faces of highest weight modules and the universal …gsd/facesuniv.pdf · max, and therefore the...

FACES OF HIGHEST WEIGHT MODULES AND

THE UNIVERSAL WEYL POLYHEDRON

GURBIR DHILLON AND APOORVA KHARESTANFORD UNIVERSITY

Abstract Let V be a highest weight module over a KacndashMoody algebra g and let conv Vdenote the convex hull of its weights We determine the combinatorial isomorphism type ofconv V ie we completely classify the faces and their inclusions In the special case whereg is semisimple this brings closure to a question studied by CellinindashMarietti [IMRN 2015]for the adjoint representation and by Khare [J Algebra 2016 Trans Amer Math Soc toappear] for most modules The determination of faces of finite-dimensional modules up tothe Weyl group action and some of their inclusions also appears in previous work of Satake[Ann of Math 1960] BorelndashTits [IHES Publ Math 1965] Vinberg [Izv Akad Nauk 1990] andCasselman [Austral Math Soc 1997]

For any subset of the simple roots we introduce a remarkable convex cone which we callthe universal Weyl polyhedron which controls the convex hulls of all modules parabolicallyinduced from the corresponding Levi factor Namely the combinatorial isomorphism type ofthe cone stores the classification of faces for all such highest weight modules as well as howfaces degenerate as the highest weight gets increasingly singular To our knowledge this coneis new in finite and infinite type

We further answer a question of Michel Brion by showing that the localization of conv Valong a face is always the convex hull of the weights of a parabolically induced module Finallyas we determine the inclusion relations between faces representation-theoretically from the set ofweights without recourse to convexity we answer a similar question for highest weight modulesover symmetrizable quantum groups

Contents

1 Introduction 22 Statement of results 33 Notation and preliminaries 54 Classification of faces of highest weight modules 105 Inclusions between faces 126 Families of Weyl polyhedra and representability 167 Faces of the universal Weyl polyhedron 178 Weak families of Weyl polyhedra and concavity 239 Closure polyhedrality and integrable isotropy 2610 A question of Brion on localization of convex hulls 2911 Highest weight modules over symmetrizable quantum groups 30References 31

Date May 15 20172010 Mathematics Subject Classification Primary 17B10 Secondary 17B67 22E47 17B37 20G42 52B20Key words and phrases Highest weight module parabolic Verma module Weyl polytope integrable Weyl

group Tits cone f -polynomial universal Weyl polyhedron quantum group

1

2 GURBIR DHILLON AND APOORVA KHARE

1 Introduction

Throughout the paper unless otherwise specified g is a KacndashMoody algebra over C with tri-angular decomposition nminus oplus h oplus n+ and Weyl group W and V is a g-module of highest weightλ isin hlowast

In this paper we study and answer several fundamental convexity-theoretic questions relatedto the weights of V and their convex hull denoted wtV and conv V respectively In contrastto previous approaches in the literature our approach is to use techniques from representationtheory to directly study wtV and deduce consequences for conv V

To describe our results and the necessary background first suppose g is of finite type and Vis integrable In this case conv V is a convex polytope called the Weyl polytope and is wellunderstood In particular the faces of such Weyl polytopes have been determined by Vinberg[26 sect3] and Casselman [6 sect3] and as Casselman notes are implicit in the earlier work of Satake[25 sect23] and BorelndashTits [3 sect1216] The faces are constructed as follows Let I index thesimple roots of g and ei fi hi i isin I the Chevalley generators of g Given J sub I denote by lJthe corresponding Levi subalgebra generated by ej fj forallj isin J and hi foralli isin I Also define thefollowing distinguished set of weights

wtJ V = wtU(lJ)Vλ (11)

where Vλ is the highest weight space The aforementioned authors showed that every face of theWeyl polytope for V is a Weyl group translate of the convex hull of wtJ V for some J sub I

To complete the classification of faces it remains to determine the redundancies in the abovedescription Formally consider the face map

FV W times 2I rarr 2wtV FV (w J) = w(wtJ V ) (12)

where 2S denotes the power set of a set S As wtJ V and its convex hull determine one an-other computing the redundancies is the same as computing the fibers of the face map Thedetermination of the fibers for the restriction FV (1minus) is implicit in the above works of Vinbergand Casselman A complete understanding of the fibers of FV (minusminus) was achieved very recentlyby CellinindashMarietti [7] for the adjoint representation and subsequently by Khare [18] for allhighest weight modules While the fibers of the face map were understood for all highest weightmodules V it was not known whether all faces were of the above form

In this paper we resolve both parts of the above problem for any highest weight module Vover a KacndashMoody algebra g While the answer takes a similar form to the known cases in finitetype the proof of the first part requires a substantially different argument as prior methods failat several crucial junctures

(i) A key input in the known cases of the classification of faces of conv V was that every faceis the maximizer of a linear functional which was deduced from polyhedrality Howevereven the polyhedrality of conv V was not known in general in finite type

(ii) We show in related work [8] that indeed conv V is always a polyhedron in finite typeHowever as we show below for g of infinite type conv V is rarely polyhedral genericallylocally polyhedral and in some cases not locally polyhedral In particular one cannot apriori appeal to a maximizing functional to understand a face

(iii) Finally even given a face cut out by a maximizing functional the possibility of the func-tional to fall outside the Tits cone makes the usual analysis inapplicable

In this paper we provide an infinitesimal analysis of the faces that avoids invocation of func-tionals completing the first part of the classification of faces As a consequence every face isa posteriori cut out by a maximizing functional lying in the Tits cone For the second part

FACES OF HIGHEST WEIGHT MODULES AND THE UNIVERSAL WEYL POLYHEDRON 3

ie computing the redundancies we give a direct representation theoretic argument which inparticular simplifies the existing arguments in finite type

Moreover we construct a remarkable convex cone which we call the universal Weyl poly-hedron Puniv whose combinatorial isomorphism class controls the convex hulls of weights ofall highest weight modules which are parabolically induced from a fixed Levi Namely (i) theclassification of faces of Puniv is equivalent to that of all such highest weight modules and (ii)inclusions of faces of Puniv control how faces of such highest weight modules degenerate as thehighest weight specializes

2 Statement of results

Let g be a KacndashMoody algebra and V a highest weight g-module We now state the first partof the classification of faces of conv V

Theorem 21 Denote by IV the integrability of V ie

IV = i isin I fi acts locally nilpotently on V

For each standard Levi subalgebra lJ sub g the locus convU(lJ)Vλ is a face of conv V Theseaccount for every face of conv V which contains an interior point in the IV dominant chamberAn arbitrary face F of conv V is in the WIV orbit of a unique such face

For g of finite type this classification was obtained by Satake [25] BorelndashTits [3] Vinberg[26] and Casselman [6] for V finite-dimensional and by Khare [17] for many infinite-dimensionalmodules Theorem 21 settles the remaining cases in finite type and is to our knowledge new ininfinite type As explained above a proof in this setting requires new ideas Interestingly ourproof which proceeds by induction on the rank of g uses the consideration of non-integrablemodules in an essential way even for the case of integrable modules

The classification of inclusion relations between faces akin to the one in finite type [7 18]is our next main result By Theorem 21 it suffices to study the fibers of the face map FV WIV times 2I rarr 2wtV defined as in (12)

Theorem 22 For each J sub I there exist sets Jmin sub J sub Jmax sub I such that

wtJ V sub wtJ prime V lArrrArr Jmin sub J primewtJ V = wtJ prime V lArrrArr Jmin sub J prime sub Jmax

In particular in the poset 2I under inclusion the fibers of the face map FV (1minus) are intervalsMoreover the fibers of FV (minus J) are left cosets of WIV capJmax in WIV Given the final assertionof Theorem 21 this determines the fibers of the map FV (minusminus)

Explicit formulas for Jmin Jmax are presented before Theorem 51 and after Remark 55respectively In particular we determine the f -polynomial of conv V cf Proposition 514Thus Theorems 21 and 22 together settle the classification problem for faces of arbitraryhighest weight modules over all KacndashMoody algebras

Modulo the action of the Weyl group Theorems 21 and 22 are essentially statements aboutthe actions of the standard Levi subalgebras on the highest weight line In this light they admita natural extension to quantum groups which we prove in Section 11

As shown below the sets Jmin Jmax and therefore the combinatorial isomorphism class ofconv V only depend on λ through i isin IV (αi λ) = 0 Motivated by this we study de-formations of the convex hulls conv V for highest weight modules with fixed integrability Tointerpolate between such convex hulls we fix J sub I and consider Weyl polyhedra which aresimply convex combinations of the convex hulls of highest weight modules with integrability J


We study a natural notion of a family of Weyl polyhedra over a convex base S and show thatsuch families admit a classifying space

Theorem 23 Consider the contravariant functor F from convex sets to sets which takes aconvex set to all families of Weyl polyhedra over it and morphisms to pullbacks of familiesThen F is representable

If Buniv represents F we obtain a universal family of Weyl polyhedra Puniv rarr Buniv Con-cretely one may take Buniv to be a parabolic analogue of the dominant chamber so thatPuniv rarr Buniv extends the usual parametrization of modules by their highest weight Westudy the convex geometry of the total space Puniv and show a tight connection between itscombinatorial isomorphism type and the previous results on the classification of faces

Theorem 24 For J sub I as above Puniv is a convex cone Write F for the set of faces ofPuniv Then F can be partitioned

F =⊔J primesubJ

FJ prime

so that each FJ prime ordered by inclusion is isomorphic to the face poset of conv V where V is anyhighest weight module with integrability J and highest weight λ with J prime = j isin J (αj λ) = 0

We in fact show more than is stated in Theorem 24 the remaining inclusions of faces aredetermined by how faces of conv V degenerate as the highest weight specializes cf Proposition714 In particular we determine the f -polynomial of Puniv cf Proposition 715 We areunaware of a precursor to Theorem 24 or its aforementioned strengthening in the literature

Theorem 24 is proved by using the projection to Buniv More precisely along the interior ofeach face of Buniv the restriction of Puniv is a lsquofibrationrsquo and we use this to construct faces ofPuniv from lsquoconstantrsquo families of faces along the fibres

Finally we explain a question of Michel Brion on localization of faces which we answer inthis paper

To do so we introduce a notion of localization in convex geometry If E is a real vector spaceand C sub E a convex subset then for c isin C to study the lsquogermrsquo of C near c one may form thetangent cone TcC given by all rays c + tv v isin E t isin Rgt0 such that for 0 lt t 1 c + tv isin CGiven a face F of C one can similarly form the lsquogermrsquo of C along F by setting LocF C to bethe intersection of the tangent cones TfCforallf isin F It may be helpful to think of tangent conesas convex analogues of tangent spaces and LocF C a convex analogue of the normal bundle of asubmanifold

Let g be of finite type λ a regular dominant integral weight and L(λ) the correspondingsimple highest weight module Brion observed that for any face F of convL(λ) LocF convL(λ)is always the convex hull of a parabolic Verma module for a Category O corresponding to thetriangular decomposition wnminuswminus1 oplus hopluswn+wminus1 for some w isinW Further if λ isin F then onecould take w = e Brion asked whether this phenomenon persists for a general highest weightmodule V

In answering Brionrsquos question we will prove the equivalence of several convexity-theoreticproperties of the convex hull of a highest weight module which we feel is of independent interestIn particular the following result explains the subtlety in classifying faces of conv V for g ofinfinite type as alluded to in the introduction

Theorem 25 Let g be indecomposable of infinite type and V a highest weight g-module Thenconv V is a polytope if and only if V is one-dimensional and is otherwise a polyhedron if andonly if IV corresponds to a Dynkin diagram of finite type If V is not one-dimensional thefollowing are equivalent


(1) The intersection of conv V with any polytope is a polytope

(2) The tangent cone TλV is closed and conv V =⋂

wisinWIV

TwλV

(3) conv V is closed(4) λ lies in the interior of the IV Tits cone

In the somewhat unrepresentative case of integrable modules in finite type Theorem 25 isstraightforward and well known see eg [14 sect23] For general g and V one of the implicationscan be deduced from work of Looijenga on root systems [22] and a theorem concerning localpolyhedrality in a similar geometric context appears in his later work [23] Otherwise to ourknowledge Theorem 25 is largely new including for all non-integrable modules in finite type

Returning to Brionrsquos question given Theorem 21 it suffices to consider faces of the formconvU(lJ)Vλ We show

Theorem 26 Suppose λ has trivial stabilizer in WIV and J sub I Writing F for the faceconvU(lJ)Vλ we have

LocF conv V = convM(λ IV cap J)

The regularity assumption appearing in Theorem 26 specializes to that of Brion for integrablemodules in finite type That such a condition is necessary to recover from conv V the convexhull of a module with less integrability is already clear from considering edges ie the case ofsl2

In Brionrsquos motivating example this convexity-theoretic localization is a combinatorial shadowof localizing the corresponding line bundle on the flag variety off of certain Schubert divisorsAs we develop in [8] for general V with dot-regular highest weight in finite type the aboveformula is the combinatorial shadow of localizing the corresponding (twisted) D-module on theflag variety

Organization of the paper After discussing preliminaries and establishing notation in Sec-tion 3 we show in Sections 4 5 6 7 9 10 in order the main results stated above Theorems21ndash26 In Section 8 we prove an analogue of Theorem 23 for a more flexible notion of Weylpolyhedra and in Section 11 we show the analogue of Theorem 22 for Uq(g)-modules

Acknowledgments We thank Michel Brion Daniel Bump and Victor Reiner for valuablediscussions The work of GD was partially supported by the Department of Defense (DoD)through the NDSEG fellowship

3 Notation and preliminaries

We now set notation and recall preliminaries and results from the companion work [8] inparticular the material in this section and Section 111 overlaps with loc cit We advise thereader to skim Subsections 32 and 35 and refer back to the rest only as needed

Write Z for the integers and QRC for the rational real and complex numbers respectivelyFor a subset S of a real vector space E write Zgt0S for the set of finite linear combinations ofS with coefficients in Zgt0 and similarly ZSQgt0SRS etc

31 Notation for KacndashMoody algebras standard parabolic and Levi subalgebrasThe basic references are [13] and [19] In this paper we work throughout over C Let I be afinite set Let A = (aij)ijisinI be a generalized Cartan matrix ie an integral matrix satisfyingaii = 2 aij 6 0 and aij = 0 if and only if aji = 0 for all i 6= j isin I Fix a realizationie a triple (h π π) where h is a complex vector space of dimension |I|+ corank(A) and


π = αiiisinI sub hlowast π = αiiisinI sub h are linearly independent collections of vectors satisfying(αi αj) = aij foralli j isin I We call π and π the simple roots and simple coroots respectively

Let g = g(A) be the associated KacndashMoody algebra generated by ei fi i isin I and hmodulo the relations

[ei fj ] = δijαi [h ei] = (h α)ei [h fi] = minus(h α)fi [h h] = 0 forallh isin h i j isin I(ad ei)

1minusaij (ej) = 0 (ad fi)1minusaij (fj) = 0 foralli j isin I i 6= j

Denote by g(A) the quotient of g(A) by the largest ideal intersecting h trivially these coincidewhen A is symmetrizable When A is clear from context we will abbreviate these to g g

In the following we establish notation for g the same apply for g mutatis mutandis Let∆+∆minus denote the sets of positive and negative roots respectively We write α gt 0 for α isin ∆+and similarly α lt 0 for α isin ∆minus For a sum of roots β =

sumiisinI kiαi with all ki gt 0 write

suppβ = i isin I ki 6= 0 Write

nminus =oplusαlt0

gα n+ =oplusαgt0

gα

Let 6 denote the standard partial order on hlowast ie for micro λ isin hlowast micro 6 λ if and only ifλminus micro isin Zgt0π

For any J sub I let lJ denote the associated Levi subalgebra generated by ei fi i isin J and hdenote li = li for i isin I For λ isin hlowast write LlJ (λ) for the simple lJ -module of highest weight λWriting AJ for the principal submatrix (aij)ijisinJ we may (non-canonically) realize g(AJ) = gJas a subalgebra of g(A) Now write πJ ∆

+J ∆

minusJ for the simple positive and negative roots of

g(AJ) in hlowast respectively (note these are independent of the choice of realization) Finally wedefine the associated Lie subalgebras u+

