Characters of Highest Weight Modules and Integrability

CHARACTERS OF HIGHEST WEIGHT MODULES AND INTEGRABILITY

GURBIR DHILLON AND APOORVA KHARE

Abstract. We give the first positive formulas for the weights of every simple highest weight module L(λ) over an arbitrary Kac–Moody algebra. These formulas affirmatively answer a question of Daniel Bump on simple characters. Under a mild condition on the highest weight, we also express the weights of L(λ) as an alternating sum similar to the Weyl–Kac character formula. To obtain these results, we show the following data attached to a highest weight module are equivalent: (i) its integrability, (ii) the convex hull of its weights, (iii) the Weyl group symmetry of its character, and (iv) when a localization theorem is available, its behavior on certain codimension one Schubert cells. We further determine precisely when the above datum determines the weights themselves, which answers a question of Lepowsky. Moreover, we use condition (iv) to relate localizations of the convex hull of the weights with the introduction of poles of the corresponding D-module on certain divisors, which answers a question of Brion. Many of these results are new even in finite type. We prove similar assertions for highest weight modules over a symmetrizable quantum group.

Contents 1. Introduction 1 2. Statements of results 2 3. Preliminaries and notations 7 4. A classiﬁcation of highest weight modules to ﬁrst order 9 5. A question of Bump and weight formulas for non-integrable simple modules 12 6. A question of Lepowsky on the weights of highest weight modules 13 7. The Weyl–Kac formula for the weights of simple modules 16 8. A question of Brion on localization of D-modules and a geometric interpretation of integrability 18 9. Highest weight modules over symmetrizable quantum groups 21 References 25

1. Introduction A fundamental problem in the representation theory of semisimple and Kac–Moody Lie algebras is to compute characters of simple highest weight modules. In all cases where they are known even conjecturally, the solutions proceed roughly by (i) expressing simple

Date: August 10, 2018. 2010 Mathematics Subject Classification. Primary: 17B10; Secondary: 17B67, 22E47, 17B37, 20G42. Key words and phrases. Highest weight module, parabolic Verma module, integrable Weyl group, Tits cone, Weyl–Kac formula, weight formula, Beilinson–Bernstein localization, quantum group. 1 2 GURBIR DHILLON AND APOORVA KHARE characters as linear combinations of Verma characters. The relevant block of Category O is then (ii) identified as a categorification of a module for the Hecke algebra. The two bases of simple and Verma characters correspond to two canonical bases for the module, whose change of basis is determined via the combinatorics of the Coxeter system. The subtlety in steps (i) and (ii) is already apparent for regular blocks for affine Lie algebras. At noncritical levels, as in finite type, one takes bona fide Verma modules, and the relevant Hecke module is the regular representation. At the critical level, one replaces each Verma module with a quotient, the restricted Verma module, and the desired Hecke module is the periodic module. To our knowledge, there is not even a conjectural description of critical blocks beyond affine type along the lines of (i) and (ii). Another issue of longstanding interest is to obtain manifestly positive formulas for simple characters. I.e., in step (i) the coefficients of Verma characters generically come with signs, as visible in the Weyl–Kac and Kazhdan–Lusztig character formulas, and one would like alternative formulae for simple characters without this issue. This problem is solved via crystal bases for regular dominant and regular anti-dominant highest weights, but is wide open in general. With these motivations, in this paper we address the simpler question of determining not the character of a simple highest weight module but rather its weights. I.e., one can ask for the eigenvalues of the Cartan’s action, but not the exact multiplicities. We give multiple formulas for weights of an arbitrary simple highest weight module for any Kac–Moody algebra. In light of the above discussion, we emphasize first that the formulas hold in the cases where there are no predictions for characters in the spirit of (i) and (ii), and moreover are remarkably uniform across different levels. We emphasize second that the formulas are visibly positive. After conversations with experts, it seems the formulas described in this paper likely did not appear earlier, even as folkore. This is rather remarkable, considering both the accessibility and naturality of the question and the transparency of its solution. The main inputs into our formulas can be packaged as a novel equivalence of several invariants of general highest weight modules. We also apply this equivalence to answer questions of Daniel Bump and James Lepowsky on the weights of highest weight modules, and a question of Michel Brion concerning the corresponding D-modules on the flag variety. Several more applications, including a conjectural identity in the combinatorics of affine root systems, are described below. In the companion paper [11], we further apply the equivalence to obtain a complete combinatorial description of the convex hull of the character of an arbitrary highest weight module.

