L1, . . . ,Lh) is unmixing �

Proof of Theorem 2.3. By Lemmata 2.6 and 2.7, the induction L1 ◦ · · · ◦ Lh has a uniquesimple quotient. It remains to show that every simple appears in this way for aunique sequence of semi-cuspidals.

Consider the maximum argument

cmax := maxeiL,0

arg(c(top(i)))

of the top of any word in the character of L; let ν1 be the element of the root latticegreatest height such that cmax is achieved by a prefix of weight ν1. We’ll prove the

19


result by induction on the height of ν − ν1. If ν = ν1, then L is semi-cuspidal, and weare done.

By assumption, Resνν1,ν−ν1L , 0, and so this has a simple submodule L′ � L′′.

Furthermore, L′ is semi-cuspidal (by the definition of ν1) and (L′,L′′) is an unmixingpair (by arguments as in the proof of Lemma 2.7). In particular, L is the unique simplequotient of L′◦L′′ and the quotient map induces an isomorphism Resνν1,ν−ν1

L � L′�L′′.By the inductive assumption, L′′ has a unique semi-cuspidal decomposition L′′ =

A(L1, . . . ,Lh); thus we have that L = A(L′,L1, . . . ,Lh), so this proves that every simpleis of the desired form.

On the other hand, if L = A(L′1, . . . ,L′

p) for some other cuspidal simples withwt(L′1) > · · ·wt(L′p), we cannot have wt(L′1) > ν1, by the maximality of the argument ofν1. Then we must also have words in the characters of L′i must have prefixes whoseweights add to ν1; by arguments in the proof of Lemma 2.7, this is only possible ifthis word is entirely from L′1. By symmetry, there is a word in the character of L′

whose weight in wt(L′1), so we must have wt(L′1) = ν1.Thus, Resνν1,ν−ν1

L � L′1 � A(L′2, · · · ,L′

p); consequently, L′i � L′ and L′′ = A(L′2, · · · ,L′

p).Applying the inductive hypothesis, the uniqueness follows. �

Definition 2.8 For a fixed charge c and simple L , we call the associated simples(L1, . . . ,Lh) for a fixed charge the c-semi-cuspidal decomposition of L.

Corollary 2.9 Fix a charge c. The number of c-semi-cuspidal representations ofweight ν is the sum

∑ν=β1+···+βn

arg c(βi)=arg c(ν)

n∏i=1

mβi

of the product of the root multiplicities over the distinct ways ν = β1 + · · · + βn ofwriting ν as a sum of positive roots β∗ which all satisfy arg c(βi) = arg c(ν).

Proof. We proceed by induction on ρ∨(ν). If ν is a simple root, then the statement isobvious, providing the base case.

In general, the dimension of U(n)ν is the number of ways of writing ν as a sum ofmultiples of positive roots, so this is the number of simple representations of R(ν). Bythe inductive assumption and Theorem 2.3, the number of simple representations ofR(ν) that have a semi-cuspidal decomposition with at least two parts is the numberof ways of writing ν as a sum of multiples of roots α where the arguments arg c(α)are not all equal. Thus the number of semi-cuspidal simple representations of R(ν) isthe number of ways of writing ν as a sum of multiples of positive roots all of whichhave the same argument. �

20


Corollary 2.10 If g is finite type and c is a generic charge (i.e. a charge such thatarg c(α) , arg c(β) for all α , β ∈ ∆+), then there is a unique cuspidal representationLα of R(α) for each positive root α, and no others.

Remark 2.11 The finite-type case of Theorem 2.3 (and thus Corollary 2.10) have beenshown independently by McNamara [McN, 3.1]; this has been extended to affinetype by Kleshchev in [Kle].

Proof of Corollary 2.10. By Corollary 2.9 the only ν for which there is a semi-cuspidalrepresentation are ν = kα for some k ≥ 1 and α ∈ ∆+, and in all these cases thereis only one isomorphism class of semi-cuspidal representation. The semi-cuspidalrepresentation Lα of dimension α must in fact be cuspidal, since there is no elementof the root lattice on the line from 0 to α. �

Remark 2.12 For minimal roots (i.e. rootsα such that xα is not a root for any 0 < x < 1;see section 1.2), the same arguments used in the proof of Corollary 2.10 shows that theroot multiplicity coincides with the number of cuspidal representations. However,this is not always the case for example, Section 3.4 gives an example where this isfalse for sl2 with ν = 2δ.

We also have the following generalized notion of cuspidal representation wherewe allow any convex order on ∆min

+ , not just those coming from charges.

Definition 2.13 Fix a pair (α,�) of a minimal positive root and a convex order on ∆min+ .

We say a simple representation L of R(α) is �-(semi)-cuspidal if L is c-(semi)-cuspidalfor some (α,�)-compatible charge c (see Definition 1.9).

Proposition 2.14 A simple representation L of R(α) for some positive root α is �-(semi)-cuspidal if and only if L is c-(semi)-cuspidal for all (α,�)-compatible chargesc.

Proof. Assume that L is �-cuspidal, and let c be the (α,�) compatible charge fromDefinition 2.13. Let c′ be another (α,�)-compatible charge, and assume L is notcuspidal for c′. Thus, there exists β with β >c′ α such that Resαβ;α−β L , 0.

Since c′ is (α,�)-compatible this implies that β � α. Since c is also (α,�) compatible,this implies β >c α as well. But L is c-cuspidal, so Resαβ;α−β L , 0 is a contradiction.Thus L is in fact cuspidal for c′ as well. The same argument carries through forsemi-cuspidality. �

21


Corollary 2.15 For any convex order � on ∆min+ , the number of �-semi-cuspidal

representations of weight ν is the sum∑ν=β1+···+βn

ν=aβi

n∏i=1

mβi

of the product of the root multiplicities over the distinct ways ν = β1 + · · · + βn ofwriting ν as a sum of positive roots β∗ parallel to ν. In particular, if g is finite typethen there is a unique �-cuspidal representation Lα of R(α) for each positive root α,and no others.

Proof. This follows immediately from Proposition 2.9 and Corollary 2.10 using some(α,�)-compatible charge c. �

2.2. Lusztig data in KLR. Fix a convex order �. We now strengthen Theorem 2.3 towork for an arbitrary convex order (as opposed to just convex orders coming fromcharges). This gives a generalization of the notion of Lusztig data commonly used infinite type.

Lemma 2.16 Fix a convex order �. Any h-tuple L1, . . . ,Lh of �-semi-cuspidal repre-sentations with wt(L1) � · · · � wt(Lh) is unmixing.

Proof. We proceed by induction on h. For k = 1, . . . , h, let βk be the minimal rootparallel to wt(Lk). Fix a (β1,�) compatible charge. Let νi = wt(Li) and ν = ν1 + · · ·+ νh.Assume that

Resνν1,ν−ν1(L1 ◦ · · · ◦ Lh) , L1 � (L2 ◦ · · · ◦ Lh).

Then there would have to be a word j with wt(j) a multiple of β1, such that j was ashuffle of prefixes of words that appear in the character of Lk as k ranges from 1 to h.Since each Lk is �-cuspidal and β1 � βk for each k > 1, this is impossible. Thus, wemust have that

Resνν1,ν−ν1(L1 ◦ · · · ◦ Lh) = L1 � (L2 ◦ · · · ◦ Lh).

By induction, we have already assumed that (L2, . . . ,Lh) is unmixing, so the resultfollows. �

Theorem 2.17 Fix a convex order �. If L1, . . . ,Lh is an any tuple of �-semi-cuspidalrepresentations with wt(L1) � · · · � wt(Lh), then the induction

L1 ◦ · · · ◦ Lh

has a unique simple quotient. Furthermore, every gradable simple appears this wayfor a unique sequence of semi-cuspidal representations.

Proof. By Lemmata 2.6 and 2.16, it is clear that L1◦· · ·◦Lh has a unique simple quotient.Now we must show that every simple L is of this form of a unique h-tuple L1, . . . ,Lh.

22


Consider the root α ∈ ∆min+ be greatest in the order � subject to the condition that

Resνmα,ν−mα L , 0 for some integer m; let m be the maximal integer for which this holds.Now we induct on the height of ν − mα. If ν = mα, then L is semi-cuspidal, and weare done.

The induction step is exactly as in the proof of Theorem 2.3. We must have thatthe restriction Resνmα,ν−mα L � L1 � L′ is an outer tensor product with L1 simple andsemi-cuspidal and L simple. By the inductive hypothesis, the simple L′ has a semi-cuspidal decomposition L′ = A(L2, . . . ,Lh) with wt(L1) � wt(L j) for all 2 ≤ j ≤ h, so Lis of the desired form. For any other such decomposition L = A(L′1, . . . ,L

′

p), symmetryconsiderations show that L1 � L′1, and the result follows from applying inductionagain. �

Remark 2.18 Theorem 2.17 is a generalization of [KR, Theorem 7.2], which givesexactly the same sort of description of all simple modules, but only applies to theconvex orders arising from Lyndon words.