J uminusJ n

+J n

minusJ by

uplusmnJ =oplus

αisin∆plusmn∆plusmnJ

gα nplusmnJ =oplusαisin∆plusmnJ

gα

and pJ = lJ oplus u+J to be the associated parabolic subalgebra

32 Weyl group parabolic subgroups Tits cone Write W for the Weyl group of ggenerated by the simple reflections si i isin I and let ` W rarr Zgt0 be the associated lengthfunction For J sub I let WJ denote the parabolic subgroup of W generated by sj j isin J

Write P+ for the dominant integral weights ie micro isin hlowast (αi micro) isin Zgt0foralli isin I Thefollowing choice is non-standard Define the real subspace hlowastR = micro isin hlowast (αi micro) isin R foralli isin INow define the dominant chamber as D = micro isin hlowast (αi micro) isin Rgt0 foralli isin I sub hlowastR and the Titscone as C =

⋃wisinW wD

Remark 31 In [19] and [13] the authors define hlowastR to be a real form of hlowast This is smallerthan our definition whenever the generalized Cartan matrix A is non-invertible and has theconsequence that the dominant integral weights are not all in the dominant chamber unlikefor us This is a superficial difference but our convention helps avoid constantly introducingarguments like [19 Lemma 832]

We will also need parabolic analogues of the above For J sub I define hlowastR(J) = micro isin hlowast (αj micro) isin Rforallj isin J the J dominant chamber as DJ = micro isin hlowast (αj micro) isin Rgt0 forallj isin Jand the J Tits cone as CJ =

⋃wisinWJ

wDJ Finally we write P+J for the J dominant integral

weights ie micro isin hlowast (αj micro) isin Zgt0forallj isin J The following standard properties will be usedwithout further reference in the paper the reader can easily check that the standard argumentscf [19 Proposition 142] apply in this setting mutatis mutandis


Proposition 32 For g(A) with realization (h π π) let g(At) be the dual algebra with realiza-tion (hlowast π π) Write ∆+

J for the positive roots of the standard Levi subalgebra ltJ sub g(At)

(1) For micro isin DJ the isotropy group w isinWJ wmicro = micro is generated by the simple reflectionsit contains

(2) The J dominant chamber is a fundamental domain for the action of WJ on the J Titscone ie every WJ orbit in CJ meets DJ in precisely one point

(3) CJ = micro isin hlowastR(J) (α micro) lt 0 for at most finitely many α isin ∆+J in particular CJ is a

convex cone(4) Consider CJ as a subset of hlowastR(J) in the analytic topology and fix micro isin CJ Then micro is an

interior point of CJ if and only if the isotropy group w isinWJ wmicro = micro is finite

We also fix ρ isin hlowast satisfying (αi ρ) = 1foralli isin I and define the dot action of W via w middot micro =w(micro+ρ)minusρ Recall this does not depend on the choice of ρ Indeed for any w isinW a standardinduction on `(w) shows

w(ρ)minus ρ =sum

αgt0wαlt0

w(α) (33)

We define the dot dominant chamber to be D = micro isin hlowast (αi micro+ ρ) isin Rgt0 foralli isin I and the

dot Tits cone to be⋃wisinW w middot D It is clear how to similarly define their parabolic analogues

the J dot dominant chamber and the J dot Tits cone Now Proposition 32 holds for the dotaction mutatis mutandis

33 Representations integrability and parabolic Verma modules Given an h-moduleM and micro isin hlowast write Mmicro for the corresponding simple eigenspace of M ie Mmicro = m isin M hm = (h micro)m forallh isin h and write wtM = micro isin hlowast Mmicro 6= 0

Now recall V is a highest weight g-module with highest weight λ isin hlowast For J sub I writewtJ V = wtU(lJ)Vλ ie the weights of the lJ -module generated by the highest weight line Wesay V is J integrable if fj acts locally nilpotently on V forallj isin J Let IV denote the maximal Jfor which V is J integrable ie

IV = i isin I (αi λ) isin Zgt0 f(αiλ)+1i Vλ = 0 (34)

We will call WIV the integrable Weyl groupWe next remind the basic properties of parabolic Verma modules over KacndashMoody algebras

These are also known in the literature as generalized Verma modules eg in the original papersby Lepowsky (see [20] and the references therein)

Fix λ isin hlowast and a subset J of IL(λ) = i isin I (αi λ) isin Zgt0 The parabolic Verma moduleM(λ J) co-represents the following functor from g -mod to Set

M m isinMλ n+m = 0 fj acts nilpotently on mforallj isin J (35)

When J is empty we simply write M(λ) for the Verma module Explicitly

M(λ J) M(λ)(f(αj λ)+1j M(λ)λforallj isin J) (36)

Finally it will be useful to us to recall the following result on the weights of integrable highestweight modules ie those V with IV = I

Proposition 37 ([13 sect112 and Proposition 113(a)]) For λ isin P+ micro isin hlowast say micro is non-degenerate with respect to λ if micro 6 λ and λ is not perpendicular to any connected componentof supp(λminus micro) Let V be an integrable module of highest weight λ

(1) If micro isin P+ then micro isin wtV if and only if micro is non-degenerate with respect to λ(2) If the sub-diagram on i isin I (αi λ) = 0 is a disjoint union of diagrams of finite type

then micro isin P+ is non-degenerate with respect to λ if and only if micro 6 λ


(3) wtV = (λminus Zgt0π) cap conv(Wλ)

34 Convexity Let E be a finite-dimensional real vector space (we will take E = hlowast) For asubset X sub E write convX for its convex hull and aff X for its affine hull For an h-moduleM for brevity we write convM aff M for conv wtM aff wtM respectively A collection ofvectors v0 vn is said to be affine independent if their affine hull has dimension n A closedhalf space is a locus of the form ζminus1([tinfin)) for ζ isin Elowast t isin R For us a polyhedron is a finiteintersection of closed half spaces and a polytope is a compact polyhedron The following wellknown alternative characterization of polyhedra will be useful to us

Proposition 38 (WeylndashMinkowski) A subset X sub E is a polyhedron if and only if it can bewritten as the Minkowski sum X = conv(X prime) +Rgt0X primeprime for two finite subsets X prime X primeprime of E If Xis a polyhedron that does not contain an affine line then X prime X primeprime as above of minimal cardinalityare unique up to rescaling vectors in X primeprime by Rgt0

Let C sub E be convex We endow aff C with the analytic topology and denote by relintC theinterior of C viewed as a subset of aff C this is always non-empty For brevity we will oftenrefer to c isin relintC as an interior point of C For a point c isin C we denote the tangent cone atc by TcC recall this is the union of the rays c + Rgt0(cprime minus c) forallcprime isin C Given a highest weightmodule V and micro isin wtV we will write TmicroV for Tmicro(conv wtV ) for convenience

A convex subset F sub C is called a face if whenever a convex combinationsum

i tici of pointsci 1 6 i 6 n of C lies in F all the points ci lie in F Two convex sets CC prime sub E are said to becombinatorially isomorphic if there is a dimension-preserving isomorphism of their face posetsordered by inclusion

If ζ is a real linear functional and ζ|C has a maximum value m the locus c isin C (ζ c) = mis an exposed face All exposed faces are faces but the reverse implication does not hold ingeneral The following simple lemmas will be used later

Lemma 39 If F is a face of C then F = C cap aff F

Lemma 310 If F1 F2 are faces of C and F1 contains a point of relintF2 then F1 containsF2 In particular if relintF1 cap relintF2 6= empty then F1 = F2

Lemma 311 If F is a face of C and λ isin F then TλF is a face of TλC

By a rational structure on E we mean the choice of a rational subspace PQ sub E such that

PQ otimesQ R simminusrarr E Let us fix a rational structure P on E = hlowast such that Qπ sub PQ With thechoice of rational structure one can make sense of rational vectors rational hyperplanes andhalf-spaces and by taking finite intersections of rational half-spaces rational polyhedra Wehave the following

Proposition 312 (Basic properties of rational polyhedra)

(1) The affine hull of a rational polyhedron is rational(2) A cone is rational if and only if it is generated by rational vectors(3) A polytope is rational if and only if its vertices are rational points(4) A polyhedron is rational if and only if it is the Minkowski sum of a rational polytope and

a rational cone(5) Every face of a rational polyhedron is a rational polyhedron(6) Every point in a rational polyhedron is a convex combination of rational points

Proof (1)ndash(5) may be found in [5 Proposition 169] For the last point by (5) and the supportinghyperplane theorem it suffices to consider an interior point by (1) it suffices to note that if Uis an open set in Rn containing a point q we can find qi isin U cap Qn such that q is a convexcombination of the qi one may in fact choose them so that aff qi = Rn


35 Results from previous work on highest weight modules We now recall results fromrecent work [8] on the structure of wtV for an arbitrary highest weight g-module V The firstresult explains how every module V is equal up to ldquofirst orderrdquo to its parabolic Verma coverM(λ IV ) In particular we obtain the convex hull of wtV

Theorem 313 The following data are equivalent

(1) IV the integrability of V (2) conv V the convex hull of the weights of V (3) The stabilizer of conv V in W

In particular the convex hull in (2) is always that of the parabolic Verma module M(λ IV ) andthe stabilizer in (3) is always the parabolic subgroup WIV

The argument for the equality of convM(λ IV ) and conv V involves two ingredients that arerepeatedly used in this paper The first is to study the weights of M(λ IV ) via restriction tothe Levi subalgebra lIV corresponding to IV The representation decomposes into a direct sumcorresponding to cosets of hlowast modulo translation by ∆IV the roots of lIV and it transpires eachlsquoslicersquo has easily understood weights

Proposition 314 (Integrable Slice Decomposition [8]) For J sub IL(λ) we have

wtM(λ J) =⊔

microisinZgt0(ππJ )

wtLlJ (λminus micro) (315)

where LlJ (ν) denotes the simple lJ -module of highest weight ν In particular wtM(λ J) lies inthe J Tits cone (cf Section 32) Moreover

wtM(λ J) = (λ+ Zπ) cap convM(λ J) (316)

We include two illustrations of Equation (315) in Figure 31

λminusα1 minus α2

minusα1

minusα3

g =

α1 α2 α3

α1 2 minus1 0α2 minus1 2 minus1α3 0 minus1 2

J = 1 2

λ

minusα0

minusα1

minusα2

g =

α0 α1 α2

α0 2 minus2 minus1α1 minus2 2 0α2 minus1 0 2

J = 0 1

Figure 31 Integrable Slice Decomposition with finite and infinite integrabilityNote we only draw four of the infinitely many rays originating at λ in the right-hand illustration


Next using Proposition 314 we obtain a formula for convM(λ IV ) via lsquogeneratorsrsquo ie theconvex hull of an explicit set of points and rays which also lie in conv V

Proposition 317 (Ray Decomposition [8])

conv V = conv⋃

wisinWIV iisinIIV

w(λminus Zgt0αi) (318)

When IV = I by the right-hand side we mean conv⋃wisinW wλ Moreover each ray w(λminusRgt0αi)

is a face of conv V

We include two illustrations of Proposition 317 in Figure 32 To our knowledge Proposition317 was not known in either finite or infinite type prior to [8]


minusα1

minusα3

λ

minusα0

minusα1

minusα2

Figure 32 Ray Decomposition with finite and infinite integrability here gand J are as in Figure 31

4 Classification of faces of highest weight modules

Let V be a g-module of highest weight λ For a standard Levi subalgebra l we introduce thenotation wtl = conv wtU(l)Vλ and its translates by the integrable Weyl group wwtl forallw isinWIV We first check these are indeed faces of conv V

Lemma 41 wwtl is an (exposed) face of conv V for any standard Levi l and w isinWIV

Proof It suffices to consider w = 1 ie wtl Write l = lJ for some J sub I and chooselsquofundamental coweightsrsquo ωi in the real dual of hlowast satisfying (ωi αj) = δij foralli j isin I Then it isclear that

sumiisinIJ ωi is maximized on wt l as desired

The non-trivial assertion is that the converse to Lemma 41 also holds which is the mainresult of this section

Theorem 42 Every face of conv V is of the form wwtl for some standard Levi l and w isinWIV

To prove this we first examine the orbits of wt l under WIV in the spirit of [26]

Proposition 43 (Action of WIV on faces) Let l lprime be standard Levi subalgebras and w isinWIV

(1) wtl has an interior point in the IV dominant chamber(2) If wwtl has an interior point in the IV dominant chamber then wwtl = wtl


(3) If the WIV orbits of wtlwtlprime are equal then wtl = wtlprime

Proof

(1) Let ν be an interior point of wtl and write l = lJ for some J sub I Setting J prime = IV capJ itis easy to see WJ prime is the integrable Weyl group of U(lJ)Vλ In particular by Proposition314 it lies in the IV Tits cone whence acting by WJ prime we may assume ν lies in the J prime

dominant chamber ie (αj ν) gt 0forallj isin J prime Since ν lies in λminus Rgt0πJ it follows for alli isin IV J that (αi ν) gt 0 Thus ν lies in the IV dominant chamber

(2) The proof of (1) showed that in fact any interior point of wtl may be brought via WJ prime

into the IV dominant chamber Let ν prime be an interior point of wwtl which is IV dominantThen wminus1ν prime is an interior point of wtl pre-composing w with an element of WJ prime we mayassume wminus1ν prime is IV dominant But since the IV dominant chamber is a fundamentaldomain for the IV Tits cone (cf Proposition 32) it follows wminus1ν prime = ν prime As the faceswtl wwtl have intersecting interiors by Lemma 310 they coincide

(3) This is immediate from the first two assertions

Proof of Theorem 42 We proceed by induction on |I| ie the number of simple roots So wemay assume the result for all highest weight modules over all proper Levi subalgebras

If F is a face of conv V by the Ray Decomposition 317 F contains wλ for some w isin WIV Replacing F by wminus1F we may assume F contains λ Passing to tangent cones it follows byLemma 311 that TλF is a face of TλV We observe that the latter has representation theoreticmeaning

Lemma 44 Set J = IV cap λperp where λperp = i isin IV (αi λ) = 0 Then TλV = convM(λ J)

Proof Composing the surjections M(λ J) rarr M(λ IV ) rarr V we have TλV sub TλM(λ J) Onthe other hand by the Ray Decomposition 317

convM(λ J) = conv⋃

wisinWJ iisinIJ

w(λminus Zgt0αi) = TλM(λ J)

which evidently lies in TλV

It may be clarifying for the reader in the remainder of the argument to keep in mind theright-hand picture in Figure 31

We now reduce to proving Theorem 42 for M(λ J) with J = IV cap λperp as above Indeedassuming this holds by Lemma 311 we may write TλF = wwtlM(λ J) for w isin WJ Againreplacing F with wminus1F we may assume TλF = wtlM(λ J) By a similar argument to Lemma44 Tλ wtl V = wtl TλV = TλF It follows that aff F = aff wtl whence F = wtl by Lemma 39

It remains to prove Theorem 42 for V = M(λ J) We may assume J is a proper subset ofI otherwise V is a 1-dimensional module and the theorem is tautological Let F be a face ofconv V We first claim that F contains a relative interior point v0 such that v0 isin λ minus Qgt0πIndeed let V prime = conv V minusλ F prime = Fminusλ Then by the Ray Decomposition 317 V prime is the convexhull of a collection of rational rays ri = Rgt0pi pi isin PQ using the rational structure PQ sub hlowast

fixed in Section 34 If F prime is not the cone point 0 for which the theorem is tautological it isthe convex hull of those rays ri whose relative interiors it contains In particular F prime is again aconvex hull of rational rays rprimei It follows that aff F prime = P primeotimesQ R where P prime sub PQ is the Q-span ofthe rprimei Since relintF prime is open in aff F prime it is now clear it contains a rational point as desired

We next reduce to the case of an interior point in λ minus Zgt0π To see why note that for anyn isin Zgt0 dilating hlowast by a factor of n gives an isomorphism dn convM(λ J)rarr convM(nλ J)eg by the Ray Decomposition 317 Moreover dn sends wwtlM(λ J) to wwtlM(nλ J) forall standard Levi subalgebras l and w isinWJ