Organization of the paper. In Section 2, we describe our results and mention two questions that may warrant further consideration. After introducing notation in Section 3, we prove in Section 4 the main result. In Sections 5–8, we develop diﬀerent applications of the main result. Finally, in Section 9, we discuss extensions of the previous sections to quantized enveloping algebras.

2. Statements of results Throughout the paper, unless otherwise speciﬁed g is a Kac–Moody algebra over C with triangular decomposition g = n− ⊕ h ⊕ n+, V is a g-module of highest weight λ ∈ h∗, and L(λ) is the simple module with highest weight λ. The ordering of this section is parallel to the organization of the paper and we refer the reader to Section 3 for notation. CHARACTERS OF HIGHEST WEIGHT MODULES AND INTEGRABILITY 3

2.1. A classification of highest weight modules to first order. As alluded to in the introduction, our weight formulas for simple modules emerge from a more general study of highest weight modules. The basic ingredient, which also underlies our answers to the questions of Brion, Bump, and Lepowsky, is the identification of several a priori different invariants of a general highest weight module. Let I index the simple roots of g, with corresponding simple lowering operators fi, i ∈ I. Theorem 2.1. The following invariants of a highest weight module V are equivalent, i.e. determine one another:

(1) IV , the integrability of V , i.e. IV = {i ∈ I : fi acts locally nilpotently on V }. (2) conv V , the convex hull of the weights of V . (3) The stabilizer of the character of V in the Weyl group W .

In particular, the convex hull in (2) is always that of the parabolic Verma module M(λ, IV ), and the stabilizer in (3) is always the parabolic subgroup WIV . To our knowledge, the equivalences of Theorem 2.1 are new, even for g of finite type. Moreover, when a localization theorem is available, we describe an equivalent geometric invariant of the perverse sheaf corresponding to V in Proposition 2.13 below. Let us indicate the sense in which the datum of Theorem 2.1 can be thought of as a ‘first- order’ invariant of V . Algebraically, from (1), we see that this datum is sensitive only to the action of each simple lowering operator fi on the highest weight line. Categorically, this is the same as the Jordan–Höldercontent [V : L(si · λ)], for those i ∈ I with si · λ 6 λ. 1 Geometrically, when localization is available, the same si · λ index certain Schubert divisors in the support of V . As we will see in Proposition 2.13, this datum records the generic behavior of V along these divisors, and in particular is insensitive to any phenomenon in codimension at least two. In the proof of Theorem 2.1, the difficult implication is that the natural map M(λ, IV ) ⊗ Vλ → V induces an equality on convex hulls of weights, where Vλ is the highest weight line of V . To our knowledge this implication, and in particular the theorem, is new for g of both finite and infinite type. It is somewhat surprising that this theorem was not previously known, and we shall see it can be applied fruitfully to several experts’ questions in highest weight representation theory. 2.2. A question of Bump and weight formulas for non-integrable simple modules. In this section we obtain three positive formulas for the weights of simple highest weight modules, wt L(λ), corresponding to the manifestations (1)–(3) of the datum of Theorem 2.1. The first formula uses restriction to the maximal standard Levi subalgebra whose action is integrable:

Proposition 2.2. Write l for the Levi subalgebra corresponding to IL(λ), and write Ll(ν) for ∗ the simple l module of highest weight ν ∈ h . Denote by π the simple roots of g, and πIL(λ) the simple roots of l. Then: G wt L(λ) = wt Ll(λ − µ). (2.3) µ∈ >0π\π Z IL(λ)

1Throughout the paper, we use the shorthand ‘when localization is available’ to mean ‘using the equivalence of categories afforded by Beilinson–Bernstein localization, when it is a theorem for the relevant Kac–Moody algebra and block of Category O’. For example, there are equivalences between regular integral blocks of Category O for g of finite type (or untwisted affine type at negative level) and Schubert constructible perverse sheaves on the (affine) flag variety. 4 GURBIR DHILLON AND APOORVA KHARE