2.3. Saito reflections on KLR. In this section, we discuss how the Saito reflectionfrom Section 1.1 works when the underlying set of B(−∞) is identified withKLR, andspecifically how it interacts with the operation of induction.

Note that if ϕ∗i (L1) = ϕ∗i (L2) = 0, then any composition factor L of L1 ◦ L2 is alsohas ϕ∗i (L) = 0 by [KL09, 2.18]; thus, we can consider the action of Saito reflections onthese simples. Now assume that (L1,L2) are an unmixing pair (see Definition 2.5), soL1 ◦ L2 has a unique simple quotient L.

Lemma 2.19 Assume that ϕ∗i (L1) = ϕ∗i (L2) = 0. Then we have that (e∗i )nL is the unique

simple quotient of

L(n) =

{(e∗i )

nL1 ◦ L2 n ≤ ε∗i (L1)(e∗i )

εi(L1)L1 ◦ (e∗i )n−εi(L1)L2 ε∗i (L1) < n

Proof. Since there are no words in the character of L1 or L2 beginning with i, the triple(Ln

i ,L1,L2) is unmixing. By Lemma 2.6, the inductionLni ◦L1 ◦L2 has a unique simple

quotient.Thus, if we define a surjective map Li ◦ L(n−1)

→ L(n), this will show by inductionthat L(n) has unique simple quotient, and that this quotient agrees with (e∗i )

nL.If n ≤ ε∗i (L1), then the map is the obvious one. If n > ε∗i (L1), then we use the fact that

Li ◦ (e∗i )ε∗i (L1)L1 � (e∗i )

ε∗i (L1)L1 ◦ Li,

so we have thatLi ◦ L(n−1) � (e∗i )

ε∗i (L1)L1 ◦ Li ◦ (e∗i )n−1−ε∗i (L1)L2

which has an obvious surjective map to L(n). �

23


Lemma 2.20 If (L1,L2) is an unmixing pair such that ϕ∗i (L1) = ϕ∗i (L2) = 0, and(σi(L1), σi(L2)) is also an unmixing pair, then σi(A(L1,L2)) = A(σi(L1), σi(L2)).

Proof. Let L = A(L1,L2) and L′ = A(σi(L1), σi(L2)); note that these are both simple.It follows from Proposition 1.4 that, for any element of B(∞) with f ∗i (M) = 0,

ϕ∗i ((e∗

i )nM) + ϕi((e∗i )

nM) − 〈wt((e∗i )nM), α∨i 〉 = max(0, εi(M) − n).

Applying Proposition 1.4(iii-v) again gives

(5) f ni (e∗i )

εi(M)M � (e∗i )εi(M) f n

i M and eni (e∗i )

εi(M)M � (e∗i )εi(M)+nM.

By Lemma 2.19, (e∗i )εi(L1)+εi(L2)L is the unique simple quotient of (e∗i )

εi(L1)L1 ◦ (e∗i )εi(L2)L2,

and eϕi(L1)+ϕi(L2)i L′ is the unique simple quotient of eϕi(L1)

i σiL1◦eϕi(L2)i σiL2. By the definition

of Saito reflection (Definition 1.5),

eϕi(L j)i σiL j � eϕi(L j)

i (e∗i )εi(L j) fϕi(L j)

i L j � eϕi(L j)i fϕi(L j)

i (e∗i )εi(L j)L j � (e∗i )

εi(L j)L j,

where the middle step uses (5). Thus

(6) (e∗i )εi(L1)+εi(L2)L = (ei)ϕi(L1)+ϕi(L2)L′.

It follows that

σiL � fϕi(L)i (e∗i )

εi(L)L by (5)

� fϕi(L1)+ϕi(L2)i eϕi(L1)+ϕi(L2)−ϕi(L)

i (e∗i )εi(L)L

� fϕi(L1)+ϕi(L2)i eεi(L1)+εi(L2)−εi(L)

i (e∗i )εi(L)L by additivity of weights

� fϕi(L1)+ϕi(L2)i (e∗i )

εi(L1)+εi(L2)L by (5)

� fϕi(L1)+ϕi(L2)i eϕi(L1)+ϕi(L2)

i L′ by (6)

= L′.

This completes the proof. �

Proposition 2.21 Fix a simple module L with f ∗i L = 0, and let (L1, . . . ,Lh) be its thesemi-cuspidal decomposition with respect to a fixed convex order �with αi greatest.Then the semi-cuspidal decomposition of σ(L) for �si is (σiL1, . . . , σiLh). In particular,the operation σi defines a bijection between semi-cuspidal modules for�with f ∗i L = 0and semi-cuspidal modules for �si with fiL = 0. The inverse of this bijection is σ∗i .

Proof. The proof is by induction, fixing L, and assuming the proposition for anysimple which has a smaller maximal height in its semi-cuspidal decomposition, or asmaller number of components.

If h > 1, let L′ = A(L2, . . . ,Lh). Then (L1,L′) is an unmixing pair, and L = A(L1,L′).By induction, the modules σiL j are semi-cuspidal and σ(L′) = A(σiL2, . . . , σiLh). Thus,σi(L1) and σi(L′) are again an unmixing pair, and by Lemma 2.20, we have thatσi(L) = A(σi(L1), σi(L′)). Thus, we have that σi(L) = A(σiL1, . . . , σiLh).

24


If L is semi-cuspidal, then we need only establish that σiL is again semi-cuspidal.Since the sets of semi-cuspidals for � of weight ν and for �si of weight siν have thesame number, and we know σ is a bijection between the set of simples L′with f ∗i L′ = 0and those with fiL′ = 0, we can instead show if L′ has the same weight as L but isnot semi-cuspidal, then σiL′ is not semi-cuspidal. But in that case L′ has a smallermaximal height in its cuspidal decomposition then L, so σiL = A(σiL1, . . . , σiLh) byinduction. �

Put another way, we have that:

Corollary 2.22 Assume � is a convex order such that αi is greatest and that L1, . . . ,Lh

are �-semi-cuspidal representations with αi � wt(L1) � · · · � wt(Lh). Then

σiA(L1, . . . ,Lh) � A(σiL1, . . . , σiLh).

Remark 2.23 As was recently explained by Kato [Kat], in symmetric type there arein fact equivalences of categories

(7){

L : Fi(L) = 0}↔

{L : F∗i (L) = 0

}which induce Saito reflections on the set of simples. Kato’s proof uses the geometryof quiver varieties, which is why it is only valid in symmetric type, but it seems likelythat there is an algebraic version of Kato’s functor as well, which should extend hisresult to all symmetrizable types. We feel this should give an alternative and perhapsmore satisfying explanation for Proposition 2.21 and Corollary 2.22.

Corollary 2.22 is a very important technical tool for us. In particular it allows us toreduce questions about cuspidal representations to the case where the root is simple,using the following.

Lemma 2.24 Fix a simple L and a convex order �, and assume the semi-cuspidaldecomposition of L is L = A(L1, . . . ,Lh).

Assume L1 = L nα for some real root α. If g is finite type or is affine with α � δ, then

there is a finite sequence σi1 , . . . , σik of Saito reflections such that sik · · · si1α is a simpleroot αm, for each j we have ϕ∗j(σi j−1 · · · σi1L) = 0, and

σik · · · σi1L = A(L nαm, . . . , σik · · · σi1Lh).

If instead Lh = Lpβ for some real root β and g is finite type or affine with β ≺ δ, then

a similar list of dual Saito reflections σ∗i1 , . . . , σ∗

ihexists with

σ∗ih · · · σ∗

i1L = A(σ∗ik · · · σ∗

i1L1, . . . ,Lpα`).

Proof. The two statements are swapped by the Kashiwara involution, so we needonly prove the first. We proceed by induction on the number of positive roots η � α(which is thus finite). The case when α is greatest with respect to � (and hence is

25


simple) is trivially true; so assume that for some k ≥ 1 the statement is known for allcharges c and all positive roots α with at most k − 1 positive roots η � α.

Fix c and αwith exactly k roots� α. Let αi1 be the greatest root (which is necessarilysimple). Then ϕ∗i1(L) = 0, since Lαi1

does not appear in its cuspidal decomposition,and so we can apply Corollary 2.22. This reduces to the same questions with �si , andsi(α) and σi1L. Furthermore, the positive roots β �si si1α are exactly those of the formβ = siβ′ for β′ � α with β′ , αi1 , so there are one fewer of these then for α and �, andthe induction proceeds until we have found the desired sequence of i•’s.