With these reductions let v0 be an interior point of F lying in λminusZgt0π Since v0 lies in theJ Tits cone by Proposition 314 replacing F by wF for some w isin WJ we may assume v0 is inthe J dominant chamber Write

v0 = λminussumjisinJ prime

njαj minussumiisinIprime

niαi where J prime sub J I prime sub I J and nj ni isin Zgt0forallj isin J prime i isin I prime

Write l for the Levi subalgebra associated to J prime t I prime we claim that v0 is also a interior pointof wtl which will complete the proof by Lemma 310 We address this claim in two steps byanalyzing first the integrable directions J prime and then incorporating I prime

Consider the integrable slice convLlJ (λprime) where λprime = λminussum

iisinIprime niαi We first claim v0 is aninterior point of wtlJprime LlJ (λprime) Recalling that J is a proper subset of I the claim follows fromthe following lemma applied to the lJ prime-module generated by LlJ (λprime)λprime

Lemma 45 Assume Theorem 42 is true for g Suppose a point v0 of conv V (i) lies in the IVdominant chamber and (ii) is of the form λminus

sumiisinI ciαi ci gt 0 foralli isin I Then v0 is an interior

point of conv V

Proof Iteratively applying the supporting hyperplane theorem we may assume v0 is an interiorpoint of a face F By Theorem 42 F = wwtl for some standard Levi subalgebra l ( g andw isinWIV By assumption (i) and Proposition 43 we may take w = 1 and now (ii) yields l = gcompleting the proof

Next it is easy to see that dim aff wtlJprime LlJ (λprime) = |J prime| Indeed the inequality 6 always holdsFor the reverse inequality recall that v0 = λprimeminus

sumjisinJ prime njαj nj gt 0forallj isin J prime Consider a lowering

operator D of weight minussum

jisinJ prime njαj in U(nminusJ prime) which does not annihilate the highest weight line

of V = M(λ J) and whose existence is guaranteed by Equations (315) and (316) Usingthe surjection from the tensor algebra generated by fj j isin J prime onto U(nminusJ prime) there exists a simpletensor in the fj of weight minus

sumjisinJ prime njαj that does not annihilate the highest weight line Viewing

the simple tensor as a composition of simple lowering operators it follows that the weight spacesit successively lands in are nonzero Since nj gt 0 forallj isin J prime this proves the reverse inequality

Now choose affine independent p0 p|J prime| in wtlJprime LlJ (λprime) whose convex hull contains v0 asan interior point Write each as a convex combination of WJ prime(λ

prime) ie

pk =sumj

tkjwprimej(λprime) with wprimej isinWJ prime tkj isin Rgt0

sumj

tkj = 1 0 6 k 6 |J prime|

As affine independence can be checked by the non-vanishing of a determinant we may pick acompact neighborhood B around

sumiisinIprime niαi in Rgt0πIprime such that for each b isin B the points

pk(b) =sum

j tkjwprimej(λ minus b) are again affine independent Writing Σ(b) = convpk(b) 0 6 k 6

|J prime| and Σ(B) = cupbisinBΣ(b) is it easy to see we have a homeomorphism Σ(B) σ timesB whereσ denotes the standard |J prime|-simplex In particular v0 is an interior point of Σ(B) Notingthat Σ(B) sub wtl and dim aff wtl 6 dim aff Σ(B) = |J prime| + |I prime| it follows that dim aff Σ(B) =dim aff wtl and hence v0 is an interior point of wtl as desired

5 Inclusions between faces

Having shown in Section 4 that every face is of the form wwtl to complete their classificationit remains to address their inclusion relations and in particular their redundancies We firstignore the complications introduced by the Weyl group ie classify when wtl sub wtlprime for Levisubalgebras l lprime

To do so we introduce some terminology For J sub I call a node j isin J active if either (i)j isin IV or (ii) j isin IV and (αj λ) isin Rgt0 and otherwise call it inactive Notice that the active


nodes are precisely those j isin J for which λ minus αj isin wtV Decompose the Dynkin diagramcorresponding to J as a disjoint union of connected components Cβ and call a componentCβ active if it contains an active node Finally let Jmin denote the nodes of all the activecomponents The above names are justified by the following theorem

Theorem 51 Let l lprime be standard Levi subalgebras and J J prime the corresponding subsets of Irespectively Then wtl sub wtlprime if and only if Jmin sub J prime

Proof First by the PBW theorem and V = U(nminus)Vλ

U(l)Vλ =oplus

microisinλ+ZπJ

Vmicro (52)

so wtl = conv V cap (λ+ RπJ) Accordingly it suffices to check that

aff wtl = λ+ RπJmin (53)

To see sup for any node i isin Jmin we may choose a sequence i1 in = i of distinct nodes inJmin such that (i) i1 is active and (ii) ij is inactive and shares an edge with ijminus1 forallj gt 2 Thenby basic sl2 representation theory it is clear the weight spaces corresponding to the followingsequence of weights are nonzero

λminus αi1 λminus (αi1 + αi2) λminussum

16j6n

αij (54)

It is now clear that aff wtl contains λ+ Rαi proving sup as desiredFor the inclusion sub by assumption the simple coroots corresponding to J Jmin act by 0

on λ and are disconnected from Jmin in the Dynkin diagram In particular as Lie algebrasnminusJ = nminusJmin

oplus nminusJJmin whence by the PBW theorem

U(l)Vλ = U(nminusJmin)otimesC U(nminusJJmin

)Vλ

and the augmentation ideal of U(nminusJJmin) annihilates Vλ as desired

Remark 55 Equation (52) immediately implies a result in [18] namely that wtl = wtlprime ifand only if U(l)Vλ = U(lprime)Vλ

As we have seen J Jmin doesnrsquot play a role in wtl1 Motivated by this let

Jd = j isin IV (αj λ) = 0 and j is not connected to Jmin (56)

and set Jmax = Jmin t Jd Here we think of the subscript d as standing for lsquodormantrsquo An easyadaptation of the argument of Theorem 51 in particular Equation (53) shows the following

Theorem 57 Let l lprime be standard Levi subalgebras and J J prime the corresponding subsets of IThen wtl = wtlprime if and only if Jmin sub J prime sub Jmax

Remark 58 We now discuss precursors to Theorem 57 The sets Jmin J sub I appear inSatake [25 sect23] BorelndashTits [3 sect1216] Vinberg [26 sect3] and Casselman [6 sect3] for integrablemodules in finite type To our knowledge Jmax first appears in work of CellinindashMarietti [7] andsubsequent work by Khare [18] again in finite type

Finally we comment on the relationship of the language in Theorems 51 and 57 to the twoprevious approaches (i) that of Vinberg [26] and Khare [18] and (ii) that of Satake [25] BorelndashTits [3] Casselman [6] CellinindashMarietti [7] and LindashCaondashLi [21] Ours is essentially an averageof the two different approaches in our terminology unlike in [18] we do not distinguish between

1In fact an easy adaptation of the proof of Theorem 51 shows the derived subalgebra of gJJminacts trivially

on U(l)Vλ


integrable and non-integrable lsquoactiversquo nodes which simplifies the subsets J1ndashJ6 of Definition 31of loc cit to our Jmin and Jmax Similarly if we extend the Dynkin diagram of g by adding anode i that is connected to exactly the active nodes then we may rephrase our results alongthe lines of [3] [6] [7] [21] [25] as follows standard parabolic faces of conv V are in bijectionwith connected subsets of the extended Dynkin diagram that contain i

Corollary 59 Let L(λ) be a simple module and lJ a standard Levi subalgebra The restrictionof L(λ) to lJ is simple if and only if Imin sub J

Corollary 510 Let V be a highest weight module and lJ a standard Levi subalgebra Therestriction of V to lJ is a highest weight module if and only if Imin sub J

We are ready to incorporate the action of the integrable Weyl group Of course to understandwhen wprimewtlprime sub wwtl we may assume wprime = 1

Proposition 511 Set JmaxV = IV cap Jmax and Jmin

V = IV cap Jmin Given w isin WIV we havewtlprime sub wwtl if and only if (i) wtlprime sub wtl and (ii) the left coset wWJmin

Vintersects the integrable

stabilizer StabWIV(wtlprime) non-emptily

Proof One implication is tautological For the reverse implication suppose wtlprime sub wwtl Ap-plying Theorem 42 to gJmin

V we have wtlprime = wwprimewtlprimeprime for some wprime isin WJmin

V lprimeprime sub l Then by

Proposition 43 wtlprime = wtlprimeprime = wwprimewtlprimeprime

Our last main result in this section computes the stabilizers of the parabolic faces of conv V under the action of the integrable Weyl group

Theorem 512 Retaining the notation of Proposition 511 we have

StabWIV(wtl) = WJmax

VWJmin

VtimesWJmax

V JminV (513)

Proof The inclusion sup is immediate For the inclusion sub write F for the face wtl and D forthe IV dominant chamber DIV Then by an argument similar to Proposition 43 we have that

F =⋃

wisinWJminV

(F cap wD)

Suppose that w isinWIV stabilizes F Then we have

F =⋃

wisinWJminV

(F cap wwD)

As F capD cap wwD is a face of F capD for all w isinWJminV

it follows by dimension considerations

that F cap D = F cap D cap wwD for some w isin WJminV

So replacing w by ww we may assume w

fixes F capD identically It suffices to show this implies w isinWJd as defined in Equation (56) Tosee this for i isin Jmin choose as in the proof of Theorem 51 a sequence i1 in = i of distinctij isin Jmin such that (i) i1 is active and (ii) ij is inactive and shares an edge with ijminus1 forallj gt 2Write K(i) = ik16k6n it follows from Proposition 43(1) that wtK(i) has an interior point viin D Write vi = λ minus

sumnk=1 ckαik and let K prime = ik ck gt 0 Then applying Lemma 45 for

gKprime it follows that vi is an interior point of wtKprime whence wtKprime = wtK(i) by Lemma 310 ButK(i) = K(i)min so K prime = K(i) and we have ck is nonzero 1 6 k 6 n Since w fixes vi it followsby Proposition 32 that w isinWJ(i) where we write

J(i) = IV cap vperpi = j isin IV (αj λminussumk

ckαik) = 0


Since i isin Jmin was arbitrary by the observation WKprime capWKprimeprime = WKprimecapKprimeprime forallK primeK primeprime sub I wehave

w isinWK where K = k isin IV (αk λ) = 0 (αk αi) = 0 foralli isin JminBut this is exactly Jd as desired

We now deduce the coefficients of the f -polynomial of the ldquopolyhedralrdquo set conv V althoughas we presently discuss conv V is rarely a polyhedron in infinite type The following result is aconsequence of Theorems 42 and 512 Proposition 43 and Equation (53)

Proposition 514 Write fV (q) for the f -polynomial of conv V ie fV (q) =sum

iisinZgt0 fiqi

where fi isin Zgt0 cup infin is the number of faces of dimension i Then we have

fV (q) =sumwtlJ

[WIV WIV capJmax ]q|Jmin| (515)

where wtlJ ranges over the (distinct) standard parabolic faces of V and we allow fV (q) to havepossibly infinite coefficients

Proposition 514 was previously proved by Cellini and Marietti [7] for the adjoint representa-tion when g is simple and by the second named author [18] for g finite-dimensional and mostV Thus Proposition 514 addresses the remaining cases in finite type as well as all cases for gof infinite type

Example 516 In the special case of λ isin P+ regular the formula yields

fV (q) =sumJsubI

[W WJ ]q|J | (517)

Example 518 More generally if λ is IV regular then writing Vtop for the IV integrable sliceconvU(lIV )Vλ we have

fV (q) = fVtop(q) middot (1 + q)|I|minus|IV | (519)

where fVtop(q) is as in (517)

Example 518 is a combinatorial shadow of the fact that for λ IV regular the standardparabolic faces of conv V are determined by their intersection with the top slice and theirextremal rays at λ along non-integrable simple roots

Figure 51 shows an example in type A3 namely a permutohedron In this case the f -polynomial equals q3 + 14q2 + 36q + 24 and on facets we get 14 = 4 + 4 + 6 corresponding tofour hexagons each from 1 2 and from 2 3 and six squares from 1 3

Figure 51 A3-permutohedron (ie λ isin P+ regular)


6 Families of Weyl polyhedra and representability

Let g be a KacndashMoody algebra and p a parabolic subalgebra with Levi quotient l Considerindg

p L for L an integrable highest weight l module This induced module still has a weightdecomposition with respect to a fixed Cartan of l and one can study the convex hulls of theweights of these modules as L varies

Less invariantly if l corresponds to a subset J sub I of the simple roots the resulting convexhulls are precisely those of the parabolic Verma modules with integrability J or equivalently ofall highest weight modules with integrability J

We now introduce a family of convex shapes interpolating between the above convex hullsIn particular for the remainder of this section we fix a subset of the simple roots J sub I

Definition 61 Say a subset P sub hlowast is a Weyl polyhedron if it can be written as a convexcombination

P =sumi

ti conv Vi ti gt 0sumi

ti = 1

where Vi are highest weight modules with integrability J

We will see shortly that Weyl polyhedra are always polyhedra if and only if the correspondingLevi is finite dimensional so this is strictly speaking abuse of notation

Having introduced lsquoreal analoguesrsquo of convex hulls of parabolically induced modules we cannow let convex hulls vary in families By a convex set S we simply mean a subset of some realvector space which is convex is the usual sense

Definition 62 Let S be a convex set Define an S family of Weyl polyhedra to be a subsetP sub S times hlowast satisfying

(1) Under the natural projection π P rarr S the fibers Fs = πminus1s are Weyl polyhedra forall s isin S

(2) For s0 s1 isin S 0 6 t 6 1 we have

tFs0 + (1minus t)Fs1 = Fts0+(1minust)s1

We remind that a morphism of convex sets S rarr Sprime is a map of sets which commutes withtaking convex combinations With this we have that families of Weyl polyhedra pull back

Lemma 63 If φ Sprime rarr S is a morphism of convex sets and P rarr S is an S family of Weylpolyhedra then the base change φlowastP = P timesS Sprime is an Sprime family of Weyl polyhedra

It follows that the assignment F S S families of Weyl polyhedra φ φlowast defines acontravariant functor from convex sets to sets We now construct what could be loosely thoughtof as a convexity-theoretic Hilbert scheme for Weyl polyhedra

Theorem 64 The functor F is representable

From the theorem we obtain a unique up to unique isomorphism base space B and universalfamily Puniv rarr B In the subsequent section we will show that the total space of Puniv is aremarkable convex cone whose combinatorial isomorphism type stores the classification of facesobtained in the previous sections

Proof We need to show there exists a convex set B and bijections for every convex set S

S families of Weyl polyhedra Hom(SB) (65)

which intertwine pullback of families and precomposition along morphisms S rarr SprimeTo do so we introduce in the following lemma a Ray Decomposition for Weyl polyhedra akin

to Proposition 317


Lemma 66 For λ isin DJ ie (αj λ) isin Rgt0forallj isin J set

P (λ J) = conv⋃

wisinWJ iisinIJ

w(λminus Rgt0αi) (67)

If J = I by the RHS we mean conv⋃wisinW wλ Then for any finite collection λi isin DJ ti gt

0sum

i ti = 1 we have sumi

tiP (λi J) = P (sumi

tiλi J) (68)

The lemma is proved by direct calculationNotice that a highest weight module V with integrability J and highest weight λ has conv V =

P (λ J) It follows from Equation (68) that Weyl polyhedra are precisely the shapes P (λ J)for λ isin DJ As these have a unique vertex in the J dominant chamber namely λ they inparticular are distinct and thus we have a naive parametrization of Weyl polyhedra by theirhighest weights

Introduce the tautological family PJ rarr DJ where we define

PJ = (λ ν) λ isin DJ ν isin P (λ J) (69)

A direct calculation shows that PJ is a DJ family of Weyl polyhedraWe now show that DJ is the moduli space that represents F with universal family Puniv = PJ

Indeed given a morphism of convex sets φ S rarr DJ we associate to it the pullback of thetautological family φlowastPJ In the reverse direction given an S family of Weyl polyhedra ifthe fiber over s isin S is P (λ(s) J) then the map s 7rarr λ(s) is a morphism of convex sets byEquation (68) One can directly verify these assignments are mutually inverse and intertwineprecomposition and pullback of families under a convex morphism S rarr Sprime