The second formula explicitly shows the relationship between wt L(λ) and its convex hull conv L(λ): Proposition 2.4. ∗ wt L(λ) = conv L(λ) ∩ {µ ∈ h : µ 6 λ}. (2.5) The third formula uses the Weyl group action, and a parabolic analogue P + of the IL(λ) dominant chamber introduced in Section 3.3:

Proposition 2.6. Suppose λ has finite stabilizer in WIL(λ) , the Weyl group of l. Then: + wt L(λ) = WI {µ ∈ P : µ λ}. (2.7) L(λ) IL(λ) 6 We remind that the assumption that λ has finite stabilizer is very mild, as recalled in Propo- sition 3.2(4). The question of whether Proposition 2.4 holds, namely, whether the weights of simple highest weight modules are no finer an invariant than their hull, was raised by Daniel Bump. Propositions 2.2, 2.4, and 2.6 are well known for integrable L(λ), and Propositions 2.2 and 2.4 were proved by the second named author in finite type [28]. The remaining cases, in particular Proposition 2.6 in all types, are to our knowledge new. The obtained descriptions of wt L(λ) are particularly striking in infinite type. When g is affine the formulae are insensitive to whether λ is critical or non-critical. In contrast, critical level modules, which figure prominently in the Geometric Langlands program [1, 14, 15], behave very differently from noncritical modules, even at the level of characters. When g is symmetrizable we similarly obtain weight formulae at critical λ. The authors are unaware of even conjectural formulae for ch L(λ) in this case. Finally, when g is non-symmetrizable, it is unknown how to compute weight space multiplicities even for integrable L(λ). Thus for g non-symmetrizable, our formulae provide as much information on ch L(λ) as one could hope for, given existing methods. We obtain the above three formulae using the following theorem. For V a general highest weight module, it gives a necessary and sufficient condition for M(λ, IV ) → V to induce an equality of weights, in terms of the action of a certain Levi subalgebra. Define the potential p integrability of V to be IV := IL(λ) \ IV . To justify the terminology, note that these are precisely the simple directions which become integrable in quotients of V .

Theorem 2.8. Let V be a highest weight module, Vλ its highest weight line. Let l denote the p Levi subalgebra corresponding to IV . Then wt V = wt M(λ, IV ) if and only if wt U(l)Vλ = 0 wt U(l)M(λ, IV )λ, i.e. wt U(l)Vλ = λ − > π p . Z IV In particular, if V = L(λ), then wt L(λ) = wt M(λ, IL(λ)). Theorem 2.8 is new in both finite and infinite type. 2.3. A question of Lepowsky on the weights of highest weight modules. In this section, we revisit the problem encountered in the proofs of Propositions 2.2, 2.4, 2.6, namely under what circumstances does the datum of Theorem 2.1 determine wt V . It was noticed by the second named author in [28] that wt V is a finer invariant than p conv V whenever the potential integrability IV contains orthogonal roots. The prototypical example here is for g = sl2 ⊕ sl2, and V = M(0)/M(s1s2 · 0). James Lepowsky asked whether this orthogonality is the only obstruction to determining wt V from its convex hull. In finite type, this would reduce to a question in types A2,B2, and G2. We now answer this question affirmatively for g an arbitrary Kac–Moody algebra. CHARACTERS OF HIGHEST WEIGHT MODULES AND INTEGRABILITY 5

∗ Theorem 2.9. Fix λ ∈ h and J ⊂ IL(λ). Every highest weight module V of highest weight λ and integrability J has the same weights if and only if IL(λ) \ J is complete, i.e. (ˇαi, αj) 6= 0, ∀i, j ∈ IL(λ) \ J. We are not aware of a precursor to Theorem 2.9 in the literature, even in ﬁnite type.

2.4. The Weyl–Kac formula for the weights of simple modules. We prove the following identity, whose notation we will explain below.