By Proposition 2.21, the modules (σik · · · σi1L1, . . . , σik · · · σi1Lh) are the semi-cuspidaldecomposition of L with respect to the convex order �sik ···si1 = (· · · (�si1 )si2 · · · )sik . Sincesik · · · si1αi1 = αm, we have that σik · · · σi1L1 � L n

αm. �

3. KLR polytopes andMV polytopes

Having developed this combinatorics for understanding representations of KLRalgebras, we now turn to encoding this information in polytopes.

3.1. KLR polytopes.

Definition 3.1 For each simple L, the character polytope PL is the convex hull of theweights ν′ such that Resνν′,ν−ν′ L , 0.

Remark 3.2 Recalling the definition of the character ch(L) of L from Remark 1.32, wecan think of every word i appearing in ch(L) as a path in h∗; the polytope PL can alsobe described as the convex hull of all these paths. This explains our terminology.

The polytopes PL live in the dual of the Cartan h∗, which has a natural heightfunction ρ∨. This orients each edge of the polytope, and gives every face F a highestvertex vt and lowest vertex vb. We associate a KLR algebra RF to each face F byRF := R(vt − vb). Thus for each face F of PL, we have the subalgebra of R(ν) given byR(vb) ⊗ RF ⊗ R(ν − vt), and can consider the restriction functor ResνF restricting to thissubalgebra.

Proposition 3.3 For any simple L and face F of PL, the restriction ResνF L is simple andthus the outer tensor of three simples L′ � LF � L′′.

Proof. Choose a function φ that obtains its minimum on PL exactly on F, and considerthe charge ρ∨ + iϕ; the simple L is the unique simple quotient of an increasinginduction of semi-cuspidals L1 ◦ · · · ◦ Lh; let k be smallest index where ϕ(wt(Lk)) = 0and m the largest such index. Let L′ be the simple quotient of L1 ◦ · · · ◦ Lk−1, let L f bethe simple quotient of Lk ◦ · · · ◦ Lm, and let L′′ be the simple quotient of Lm+1 ◦ · · · ◦ Lh.

26


So, the module L′ ◦LF ◦L′′ is a quotient of L1 ◦ · · · ◦Lh, and thus has a unique simplequotient, which is L. On the other hand, ResνF(L′ ◦ LF ◦ L′′) = L′ � LF � L′′, so L mustalso restrict to this same module. �

Definition 3.4 Fix L ∈ KLR. The KLR polytope PL of L is the polytope PL along withthe data of the isomorphism class of the semi-cuspidal representation LE associatedto each edge E of PL as above. We denote by PKLR the set of all KLR polytopes.

Remark 3.5 The representations which can appear as labels in PL are not arbitrary;they must be semi-cuspidal for any charge which includes that edge in its walk.

Proposition 3.6 Every edge of PL is parallel to a positive root of g. That is, PL is apseudo-Weyl polytope.

Proof. For any edge E, we can pick a generic function ϕwhich achieves its maximumon PL exactly on E. Since at most one element of ∆min

+ is parallel to E, a generic suchϕ produces a charge generic in the usual sense: it induces a total order on the wordsappearing in simple representations of RE. Furthermore, LE is semi-cuspidal for thischarge so by Corollary 2.9 is a multiple of a positive root. �

Remark 3.7 In finite type, there is exactly one semi-cuspidal simple representationof R(kα) for each positive root α and k ≥ 1, so the decoration is superfluous. The KLRpolytope PL is completely determined by the character polytope PL, and PKLR can bethought of as simply a set of pseudo-Weyl polytopes.

Recall that that, as in Lemma 1.15, each convex order � in Lemma 1.15 defines apath P�L . through PL. We obtain a list simple modules of simple modules L1, . . . ,Lh

with wt(L1) � · · · � wt(Lh) by taking the modules corresponding to the edges in P�.

Proposition 3.8 For any simple L,we have L = A(L1, . . . ,Lh), where L1, . . . ,Lh are asdescribed above.

Proof. We induct on h, the case when L is �-semi-cuspidal being obvious. Let E be thelast edge in P�, and consider ResνE L; this is of the form L′ � Lh. Obviously the edgesin PL′ and PL coincide along the walk corresponding to � up to but not includingE. Thus, L1, . . . ,Lh−1 are the simples associated to this walk for L′ by the algorithmabove, and by the inductive assumption, L′ = A(L1, . . . ,Lh−1). Thus, A(L1, . . . ,Lh) isthe unique simple quotient of L′ ◦ Lh. Of course, L is also a quotient of this moduleby Frobenius reciprocity, so these simples coincide. �

Proposition 3.8 has the following immediate consequences:

27


Corollary 3.9 For any �, the polytope PL with the labeling of just its edges along P�

uniquely determines the simple L. In particular, the map L 7→ PL defines a bijectionKLR → PKLR. �

Corollary 3.10 The function sending a labelled polytope to the list of semi-cuspidalrepresentations attached to P� is a bijection from PKLR to the set of ordered lists ofsemi-cuspidal representations, for any convex order. �

Since the map which takes L to PL is injective, the crystal structure onKLR gives riseto a crystal structure on PKLR. Using Corollary 3.9, we can now give a combinatorialdescription of the resulting crystal operators.

Proposition 3.11 To apply the operator fi to P ∈ PKLR, we choose a convex order withαi lowest, and read the path determined by that order to obtain a list of semi-cuspidalrepresentations L1, · · · ,Lh corresponding to increasing roots in that order. If Lh = L k

ifor some k ≥ 1, then

fiP = PA(L1,...,Lh−1,Lk−1

i ).

If Lh � L ki , then fiP = 0.

Proof. If Lh � L ki , then A(L1, . . . ,Lh) is a quotient of L1 ◦ · · · ◦ Lh, whose character is a

quantum shuffle of words not ending in i, and thus only contains words not ending ini. Thus, fiA(L1, . . . ,Lh) = 0 follows immediately. On the other hand A(L1, . . . ,Lh−1,L k

i )is a quotient of

A(L1, . . . ,Lh−1) ◦L ki � (A(L1, . . . ,Lh−1) ◦L k−1

i )) ◦ Li → A(L1, . . . ,Lh−1,Lk−1

i ) ◦Li,

and thus by definition is eiA(L1, . . . ,Lh−1,L k−1i ) = A(L1, . . . ,Lh) and we are done. �

We also have the following, which is simply a restatement of Corollary 2.22 in thelanguage of polytopes.

Proposition 3.12 To apply a Saito reflection functor σi to a polytope P ∈ PKLR withf ∗i P = 0, choose a convex order with αi greatest and let L1, · · · ,Lh be as before. Then

σiP = PA(σiL1,...,σiLh).

�

Comparing Propositions 3.11 and 3.12 with the definition of crytal theoretic Lusztigdata 1.11, it is immediate that:

Corollary 3.13 For any simple L with corresponding element b in B(−∞) and anyconvex order �, the geometric Lusztig data aα(PL) from Definition 1.16 agrees withthe crystal-theoretic Lusztig data aα(b) from Definition 1.11 for all real roots which

28


are greater than any imaginary root, and with the dual data a∗

α(b) for all real rootswhich are less than any imaginary root. �

Proof of Theorem A. Two pseudo-Weyl polytopes for a finite dimensional root systemcoincide if and only if their Lusztig data are identical for every convex order. ByCorollary 1.20 the Lusztig data of the MV polytope correspnding to b is given by thecrystal-theoretic Lusztig data a•(b), and that of the KLR polytope PL is given by a•(L)by definition. Thus Corollary 3.13 shows that these polytopes coincide. �

We can generalize this theorem a little bit; fix a charge c such that every imaginaryroot α has arg(c(α)) < π/2; we can, of course, symmetrically deal with the faceswhere arg(c(α)) > π/2 instead. While this might seem like a strange and restrictivecondition, it has a natural interpretation. These are precisely the faces parallel to aface of the weight polytope of a lowest weight representation (or highest weight, forthe reversed condition). In affine type, we can split faces into two groups: eitherthey have this form, or they are parallel to δ. In hyperbolic type, the situation will beconsiderably more complicated.

Proposition 3.14 For each L ∈ KLR, the face F of PL defined by c is an MV polytopefor gc.