Remark 610 Notice that both sides of Equation (65) are naturally convex sets (given bytaking convex combinations fiberwise and pointwise respectively) and in fact the isomorphismis one of convex sets

Example 611 For a simplex S with vertices v0 vn an S family of Weyl polyhedra is thesame datum as the fibers at the vertices ie Weyl polyhedra P0 Pn Thus in this case bothsides of Equation (65) indeed reduce to Dn+1

J

Example 612 We could just as well have parametrized Weyl polyhedra by their unique vertexin another fixed chamber of the J Tits cone Writing the chamber as wDJ w isin WJ theisomorphism of convex sets wDJ DJ corresponding to Theorem 64 is simply application ofwminus1

We conclude this section with an illustration of a universal family of Weyl polyhedra seeFigure 61

7 Faces of the universal Weyl polyhedron

In the previous section we began studying families of Weyl polyhedra P rarr S In this sectionwe study faces of such families ie of the total spaces P which indeed are convex Notice thatjust as families pull back so do faces

Lemma 71 Let φ Sprime rarr S be a morphism of convex sets P rarr S a family of Weyl polyhedraand F rarr P a face of the total space Then the pullback φlowastF is a face of φlowastP

We motivate what follows with a very natural face of any family As before we work withWeyl polyhedra corresponding to a fixed J sub I


Figure 61 Schematic of the universal family of Weyl polyhedra for g = sl3and J = I The fibers P (λ J) are illustrated at some points λ along the relativeinteriors of each face in the dominant chamber D = DI

Example 72 For j isin J consider the corresponding root hyperplane Hj in hlowast If P rarr S is afamily of Weyl polyhedra then the union of the fibers whose highest weight lies in Hj is a faceof P

Notice in the above example that this construction commutes with pullback of families In thisway one can study lsquocharacteristic facesrsquo ie the assignment to every family of Weyl polyhedraP of a face F sub P which commutes with pullback As the intersection of each face with a givenfiber is a face of the fiber one may think of characteristic faces as controlling how the faces ofconv V deform as one deforms the highest weight

As families of Weyl polyhedra admit a moduli space ie Theorem 64 characteristic faces arethe same data as faces of Puniv which we call the universal Weyl polyhedron In this section wewill determine Puniv up to combinatorial isomorphism ie enumerate all faces as well as theirdimensions and inclusions

The main ingredients will be (i) an extension of the classification of faces of conv V fromSections 4 5 to all Weyl polyhedra and (ii) a stratification of the base DJ along which Punivis lsquoconstantrsquo The latter will in particular strengthen the consequence of Section 5 that theclassification of faces of conv V only depends on the singularity of the highest weight whichfollowed from the definitions of Jmin Jmax

71 Faces of Weyl polyhedra In this subsection we show the classification of faces ofP (λ J) It is exactly parallel to that of convM(λ J) and we elaborate whenever the proofsdiffer

For J prime sub I with associated Levi subalgebra l consider the functionalsum

iisinIJ prime ωi as in Lemma

41 As P (λ J) sub λ minus Rgt0πsum

iisinIJ prime ωi attains a maximum on P (λ J) and write wtl for the

corresponding exposed face As P (λ J) is WJ invariant similarly wwtl is a face of P (λ J) forw isinWJ

Theorem 73 Every face of P (λ J) is of the form wwtl for w isin WJ and a standard Levialgebra l

Proof This is proved similarly to Theorem 42 The analogue of Lemma 44 may be proved asfollows


Lemma 74 TλP (λ J) = P (λ J cap λperp) where λperp = i isin I (αi λ) = 0

Proof The inclusion supe is as in Lemma 44 For the inclusion sube we prove by induction on `(w)that w(λ minus tαi) isin P (λ J cap λperp) for t gt 0 i isin Jw isin WJ The base case of w = 1 is clear Forthe induction step write w = sjw with `(w) lt `(w) j isin J If j isin λperp we use the inductive

hypothesis for w and the sj invariance of P (λ J cap λperp) If j isin λperp since λ is J-dominant wehave

w(λminus tαi) = w(λminus tαi)minus tprimeαj tprime isin Rgt0

The proof concludes by noting that P (λ J cap λperp) is a cone with vertex λ and contains λ minusRgt0αj

Following the notation of the proof of Theorem 42 the only argument which requires non-trivial modification is the equality dim aff wtlJprime LlJ (λprime) = |J prime| This however is a consequence ofthe following Theorem 75

We next determine the inclusions between standard parabolic faces For J prime sub I the notionsof active and inactive nodes as well as J primemin J

primemax are as in Section 5 provided one replaces IV

with J in the definitions With this we have

Theorem 75 Let lprime lprimeprime be standard Levi subalgebras and J prime J primeprime the corresponding subsets of Irespectively Then wtlprime sub wtlprimeprime if and only if J primemin sub J primeprime

Proof It suffices to check that aff wtlprime = λ + RπJ primemin The inclusion sup is proved as in Theorem

51 replacing λ minus αi1 with λ minus ε1αi1 for some ε1 gt 0 λ minus αi1 minus αi2 with λ minus ε1αi1 minus ε2αi2 forsome ε2 gt 0 and so on For the inclusion sub we will show a lsquoRay Decompositionrsquo for each face

wtlprime = conv⋃

wisinWJcapJprime iisinJ primeJ


Having shown this we will be done as we may factor WJcapJ prime WJcapJ primemintimesWJ primeJ primemin

and the

latter parabolic factor acts trivially on λminusRgt0αiforalli isin J primeJ To see Equation (76) the inclusionsup is straightforward For the inclusion sub it suffices to see that if w(λ minus tαi) lies in wtlprime forsome w isinWJ t gt 0 and i isin I J then (i) i isin J prime J and (ii) w(λminus tαi) = w(λminus tαi) for somew in WJcapJ prime But (i) follows from the linear independence of simple roots and (ii) follows fromnoting that w(λminus tαi) lies in the J cap J prime Tits cone and λminus tαi lies in the J dominant chamberwhich is a fundamental domain for the action of WJ cf Proposition 32(2)

The remaining results of Section 5 namely Theorem 57 Proposition 511 Theorem 512 andProposition 514 hold for P (λ J) mutatis mutandis with similar proofs In particular the faceposet of P (λ J) depends on λ isin DJ only through J cap λperp proving Theorem 24(2)

72 Strong combinatorial isomorphism and strata of the Weyl chamber Before con-tinuing let us sharpen our observation above that the combinatorial isomorphism class of P (λ J)depends only on the lsquointegrable stabilizerrsquo of λ To do so we first remind the face structure ofDJ

Lemma 77 Faces of DJ are in bijection with subsets of J under the assignment

K sub J perpK = λ isin DJ (αk λ) = 0 forallk isin K

Moreover this bijection is an isomorphism of posets where we order faces by inclusion andsubsets of J by reverse inclusion


Let us call the interior of a face of DJ a stratum of DJ With this our earlier observation isthat the combinatorial isomorphism class of P (λ J) is constant on strata of DJ

To explain the sharpening we recall a notion from convexity cf [1] Two convex sets CC prime

in a real vector space E are said to be strongly combinatorially isomorphic if for all functionalsφ isin Elowast we have (i) φ attains a maximum value on C if and only if φ attains a maximumvalue on C prime and (ii) if the maxima exist writing F F prime for the corresponding exposed faces ofCC prime we have dimF = dimF prime Informally strong combinatorial isomorphism is combinatorialisomorphism plus lsquocompatiblersquo embeddings in a vector space

Proposition 78 Fix λ λprime isin DJ Then

(1) If P (λ J) and P (λprime J) are strongly combinatorially isomorphic then λ and λprime lie in thesame stratum of DJ

(2) If λ λprime have finite stabilizer in WJ then the converse to (1) holds ie if they lie in thesame stratum DJ then P (λ J) and P (λprime J) are strongly combinatorially isomorphic

Proof To see (1) if P (λ J) and P (λprime J) are strongly combinatorially isomorphic using thefunctionals

sumi 6=j ωi j isin J shows that λ λprime lie in the interior of the same face of DJ To see

(2) let φ be a functional which is bounded on P (λ J) By applying some w isin WJ to φwe may assume Fφ contains λ By the assumption of finite stabilizer in WJ we may further

assume (φ αj) gt 0 for all j isin J cap λperp Since λ minus εαi isin P (λ J) for some ε gt 0 i isin J cap λperpit follows for P (λ J) P (λprime J) that φ cuts out the standard parabolic face corresponding toi isin I (φ αi) = 0 As by assumption λprime λ lie in the same faces of the J dominant chamberthe equality of dimensions follows from Theorem 75

Remark 79 It would be interesting to know if Proposition 78(2) holds without the assump-tion of finite integrable stabilizer Note also that the proof contains a short argument for theclassification of faces for P (λ J) when λ has finite integrable stabilizer

73 Faces of the universal Weyl polyhedron We now determine the face poset of Puniv =PJ More generally in this subsection we do so for P rarr S an arbitrary family of Weyl polyhedrawhere S admits a convex embedding into a finite dimensional real vector space

Write F for the face poset of S By our assumption on S it is stratified by the relativeinteriors of its faces

S =⊔FisinF

SF

We first note that with respect to this stratification P is lsquoconstantrsquo on strata

Lemma 710 Let F isin F Then for all s sprime isin SF λ(s)perp = λ(sprime)perp where as before λ(s) denotesthe highest weight of the fiber over s

In particular the fibers over SF are combinatorially isomorphic

Proof Under the corresponding convex morphism F rarr DJ if an interior point of F lands inthe interior of a face of DJ then the interior of F lands in the interior of that face

We now produce lsquotautologicalrsquo faces of P by taking families of standard parabolic faces alongthe closure of a stratum

Proposition 711 For F isin F and a standard Levi l of g the locus

wtlF = (s x) s isin F x isin wtl P (λ(s) J)

is a face of P


Proof The pullback of the face F along the projection π P rarr S yields a face πlowastF of PAs in Lemma 41 consider the exposed face of πlowastF corresponding to the vanishing locus of(s x) 7rarr

sumi 6isinJ(ωi λ(s)minus x) to obtain the desired face wtlF of P

Notice that WJ acts on P via the usual action on the second factor Thus we obtain morefaces wwtlF of P ndash in fact all of them

Theorem 712 Every face of P is of the form wwtlF for some w isinWJ standard Levi l andF isin F

Proof For any face of P choose an interior point (s x) s isin S x isin P (λ(s) J) By acting byWJ we may assume that x is J dominant By Theorem 73 and the analogue of Proposition 43x is an interior point of wtl P (λ(s) J) for a standard Levi l For some F isin F s isin SF As in theproof of Theorem 42 we produce a ldquoproductrdquo neighborhood of (s x) in wtlF in the followingmanner Take a relatively open simplex in wtl P (λ(s) J) containing x Writing its vertices asconvex combinations of the terms in the Ray Decomposition (76) in an open neighborhood ofs in SF the corresponding convex combinations are affine independent By dimension counting(s x) is an interior point of wtlF and we are done by Lemma 310

Remark 713 Theorem 712 shows that every face of an S family of Weyl polyhedra is againa family of Weyl polyhedra provided one passes to (i) a face of the base and to (ii) a Levisubalgebra of g

We next deduce the structure of the face poset of P from our prior analysis of the face posetsof the fibers

Proposition 714 Let J prime J primeprime sub I with associated Levi subalgebras lprime lprimeprime and let F prime F primeprime isin F Below we calculate J primemin and J primemax for s lying in the stratum of S corresponding to F prime cf Lemma710 Then

(1) wtlprimeF prime sub wtlprimeprimeF primeprime if and only if (i) F prime sub F primeprime and (ii) J primemin sub J primeprime(2) wtlprimeF prime = wtlprimeprimeF primeprime if and only if (i) F prime = F primeprime and (ii) J primemin sub J primeprime sub J primemax(3) For w isin WJ we have wwtlprimeF prime sub wtlprimeprimeF primeprime if and only if (i) F prime sub F primeprime and (ii) for an

interior point s of F prime we have wwtlprime P (λ(s) J) sub wtlprimeprime P (λ(s) J) cf Proposition 511(4) The stabilizer of wtlprimeF prime in WJ is WJ primemaxcapJ

In particular we obtain the following formula for the f -polynomial of P

Proposition 715 For F isin F write fF (q) for the f polynomial of P (λ(s) J) s isin SF Then

fP(q) =sumFisinF

qdimF fF (q)

The above results may be specialized to the case of the universal Weyl polyhedron Puniv = PJusing the description of the face poset of the base DJ reminded in Lemma 77 Before doing sowe first determine when the universal Weyl polyhedron is indeed polyhedral

Proposition 716 PJ is a polyhedron if and only if J is of finite type

Remark 717 In fact one can show that if J is of finite type then for any family P rarr S Pis a polyhedron if and only if S is

Proof of Proposition 716 If J is not of finite type consider the tautological faces over theopen stratum of DJ If J is of finite type the polyhedrality of PJ follows from the followingobservation


Proposition 718 For J sub I arbitrary PJ is the Minkowski sum of the following rays andlines

Rgt0(0minuswαi) Rgt0(ωj wωj) R(νk νk)

where i isin I J w isinWJ j isin J the ωj satisfy (αjprime ωj) = δjjprime forallj jprime isin J and νk form a basis

of⋂jisinJ α

perpj

Returning to the specialization of the results about a general family P rarr S the f -polynomialof the universal Weyl polyhedron is given by

Proposition 719 Recalling our convention for DJ cf Remark 31 we have

fPJ (q) = q2 dimC hlowastminusrk gsumKsubJ

qminusKfP (λK J)(q) (720)

where λK is any point in the interior of the face perpK of DJ

Remark 721 Under the normal convention that the J dominant chamber DJ lies in a realform of hlowast Equation (720) applies provided one replaces the leading exponent with dimC hlowast

Example 722 Figure 71 shows the two universal Weyl polyhedra for g = sl2 In particularfPI (q) = (1 + q)2 for the first polyhedron For the second ie the case of empty integrabilityJ = empty Proposition 715 implies more generally for any g that

fPempty(q) = q2 dimC hlowastminusrk g(1 + q)rk g

0

0

λ

λ

minusλ

λ

λ

Figure 71 Universal Weyl polyhedra and their fibers over the J dominantchambers Here g = sl2 and J = I empty ie full and empty integrabilities

Of course strata of the base DJ parametrize classes of highest weights that are lsquoequisingularrsquoEquation (720) is a numerical shadow of the fact that in the classification of faces of Puniv thereis a contribution from the combinatorial isomorphism type of each class of equisingular highestweights exactly once Thus the classification of faces for Puniv is equivalent to the classificationof faces of all highest weight modules with integrability J Moreover the combinatorial isomor-phism type of Puniv ie inclusions of faces stores the data of how parabolic faces degenerate asone specializes the highest weight


8 Weak families of Weyl polyhedra and concavity

We continue to fix an arbitrary KacndashMoody algebra g and a subset J sub I of simple rootsRecall that in the definition of a family of Weyl polyhedra π P rarr S we asked that fors0 s1 isin S 0 6 t 6 1 we have

πminus1(ts0 + (1minus t)s1) = tπminus1s0 + (1minus t)πminus1s1

In this section we relax this condition and relate a more flexible notion of a family of Weylpolyhedra to certain maps to the J dominant chamber Along the way we prove new resultseven about the convex hulls of highest weight modules cf Proposition 814

Definition 81 Let S be a convex set By a weak S-family of Weyl polyhedra we mean a subsetP sub S times hlowast such that (i) under the natural projection π P rarr S for each s isin S the fiber πminus1sis a Weyl polyhedron and (ii) P is convex

See Figure 81 for an example of an S-family and a weak S-family

S

P

s s

Figure 81 Examples of an S family and a weak S family of Weyl polyhedrarespectively for g = sl2 and J = I The fiber in either case is the segmentbetween (s λ(s)) and (sminusλ(s))

As before we have

Lemma 82 If Sprime rarr S is a morphism of convex sets and P rarr S a weak S-family of Weylpolyhedra then the pullback P timesS Sprime is a weak Sprime-family of Weyl polyhedra

The goal in this section is to relate weak families to certain maps to DJ in the spirit ofTheorem 64 To this end we will need a more explicit description of the J dominant points ofP (λ J) in the spirit of Proposition 37 and [22 Proposition 24] After collecting some helpfulingredients this description will be obtained in Theorem 816