∗ Theorem 2.10. For λ ∈ h such that the stabilizer of λ in WIL(λ) is ﬁnite, we have: X eλ wt L(λ) = w . (2.11) Q (1 − e−αi ) w∈W i∈I IL(λ) P µ On the left hand side of (2.11) we mean the ‘multiplicity-free’ character ∗ e . µ∈h :L(λ)µ6=0 On the right hand side of (2.11), in each summand we take the ‘highest weight’ expansion of w(1 − e−αi )−1, i.e.:

( −wα −2wα 1 1 + e i + e i + ··· , wαi > 0, w −α := wα 2wα 3wα 1 − e i −e i − e i − e i − · · · , wαi < 0. In particular, as claimed in the introduction, each summand comes with signs but no multiplicities as in Kazhdan–Lusztig theory. Finally, we note that the denominator in (2.11) is not the usual Weyl denominator, but instead a multiplicity-free variant which runs only over the simple roots, i.e. replaces n by n/[n, n]. Precursors to Theorem 2.10 include Kass [26], Brion [8], Walton [42], Postnikov [39], and Sch¨utzer[40] for the integrable highest weight modules. As pointed out to us by Michel Brion, in ﬁnite type Theorem 2.10 follows from a more general formula for exponential sums over polyhedra, cf. Remark 7.9 below. All other cases are to our knowledge new.

2.5. A question of Brion on localization of D-modules and a geometric interpretation of integrability. In this section we use a topological manifestation of the datum in Theorem 2.1 to answer a question of Brion in geometric representation theory. Brion’s question is as follows. Let g be of finite type, λ a regular dominant integral weight, and consider conv L(λ), the Weyl polytope. For any J ⊂ I, if one intersects the tangent cones of conv L(λ) at wλ, w ∈ WJ , one obtains conv M(λ, J). Brion first asked whether an analogous formula holds for other highest weight modules. We answer this question affirmatively in [11]. Brion further observed that this procedure of localizing conv L(λ) in convex geometry is the shadow of a localization in complex geometry. More precisely, let X denote the flag 0 variety, and Lλ denote the line bundle on X with H (X, Lλ) ' L(λ). As usual, write Xw for the closure of the Bruhat cell Cw := BwB/B, ∀w ∈ W , and write w◦ for the longest element of W . Let D denote the standard dualities on Category O and regular holonomic

D-modules. Then for a union of Schubert divisors Z = ∪i∈I\J Xsiw◦ with complement U, 0 we have H (U, Lλ) ' DM(λ, J). Thus in this case localization of the convex hull could be recovered as taking the convex hull of the weights of sections of Lλ on an appropriate open set U. Brion asked whether a similar result holds for more general highest weight modules. We answer this affirmatively for a regular integral infinitesimal character. By translation, it suffices to examine the regular block O0. 6 GURBIR DHILLON AND APOORVA KHARE

Theorem 2.12. Let g be of ﬁnite type, and λ = w·−2ρ. Let V be a g-module of highest weight

λ, and write V for the corresponding D-module on X. For J ⊂ IV , set Z = ∪i∈IV \J Xsiw, 0 and write G/B = Z t U˜. Then we have: DH (U,˜ DV) is a g-module of highest weight λ and integrability J.

Let us mention one ingredient in the proof of Theorem 2.12. Let Vrh denote the perverse sheaf corresponding to V under the Riemann–Hilbert correspondence. Then the datum of Theorem 2.1 manifests as:

Proposition 2.13. For J ⊂ IL(λ), consider the smooth open subvariety of Xw given on complex points by: G UJ := Cw t Csiw. (2.14) i∈J

Letting V, Vrh be as above, upon restricting to UIL(λ) we have:

Vrh|U ' j! U [`(w)], (2.15) IL(λ) C IV where j is the open embedding UIV → UIL(λ) and j! is extension by zero in the constructible derived category. In particular, IV = {i ∈ I : the stalks of Vrh along Csiw are nonzero}, where by stalk we mean the object of Db(Vect) obtained by ∗-restriction to a point. To our knowledge Proposition 2.13 is new, though not difficult to prove once stated. The proof is a pleasing geometric avatar of the idea that M(λ, IV ) → V is an isomorphism ‘to first order’. 2.6. Highest weight modules over symmetrizable quantum groups. In the final section, we apply both the methods and the results from earlier to study highest weight modules over quantum groups, and obtain results similar to those discussed above. 2.7. Further problems. We conclude this section by calling attention to two problems suggested by our results. Problem 1. Multiplicity-free Macdonald identities. The statement of Theorem 2.10 includes the assumption that the highest weight has finite integrable stabilizer. However, the asserted identity otherwise tends to fail in interesting ways. For example: Proposition 2.16. For g of rank 2 and the trivial module L(0), we have: X 1 X w = 1 + eα, (2.17) (1 − e−α1 )(1 − e−α2 ) w∈W im α∈∆− im where ∆− denotes the set of negative imaginary roots. It would be interesting to formulate a correct version of Theorem 2.10 even for one-dimensional modules in affine type, which might be thought of as a multiplicity-free Macdonald identity. Indeed, Equation (2.17) suggests correction terms coming from imaginary roots, akin to [24]. Moreover, discussion with an expert suggests such a formula is indeed true, and we hope to return to it in subsequent work. Problem 2. From convex hulls to weights. The regular block of Category O is the prototypical example of a category of infinite-dimensional g-modules, of perverse sheaves on a stratified space, and of a highest weight category. Accordingly, it has very rich representation-theoretic, geometric, and purely categorical structure. The main theorem of the paper, Theorem 2.1, determines the meaning of the convex hull of the weights in these three guises, namely: the CHARACTERS OF HIGHEST WEIGHT MODULES AND INTEGRABILITY 7 integrability, behavior in codimension one, and the Jordan–Höldercontent [V : L(si · λ)]. It would be very interesting to obtain a similar understanding of the weights themselves.

Acknowledgments. It is a pleasure to thank Michel Brion, Daniel Bump, Ian Grojnowski, Shrawan Kumar, James Lepowsky, Sam Raskin, and Zhiwei Yun for valuable discussions. The work of G.D. is partially supported by the Department of Defense (DoD) through the NDSEG fellowship. A.K. is partially supported by Ramanujan Fellowship SB/S2/RJN-121/2017 and MATRICS grant MTR/2017/000295 from SERB (Govt. of India), by grant F.510/25/CAS- II/2018(SAP-I) from UGC (Govt. of India), and by a Young Investigator Award from the Infosys Foundation.

3. Preliminaries and notations The contents of this section are mostly standard. We advise the reader to skim Subsection 3.3, and refer back to the rest only as needed.

3.1. Notation for numbers and sums. We write Z for the integers, and Q, R, C for the rational, real, and complex numbers respectively. For a subset S of a real vector space E, we 0 0 write Z> S for the set of ﬁnite linear combinations of S with coeﬃcients in Z> , and similarly 0 ZS, Q> S, RS, etc.

3.2. Notation for Kac–Moody algebras, standard parabolic and Levi subalgebras. The basic references are [25] and [33]. In this paper we work throughout over C. Let I be a ﬁnite set, and A = (aij)i,j∈I a generalized Cartan matrix. Fix a realization (h, π, πˇ), with ∗ simple roots π = {αi}i∈I ⊂ h and corootsπ ˇ = {αˇi}i∈I ⊂ h satisfying (ˇαi, αj) = aij, ∀i, j ∈ I. Let g := g(A) be the associated Kac–Moody algebra generated by {ei, fi : i ∈ I} and h, modulo the relations:

[ei, fj] = δijαˇi, [h, ei] = (h, αi)ei, [h, fi] = −(h, αi)fi, [h, h] = 0, ∀h ∈ h, i, j ∈ I,

1−aij 1−aij (ad ei) (ej) = 0, (ad fi) (fj) = 0, ∀i, j ∈ I, i 6= j. Denote by g(A) the quotient of g(A) by the largest ideal intersecting h trivially; these coincide when 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 ∆+, ∆− denote the sets of positive and negative roots, respectively. We write α > 0 for + − P α ∈ ∆ , and similarly α < 0 for α ∈ ∆ . For a sum of roots β = i∈I kiαi with all ki > 0, write supp β := {i ∈ I : ki 6= 0}. Write