Proof. Of course, we can assume that L is semi-cuspidal with argument π/2 withoutloss of generality. By Lemma 2.24, we can find a list of reflections si1 , . . . , sik such thatfor csi1 ···sik , the roots si1 · · · sikβ j are all greater than all other roots in the correspondingconvex preorder. Thus, these roots must all be simple, and we have that si1 · · · sikFis a face of the polytope Pσi1 ···σik L by Proposition 3.12. On the other hand, σi1 · · · σikLis a representation on a KLR algebra for the finite type algebra gc

si1···sik ; we simply

don’t use any strands labeled with other roots. Thus, PL is an MV polytope for gcsi1···sik

algebra by Theorem A, and thus so is F. �

3.2. The crystal corresponding to a face. Fix a charge c. The set of real roots withargument π/2 are the real roots of some root system ∆c. Let gc be the correspondingLie algebra. Let β1, . . . , βs be the simple roots of gc. To avoid the possibility ofconfusion between the indexing set of the roots of g and those of gc, we will alwaysindex the latter with underlined numbers.

Remark 3.15 As we discuss in the Section 3.6, in general it may be better to associateto a face all roots of argument π/2, which in general is the root system of a Borcherdsalgebra of possibly infinite rank. However, our main goal here is to understand affinetype, and there defining gc as we do is more convenient.

Definition 3.16 Fix a charge c. The face crystal KLR[c] for c is the set of c-semi-cuspidal representations L of argument π/2.

29


Notice that KLR[c] consists exactly of those representations which occur as therepresentation LF associated in the previous section to the face F of PL defined by thecharge c and argument π/2, where L is some simple. This justifies the term “face” inDefinition 3.16. The term “crystal” is justified by the following:

Proposition 3.17 The operators given by

eiL = cosoc(L ◦Lβi), e∗i L = cosoc(Lβi ◦ L),

fiL = soc(HomR(ν)(R ◦Lβi ,L)), f∗i L = soc(HomR(ν)(Lβi ◦ R,L))

define a gc combinatorial bicrystal structure onKLR[c]. The additional combinatorialdata is given by

wt(L) = wt(L)|hc ,

ϕi(L) = max{n | fni (L) , 0}, ϕ∗i (L) = max{n | (f∗i )

n(L) , 0},

εi(L) = ϕβi(L) − α∨βi(wt(L)), ε∗i (L) = ϕ∗βi

(L) − α∨βi(wt(L)).

Proof. Our principal task is to show that these operators are well defined.Fix L and let ν = wt(L). For each simple root βi of ∆c, we can choose a deformations

c± of the charge c such that:

(i) for some small ε > 0, elements µ of the weight lattice with ν − µ ∈ spanZ≥0{αi}

have• arg(c±(µ)) ∈ (π/2 − ε, π/2 + ε) if and only if arg(c(µ)) = π/2,• arg(c±(µ)) > π/2 + ε if and only if arg(c(µ)) > π/2,• arg(c±(µ)) < π/2 − ε if and only if arg(c(µ)) < π/2.

(ii) the root βi is greater for >c+ and lesser for >c− than all other roots β , βi witharg(c(β)) = π/2.

Note that we cannot require these conditions for all µ if g is not finite type. Forexample, if g is affine and arg(c(δ)) = π/2, for any real root with arg(c(β)) , π/2 wemust have arg(c±(β+kδ)) ∈ (π/2−ε, π/2+ε) for k� 0. Of course, ν−β−kδ < spanZ≥0

{αi}

for k� 0, so this is not a fatal problem.For each L ∈ KLR[c], we have semi-cuspidal decompositions for c±. The conditions

on c± imply that every representation which appears in these must be itself inKLR[c].Thus, we have a surjective map · · · ◦L k+1

βi→ L ◦Lβi , and similarly for Lβi ◦ L. By

Theorem 2.3, this shows that L◦Lβi and Lβi ◦L have unique simple quotients; in fact,we can define our crystal operators by

e∗i L = A(L n+1βi, . . . ) eiL = A(. . . ,L k+1

βi).

Condition (i) of Definition 1.1 is tautological. Condition (ii) follows from theobservation that k = ϕi(L) and n = ϕ∗i (L), so indeed these increase as expected with ei

and e∗i .

30


Now, consider Resν+βi

ν,βieiL; the socle of this module is of the form L′ �Lβi for some

semi-simple module L′, and

L′ = soc(HomR(ν+βi)(R ◦Lβi , eiL)).

First of all, L′ can have no simple summands other than L, since if L′′ where such asimple, we would have a surjection L′′ ◦Lβi → L, which is impossible by Theorem2.3. On the other hand, Hom(L,L′) � Hom(L ◦Lβi , eiL) � k, so L has multiplicity 1 inL′, and so L � L′; thus condition (iii) holds for unstarred operators, and exactly thesame argument shows it for starred operators. Finally, condition (iv) is vacuous inthis case. �

The remainder of Section 3.2 is devoted to proving results on the structure of theface-crystal in special cases. These cases will be crucial to understanding the finiteand affine type situation later on.

Lemma 3.18 Fix a charge c and a simple rootαi. Ifαi is the greatest root for c then Saitoreflection σi induces a bicrystal isomorphism betweenKLR[c] andKLR[csi]. Similarly,if αi is the lowest root for c, then σ∗i induces a bi-crystal isomorphism betweenKLR[c]andKLR[csi].

Proof. Of course, we need only check this for one set of operators ei and e∗i at a time.Thus, fix βi as before.

We can choose c± as in the proof of Proposition 3.17; we also require that αi isstill maximal for both these orders. For any simple module L ∈ KLR[c], we havesemi-cuspidal decompositions for c± given by

L = A(L nβi, . . . ) = A(. . . ,L k

βi).

We have that

e∗i L = A(L n+1βi, . . . ) eiL = A(. . . ,L k+1

βi).

It follows immediately from Lemma 2.24 that these operations commute with Saitoreflection. �

We say that an element of a bicrystal is lowest weight if it is killed by all loweringKashiwara operators, both starred and unstarred.

Lemma 3.19 Assume Lh is a c-semi-cuspidal representation of argument π/2 whichis lowest weight for the bicrystal structure with wt(Lh) a null vector. Then the com-ponent generated by Lh under the crystal operators e j is the same as the componentgenerated by Lh under the e∗j.

31


Proof. The proof is by induction on the sum d(L) of the coefficients of the expressionfor wt(L)−wt(Lh) in terms of the βk. The proof is symmetric in the two structures, soit suffices to fix L = e jde jd−1 · · · e j1L

h and show that it is in the starred component of Lh.If d(L) = 1 then wt(L) − wt(Lh) = β j, and L = f jLh for some j. As in the proof of

Proposition 3.17 we can apply Saito reflections until βk = αi. In that case it followsfrom Proposition 1.4 and the fact that the whole crystal is Bg(−∞) that f jLh = f∗jL

h, sothe claim holds.

Now assume the result holds for all L′ with depth d(L′) < d. By the d = 1 case thisis isomorphic to L = e jde jd−1 · · · e

∗

j1Lh. From here it is clear that Reswt(L)

β j1 ,wt(L)−β j1L , 0; any

simple submodule of this restriction is of the form Lβ j1� L′ for some L′. Thus, we

have that e∗j1L′ = L, so f∗j1L , 0. Since we know thatKLR in isomorphic to B(−∞) as a

bi-crystal, by Proposition 1.4 we must be in one of the following two cases:(1) If j1 , jd or j1 = jd and f∗j1 f j1L = f j1 f

∗

j1L then L = e∗j1e jd f

∗

j1f jdL. The module f jdL is

manifestly in the component of the unstarred component of Lh, and thus by inductionin the starred component as well. Using the inductive hypothesis again, e jd f

∗

j1f jdL is

in the starred components of Lh, and so L is as well.(2) If j1 = jd and f∗j1L = f j1L, then f∗j1L = f j1L is in both the starred and unstarred

component of Lh. Since L = e∗jd f∗

j1L, we see that L is also in the starred component. �

Proposition 3.20 Assume Lh∈ KLR[c] is lowest weight for the bicrystal structure,

and that wt(Lh) is a null vector. Then the component generated by Lh under all e j, e∗jis isomorphic (as a bicrystal) to the infinity crystal Bgc(−∞).

Proof. By Lemma 3.19, it suffices to check conditions (ii)-(vi) of Corollary 1.4. To checkcondition (ii), consider the module Lβ j ◦ L ◦Lβi ; both eie∗jL and e∗jeiL are quotients ofthis module. Since β j and βi are simple among the roots with c-argument π/2, thereis a deformation c′ of c such that βi is lowest among the roots with arg(c(β)) = π/2,β j is greatest. Let (L1, . . . ,Lh) be the semi-cuspidal decomposition of L with respectto c′. By assumption, each Li is semi-cuspidal for c, so their weights have argumentbetween that of c′(β j) and c′(βi) (inclusive). Thus, Lβ j ◦ L1 ◦ · · · ◦ Lh ◦Lβi has a uniquesimple quotient by Theorem 2.3. Since Lβ j ◦ L ◦Lβi is a quotient of this module, thisuniqueness proves eie∗jL = e∗jeiL.