Definition 83 Let λ ν isin hlowast We say ν is J nondegenerate with respect to λ if the followingtwo conditions hold

(1) λminus ν isin Rgt0π(2) Write λ minus ν = βJ + βIJ βJ isin Rgt0πJ βIJ isin Rgt0πIJ Then for any connected

component C of suppβJ there exists c isin C such that (αc λminus βIJ) gt 0

For brevity we will write ν 6J λ if ν is J nondegenerate with respect to λ

Remark 84 One can show that 6J is the usual partial order on hlowast if and only if J is empty


Remark 85 If λ ν are J dominant and λ has finite stabilizer in WJ then condition (2) isautomatic

To relate weak families to maps to DJ we introduce the following class of maps

Definition 86 Let S be a convex set We say a map f S rarr DJ is concave if for alls1 s2 isin S 0 6 t 6 1 we have

tf(s1) + (1minus t)f(s2) 6J f(ts1 + (1minus t)s2)

ie the secant line lies below the graph in the partial order 6J

Notice that concave maps pull back under morphisms of convex sets

Lemma 87 If φ Sprime rarr S is a morphism of convex sets and f S rarr DJ a concave mapThen f φ Sprime rarr DJ is concave

It follows that the assignment S concave maps S rarr DJ is a contravariant functor fromconvex sets to sets We can now state the main result of this section

Theorem 88 There is an isomorphism of functors

weak S-families of Weyl polyhedra concave maps S rarr DJ (89)

The remainder of this section is devoted to proving Theorem 88 Note that as the terminologysuggests we have

Lemma 810 6J is a partial order on hlowast

Proof We only explain transitivity as the other properties are immediate Suppose micro 6J ν 6Jλ Write

λ = ν + βJ + βIJ βJ isin Rgt0πJ βIJ isin Rgt0πIJ

ν = micro+ γJ + γIJ γJ isin Rgt0πJ γIJ isin Rgt0πIJ

We need to show that for any connected component C of supp(βJ + γJ) there exists c isin Cwith (αc λ minus βIJ minus γIJ) gt 0 If C contains a connected component of suppβJ then we maypick c isin C with (αc λminus βIJ) gt 0 whence

(αc λminus βIJ minus γIJ) gt (αc λminus βIJ) gt 0

If C contains no components of suppβJ then we have (αc λminus βIJ) = (αc ν) forallc isin C Thusif we pick c isin C with (αc ν minus γIJ) gt 0 we have

(αc λminus βIJ minus γIJ) = (αc ν minus γIJ) gt 0

We also note the following

Lemma 811 For λ isin DJ the locus ν isin hlowast ν 6J λ is convex and forms a cone with vertexat λ

We now obtain the desired description of P (λ J) in the case J = I

Proposition 812 For a point λ in the dominant chamber DI we have

convWλ =⋃wisinW

wν isin DI ν 6I λ


Proof For the inclusion sub it suffices to show that ν 6I λ for any point ν of convWλ We firstshow wλ 6I λ by induction on `(w) The case w = 1 is trivial and for the inductive step writew = siw

prime where `(wprime) = `(w) minus 1 As λ is dominant we have wλ 6 wprimeλ and by consideringαi we have wλ 6I wprimeλ We therefore are done by induction and Lemma 810 Having shownwλ 6I λforallw isinW we are done by Lemma 811

For the inclusion sup it suffices to show that for ν isin DI ν 6I λ we have ν isin convWλ Wewill in fact show

For micro 6I η J = supp(η minus micro) if micro and η are J dominant then micro isin convWJ(η) (813)

Let C be a connected component of J and write ηminusmicro = βC+βJC where βC isin Rgt0πC βJC isinRgt0πJC If we can show that ηminusβC lies in convWC(η) then ηminusβC micro again satisfy the conditionsof Equation (813) so we are done by induction on the cardinality of J

So it remains to show Equation (813) when J is connected We break into two cases Ifthere exists j isin J with (αj micro) gt 0 then the desired claim is shown in the second and thirdparagraph of [22 Proposition 24]

If (αj micro) = 0forallj isin J we argue as follows Since micro 6I η for some j0 isin J we have (αj η) gt 0Thus η minus micro is a positive real combination of simple roots satisfying (αj η minus micro) gt 0 forallj isin J (αj0 ηminusmicro) gt 0 whence J is of finite type It therefore remains to show that if g is a simple Liealgebra λ a real dominant weight then 0 isin convW (λ) But this follows from W -invariance

0 =1

|W |sumwisinW

wλ

To obtain the desired description of P (λ J) for general J we need one more preliminarywhich is the convex analogue of the Integrable Slice Decomposition 314

Proposition 814 For any J sub I and λ isin DJ we have

P (λ J) =⊔

microisinRgt0IJ

convWJ(λminus micro) (815)

Proof The inclusion sup is straightforward from the definition of P (λ J) For the inclusion subwrite an element of the left hand side as

ν =summ

tmwm(λminus rmαim) tm gt 0summ

tm = 1 rm gt 0 im isin I J

By WJ invariance we may assume ν is J dominant whence by Proposition 812 applied to gJit suffices to show ν is J nondegenerate with respect to λminus

summ tmrmαim This in turn follows

by using that for each m wm(λminus rmαim) is J nondegenerate with respect to λminus rmαim as wasshown in the proof of Proposition 812

Finally we obtain the desired alternative description of P (λ J) in terms of 6J

Theorem 816 For any J sub I λ isin DJ we have

P (λ J) =⋃

wisinWJ

wν isin DJ ν 6J λ (817)

Proof This is immediate from Propositions 812 and 814

Proof of Theorem 88 The maps between the two sides will be exactly as in Theorem 64 Weneed only to see that the property that the total space of a family P be convex is equivalent


to concavity of the corresponding map φP to DJ Let s0 s1 be two distinct points of S andP (λ0 J) P (λ1 J) the corresponding fibers A direct calculation shows

conv (s0 times P (λ0 J) t s1 times P (λ1 J)) =⊔

06t61

(ts0 + (1minus t)s1)times P (tλ0 + (1minus t)λ1 J)

Therefore the condition that the total space is convex is that P (tλ0 + (1minus t)λ1) sub P (λ(ts0 +(1minus t)s1) J) That this is equivalent to the concavity of φP follows from the following Lemma818

Lemma 818 Let micro ν isin DJ Then P (micro J) sub P (ν J) if and only if micro 6J ν

We note that for g of finite type and micro ν dominant integral weights Lemma 818 can befound in [16 Proposition 22]

Proof If P (micro J) sub P (ν J) in particular micro isin P (ν J) whence we are done by Theorem 816Conversely suppose micro 6J ν By WJ invariance it suffices to show that every J dominant weightη of P (micro J) lies in P (ν J) To see this again by Theorem 816 η 6J micro whence η 6J ν byLemma 810 and so η isin P (ν J) by Theorem 816

Remark 819 Using results from related work [8] specifically Propositions 37 and 314 onesimilarly obtains a description of the weights of parabolic Verma modules M(λ J) Let us write6primeJ for the variant on 6J given by replacing Rgt0π by Zgt0π in Definition 83 Then

wtM(λ J) =⋃

wisinWJ

wν isin P+J ν 6primeJ λ (820)

By another result of loc cit for L(λ) a simple highest weight module and IL(λ) = i isin I

(αi λ) isin Zgt0 we have

wtL(λ) =⋃

wisinWIL(λ)

wν isin P+IL(λ)

ν 6primeIL(λ)λ (821)

9 Closure polyhedrality and integrable isotropy

In this section we turn to the question of polyhedrality of conv V As we have seen conv V =convM(λ IV ) so it suffices to consider parabolic Verma modules If the Dynkin diagram Iof g can be factored as a disconnected union I = I prime t I primeprime in the obvious notation we havecorresponding factorizations

M(λprime oplus λprimeprime J prime t J primeprime) M(λprime J prime)M(λprimeprime J primeprime)

so it suffices to consider g indecomposable As is clear from the Ray Decomposition 317 in finitetype conv V is always a polyhedron so throughout this section we take g to be indecomposableof infinite type

We first show that conv V is almost never a polytope Let us call V trivial if [g g] acts by 0ie V is one-dimensional

Proposition 91 conv V is a polytope if and only if V is trivial

Proof One direction is tautological For the reverse if conv V is a polytope then V is integrableAssume first that V is simple then V is the inflation of a g-module where g is as in Section 31Now recall that if i is an ideal of g then either i is contained in the (finite-dimensional) centerof g or contains [g g] Since the kernel of grarr End(V ) has finite codimension the claim followsFor the case of general integrable V we still know V has a unique weight by Proposition 37Now the result follows by recalling that [g g] cap h is spanned by αi i isin I


Even if we relax the assumption of boundedness it transpires that conv V is only a polyhedronin fairly degenerate circumstances

Proposition 92 Let V be a non-trivial module Then conv V is a polyhedron if and only ifIV corresponds to a Dynkin diagram of finite type

Proof If IV corresponds to a Dynkin diagram of finite type we are done by the Ray Decomposi-tion 317 Now suppose IV is not a diagram of finite type If the orbit WIV λ is infinite we deducethat conv V has infinitely many 0-faces whence it is not a polyhedron If instead the orbit isfinite we deduce that the lIV module of highest weight λ is finite-dimensional whence by thesame reasoning as Proposition 91 the orbit is a singleton In this case by the non-triviality ofV and indecomposability of g we may choose i isin I IV which is connected to a component of IVof infinite type It follows by the previous case that the orbit of λminus αi is infinite Hence by theRay Decomposition 317 conv V has infinitely many 1-faces whence it is not a polyhedron

In particular we have shown

Corollary 93 For any V in finite type conv V is the Minkowski sum of convWIV (λ) and thecone Rgt0WIV (πIIV )

This corollary extends the notion of the Weyl polytope of a finite-dimensional simple repre-sentation to the Weyl polyhedron of any highest weight module V in finite type

Recall that a set X sub hlowast is locally polyhedral if its intersection with every polytope is apolytope It turns out that for generic V conv V is locally polyhedral as we show in the maintheorem of this section

Theorem 94 Let V be a non-trivial module The following are equivalent

(1) conv V is locally polyhedral


wisinWIV

TwλV

(3) conv V is closed(4) The stabilizer of λ in WIV is finite(5) λ lies in the interior of the IV Tits cone

Remark 95 The equivalence of (4) and (5) was already recalled in Proposition 32 andwe include (5) only to emphasize the genericity of these conditions In the integrable caseie IV = I Borcherds in [2] considers the very same subset of the dominant chamber which hecalls CΠ and writes

The Tits cone can be thought of as obtained from WΠCΠ by ldquoadding some bound-ary componentsrdquo

Proof of Theorem 94 We may assume V is a parabolic Verma module As remarked above (4)and (5) are equivalent We will show the following sequence of implications

(1) =rArr (3) =rArr (4) =rArr (2) =rArr (3) and (2)minus (5) =rArr (1)

It is easy to see (1) implies (3) by intersecting with cubes of side-length tending to infinity Wenow show the equivalence of (2) (3) and (4) The implication (2) implies (3) is tautologicalTo show that (3) implies (4) write J = j isin IV (αj λ) = 0 as in Lemma 44 and recall byProposition 32 that the stabilizer of λ isWJ Suppose thatWJ is infinite and choose a connectedsub-diagram J prime sub J of infinite type By the non-triviality of V and the indecomposability of gwe may choose i isin I J which shares an edge with J prime It follows that λminusαi is a weight of V andU(lJ prime)Vλminusαi is a non-trivial integrable highest weight lJ prime-module We may choose an imaginarypositive root δ of lJ prime with supp δ = J prime cf [13 Theorem 56 and sect512] By Proposition 37


λ minus αi minus Zgt0δ lies in wtV It follows that λ minus Zgt0δ lies in the closure of conv V but cannotlie in conv V by Equation (316) It may be clarifying for the reader to visualize this argumentusing Figure 31

We next show that (4) implies (2) By Lemma 44 and the Ray Decomposition 317 ourassumption on λ implies the tangent cone TλV is a polyhedral cone whence closed It thereforeremains to show

conv V =⋂

wisinWIV

TwλV (96)

The inclusion sub is tautological For the reverse writing D for the IV dominant chamber we firstshow TλV capD sub conv V ie the left-hand side coincides with the intersection of the right-handside with the IV Tits cone Since (αj λ) isin Zforallj isin IV it is straightforward to see that thetranslate (TλV capD)minusλ is a rational polyhedron (cf Proposition 34) In particular it suffices toconsider v isin TλV capD of the form v isin λminusQgt0π As both sides of Equation (96) are dilated by nwhen we pass from M(λ IV ) to M(nλ IV ) for any n isin Zgt0 it suffices to consider v isin TλV capDof the form v isin λminusZgt0π Writing v = λminusmicroprimeminusmicroprimeprime where microprime isin Zgt0πIV micro

primeprime isin Zgt0πIIV it sufficesto see that v is non-degenerate with respect to λ minus microprimeprime But this follows since the stabilizer ofλminus microprimeprime is again of finite type by Proposition 37(2)

It remains to show the right-hand side lies in the IV Tits cone Looijenga has shown in [22Corollary 25] that if K sub D is a compact subset of the interior of the IV Tits cone thenconvWIV (K) is closed To apply this in our situation introduce the following lsquocutoffsrsquo of theRay Decomposition 317

Kt =⋃

iisinIIV

λminus [0 t]αi t gt 0

By filtering hlowast by half spaces of the form (microminus) gt t where micro =sum

iisinIIV ωi it suffices to know

that convWIV (Kt) is closed for all t gt 0 but this follows from Looijengarsquos resultContinuing with the proof if the right-hand side of Equation (96) contained a point x outside

of the IV Tits cone consider the line segment ` = [λ x] The intersection of ` with the IV Titscone is closed and convex whence of the form [λ y] But now y is a point of conv V which lieson the boundary of the IV Tits cone In particular conv V is not contained in the interior ofthe IV Tits cone which contradicts (5)

It remains to see the equivalent conditions (2)ndash(5) imply (1) For this we will introduce aconvexity-theoretic analogue of the principal gradation cf [13 sect108] Choose ρ in the real dualof hlowast satisfying (ρ αi) = 1 foralli isin I and for A sub R write HA = ρminus1(A) In particular writeH[kinfin) for the half-space ρminus1([kinfin)) and V[kinfin) = (conv V ) capH[kinfin) It suffices to check thatV[kinfin) is a polytope for all k isin R Fix k isin R which we may assume to satisfy k 6 ρ(λ) andnote wtV cap H[kinfin) consists of finitely many points Let vi 1 6 i 6 n be an enumeration ofWIV (λ)capH(kinfin) For each vi by (4) and the Ray Decomposition 317 let fij denote the finitelymany 1-faces containing vi For example by Theorem 42 if vi = λ these 1-faces are

wλminus Rgt0αi and w[λ sj(λ)] where w isin StabWIV(λ) i isin I IV j isin IV

Here [a b] denotes the closed line segment joining a and b Each fij intersects H[kinfin) in a linesegment we define eij to be the other endpoint ie [vi eij ] = fij capH[kinfin) By removing someeij we may assume eij 6= vkforalli j k It suffices to show

conv V[kinfin) = convvi eij

Writing Hk for the affine hyperplane ρminus1(k) we claim that Hk cap conv V = conv eij We firstobserve that the intersection is a bounded subset of Hk Indeed it suffices to check Hk cap TλVis bounded but TλV by Lemma 44 is of the form convλminus Rgt0rk16k6m with ρ(rk) gt 0 1 6


k 6 m So Hk cap conv V is bounded and closed by (2) whence it is the convex hull of its 0-facescf [10 sect245] But if v is an interior point of a d-face of conv V d gt 2 then v cannot be a0-face of a hyperplane section Hence the 0-faces of Hk cap conv V are found by intersecting the0-faces and 1-faces of conv V with Hk which is exactly the collection eij