− M + M n := gα, n := gα. α<0 α>0 ∗ ∗ Let 6 denote the standard partial order on h , i.e. for µ, λ ∈ h , µ 6 λ if and only if 0 λ − µ ∈ Z> π. For any J ⊂ I, let lJ denote the associated Levi subalgebra generated by {ei, fi : i ∈ J} ∗ and h. For λ ∈ h write LlJ (λ) for the simple lJ -module of highest weight λ. Writing AJ for the principal submatrix (ai,j)i,j∈J , we may (non-canonically) realize g(AJ ) =: gJ as a + − subalgebra of g(A). Now write πJ , ∆J , ∆J for the simple, positive, and negative roots of ∗ g(AJ ) in h , respectively (note these are independent of the choice of realization). Finally, 8 GURBIR DHILLON AND APOORVA KHARE

+ − + − we define the associated Lie subalgebras uJ , uJ , nJ , nJ by: ± M ± M uJ := gα, nJ := gα, ± ± ± α∈∆ \∆J α∈∆J + and pJ := lJ ⊕ uJ to be the associated parabolic subalgebra. 3.3. Weyl group, parabolic subgroups, Tits cone. Write W for the Weyl group of g, 0 generated by the simple reflections {si, i ∈ I}, and let ` : W → Z> be the associated length function. For J ⊂ I, let WJ denote the parabolic subgroup of W generated by {sj, j ∈ J}. + ∗ 0 Write P for the dominant integral weights, i.e. {µ ∈ h : (ˇαi, µ) ∈ Z> , ∀i ∈ I}. The following choice is non-standard. Define the real subspace h∗ := {µ ∈ h∗ : (ˇα , µ) ∈ , ∀i ∈ I}. R i R ∗ 0 ∗ Now define the dominant chamber as D := {µ ∈ h : (ˇαi, µ) ∈ R> , ∀i ∈ I} ⊂ h , and the S R Tits cone as C := w∈W wD. Remark 3.1. In [33] and [25], the authors define h∗ to be a real form of h∗. This is smaller R than our definition whenever the generalized Cartan matrix A is non-invertible, and has the consequence that the dominant integral weights are not all in the dominant chamber, unlike for us. This is a superficial difference, but our convention helps avoid constantly introducing arguments like [33, Lemma 8.3.2]. We will also need parabolic analogues of the above. For J ⊂ I, define h∗ (J) := {µ ∈ h∗ : R ∗ 0 (ˇαj, µ) ∈ R, ∀j ∈ J}, the J dominant chamber as DJ := {µ ∈ h : (ˇαj, µ) ∈ R> , ∀j ∈ J}, and the J Tits cone as C := S wD . Finally, we write P + for the J dominant integral J w∈WJ J J ∗ 0 weights, i.e. {µ ∈ h : (ˇαj, µ) ∈ Z> , ∀j ∈ J}. The following standard properties will be used without further reference in the paper: Proposition 3.2. For g(A) with realization (h, π, πˇ), let g(At) be the dual algebra with real- ∗ ˇ + t t ization (h , π,ˇ π). Write ∆J for the positive roots of the standard Levi subalgebra lJ ⊂ g(A ). (1) For µ ∈ DJ , the isotropy group {w ∈ WJ : wµ = µ} is generated by the simple reflections it contains. (2) The J dominant chamber is a fundamental domain for the action of WJ on the J Tits cone, i.e., every WJ orbit in CJ meets DJ in precisely one point. (3) C = {µ ∈ h∗ (J) : (ˇα, µ) < 0 for at most finitely many αˇ ∈ ∆ˇ +}; in particular, C J R J J is a convex cone. (4) Consider C as a subset of h∗ (J) in the analytic topology, and fix µ ∈ C . Then µ is J R J an interior point of CJ if and only if the isotropy group {w ∈ WJ : wµ = µ} is finite. Proof. The reader can easily check that the standard arguments, cf. [33, Proposition 1.4.2], apply in this setting mutatis mutandis. ∗ We also fix ρ ∈ h satisfying (ˇαi, ρ) = 1, ∀i ∈ I, and define the dot action of W via w · µ := w(µ + ρ) − ρ; this does not depend on the choice of ρ. 3.4. Representations, integrability, and parabolic Verma modules. Given an h- ∗ module M and µ ∈ h , write Mµ for the corresponding simple eigenspace of M, i.e. Mµ := ∗ {m ∈ M : hm = (h, µ)m ∀h ∈ h}, and write wt M := {µ ∈ h : Mµ 6= 0}. Let V be a highest weight g-module with highest weight λ ∈ h∗. For J ⊂ I, we say V is J integrable if fj acts locally nilpotently on V , ∀j ∈ J. The following standard lemma may be deduced from [33, Lemma 1.3.3] and the proof of [33, Lemma 1.3.5]. Lemma 3.3. CHARACTERS OF HIGHEST WEIGHT MODULES AND INTEGRABILITY 9