Each of the conditions (iii)–(vi) only involves a single simple root βi. Furthermore,each of these conditions holds before Saito reflection if and only it holds after byCorollary 2.22. Thus, as in the proof that operators are well-defined (Proposition3.17), we can apply Saito reflections until βi is simple, in which case these conditionsfollow from the isomorphism ofKLR with B(−∞) for g. �

32


Corollary 3.21 Fix a charge c such that every root with argument π/2 is real (so inparticular gc is of finite type). Then KLR[c] with the operators from Proposition 3.17is isomorphic as a bicrystal to Bgc(−∞).

Proof. The trivial representation satisfies the conditions of Proposition 3.20, so gener-ates a copy of Bgc(−∞). Since all roots of gc are real, the number of these representa-tions of a given weight is exactly the Kostant partition function of gc, so this exhaustsKLR[c]. �

Corollary 3.22 If both g and gc are of affine type, then KLR[c] is isomorphic as abicrystal to a direct sum of copies of Bgc(−∞), all lowest weight elements Lh havewt(Lh) = kδ for some k, and the number of lowest weight elements of weight kδ is thenumber of q-multipartitions of k, where q = r − s = rk g − rk gc.

Proof. By Proposition 3.20 it suffices to show that all lowest weight elements for theunstarred gc crystal structure are also lowest wight for the starred gc crystal structure,that they must have weight kδ, and that there are the right number for each k. Weproceed by induction on weight of (potential) lowest weight elements. In weight0, there is one lowest weight element (the trivial representation), and it satisfies allthe conditions. So, assume these conditions hold for all lowest weight elements ofdepth ≤ kδ for some k ≥ 0. By Proposition 3.20, each of these lowest weight elementsgenerate a copy of Bgc(−∞). The generating function for the number of c-stable repsof argument π/2 is exactly

a(t) =∏α∈∆c

1(1 − tα)dim gα

.

By comparing with the Kostant partition function

b(t) =∏α∈∆c

1(1 − tα)dim(gc)α

for gc, we see thatb(t)a(t)

=∏k≥1

1(1 − tkδ)q

is just the generating function of the number of q-multipartitions with variable tδ.We know that at kδ and below, the number of lowest weight elements of the bicrystalstructure exactly matches this partition function. Thus, we see that the copies ofBgc(−∞) exhaust all c-stable elements of depth less than (k + 1)δ, and miss exactly thenumber of q-multipartitions of k+1 in that depth. These must all be lowest weight forboth structures, and satisfy the necessary conditions, so the induction proceeds. �

Proposition 3.23 Assume g is of affine type. Fix M,N ∈ KLR[c]. Assume M is lowestweight for the gc crystal structure, and N is in the component generated by the trivial

33


representation. Then M ◦N = N ◦M, this module is irreducible, and N 7→M ◦N is abicrystal isomorphism between the component of the trivial representation and thatof M.

Lemma 3.24 With the notation of Proposition 3.23, M ◦N has a unique simple quo-tient, and the map N 7→ A(M,N) commutes with crystal operators.

Proof. For any list of weights ν1, . . . , νm, let eν1,...,νm be the idempotent that projects toall sequences which consist of a block of strands summing to ν1, a block summing toν2, etc.

Choose any infinite list of nodes j1, j2, . . . in the Dynkin diagram of gc in which eachnode appears infinitely many times (for example, we can pick an order and cycle). Inthe crystal Bgc(−∞), the module N has a string parameterization N = ea1

j1ea2

j2· · · ea`

j`L∅,

where a1 is maximal number of times f j1 can be applied to N before it becomes 0, a2

the maximal number of times f j2 can be applied to fa1j1

N, etc.

For any Lustzig datum a, define ea = ea`β j` ,...,a1β j1. Let La denote the quotient of

La`j`◦ L

a`−1j`−1◦ · · · ◦ L

a1j1

by all ea′(La`j`◦ L

a`−1j`−1◦ · · · ◦ L

a1j1

) for a′ > a in lexicographic order.

By the definition of string parameterization, N is a quotient of La.Each of the roots β j is lowest, so L j`−1 is necessarily cuspidal for c, not just semi-

cuspidal. Thus, the space eaLa is spanned by diagrams which permute the simpleterms of the induction. But any diagram that gives a non-trivial permutation mustfactor through the image of an idempotent ea′ which we have killed (compare withthe argument in [KL09, 3.7]). Thus eaLa is just a copy of La`

j`� · · · � La1

j1, which is

simple. Now, we use the standard argument to show that La has unique simplequotient—any proper submodule is killed by ea, so their sum is as well, and thus isstill proper.

Since M is lowest weight for the gc-crystal structure, ewt(M)−β j,β jM = 0 for all j. Thesame argument as above then shows that ekδ,a(M ◦ La) is simple, and therefore M ◦ La

has a unique simple quotient as well. But M ◦N is a quotient of M ◦ La, so it also hasa unique simple quotient.

It is also clear that ea1j1

ea2j2· · · ea`

j`M is a quotient of M ◦ La, so this must agree with the

unique simple quotient of M ◦ N, and hence the map N → A(M,N) commutes withthe ordinary crystal operators. �

Remark 3.25 The reader may notice the resemblance of the above argument to thatwe used earlier based on the unmixing property; unfortunately, neither (M,La) nor(M,La`

j`, . . . ,La1

j1) is actually unmixing, so we must use this slightly more elaborate

argument.

34


Proof of Proposition 3.23. The character of M ◦ N is the shuffle of the characters of Mand N. Since M � N is simple and is killed in no simple quotient, it is contained inthe cosocle (which by Lemma 3.24 is simple). By [LV11, 2.2], the induction M ◦N isisomorphic to the coinduction coind(N �M), so there is an injection from N �M intothe socle of M ◦N.

If i is a non-trivial word in the character of M, then the weight of any prefix ip iseither <c δ or is a multiple of δ. In particular, any word in the character of M ◦ Nwhich begins with blocks that step along βi1 , . . . , βid for an arbitrary sequence i1, . . . , id,where d =

∑a j is the depth of N, must lie in the socle, since the multiplicities of such

sequences is the same for N �M and M ◦N.On the other hand, the cosocle is obtained from M by applying d crystal operators

to N, so it can also be obtained by applying d starred crystal operations by Corollary3.22. Thus, there is at least one word in the character of the cosocle which beginswith d =

∑ai many steps along βi for various i. It follows that the natural map from

the socle to the cosocle must be non-zero, thus an isomorphism. Thus M◦N is in factsimple.

Notice also that the natural map from N ◦M to the socle of M◦N must be non-zeroand thus an isomorphism. Hence N ◦M 'M ◦N.

We have already established that N → A(M,N) = M ◦ N is a crystal isomorphismfor the unstarred operators; the symmetric argument for N ◦M establishes that it isfor the starred operators as well. �

3.3. Affine polytopes. Outside of finite type, the conventional definition of MVpolytope fails, although as shown in [BKT], an alternate geometric definition can beextended to symmetric affine type. We propose to use the decorated polytopes PL asthe “general type MV polytopes.” This construction is not completely combinatorial,as the decoration consists of various representations of KLR algebras. However, inaffine type, we can extract purely combinatorial objects.

For the rest of this section fix g of affine type with rank r + 1. As usual, label thesimple roots of g by α0, . . . , αr with α0 being the distinguished vertex as in [Kac90].We first prove some technical results concerning the structure of the semi-cuspidalrepresentations of KLR algebras with weight a multiple of δ. These will allow us toprecisely define the partitions πγ associated with a simple L in the introduction.

Consider the projection p : αi → αi for i , 0, δ → 0 from affine root space to theroot space for gfin, the Lie algebra attached to the Dynkin diagram with the 0 noderemoved, where we use α to denote roots in the finite type root system. In all casesother then A(2)

2n the image of this map is exactly the set of finite type roots along with0 (this can be seen by checking that p sends the simple affine roots to a set of finitetype roots including all the simples, and using the affine Weyl group). For A(2)

2n , theimage also contains α/2 for each of the long roots α in the finite type root system.

35


For each chamber coweightγ = θω∨i in the finite type root system (i.e. each elementin the Weyl group orbit of a fundamental coweight), define a charge cγ by

cγ(α) = 〈γ, p(α)〉 + iρ∨(α).

The set of roots with argument π/2 for cγ is a rank r affine sub-root system. Hence,for any L, cγ defines a vertical face of PL, and this is generically dimension r.