We are ready to show that V[kinfin) = convvi eij The inclusion sup is immediate For theinclusion sub write v isin V[kinfin) as a convex combination

v =sumrprimeisinRprime

trprimerprime +

sumrprimeprimeisinRprimeprime

trprimeprimerprimeprime

where rprime rprimeprime are points along the rays occurring in the Ray Decomposition 317 satisfying ρ(rprime) gtk gt ρ(rprimeprime) respectively For rprime isin Rprime by construction we can write rprime as a convex combination ofvi eij Observe Rprime must be non-empty else ρ(v) lt k If Rprimeprime is also non-empty define the partialaverages

vprime =1sum

sprimeisinRprime tsprime

sumrprimeisinRprime

trprimerprime vprimeprime =

1sumsprimeprimeisinRprimeprime tsprimeprime

sumsprimeprimeisinRprimeprime


The line segment [vprime vprimeprime] is not a single point as ρ(vprimeprime) lt k Writing vprimeprimeprime for its intersection withHk we have v lies in [vprime vprimeprimeprime] But we already saw vprime lies in convvi eij and vprimeprimeprime isin conveij bythe preceding paragraph hence v isin convvi eij as desired

Remark 97 It would be interesting to know whether the intersection formula in Theorem94(2) holds without finite stabilizers

As an application of Theorem 94 we provide a half-space representation of conv V for generichighest weights valid in any type

Proposition 98 When it is closed conv V coincides with the intersection of the half-spaces

Hiw = micro isin λminus Rπ (wλminus microwωi) gt 0 forallw isinWIV i isin I

The above half-space representation extends results in finite type proved in [7 18] moreoverour proof uses tangent cones and is thus different from loc cit

Proof One inclusion is clear since wtV is contained in each half-space Hiw by WIV -invarianceConversely by Theorem 94 we need to show that the intersection⋂

iisinI

⋂wisinWIV

Hiw

is contained in the tangent cone TλV Since λ has finite stabilizer in WIV the cone TλV is aconvex polyhedron (eg by Lemma 44) Hence it is the intersection of the half-spaces that defineits facets Now observe that each of these half-spaces appears in the above intersection

10 A question of Brion on localization of convex hulls

We now return to the case of an arbitrary KacndashMoody Lie algebra g and answer the questionof Brion [4] mentioned in Section 2

Theorem 101 Suppose g is arbitrary Fix λ V with (αi λ) gt 0 for all i isin IV Given astandard face F = wtlJ for some J sub I and setting J prime = IV cap J we have

LocJ conv V =⋂

wisinWJprime

TwλV = convM(λ J prime) (102)

Note that the localization of conv V at other faces F is computed by Theorem 42


Proof Given distinct points a b in any convex set C in a real vector space if the line segment[a b] can be slightly extended in C past b namely a + (1 + ε)(b minus a) isin C for some ε gt 0 thenTaC sub TbC This assertion can be checked by a computation in the plane

By the above observation and the Ray Decomposition 317 LocJ conv V equals the intersectionover the tangent cones at the vertices in the face ie WJ prime(λ) Since λ is IV -regular thesecoincide with the tangent cones for M(λ J prime) at these points cf Lemma 44 We are then doneby Theorem 94(2)

Remark 103 While the results in this paper are developed for g-modules they also holdfor g-modules where g is as in Section 31 That the results cited from the literature and ourprevious work [8] apply for g was explained in loc cit With this the arguments in the presentpaper apply verbatim for g except for Propositions 91 and 92 which were shown for both gand g

11 Highest weight modules over symmetrizable quantum groups

The goal of this section is to prove Theorems 51 and 57 for modules over quantum groupsUq(g) for g a KacndashMoody algebra Given a generalized Cartan matrix A as for g = g(A)to write down a presentation for the algebra Uq(g) via generators and explicit relations oneuses the symmetrizability of A When g is non-symmetrizable even the formulation of Uq(g) issubtle and is the subject of active research [15 9] In light of this we restrict to Uq(g) where gis symmetrizable

111 Notation and preliminaries We begin by reminding standard definitions and notationFix g = g(A) for A a symmetrizable generalized Cartan matrix Fix a diagonal matrix D =diag(di)iisinI such that DA is symmetric and di isin Zgt0 foralli isin I Let (h π π) be a realization of Aas before further fix a lattice Por sub h with Z-basis αi βl i isin I 1 6 l 6 |I| minus rk(A) such thatPorotimesZC h and (βl αi) isin Z foralli isin I 1 6 l 6 |I|minusrk(A) Set P = λ isin hlowast (Por λ) sub Z to bethe weight lattice We further retain the notations ρ b h lJ P

+J M(λ J) IL(λ) from previous

sections note we may and do choose ρ isin P We normalize the Killing form (middot middot) on hlowast to satisfy(αi αj) = diaij for all i j isin I

Let q be an indeterminate Then the corresponding quantum KacndashMoody algebra Uq(g) is a

C(q)-algebra generated by elements fi qh ei i isin I h isin Por with relations given in eg [11

Definition 311] Among these generators are distinguished elements Ki = qdiαi isin qPor Alsodefine Uplusmnq to be the subalgebras generated by the ei and the fi respectively

A weight of the quantum torus Tq = C(q)[qPor] is a C(q)-algebra homomorphism χ Tq rarr

C(q) which we identify with an element microq isin (C(q)times)2|I|minusrk(A) given an enumeration of αi i isin I

We will abuse notation and write microq(qh) for χmicroq(q

h) There is a partial ordering on the set of

weights given by qminusνmicroq 6 microq for all weights ν isin Zgt0πGiven a Uq(g)-module V and a weight microq the corresponding weight space of V is

Vmicroq = v isin V qhv = microq(qh)v forallh isin Por

Denote by wtV the set of weights microq Vmicroq 6= 0 We say a Uq(g)-module is highest weight ifthere exists a nonzero weight vector which generates V and is killed by ei foralli isin I If we writeλq for the highest weight the integrability of V equals

IV = i isin I dimC(q)[fi]Vλq ltinfin (111)


112 Inclusions between faces of highest weight modules For highest weight modulesover quantum groups notice convex hulls only make their appearance after specializing at q = 1[24] Nonetheless the above results give information on the action of quantized Levi subalgebras

wtJ V = wtUq(lJ)Vλq = (wtV ) cap qminusZgt0πJλq

where Uq(lJ) is the subalgebra of Uq(g) generated by qPor

and ej fj j isin J To prove the analogues of Theorems 51 and 57 over quantum groups we first define the sets

Jmin Jmax in the quantum setting For J sub I call a node j isin J active if either (i) j isin IV or (ii)j isin IV and λq(q

αj ) 6= plusmn1 Now decompose the Dynkin diagram corresponding to J as a disjointunion of connected components and let Jmin denote the union of all components containing anactive node Finally define

Jd = j isin IV λq(qαj ) = plusmn1 and j is not connected to Jmin

Jmax = Jmin t Jd(112)

Theorem 113 Let λq be arbitrary Fix subsets J J prime sub I

(1) wtJ V sub wtJ prime V if and only if Jmin sub J prime(2) wtJ V = wtJ prime V if and only if Jmin sub J prime sub Jmax

The arguments of Theorems 51 and 57 apply verbatim where one uses the representationtheory of highest weight Uq(sl2)-modules instead of that of U(sl2) cf [12]

In particular the two applications to branching Corollaries 59 and 510 also have quantumcounterparts

References

[1] Alexander Barvinok Integer points in polyhedra Zurich Lectures in Advanced Mathematics European Math-ematical Society (EMS) Zurich 2008

[2] Richard E Borcherds Coxeter groups Lorentzian lattices and K3 surfaces Int Math Res Not IMRN(19)1011ndash1031 1998

[3] Armand Borel and Jacques Tits Groupes reductifs Inst Hautes Etudes Sci Publ Math 27(1)55ndash150 1965[4] Michel Brion Personal communication 2013[5] Winfried Bruns and Joseph Gubeladze Polytopes rings and K-theory Springer Monographs in Mathematics

Springer Dordrecht 2009[6] William A Casselman Geometric rationality of Satake compactifications In Algebraic groups and Lie groups

volume 9 of Austral Math Soc Lect Ser pages 81ndash103 Cambridge Univ Press Cambridge 1997[7] Paola Cellini and Mario Marietti Root polytopes and Borel subalgebras Int Math Res Not IMRN

(12)4392ndash4420 2015[8] Gurbir Dhillon and Apoorva Khare Characters of highest weight modules and integrability Preprint

arXiv160609640 2016[9] Xin Fang Non-symmetrizable quantum groups defining ideals and specialization Publ Res Inst Math

Sci 50(4)663ndash694 2014[10] Branko Grunbaum Convex polytopes volume 221 of Graduate Texts in Mathematics Springer-Verlag New

York second edition 2003 Prepared and with a preface by Volker Kaibel Victor Klee and Gunter M Ziegler[11] Jin Hong and Seok-Jin Kang Introduction to quantum groups and crystal bases volume 42 of Graduate

Studies in Mathematics American Mathematical Society Providence RI 2002[12] Jens C Jantzen Lectures on quantum groups volume 6 of Graduate Studies in Mathematics American

Mathematical Society Providence RI 1996[13] Victor G Kac Infinite-dimensional Lie algebras Cambridge University Press Cambridge third edition

1990[14] Joel Kamnitzer Mirkovic-Vilonen cycles and polytopes Ann of Math (2) 171(1)245ndash294 2010[15] Masaki Kashiwara On crystal bases In Representations of groups (Banff AB 1994) volume 16 of CMS

Conf Proc pages 155ndash197 Amer Math Soc Providence RI 1995[16] David A Kazhdan Michael Larsen and Yakov Varshavsky The Tannakian formalism and the Langlands

conjectures Algebra Number Theory 8(1)243ndash256 2014


[17] Apoorva Khare Faces and maximizer subsets of highest weight modules J Algebra 45532ndash76 2016[18] Apoorva Khare Standard parabolic subsets of highest weight modules Trans Amer Math Soc published

online (DOI 101090tran6710) 2016[19] Shrawan Kumar KacndashMoody groups their flag varieties and representation theory volume 204 of Progress

in Mathematics Birkhauser Boston Inc Boston MA 2002[20] James Lepowsky A generalization of the BernsteinndashGelfandndashGelfand resolution J Algebra 49(2)496ndash511

1977[21] Zhou Li Yoursquoan Cao and Zhenheng Li Cross-section lattices of J -irreducible monoids and orbit structures

of weight polytopes Preprint arXiv14116140 2014[22] Eduard Looijenga Invariant theory for generalized root systems Invent Math 61(1)1ndash32 1980[23] Eduard Looijenga Discrete automorphism groups of convex cones of finite type Compos Math

150(11)1939ndash1962 2014[24] George Lusztig Quantum deformations of certain simple modules over enveloping algebras Adv Math

70(2)237ndash249 1988[25] Ichiro Satake On representations and compactifications of symmetric Riemannian spaces Ann of Math (2)

7177ndash110 1960[26] Ernest B Vinberg Some commutative subalgebras of a universal enveloping algebra Izv Akad Nauk SSSR

Ser Mat 54(1)3ndash25 221 1990

Department of Mathematics Stanford University Stanford CA 94305 USAE-mail address GD gsdstanfordedu AK kharestanfordedu

1 Introduction


Organization of the paper

Acknowledgments


31 Notation for KacndashMoody algebras standard parabolic and Levi subalgebras

32 Weyl group parabolic subgroups Tits cone

33 Representations integrability and parabolic Verma modules

34 Convexity

35 Results from previous work on highest weight modules





71 Faces of Weyl polyhedra

72 Strong combinatorial isomorphism and strata of the Weyl chamber







112 Inclusions between faces of highest weight modules

References


1 Introduction

Throughout the paper unless otherwise specified g is a KacndashMoody algebra over C with tri-angular decomposition nminus oplus h oplus n+ and Weyl group W and V is a g-module of highest weightλ isin hlowast

In this paper we study and answer several fundamental convexity-theoretic questions relatedto the weights of V and their convex hull denoted wtV and conv V respectively In contrastto previous approaches in the literature our approach is to use techniques from representationtheory to directly study wtV and deduce consequences for conv V

To describe our results and the necessary background first suppose g is of finite type and Vis integrable In this case conv V is a convex polytope called the Weyl polytope and is wellunderstood In particular the faces of such Weyl polytopes have been determined by Vinberg[26 sect3] and Casselman [6 sect3] and as Casselman notes are implicit in the earlier work of Satake[25 sect23] and BorelndashTits [3 sect1216] The faces are constructed as follows Let I index thesimple roots of g and ei fi hi i isin I the Chevalley generators of g Given J sub I denote by lJthe corresponding Levi subalgebra generated by ej fj forallj isin J and hi foralli isin I Also define thefollowing distinguished set of weights

wtJ V = wtU(lJ)Vλ (11)

where Vλ is the highest weight space The aforementioned authors showed that every face of theWeyl polytope for V is a Weyl group translate of the convex hull of wtJ V for some J sub I

To complete the classification of faces it remains to determine the redundancies in the abovedescription Formally consider the face map

FV W times 2I rarr 2wtV FV (w J) = w(wtJ V ) (12)

where 2S denotes the power set of a set S As wtJ V and its convex hull determine one an-other computing the redundancies is the same as computing the fibers of the face map Thedetermination of the fibers for the restriction FV (1minus) is implicit in the above works of Vinbergand Casselman A complete understanding of the fibers of FV (minusminus) was achieved very recentlyby CellinindashMarietti [7] for the adjoint representation and subsequently by Khare [18] for allhighest weight modules While the fibers of the face map were understood for all highest weightmodules V it was not known whether all faces were of the above form

In this paper we resolve both parts of the above problem for any highest weight module Vover a KacndashMoody algebra g While the answer takes a similar form to the known cases in finitetype the proof of the first part requires a substantially different argument as prior methods failat several crucial junctures

(i) A key input in the known cases of the classification of faces of conv V was that every faceis the maximizer of a linear functional which was deduced from polyhedrality Howevereven the polyhedrality of conv V was not known in general in finite type

(ii) We show in related work [8] that indeed conv V is always a polyhedron in finite typeHowever as we show below for g of infinite type conv V is rarely polyhedral genericallylocally polyhedral and in some cases not locally polyhedral In particular one cannot apriori appeal to a maximizing functional to understand a face

(iii) Finally even given a face cut out by a maximizing functional the possibility of the func-tional to fall outside the Tits cone makes the usual analysis inapplicable

In this paper we provide an infinitesimal analysis of the faces that avoids invocation of func-tionals completing the first part of the classification of faces As a consequence every face isa posteriori cut out by a maximizing functional lying in the Tits cone For the second part






















F =⊔J primesubJ

FJ prime












wisinWIV

TwλV























nminus =oplusαlt0

gα n+ =oplusαgt0

gα



+J ∆



J uminusJ n

+J n

minusJ by

uplusmnJ =oplus



gα




⋃wisinW wD



⋃wisinWJ












w(ρ)minus ρ =sum

αgt0wαlt0

w(α) (33)









































wtM(λ J) =⊔







minusα1

minusα3

g =

α1 α2 α3


J = 1 2

λ

minusα0

minusα1

minusα2

g =

α0 α1 α2


J = 0 1





conv V = conv⋃

wisinWIV iisinIIV



is a face of conv V



minusα1

minusα3

λ

minusα0

minusα1

minusα2














Proof











wisinWJ iisinIJ


















point of conv V










prime) ie

pk =sumj


sumj




pk(b) =sum










U(l)Vλ =oplus

microisinλ+ZπJ

Vmicro (52)





16j6n

αij (54)





)Vλ










on U(l)Vλ

















VWJmin

VtimesWJmax

V JminV (513)


F =⋃

wisinWJminV

(F cap wD)


F =⋃

wisinWJminV

(F cap wwD)









ckαik) = 0






iisinZgt0 fiqi


fV (q) =sumwtlJ





fV (q) =sumJsubI

[W WJ ]q|J | (517)














P =sumi


ti = 1
















to Proposition 317



P (λ J) = conv⋃

wisinWJ iisinIJ



0sum



tiλi J) (68)









J


































wtlprime = conv⋃




and the





















S =⊔FisinF

SF








is a face of P














fP(q) =sumFisinF

qdimF fF (q)









of⋂jisinJ α

perpj









0

0

λ

λ

minusλ

λ

λ










S

P

s s


As before we have
































convWλ =⋃wisinW










0 =1

|W |sumwisinW

wλ



P (λ J) =⊔

microisinRgt0IJ



ν =summ








P (λ J) =⋃

wisinWJ







06t61







wtM(λ J) =⋃

wisinWJ




wtL(λ) =⋃

wisinWIL(λ)

wν isin P+IL(λ)




















wisinWIV

TwλV










conv V =⋂

wisinWIV

TwλV (96)