(1) If V is of highest weight λ, then V is J integrable if and only if fj acts nilpotently on the highest weight line Vλ, ∀j ∈ J. (2) If V is J integrable, then ch V is WJ -invariant.

Let IV denote the maximal J for which V is J integrable, i.e.,

>0 (ˇαi,λ)+1 IV = {i ∈ I : (ˇαi, λ) ∈ Z , fi Vλ = 0}. (3.4)

We will call WIV the integrable Weyl group. We next remind the basic properties of parabolic Verma modules over Kac–Moody algebras. These are also known in the literature as generalized Verma modules, e.g. in the original papers by Lepowsky (see [35] and the references therein). ∗ >0 Fix λ ∈ h and a subset J of IL(λ) = {i ∈ I : (ˇαi, λ) ∈ Z }. The parabolic Verma module M(λ, J) co-represents the following functor from g -mod to Set: + M {m ∈ Mλ : n m = 0, fj acts nilpotently on m, ∀j ∈ J}. (3.5) When J is empty, we simply write M(λ) for the Verma module. From the definition and Lemma 3.3, it follows that M(λ, J) is a highest weight module that is J integrable. More generally, for any subalgebra lJ ⊂ s ⊂ g, the module Ms(λ, J) will co-represent the functor from s -mod to Set: + M {m ∈ Mλ :(n ∩ s)m = 0, fj acts nilpotently on m, ∀j ∈ J}. (3.6) We will only be concerned with s equal to a Levi or parabolic subalgebra. In accordance with the literature, in the case of “full integrability” we define Lmax(λ) := M(λ, I) for λ ∈ P +, and similarly Lmax(λ) := M (λ, J) and Lmax(λ) := M (λ, J) for λ ∈ P +. Finally, write pJ pJ lJ lJ J L (λ) for the simple quotient of Lmax(λ). lJ lJ + Proposition 3.7 (Basic formulae for parabolic Verma modules). Let λ ∈ PJ . (ˇαj ,λ)+1 (1) M(λ, J) ' M(λ)/(fj M(λ)λ, ∀j ∈ J). + + (2) Given a Lie subalgebra lJ ⊂ s ⊂ g, define sJ := lJ ⊕ (uJ ∩ s). Then:

s lJ max M (λ, J) ' Ind + Res L (λ), (3.8) s s + lJ J sJ + where we view lJ as a quotient of sJ via the short exact sequence: + + 0 → uJ ∩ s → sJ → lJ → 0. In particular, for s = g:

>0 + + wt M(λ, J) = wt LlJ (λ) − Z (∆ \ ∆J ). (3.9)

+ lJ We remark that some authors use the term inﬂation (from lJ to sJ ) in place of Res + . sJ

4. A classification of highest weight modules to first order In this section we prove the main result of the paper, Theorem 2.1, except for the perverse sheaf formulation, which we address in Section 8. The two tools we use are: (i) an “Integrable Slice Decomposition” of the weights of parabolic Verma modules M(λ, J), which extends a previous construction in [28] to representations of Kac–Moody algebras; and (ii) a “Ray Decomposition” of the convex hull of these weights, which is novel in both ﬁnite and inﬁnite type for non-integrable highest weight modules. 10 GURBIR DHILLON AND APOORVA KHARE