Let ∆fin be the root system for gfin and let ∆fin;γ be the sub-root-system of ∆fin onwhich γ vanishes. Fix a basis Π = {η1, . . . , ηr−1} for ∆fin;γ. There is a unique ηr ∈ ∆fin

whose addition makes this a base of ∆fin and such that 〈γ, ηr〉 = 1. Explicitly, ηr is theunique root with 〈γ, ηr〉 = 1 such that ηr − ηi is never a root.

Let cΠ be a charge such that the roots sent toπ/2 are exactly the linear combinationsof p−1(ηr) and δ, and such that, for all 1 ≤ i ≤ r − 1, the positive roots in p−1(ηi) are>cΠ

δ. Clearly gcΠis rank 2 affine, and thus is of type A(1)

1 or A(2)2 . The positive cone

for g defines simple roots for gcΠ, which we denote by β1 and β0; we always choose

the labeling so that 〈γ, p(β1)〉 < 0 and thus β1 > β0. For i = 0, 1, define `i =|βi|√

2(which

is always 1 or 2).

Definition 3.26 Let Lλ;γ be the element of the lowest weight crystal for gΠ generatedby the trivial module with Lusztig datum λ for the ordering β1 > β0, as defined in[BDKT]. Explicitly, one can easily show using the combinatorics in [BDKT] that

Lλ;γ = e`1λ11 (e∗0)`0λ1(e∗1)`1λ2e`0λ2

0 e`1λ31 (e∗0)`0λ3 · · ·L∅.

Lemma 3.27 If M is a simple module of weight nδwhich is semi-cuspidal for both cγand cΠ and which is in the crystal component of L∅ for cΠ, then M � Lλ;γ for some λ.

Proof. As in the proof of Proposition 3.17, we can use Saito reflections to reduce tothe case where β0 is a simple root.

Consider a representation M which is cΠ-cuspidal of weight nδ. By Theorem1.26 (see also Remark 1.27), M is of the form Lλ;γ if and only if its real Lusztig dataa(m+1)β0+mβ1(M) (as defined in Definition 1.11) with respect to the order β1 > β0 is alwaystrivial. Thus it suffices to prove that if M is semi-cuspidal and in the component ofL∅ for cΠ , and M has non-trivial Lusztig data of the form a(m+1)β0+mβ1(M) for somem ≥ 0, then M is not semi-cuspidal for cγ.

We proceed by induction on the smallest integer m such that a(m+1)β0+mβ1(M) , 0,proving the statement for all γ simultaneously. If m = 0 the statement is clear, givingthe base case of the induction.

So assume m > 0. Consider σ∗0M. By Corollary 2.22, this must be semi-cuspidal for

the charge cs0

Π. The face-crystalKLR[cs0

Π] is still rank-2 affine, with simple roots β0 and

β1, and the Lusztig data of σ∗0M for the order β1 < β0 are given by aα(σ∗0M) = as0α(M)

36


for α , β0. Note that

s0((m + 1)β0 + mβ1) = (m − 1)β0 + mβ1.

Since our inductive assumption covered all chamber weights, we are assuming thatσ∗0M is not semi-cuspidal for cs0γ. But then applying Corollary 2.22 again it is clearthat M is not semi-cuspidal for cγ. This completes the proof. �

Proposition 3.28 The modules Lπ;γ are a complete, irredundant list of lowest-weightsemi-cuspidal modules of argument π/2 for cγ, and this labeling is independent ofthe choice of base in gcγ .

Lemma 3.29 Proposition 3.28 holds when γ = ω∨i is a fundamental coweight, andthe base Π = {η j} is given by the simple roots excluding αi.

Proof. In this case, the lowest-weight semi-cuspidal modules of argument π/2 for cω∨iare precisely the semi-cuspidal modules of argument π/2 which are killed by f j forj , i, 0.

Now, consider a an irreducible representation L which is semi-cuspidal of argumentπ/2 and lowest-weight inKLR[cω∨i ]. If L is were not cΠ-semi-cuspidal, then there mustbe a cΠ-cuspidal Q whose weight is a real root α <cΠ

δ such that L is a quotient ofQ′ ◦Q for some simple Q′. Since α <cΠ

δ, p(α) is a positive multiple of a negative rootin gfin, and so α ≤cω∨i

δ. Since L is cω∨i -semi-cuspidal, we must have that α ≥cω∨iδ, so

α =cω∨iδ, or equivalently α has argument π/2 for cω∨i . Since L is cω∨i -semi-cuspidal,

this implies Q is cω∨i -semi-cuspidal as well, and since L is lowest weight for gcω∨iso is

Q; this is impossible if α is a real root by Corollary 3.22. Thus, L is cΠ-semi-cuspidal.As in Proposition 3.23, there exist canonical representations with M lowest-weight

and N in the component of the identity for gΠ such that L = M ◦ N = N ◦M. Thus,we must have that M and N are both semi-cuspidal and lowest-weight for gcω∨i

. Therepresentation M is killed

• by fi since it is lowest-weight inKLR[cΠ],• by f0 since it is semi-cuspidal for g and α0 is the lowest root for this order, and• by all other f j’s since it is lowest-weight forKLR[cω∨i ].

Thus, we must have M = L∅, and L = N.Since L is semi-cuspidal for cω∨i , Lemma 3.27 implies that the Lusztig datum of L

for the action of gΠ ordering β1 > β0 must be purely imaginary. Since the number oflowest-weight cuspidals for cω∨i of each weight is the same as the number of simplesin the component of the trivial with imaginary Lusztig data, these sets of simplesmust coincide. This establishes the result for γ = ω∨i , with the obvious choice ofbase. �

Proof of Proposition 3.28. We reduce all other cases to that covered in Lemma 3.29.

37


If γ = ω∨i but we have chosen a different base Π′ = {ηi}′ of gfin;γ, then we can find

an element w = si1 · · · sik of the Weyl group Wfin;γ such that wηi = η′i . Applying theSaito reflections σi1 · · · σik to Lπ;γ leaves Lπ;γ unchanged (since it is killed by fim andf ∗im and has weight a multiple of δ), and also sends Lπ;γ as defined using {ηi} to Lπ;γ

as defined using the {η′i}. Thus Lπ;γ is independent of this choice.Now consider a general chamber coweight γ. If γ is not a fundamental coweight,

then there must be a simple root αi for i > 0 such that 〈γ, p(αi)〉 < 0, so that theargument of αi with respect to cγ is greater than π/2. Notice that αi , β0, since〈γ, p(β0)〉 > 0 Thus ϕ∗i (Lπ;γ) = 0, so we can apply σi. If αi , β1 then by Lemma 3.18applying σi to all cuspidal modules for cγ defines an isomorphism of crystals to thesame set-up for csiγ, which is negative on one fewer positive root in the finite typesystem than γ; in particular it sends Lπ;γ to Lπ;siγ. If αi = β1, the same fact followsfrom the known action of Saito reflections on B(∞) for affine rank 2 Lie algebras.By induction, we may reduce to the case where γ is a fundamental coweight, so theresult follows by Lemma 3.29. �

Fix a generic charge c such that c(δ) ∈ iR>0. This defines a positive system in thefinite type root system, where we say α is positive if p−1(α) �c δ. Let χ1, . . . , χr be thecorresponding set of simple roots and γ1, . . . , γr the dual set of coweights. For eachr-tuples of partitions π = (πγ1 , . . . , πγr), define

(8) L (π) = Lπγ1 ;γ1 ◦Lπγ2 ;γ2 ◦ · · · ◦Lπγr ;γr .

Remark 3.30 These agree with Kleshchev’s imaginary modules [Kle, §4.3]. Note thatin contrast to Kleshchev, we have a canonical labeling of these by multipartitions.

Lemma 3.31 The module L (π) is irreducible and independent of the ordering onsimple roots. As π ranges over multipartitions with n boxes, these modules are alldistinct and a complete list of c-semi-cuspidal representations of R(nδ).

Proof. We induct on the largest j such that πγ j , ∅. Note that for the face defined bythe common maxima of γk for 1, . . . , j− 1, the module Lπγ1 ;γ1 ◦Lπγ2 ;γ2 ◦ · · · ◦Lπ

γ j−1 ;γ j−1

is a lowest weight simple for the crystal of the face by the inductive hypothesis.Furthermore, by definition, Lπ

γ j ;γ jis in the component of the identity for the face

crystal. Thus, Lπγ1 ;γ1 ◦ Lπγ2 ;γ2 ◦ · · · ◦ Lπγ j ;γ j

is irreducible by Proposition 3.23. Theindependence of ordering immediately follows from the irreducibility.