Kt =⋃

iisinIIV















trprimerprime +




vprime =1sum


sumrprimeisinRprime












iisinI

⋂wisinWIV

Hiw





LocJ conv V =⋂

wisinWJprime




































References
























Ser Mat 54(1)3ndash25 221 1990


1 Introduction



Acknowledgments





34 Convexity















References






















F =⊔J primesubJ

FJ prime












wisinWIV

TwλV























nminus =oplusαlt0

gα n+ =oplusαgt0

gα



+J ∆



J uminusJ n

+J n

minusJ by

uplusmnJ =oplus



gα




⋃wisinW wD



⋃wisinWJ












w(ρ)minus ρ =sum

αgt0wαlt0

w(α) (33)









































wtM(λ J) =⊔







minusα1

minusα3

g =

α1 α2 α3


J = 1 2

λ

minusα0

minusα1

minusα2

g =

α0 α1 α2


J = 0 1





conv V = conv⋃

wisinWIV iisinIIV



is a face of conv V



minusα1

minusα3

λ

minusα0

minusα1

minusα2














Proof











wisinWJ iisinIJ


















point of conv V










prime) ie

pk =sumj


sumj




pk(b) =sum










U(l)Vλ =oplus

microisinλ+ZπJ

Vmicro (52)





16j6n

αij (54)





)Vλ










on U(l)Vλ

















VWJmin

VtimesWJmax

V JminV (513)


F =⋃

wisinWJminV

(F cap wD)


F =⋃

wisinWJminV

(F cap wwD)









ckαik) = 0






iisinZgt0 fiqi


fV (q) =sumwtlJ





fV (q) =sumJsubI

[W WJ ]q|J | (517)














P =sumi


ti = 1
















to Proposition 317



P (λ J) = conv⋃

wisinWJ iisinIJ



0sum



tiλi J) (68)









J


































wtlprime = conv⋃




and the





















S =⊔FisinF

SF








is a face of P














fP(q) =sumFisinF

qdimF fF (q)









of⋂jisinJ α

perpj









0

0

λ

λ

minusλ

λ

λ










S

P

s s


As before we have
































convWλ =⋃wisinW










0 =1

|W |sumwisinW

wλ



P (λ J) =⊔

microisinRgt0IJ



ν =summ








P (λ J) =⋃

wisinWJ







06t61







wtM(λ J) =⋃

wisinWJ




wtL(λ) =⋃

wisinWIL(λ)

wν isin P+IL(λ)




















wisinWIV

TwλV










conv V =⋂

wisinWIV

TwλV (96)




Kt =⋃

iisinIIV















trprimerprime +




vprime =1sum


sumrprimeisinRprime












iisinI

⋂wisinWIV

Hiw





LocJ conv V =⋂

wisinWJprime




































References
























Ser Mat 54(1)3ndash25 221 1990


1 Introduction



Acknowledgments





34 Convexity















References




wisinWIV

TwλV























nminus =oplusαlt0

gα n+ =oplusαgt0

gα



+J ∆



J uminusJ n

+J n

minusJ by

uplusmnJ =oplus



gα




⋃wisinW wD



⋃wisinWJ












w(ρ)minus ρ =sum

αgt0wαlt0

w(α) (33)









































wtM(λ J) =⊔







minusα1

minusα3

g =

α1 α2 α3


J = 1 2

λ

minusα0

minusα1

minusα2

g =

α0 α1 α2


J = 0 1





conv V = conv⋃

wisinWIV iisinIIV



is a face of conv V



minusα1

minusα3

λ

minusα0

minusα1

minusα2














Proof











wisinWJ iisinIJ


















point of conv V










prime) ie

pk =sumj


sumj




pk(b) =sum










U(l)Vλ =oplus

microisinλ+ZπJ

Vmicro (52)





16j6n

αij (54)





)Vλ










on U(l)Vλ

















VWJmin

VtimesWJmax

V JminV (513)


F =⋃

wisinWJminV

(F cap wD)


F =⋃

wisinWJminV

(F cap wwD)









ckαik) = 0






iisinZgt0 fiqi


fV (q) =sumwtlJ





fV (q) =sumJsubI

[W WJ ]q|J | (517)














P =sumi


ti = 1
















to Proposition 317



P (λ J) = conv⋃

wisinWJ iisinIJ



0sum



tiλi J) (68)









J


































wtlprime = conv⋃




and the





















S =⊔FisinF

SF








is a face of P














fP(q) =sumFisinF

qdimF fF (q)









of⋂jisinJ α

perpj









0

0

λ

λ

minusλ

λ

λ










S

P

s s


As before we have
































convWλ =⋃wisinW










0 =1

|W |sumwisinW

wλ



P (λ J) =⊔

microisinRgt0IJ



ν =summ








P (λ J) =⋃

wisinWJ







06t61







wtM(λ J) =⋃

wisinWJ




wtL(λ) =⋃

wisinWIL(λ)

wν isin P+IL(λ)




















wisinWIV

TwλV










conv V =⋂

wisinWIV

TwλV (96)




Kt =⋃

iisinIIV















trprimerprime +




vprime =1sum


sumrprimeisinRprime












iisinI

⋂wisinWIV

Hiw





LocJ conv V =⋂

wisinWJprime




































References
























Ser Mat 54(1)3ndash25 221 1990


1 Introduction



Acknowledgments





34 Convexity















References










nminus =oplusαlt0

gα n+ =oplusαgt0

gα



+J ∆



J uminusJ n

+J n

minusJ by

uplusmnJ =oplus



gα




⋃wisinW wD



⋃wisinWJ












w(ρ)minus ρ =sum

αgt0wαlt0

w(α) (33)









































wtM(λ J) =⊔







minusα1

minusα3

g =

α1 α2 α3


J = 1 2

λ

minusα0

minusα1

minusα2

g =

α0 α1 α2


J = 0 1





conv V = conv⋃

wisinWIV iisinIIV



is a face of conv V



minusα1

minusα3

λ

minusα0

minusα1

minusα2














Proof











wisinWJ iisinIJ


















point of conv V










prime) ie

pk =sumj


sumj




pk(b) =sum










U(l)Vλ =oplus

microisinλ+ZπJ

Vmicro (52)





16j6n

αij (54)





)Vλ










on U(l)Vλ

















VWJmin

VtimesWJmax

V JminV (513)


F =⋃

wisinWJminV

(F cap wD)


F =⋃

wisinWJminV

(F cap wwD)









ckαik) = 0






iisinZgt0 fiqi


fV (q) =sumwtlJ





fV (q) =sumJsubI

[W WJ ]q|J | (517)














P =sumi


ti = 1
















to Proposition 317



P (λ J) = conv⋃

wisinWJ iisinIJ



0sum



tiλi J) (68)









J


































wtlprime = conv⋃




and the





















S =⊔FisinF

SF








is a face of P














fP(q) =sumFisinF

qdimF fF (q)









of⋂jisinJ α

perpj









0

0

λ

λ

minusλ

λ

λ










S

P

s s


As before we have
































convWλ =⋃wisinW










0 =1

|W |sumwisinW

wλ



P (λ J) =⊔

microisinRgt0IJ



ν =summ








P (λ J) =⋃

wisinWJ







06t61







wtM(λ J) =⋃

wisinWJ




wtL(λ) =⋃

wisinWIL(λ)

wν isin P+IL(λ)




















wisinWIV

TwλV










conv V =⋂

wisinWIV

TwλV (96)




Kt =⋃

iisinIIV















trprimerprime +




vprime =1sum


sumrprimeisinRprime












iisinI

⋂wisinWIV

Hiw





LocJ conv V =⋂

wisinWJprime




































References
























Ser Mat 54(1)3ndash25 221 1990


1 Introduction



Acknowledgments





34 Convexity















References










w(ρ)minus ρ =sum

αgt0wαlt0

w(α) (33)









































wtM(λ J) =⊔







minusα1

minusα3

g =

α1 α2 α3


J = 1 2

λ

minusα0

minusα1

minusα2

g =

α0 α1 α2


J = 0 1





conv V = conv⋃

wisinWIV iisinIIV



is a face of conv V



minusα1

minusα3

λ

minusα0

minusα1

minusα2














Proof











wisinWJ iisinIJ


















point of conv V










prime) ie

pk =sumj


sumj




pk(b) =sum










U(l)Vλ =oplus

microisinλ+ZπJ

Vmicro (52)





16j6n

αij (54)





)Vλ










on U(l)Vλ

















VWJmin

VtimesWJmax

V JminV (513)


F =⋃

wisinWJminV

(F cap wD)


F =⋃

wisinWJminV

(F cap wwD)









ckαik) = 0






iisinZgt0 fiqi


fV (q) =sumwtlJ





fV (q) =sumJsubI

[W WJ ]q|J | (517)














P =sumi


ti = 1
















to Proposition 317



P (λ J) = conv⋃

wisinWJ iisinIJ



0sum



tiλi J) (68)









J


































wtlprime = conv⋃




and the





















S =⊔FisinF

SF








is a face of P














fP(q) =sumFisinF

qdimF fF (q)









of⋂jisinJ α

perpj









0

0

λ

λ

minusλ

λ

λ










S

P

s s


As before we have
































convWλ =⋃wisinW










0 =1

|W |sumwisinW

wλ



P (λ J) =⊔

microisinRgt0IJ



ν =summ








P (λ J) =⋃

wisinWJ







06t61







wtM(λ J) =⋃

wisinWJ




wtL(λ) =⋃

wisinWIL(λ)

wν isin P+IL(λ)




















wisinWIV

TwλV










conv V =⋂

wisinWIV

TwλV (96)




Kt =⋃

iisinIIV















trprimerprime +




vprime =1sum


sumrprimeisinRprime












iisinI

⋂wisinWIV

Hiw





LocJ conv V =⋂

wisinWJprime




































References
























Ser Mat 54(1)3ndash25 221 1990


1 Introduction



Acknowledgments





34 Convexity















References

























wtM(λ J) =⊔







minusα1

minusα3

g =

α1 α2 α3


J = 1 2

λ

minusα0

minusα1

minusα2

g =

α0 α1 α2


J = 0 1





conv V = conv⋃

wisinWIV iisinIIV



is a face of conv V



minusα1

minusα3

λ

minusα0

minusα1

minusα2














Proof











wisinWJ iisinIJ


















point of conv V










prime) ie

pk =sumj


sumj




pk(b) =sum










U(l)Vλ =oplus

microisinλ+ZπJ

Vmicro (52)





16j6n

αij (54)





)Vλ










on U(l)Vλ

















VWJmin

VtimesWJmax

V JminV (513)


F =⋃

wisinWJminV

(F cap wD)


F =⋃

wisinWJminV

(F cap wwD)









ckαik) = 0






iisinZgt0 fiqi


fV (q) =sumwtlJ





fV (q) =sumJsubI

[W WJ ]q|J | (517)














P =sumi


ti = 1
















to Proposition 317



P (λ J) = conv⋃

wisinWJ iisinIJ



0sum



tiλi J) (68)









J


































wtlprime = conv⋃




and the





















S =⊔FisinF

SF








is a face of P














fP(q) =sumFisinF

qdimF fF (q)









of⋂jisinJ α

perpj









0

0

λ

λ

minusλ

λ

λ










S

P

s s


As before we have
































convWλ =⋃wisinW










0 =1

|W |sumwisinW

wλ



P (λ J) =⊔

microisinRgt0IJ



ν =summ








P (λ J) =⋃

wisinWJ







06t61







wtM(λ J) =⋃

wisinWJ




wtL(λ) =⋃

wisinWIL(λ)

wν isin P+IL(λ)




















wisinWIV

TwλV










conv V =⋂

wisinWIV

TwλV (96)




Kt =⋃

iisinIIV















trprimerprime +




vprime =1sum


sumrprimeisinRprime












iisinI

⋂wisinWIV

Hiw





LocJ conv V =⋂

wisinWJprime




































References
























Ser Mat 54(1)3ndash25 221 1990


1 Introduction



Acknowledgments





34 Convexity















References




conv V = conv⋃

wisinWIV iisinIIV



is a face of conv V



minusα1

minusα3

λ

minusα0

minusα1

minusα2














Proof











wisinWJ iisinIJ


















point of conv V










prime) ie

pk =sumj


sumj




pk(b) =sum










U(l)Vλ =oplus

microisinλ+ZπJ

Vmicro (52)





16j6n

αij (54)





)Vλ










on U(l)Vλ

















VWJmin

VtimesWJmax

V JminV (513)


F =⋃

wisinWJminV

(F cap wD)


F =⋃

wisinWJminV

(F cap wwD)









ckαik) = 0






iisinZgt0 fiqi


fV (q) =sumwtlJ





fV (q) =sumJsubI

[W WJ ]q|J | (517)














P =sumi


ti = 1
















to Proposition 317



P (λ J) = conv⋃

wisinWJ iisinIJ



0sum



tiλi J) (68)









J


































wtlprime = conv⋃




and the





















S =⊔FisinF

SF








is a face of P














fP(q) =sumFisinF

qdimF fF (q)









of⋂jisinJ α

perpj









0

0

λ

λ

minusλ

λ

λ










S

P

s s


As before we have
































convWλ =⋃wisinW










0 =1

|W |sumwisinW

wλ



P (λ J) =⊔

microisinRgt0IJ



ν =summ








P (λ J) =⋃

wisinWJ







06t61







wtM(λ J) =⋃

wisinWJ




wtL(λ) =⋃

wisinWIL(λ)

wν isin P+IL(λ)




















wisinWIV

TwλV










conv V =⋂

wisinWIV

TwλV (96)




Kt =⋃

iisinIIV















trprimerprime +




vprime =1sum


sumrprimeisinRprime












iisinI

⋂wisinWIV

Hiw





LocJ conv V =⋂

wisinWJprime




































References
























Ser Mat 54(1)3ndash25 221 1990


1 Introduction



Acknowledgments





34 Convexity















References



Proof











wisinWJ iisinIJ


















point of conv V










prime) ie

pk =sumj


sumj




pk(b) =sum










U(l)Vλ =oplus

microisinλ+ZπJ

Vmicro (52)





16j6n

αij (54)





)Vλ










on U(l)Vλ

















VWJmin

VtimesWJmax

V JminV (513)


F =⋃

wisinWJminV

(F cap wD)


F =⋃

wisinWJminV

(F cap wwD)









ckαik) = 0






iisinZgt0 fiqi


fV (q) =sumwtlJ





fV (q) =sumJsubI

[W WJ ]q|J | (517)














P =sumi


ti = 1
















to Proposition 317



P (λ J) = conv⋃

wisinWJ iisinIJ



0sum



tiλi J) (68)









J


































wtlprime = conv⋃




and the





















S =⊔FisinF

SF








is a face of P














fP(q) =sumFisinF

qdimF fF (q)









of⋂jisinJ α

perpj









0

0

λ

λ

minusλ

λ

λ










S

P

s s


As before we have
































convWλ =⋃wisinW










0 =1

|W |sumwisinW

wλ



P (λ J) =⊔

microisinRgt0IJ



ν =summ








P (λ J) =⋃

wisinWJ







06t61







wtM(λ J) =⋃

wisinWJ




wtL(λ) =⋃

wisinWIL(λ)

wν isin P+IL(λ)




















wisinWIV

TwλV










conv V =⋂

wisinWIV

TwλV (96)




Kt =⋃

iisinIIV















trprimerprime +




vprime =1sum


sumrprimeisinRprime












iisinI

⋂wisinWIV

Hiw





LocJ conv V =⋂

wisinWJprime




































References
























Ser Mat 54(1)3ndash25 221 1990


1 Introduction



Acknowledgments





34 Convexity















References











point of conv V










prime) ie

pk =sumj


sumj




pk(b) =sum










U(l)Vλ =oplus

microisinλ+ZπJ

Vmicro (52)





16j6n

αij (54)





)Vλ










on U(l)Vλ

















VWJmin

VtimesWJmax

V JminV (513)


F =⋃

wisinWJminV

(F cap wD)


F =⋃

wisinWJminV

(F cap wwD)









ckαik) = 0






iisinZgt0 fiqi


fV (q) =sumwtlJ





fV (q) =sumJsubI

[W WJ ]q|J | (517)














P =sumi


ti = 1
















to Proposition 317



P (λ J) = conv⋃

wisinWJ iisinIJ



0sum



tiλi J) (68)









J


































wtlprime = conv⋃




and the





















S =⊔FisinF

SF








is a face of P














fP(q) =sumFisinF

qdimF fF (q)









of⋂jisinJ α

perpj









0

0

λ

λ

minusλ

λ

λ










S

P

s s


As before we have
































convWλ =⋃wisinW










0 =1

|W |sumwisinW

wλ



P (λ J) =⊔

microisinRgt0IJ



ν =summ








P (λ J) =⋃

wisinWJ







06t61







wtM(λ J) =⋃

wisinWJ




wtL(λ) =⋃

wisinWIL(λ)

wν isin P+IL(λ)




















wisinWIV

TwλV










conv V =⋂

wisinWIV

TwλV (96)




Kt =⋃

iisinIIV















trprimerprime +




vprime =1sum


sumrprimeisinRprime












iisinI

⋂wisinWIV

Hiw





LocJ conv V =⋂

wisinWJprime




































References
























Ser Mat 54(1)3ndash25 221 1990


1 Introduction



Acknowledgments





34 Convexity















References





U(l)Vλ =oplus

microisinλ+ZπJ

Vmicro (52)





16j6n

αij (54)





)Vλ










on U(l)Vλ

















VWJmin

VtimesWJmax

V JminV (513)


F =⋃

wisinWJminV

(F cap wD)


F =⋃

wisinWJminV

(F cap wwD)









ckαik) = 0






iisinZgt0 fiqi


fV (q) =sumwtlJ





fV (q) =sumJsubI

[W WJ ]q|J | (517)














P =sumi


ti = 1
















to Proposition 317



P (λ J) = conv⋃

wisinWJ iisinIJ



0sum



tiλi J) (68)









J


































wtlprime = conv⋃




and the





















S =⊔FisinF

SF








is a face of P














fP(q) =sumFisinF

qdimF fF (q)









of⋂jisinJ α

perpj









0

0

λ

λ

minusλ

λ

λ










S

P

s s


As before we have
































convWλ =⋃wisinW










0 =1

|W |sumwisinW

wλ



P (λ J) =⊔

microisinRgt0IJ



ν =summ








P (λ J) =⋃

wisinWJ







06t61







wtM(λ J) =⋃

wisinWJ




wtL(λ) =⋃

wisinWIL(λ)

wν isin P+IL(λ)




















wisinWIV

TwλV










conv V =⋂

wisinWIV

TwλV (96)




Kt =⋃

iisinIIV















trprimerprime +




vprime =1sum


sumrprimeisinRprime












iisinI

⋂wisinWIV

Hiw





LocJ conv V =⋂

wisinWJprime




































References
























Ser Mat 54(1)3ndash25 221 1990


1 Introduction



Acknowledgments





34 Convexity















References

















VWJmin

VtimesWJmax

V JminV (513)


F =⋃

wisinWJminV

(F cap wD)


F =⋃

wisinWJminV

(F cap wwD)









ckαik) = 0






iisinZgt0 fiqi


fV (q) =sumwtlJ





fV (q) =sumJsubI

[W WJ ]q|J | (517)














P =sumi


ti = 1
















to Proposition 317



P (λ J) = conv⋃

wisinWJ iisinIJ



0sum



tiλi J) (68)









J


































wtlprime = conv⋃




and the





















S =⊔FisinF

SF








is a face of P














fP(q) =sumFisinF

qdimF fF (q)









of⋂jisinJ α

perpj









0

0

λ

λ

minusλ

λ

λ










S

P

s s


As before we have
































convWλ =⋃wisinW










0 =1

|W |sumwisinW

wλ



P (λ J) =⊔

microisinRgt0IJ



ν =summ








P (λ J) =⋃

wisinWJ







06t61







wtM(λ J) =⋃

wisinWJ




wtL(λ) =⋃

wisinWIL(λ)

wν isin P+IL(λ)




















wisinWIV

TwλV










conv V =⋂

wisinWIV

TwλV (96)




Kt =⋃

iisinIIV















trprimerprime +




vprime =1sum


sumrprimeisinRprime












iisinI

⋂wisinWIV

Hiw





LocJ conv V =⋂

wisinWJprime




































References
























Ser Mat 54(1)3ndash25 221 1990


1 Introduction



Acknowledgments





34 Convexity















References






iisinZgt0 fiqi


fV (q) =sumwtlJ





fV (q) =sumJsubI

[W WJ ]q|J | (517)














P =sumi


ti = 1
















to Proposition 317



P (λ J) = conv⋃

wisinWJ iisinIJ



0sum



tiλi J) (68)









J


































wtlprime = conv⋃




and the





















S =⊔FisinF

SF








is a face of P














fP(q) =sumFisinF

qdimF fF (q)









of⋂jisinJ α

perpj









0

0

λ

λ

minusλ

λ

λ










S

P

s s


As before we have
































convWλ =⋃wisinW










0 =1

|W |sumwisinW

wλ



P (λ J) =⊔

microisinRgt0IJ



ν =summ








P (λ J) =⋃

wisinWJ







06t61







wtM(λ J) =⋃

wisinWJ




wtL(λ) =⋃

wisinWIL(λ)

wν isin P+IL(λ)




















wisinWIV

TwλV










conv V =⋂

wisinWIV

TwλV (96)




Kt =⋃

iisinIIV















trprimerprime +




vprime =1sum


sumrprimeisinRprime












iisinI

⋂wisinWIV

Hiw





LocJ conv V =⋂

wisinWJprime




































References
























Ser Mat 54(1)3ndash25 221 1990


1 Introduction



Acknowledgments





34 Convexity















References








P =sumi


ti = 1
















to Proposition 317



P (λ J) = conv⋃

wisinWJ iisinIJ



0sum



tiλi J) (68)









J


































wtlprime = conv⋃




and the





















S =⊔FisinF

SF








is a face of P














fP(q) =sumFisinF

qdimF fF (q)









of⋂jisinJ α

perpj









0

0

λ

λ

minusλ

λ

λ










S

P

s s


As before we have
































convWλ =⋃wisinW










0 =1

|W |sumwisinW

wλ



P (λ J) =⊔

microisinRgt0IJ



ν =summ








P (λ J) =⋃

wisinWJ







06t61







wtM(λ J) =⋃

wisinWJ




wtL(λ) =⋃

wisinWIL(λ)

wν isin P+IL(λ)




















wisinWIV

TwλV










conv V =⋂

wisinWIV

TwλV (96)




Kt =⋃

iisinIIV















trprimerprime +




vprime =1sum


sumrprimeisinRprime












iisinI

⋂wisinWIV

Hiw





LocJ conv V =⋂

wisinWJprime




































References
























Ser Mat 54(1)3ndash25 221 1990


1 Introduction



Acknowledgments





34 Convexity















References



P (λ J) = conv⋃

wisinWJ iisinIJ



0sum



tiλi J) (68)









J


































wtlprime = conv⋃




and the





















S =⊔FisinF

SF








is a face of P














fP(q) =sumFisinF

qdimF fF (q)









of⋂jisinJ α

perpj









0

0

λ

λ

minusλ

λ

λ










S

P

s s


As before we have
































convWλ =⋃wisinW










0 =1

|W |sumwisinW

wλ



P (λ J) =⊔

microisinRgt0IJ



ν =summ








P (λ J) =⋃

wisinWJ







06t61







wtM(λ J) =⋃

wisinWJ




wtL(λ) =⋃

wisinWIL(λ)

wν isin P+IL(λ)




















wisinWIV

TwλV










conv V =⋂

wisinWIV

TwλV (96)




Kt =⋃

iisinIIV















trprimerprime +




vprime =1sum


sumrprimeisinRprime












iisinI

⋂wisinWIV

Hiw





LocJ conv V =⋂

wisinWJprime




































References
























Ser Mat 54(1)3ndash25 221 1990


1 Introduction



Acknowledgments





34 Convexity















References




























wtlprime = conv⋃




and the





















S =⊔FisinF

SF








is a face of P














fP(q) =sumFisinF

qdimF fF (q)









of⋂jisinJ α

perpj









0

0

λ

λ

minusλ

λ

λ










S

P

s s


As before we have
































convWλ =⋃wisinW










0 =1

|W |sumwisinW

wλ



P (λ J) =⊔

microisinRgt0IJ



ν =summ








P (λ J) =⋃

wisinWJ







06t61







wtM(λ J) =⋃

wisinWJ




wtL(λ) =⋃

wisinWIL(λ)

wν isin P+IL(λ)




















wisinWIV

TwλV










conv V =⋂

wisinWIV

TwλV (96)




Kt =⋃

iisinIIV















trprimerprime +




vprime =1sum


sumrprimeisinRprime












iisinI

⋂wisinWIV

Hiw





LocJ conv V =⋂

wisinWJprime




































References
























Ser Mat 54(1)3ndash25 221 1990


1 Introduction



Acknowledgments





34 Convexity















References















S =⊔FisinF

SF








is a face of P














fP(q) =sumFisinF

qdimF fF (q)









of⋂jisinJ α

perpj









0

0

λ

λ

minusλ

λ

λ










S

P

s s


As before we have
































convWλ =⋃wisinW










0 =1

|W |sumwisinW

wλ



P (λ J) =⊔

microisinRgt0IJ



ν =summ








P (λ J) =⋃

wisinWJ







06t61







wtM(λ J) =⋃

wisinWJ




wtL(λ) =⋃

wisinWIL(λ)

wν isin P+IL(λ)




















wisinWIV

TwλV










conv V =⋂

wisinWIV

TwλV (96)




Kt =⋃

iisinIIV















trprimerprime +




vprime =1sum


sumrprimeisinRprime












iisinI

⋂wisinWIV

Hiw





LocJ conv V =⋂

wisinWJprime




































References
























Ser Mat 54(1)3ndash25 221 1990


1 Introduction



Acknowledgments





34 Convexity















References














fP(q) =sumFisinF

qdimF fF (q)









of⋂jisinJ α

perpj









0

0

λ

λ

minusλ

λ

λ










S

P

s s


As before we have
































convWλ =⋃wisinW










0 =1

|W |sumwisinW

wλ



P (λ J) =⊔

microisinRgt0IJ



ν =summ








P (λ J) =⋃

wisinWJ







06t61







wtM(λ J) =⋃

wisinWJ




wtL(λ) =⋃

wisinWIL(λ)

wν isin P+IL(λ)




















wisinWIV

TwλV










conv V =⋂

wisinWIV

TwλV (96)




Kt =⋃

iisinIIV















trprimerprime +




vprime =1sum


sumrprimeisinRprime












iisinI

⋂wisinWIV

Hiw





LocJ conv V =⋂

wisinWJprime




































References
























Ser Mat 54(1)3ndash25 221 1990


1 Introduction



Acknowledgments





34 Convexity















References





of⋂jisinJ α

perpj









0

0

λ

λ

minusλ

λ

λ










S

P

s s


As before we have
































convWλ =⋃wisinW










0 =1

|W |sumwisinW

wλ



P (λ J) =⊔

microisinRgt0IJ



ν =summ








P (λ J) =⋃

wisinWJ







06t61







wtM(λ J) =⋃

wisinWJ




wtL(λ) =⋃

wisinWIL(λ)

wν isin P+IL(λ)




















wisinWIV

TwλV










conv V =⋂

wisinWIV

TwλV (96)




Kt =⋃

iisinIIV















trprimerprime +




vprime =1sum


sumrprimeisinRprime












iisinI

⋂wisinWIV

Hiw





LocJ conv V =⋂

wisinWJprime




































References
























Ser Mat 54(1)3ndash25 221 1990


1 Introduction



Acknowledgments





34 Convexity















References








S

P

s s


As before we have
































convWλ =⋃wisinW










0 =1

|W |sumwisinW

wλ



P (λ J) =⊔

microisinRgt0IJ



ν =summ








P (λ J) =⋃

wisinWJ







06t61







wtM(λ J) =⋃

wisinWJ




wtL(λ) =⋃

wisinWIL(λ)

wν isin P+IL(λ)




















wisinWIV

TwλV










conv V =⋂

wisinWIV

TwλV (96)




Kt =⋃

iisinIIV















trprimerprime +




vprime =1sum


sumrprimeisinRprime












iisinI

⋂wisinWIV

Hiw





LocJ conv V =⋂

wisinWJprime




































References
























Ser Mat 54(1)3ndash25 221 1990


1 Introduction



Acknowledgments





34 Convexity















References

























convWλ =⋃wisinW










0 =1

|W |sumwisinW

wλ



P (λ J) =⊔

microisinRgt0IJ



ν =summ








P (λ J) =⋃

wisinWJ







06t61







wtM(λ J) =⋃

wisinWJ




wtL(λ) =⋃

wisinWIL(λ)

wν isin P+IL(λ)




















wisinWIV

TwλV










conv V =⋂

wisinWIV

TwλV (96)




Kt =⋃

iisinIIV















trprimerprime +




vprime =1sum


sumrprimeisinRprime












iisinI

⋂wisinWIV

Hiw





LocJ conv V =⋂

wisinWJprime




































References
























Ser Mat 54(1)3ndash25 221 1990


1 Introduction



Acknowledgments





34 Convexity















References









0 =1

|W |sumwisinW

wλ



P (λ J) =⊔

microisinRgt0IJ



ν =summ








P (λ J) =⋃

wisinWJ







06t61







wtM(λ J) =⋃

wisinWJ




wtL(λ) =⋃

wisinWIL(λ)

wν isin P+IL(λ)




















wisinWIV

TwλV










conv V =⋂

wisinWIV

TwλV (96)




Kt =⋃

iisinIIV















trprimerprime +




vprime =1sum


sumrprimeisinRprime












iisinI

⋂wisinWIV

Hiw





LocJ conv V =⋂

wisinWJprime




































References
























Ser Mat 54(1)3ndash25 221 1990


1 Introduction



Acknowledgments





34 Convexity















References




06t61







wtM(λ J) =⋃

wisinWJ




wtL(λ) =⋃

wisinWIL(λ)

wν isin P+IL(λ)




















wisinWIV

TwλV










conv V =⋂

wisinWIV

TwλV (96)




Kt =⋃

iisinIIV















trprimerprime +




vprime =1sum


sumrprimeisinRprime












iisinI

⋂wisinWIV

Hiw





LocJ conv V =⋂

wisinWJprime




































References
























Ser Mat 54(1)3ndash25 221 1990


1 Introduction



Acknowledgments





34 Convexity















References












wisinWIV

TwλV










conv V =⋂

wisinWIV

TwλV (96)




Kt =⋃

iisinIIV















trprimerprime +




vprime =1sum


sumrprimeisinRprime












iisinI

⋂wisinWIV

Hiw





LocJ conv V =⋂

wisinWJprime




































References
























Ser Mat 54(1)3ndash25 221 1990


1 Introduction



Acknowledgments





34 Convexity















References




conv V =⋂

wisinWIV

TwλV (96)




Kt =⋃

iisinIIV















trprimerprime +




vprime =1sum


sumrprimeisinRprime












iisinI

⋂wisinWIV

Hiw





LocJ conv V =⋂

wisinWJprime




































References
























Ser Mat 54(1)3ndash25 221 1990


1 Introduction



Acknowledgments





34 Convexity















References





trprimerprime +




vprime =1sum


sumrprimeisinRprime












iisinI

⋂wisinWIV

Hiw





LocJ conv V =⋂

wisinWJprime




































References
























Ser Mat 54(1)3ndash25 221 1990


1 Introduction



Acknowledgments





34 Convexity















References


































References
























Ser Mat 54(1)3ndash25 221 1990


1 Introduction



Acknowledgments





34 Convexity















References










Ser Mat 54(1)3ndash25 221 1990


1 Introduction



Acknowledgments





34 Convexity















References

faces of highest weight modules and the universal …gsd/facesuniv.pdf · max, and therefore the...

Documents