Clearly all the representations L (π) are semi-cuspidal. They are all distinct, sincethe partition πγi is uniquely determined by the isomorphism type of L(π). By Corol-lary 2.9, this is the right number of semi-cuspidal representations. �

Thus, if we consider the KLR polytope PL of L, the labeling of each real edge isdetermined by its length and direction; any imaginary edge parallel to δ could belabeled with any L (π) (where we fix a charge c chosen so that the imaginary edge

38


lies in its associated walk). However, we now show that this information can beencoded in a labeling of facets rather than edges.

Fix a simple L and a finite type chamber coweight γ. Consider the semi-cuspidaldecomposition (. . . ,L2,L1,L0,L1,L2, . . . , ) of L for cγ where L0 is the component withargument π/2.

Definition 3.32 Let πγ be the partition such that the representation L0 lies in thecrystal component of Lπγ;γ.

Proposition 3.33 The representation decorating any imaginary edge E is exactlyL (πγ1 , . . . , πγr), where the γi are the chamber coweights which achieve their lowestvalue on E, and πγ is the partition associated to L and γi by Definition 3.32.

Proof. Fix a representation L. Let c be a generic charge such that E is part of thepath Pc

l . Let M be the representation in the semi-cuspidal decomposition of L for cwhose weight is a multiple of δ. Then M is also semi-cuspidal for each cγ (since thevalue of γ on the imaginary edge is a minimum on PL). We need only show thatif M = L (ξγ1 , . . . , ξγr), then the partition πγi attached to L by Definition 3.32 is ξγi .If c is a small deformation of cγ, then is clear. Thus, we need only consider howπγ changes as we deform c to cγ linearly. While the semi-cuspidal decompositionchanges, the imaginary edge that the associated walk goes along never changes.Thus, the representation in the semi-cuspidal decomposition of weight which is amultiple of δ never changes. Thus, that representation is always M. This establishesthe result. �

Thus, the KLR polytope in the sense of Definition 3.4 can be encoded as a decoratedaffine pseudo-Weyl polytope as defined in the introduction, where we decorate thefacet where γ achieves its minimum with πγ.

3.4. An example. If one were trying to naively generalize the finite type situation,it would be natural to hope that for a fixed generic charge, one could find a totallyordered set of cuspidal simples, with the number in each weight being the rootmultiplicity, such that, for L1 ≤ · · · ≤ Lk the module A(Ln1

1 , . . . ,Lnkk ) gives a complete

list of the simples. In fact, even in affine type, this will fail, as we now illustrate. Wenote that this example is also treated in [Kas, Example 3.3], but we wish to give atreatment emphasizing the features of interest to us.

We consider the case of sl2, and choose a charge where α0 <c α1. We must alsochoose the polynomial Q01(u, v), which we take to be u2 + quv + v2 for some q ∈ k(this is not a completely general choice of Q, but any choice of Q gives an algebraisomorphic this one after passing to a finite field extension).

There are exactly two semi-cuspidal representations of weight 2δ. These canbe described as L(2);ω = e2

1e20L∅ and L(1,1);ω = e1e0e1e0L∅. Consider the induction

39


L(1);ω ◦L(1);ω. This is 6-dimensional, spanned by the elements

0 1 0 1v =

0 10 1ψ2v =

0 10 1ψ3ψ2v =

0 10 1ψ1ψ2v =

0 10 1ψ3ψ1ψ2v =

0 10 1ψ2ψ3ψ1ψ2v =

where v is any non-zero element of L(1);ω �L(1);ω, which is 1-dimensional.The span H of the basis vectors other than v is a submodule (it is the kernel of a

map to L(1,1);ω). The image of the idempotent e0011 is irreducible over R(2α0)⊗R(2α1),and generates H. Thus, either

• H is irreducible or• ψ2ψ3ψ1ψ2v spans a submodule.

But,

0 10 1

ψ22ψ3ψ1ψ2v =

0 10 1= q

0 10 1= −q

Thus, if q , 0, H is irreducible and thus H � L(2);ω. Its inclusion is split, withcomplement spanned by qv + ψ2ψ3ψ1ψ2v. In particular, L(1);ω ◦L(1);ω is semi-simplewith both L(2);ω and L(1,1);ω occurring as summands. We see that neither of thesemodules can thus be cuspidal, since

ch(L(2);ω) = (q−2 + 2 + q2)w[0011] + w[0101].

If q = 0, then the behavior is quite different; in this case ψ2ψ3ψ1ψ2v spans the socleof L(1);ω ◦L(1);ω, and H is its radical. In particular, L(1);ω ◦L(1);ω is indecomposable,and a 3-step extension where a copy of L(2);ω is sandwiched between the socle andcosocle, both isomorphic to L(1,1);ω. So in particular, when q = 0, the representationL(2);ω is cuspidal, since

ch(L(2);ω) = (q−2 + 2 + q2)w[0011].

The KLR polytopes of these representations are independent of q and are given by

•

•

• (2)

•

•

•

•

(1) (1, 1)

40


If one takes the choice of parameters as in [VV11] corresponding to an Ext-algebraof perverse sheaves on the moduli of representations of a Kronecker quiver (which isalso that fixed by [BK09] in order to find a relationship to affine Hecke algebras withν = −1 or in characteristic 2), then we take q = −2. Thus, if the fieldkhas characteristic, 2, we have q , 0 and dim L(2);ω = 5 whereas if k does have characteristic 2, thenq = 0 and dim L(2);ω = 4. Under Brundan and Kleshchev’s isomorphism [BK09]between quotients of KLR algebras and cyclotomic Hecke algebras, this correspondsto the change in characters as we pass from the Hecke algebra at a root of unity tothe symmetric group, or the difference between the canonical basis and 2-canonicalbasis.

In the q = 0 case, the number of cuspidals in this example is in fact the rootmultiplicity of 2δ. One might naively hope that at q = 0 this happened more generally,but explicit calculation in more complicated examples show that it does not.

3.5. Proof of Theorem B and Theorem C. Fix any convex order �. For each affineLusztig datum there can be at most one decorated polytope satisfying the conditionsof Theorem B which has the specified Lusztig datum with respect to �. Furthermore,by Theorem 2.3 and Corollary 2.9 we can always find a simple L such that PL has thisLusztig datum. Thus, to prove Theorem B it suffices to show that each PL satisfies allthe specified conditions on 2-faces.

Every 2-face is either real or parallel to δ. The real 2-faces are themselves MVpolytopes by Proposition 3.14. Thus it remains to check that on 2-faces parallel to δalso yield affine MV polytopes (after shortening the imaginary edge). Fix a charge csuch that the roots sent to the imaginary line form a rank 2 sub-root system, and letgc be the associated rank 2 affine algebra. This defines a (possibly degenerate) 2-faceFc of any PL, and all imaginary 2-faces occur this way for some such c.

Let γ1, . . . , γr−1 be the r − 1 finite type chamber weights which define facets of PL

containing Fc for all L, and γ+, γ− the two chamber weights that define faces thatintersect Fc in vertical lines. If you deform c a small amount, then it gives a completeorder on roots, and picks out one of the two vertical edges of Fc. We can choosedeformations c± such that the set of chamber weights associated with these chargesare {γ1, . . . , γr−1, γ±}. Let β0 and β1 be the simple roots parallel to Fc with

〈γ+, β0〉 > 0 > 〈γ+, β1〉〈γ−, β1〉 > 0 > 〈γ−, β0〉.

We use the notations Lβ,Lπ;γi ,L (π) for the cuspidal representations for c+ andLβ, Lπ;γi , L (π) for c−. Similarly, we use aβ to denote the Lusztig data of a g-polytopewith respect to c+ or a gc-polytope for the order where β1 > β0 and aβ for c− or theorder where β0 > β1.

We define a map L 7→ PFL from the set of c-semi-cuspidal representations of weight

parallel to F to decorated pseudo-Weyl polytopes for gF by sending L to the polytope

41


PFL with Lusztig data given by

aβ(PFL) = aβ(PL) aβ(PF

L) = aβ(PL)

for every real root β parallel to F and

(9) aδ(PFL) = πγ+ aδ(PF

L) = πγ−

.

Let L,M,N be as in the statement of Proposition 3.23 for the charge c.

Proposition 3.34 The polytope PFL is the MV polytope (in the sense of [BDKT]) for

the crystal element N.

Proof. We must check the conditions of Theorem 1.26.(i) This is clear when L is a lowest weight object for the crystal (in which case the

weight is 0 on both sides), and it is also clear that this property is preservedby the gc crystal operators.

(ii.1-4) Using Lemma 2.24, we can find Saito reflections in B(−∞) which reduce usto the case where β0 or β1 is simple for g. Hence this is a consequence of aconsequence of Proposition 3.11 and the form of ∗ involution on B(−∞).

(iii.1-4) Again, using Lemma 2.24, we can assume that β0 or β1 is simple in g. Assumingβ0 is simple, it is clear that the Saito reflections in this root in Bgc(−∞) are therestrictions of the corresponding reflections in the full crystal B(−∞). Hencethe statements for these two reflections are a consequence of Corollary 2.22.To get the statements for the reflections in β1 we instead use Saito reflectionsin B(−∞) to reduce this to a simple root.

(iv) By definition, Lλ;γ = e`1λ1

1(e∗0)`0λ1Lλ\λ1;γ. Since this crystal element has trivial

real Lusztig data for β0 > β1, we know that

f1Lλ\λ1;γ = (f∗0)Lλ\λ1;γ = 0.

Thus, we see that

aα1(PFLλ;γ

) = `1λ1 aα0(PFLλ;γ

) = `0λ1.

Furthermore, the module Lλ\λ1;γ is semi-cuspidal of weight (n − λ1)δ for theorder β0 > β1 and by the definition (9), this means that aδ(PF

Lλ;γ) = λ \ λ1. This

establishes the final condition of Theorem 1.26. �

This establishes Theorem B.

Proof of Theorem C. It follows from Lemmas 1.17 and 1.18 that the sets of decoratedpseudo-Weyl polytopes PL and HNb are both uniquely determined by a single Lusztigdatum and relations extending the tropical Plucker relations that determine the struc-ture of 2-faces. Thus, the recursive nature of the relations for both polytopes meanswe only need to check that the KLR polytopes and HN polytopes coincide in therank 2 case. For real 2-faces, this follows from the Theorem A and [BKT, §1.5] and for

42


affine 2-faces, this follows from the match (with a transpose) of the sl2 MV polytopesdefined of [BKT] and [BDKT], which is shown in [MT]. Thus, every KLR polytope isan HN polytope with its imaginary Lusztig data transposed. This determines somebijection B(−∞) → B(−∞). This bijection preserves Lusztig data for every convexorder, is thus a crystal isomorphism, and it therefore is the identity. �

3.6. Beyond affine type. In affine type, while we can have many different semi-cuspidal representations corresponding to an imaginary root, we still have consid-erable control over the structure of these representations, as the proceeding sectionsshow. In particular, the additional structure in their polytopes can be captured in astraight-forward way by labeling facets with partitions.

In general, we expect that the structure of a 2-face should be controlled by theset of roots obtained by intersecting a 2-dimensional plane with ∆. If g is of finitetype then this set is also a finite type root system and the 2-faces are finite type MVpolytopes. In affine type, this intersection is a rank-2 affine root system, and 2-facesare essentially rank 2 affine MV polytopes. But because of the multiplicities, the sumof these root spaces is actually not an affine Lie algebra—rather, it is an infinite-rankBorcherds algebra whose Cartan matrix is obtained by adding infinitely many rowsand columns of zeroes to the rank 2 affine matrix. The structure we have observedin the 2-faces, a crystal for the affine Lie algebra together with an infinite family ofcommuting operators seems, in fact, to be a manifestation of this larger algebra, asopposed to just the affine one.

Beyond affine type, when one intersects ∆ with a 2-plane, the resulting set of realroots will generate a root system of rank at most 2. However, if there is to be ageneralization of Theorem B, it is probably necessary to consider not just this rank2 root system, but rather the entire sum of the root spaces; by [Bor95, Theorem 1],this will always be a Borcherds algebra. The corresponding Cartan matrix may havemany non-zero entries outside the Cartan matrix of the root system generated by thereal roots. Nonetheless, one could hope to define MV polytopes for this algebra, andthat the 2-faces could be matched to these. Unfortunately, even if this were possible,“reduction to rank two” would mean reduction to a Borcherds algebra of possiblyinfinite rank, leaving it debatable whether this actually improves matters; it still mayshed some rather interesting light on the structure of KLR algebras and their simplerepresentations. Some cases, such as toroidal algebras (where the Cartan matrixremains positive semi-definite) may be more tractable.

To illustrate some of the difficulties, consider the Cartan matrix

(10)

2 −2 −2−2 2 −2−2 −2 2

.This is of hyperbolic type, and the imaginary root β = α1 + α2 + α3 has multiplicity2. Fix a charge c with c(α0) = 1 + i, c(α1) = −1 + i, c(α2) = i. The only real root with

43


c(α) ∈ iR is α2 itself. Thus the real roots only generate a copy of sl2, which is already anew phenomenon as in finite and affine type the real roots corresponding to a 2-facealways generated a rank 2 root system.

Nonetheless, Proposition 3.17 shows that the semi-cuspidals of argument π/2 area combinatorial crystal for sl2. If the analogue of Corollary 3.22 held, then we wouldhave that e2 and e∗2 were identical acting on every semi-cuspidal of argumentπ/2, sincethis is the case in Bsl2(−∞). However, both e2e1e0L∅ and e∗2e1e0L∅ are 1-dimensional;the former has character w[012] and the latter w[201]. Thus, they are necessarilydistinct.

Attacking this case will require stronger techniques than we possess at the moment;the sharp-eyed reader will note that we give no direct connection between the KLRalgebra attached to a face and the lower rank KLR algebra for the root system spannedby that face. While this seems like an obvious suggestion, we see no such connection(say, a functor) at the moment, and our techniques were intended to circumvent thisabsence. Perhaps more progress can be made on the hyperbolic case if such a functorcan be found, using the KLR algebras for Borcherds algebras given in [KOP].

References

[And03] Jared Anderson, A polytope calculus for semisimple groups, Duke Math. J. 116 (2003), no. 3,567–588.

[BDKT] Pierre Baumann, Thomas Dunlap, Joel Kamnitzer, and Peter Tingley, Rank 2 affine MVpolytopes, arXiv:1202.6416.

[BK09] Jonathan Brundan and Alexander Kleshchev, Blocks of cyclotomic Hecke algebras and Khovanov-Lauda algebras, Invent. Math. 178 (2009), 451–484, arXiv:0808.2032.

[BKT] Pierre Baumann, Joel Kamnitzer, and Peter Tingley, Affine Mirkovic-Vilonen polytopes,arXiv:1110.3661.

[Bor95] Richard E. Borcherds, A characterization of generalized Kac-Moody algebras, J. Algebra 174(1995), no. 3, 1073–1079.

[Car] S. Carnahan, Is the centralizer of a torus in a Kac-Moody algebra always a Borcherds algebra?,http://mathoverflow.net/questions/94595.

[Dun10] Thomas Dunlap, Combinatorial representation theory of affine sl(2) via polytope calculus, 2010,Ph.D. Thesis, Northwestern University.

[HK02] Jin Hong and Seok-Jin Kang, Introduction to quantum groups and crystal bases, Graduate Studiesin Mathematics, vol. 42, American Mathematical Society, Providence, RI, 2002.

[Kac90] Victor G. Kac, Infinite-dimensional Lie algebras, third ed., Cambridge University Press, Cam-bridge, 1990.

[Kam07] Joel Kamnitzer, The crystal structure on the set of Mirkovic-Vilonen polytopes, Adv. Math. 215(2007), no. 1, 66–93, arXiv:math/0505398.

[Kam10] , Mirkovic-Vilonen cycles and polytopes, Ann. of Math. (2) 171 (2010), no. 1, 245–294,arXiv:math/0501365.

[Kas] Masaki Kashiwara, Notes on parameters of quiver Hecke algebras, arXiv:1204.1776.[Kas93] , The crystal base and Littelmann’s refined Demazure character formula, Duke Math. J. 71

(1993), no. 3, 839–858. MR 1240605 (95b:17019)[Kas95] , On crystal bases, Representations of groups (Banff, AB, 1994), CMS Conf. Proc.,

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[KL09] Mikhail Khovanov and Aaron D. Lauda, A diagrammatic approach to categorification of quantumgroups. I, Represent. Theory 13 (2009), 309–347, arXiv:0803.4121.

[Kle] Alexander Kleshchev, Cuspidal systems for affine Khovanov-Lauda-Rouquier algebras,arXiv:1210.6556.

[KOP] Seok-Jin Kang, Se-jin Oh, and Euiyong Park, Categorification of quantum generalized Kac-Moodyalgebras and crystal bases, arXiv:1102.5165.

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[KS97] Masaki Kashiwara and Yoshihisa Saito, Geometric construction of crystal bases, Duke Math. J.89 (1997), no. 1, 9–36, arXiv:q-alg/9606009.

[LV11] Aaron D. Lauda and Monica Vazirani, Crystals from categorified quantum groups, Adv. Math.228 (2011), no. 2, 803–861, arXiv:0909.1810.

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[MT] Dinakar Muthiah and Peter Tingley, PBW bases and MV polytopes for rank 2 affine algebras,Preprint, arXiv:1209.2205v1.

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