4.1. The Integrable Slice Decomposition. The following is the technical heart of this section. Proposition 4.1 (Integrable Slice Decomposition). G wt M(λ, J) = wt LlJ (λ − µ). (4.2) 0 µ∈Z> (π\πJ ) In particular, wt M(λ, J) lies in the J Tits cone (cf. Section 3.3). In proving Proposition 4.1 and below, the following results will be of use to us: Proposition 4.3 ([25, §11.2 and Proposition 11.3(a)]). For λ ∈ P +, µ ∈ h∗, say µ is non- degenerate with respect to λ if µ 6 λ and λ is not perpendicular to any connected component of supp(λ − µ), cf. Section 3.2 for notation. Let V be an integrable module of highest weight λ. (1) If µ ∈ P +, then µ ∈ wt V if and only if µ is non-degenerate with respect to λ. (2) If the sub-diagram on {i ∈ I : (ˇαi, λ) = 0} is a disjoint union of diagrams of finite + type, then µ ∈ P is non-degenerate with respect to λ if and only if µ 6 λ. 0 (3) wt V = (λ − Z> π) ∩ conv(W λ). Proposition 4.3 is explicitly stated in [25] for g, but also holds for g. To see this, note that 0 wt V ⊂ (λ − Z> π) ∩ conv(W λ), and the latter are the weights of its simple quotient L(λ), which is inflated from g. Proof of Proposition 4.1. The disjointness of the terms on the right-hand side is an easy consequence of the linear independence of simple roots. We first show the inclusion ⊃. Recall the isomorphism of Proposition 3.7:

(ˇαj ,λ)+1 M(λ, J) ' M(λ)/(fj M(λ)λ, ∀j ∈ J). S It follows that the weights of ker(M(λ) → M(λ, J)) are contained in j∈J {ν 6 sj · λ}. >0 Hence λ − µ is a weight of M(λ, J), ∀µ ∈ Z πI\J . Any nonzero element of M(λ, J)λ−µ generates an integrable highest weight lJ -module. As the weights of all such modules coincide by Proposition 4.3(3), this shows the inclusion ⊃. >0 We next show the inclusion ⊂. For any µ ∈ Z πI\J , the ‘integrable slice’ M Sµ := M(λ, J)ν ν∈λ−µ+ZπJ lies in Category O for lJ , and is furthermore an integrable lJ -module. It follows that the weights of M(λ, J) lie in the J Tits cone. >0 >0 Let ν be a weight of M(λ, J), and write λ − ν = µJ + µI\J , µJ ∈ Z πJ , µI\J ∈ Z πI\J .

We need to show ν ∈ wt LlJ (λ − µI\J ). By WJ -invariance, we may assume ν is J dominant. By Proposition 4.3, it suﬃces to show that ν is non-degenerate with respect to (λ − µI\J ). To see this, using Proposition 3.7 write: n X + + ν = λ − µL − βk, where λ − µL ∈ wt LlJ (λ), βk ∈ ∆ \ ∆J , 1 6 k 6 n. k=1 The claimed nondegeneracy follows from the fact that λ − µL is nondegenerate with respect to λ and that the support of each βk is connected. As an immediate consequence of the Integrable Slice Decomposition 4.1, we present a family of decompositions of wt M(λ, J), which interpolates between the two sides of Equation (4.2): CHARACTERS OF HIGHEST WEIGHT MODULES AND INTEGRABILITY 11

Corollary 4.4. For subsets J ⊂ J 0 ⊂ I, we have: G wt M(λ, J) = wt M (λ − µ, J), (4.5) lJ0 0 µ∈Z> (π\πJ0 ) where M (λ, J) was deﬁned in Equation (3.6). lJ0 As a second consequence, Proposition 4.3(2) and the Integrable Slice Decomposition 4.1 yield the following simple description of the weights of most parabolic Verma modules.

Corollary 4.6. Suppose λ has ﬁnite WJ -isotropy. Then, [ + wt M(λ, J) = w{ν ∈ PJ : ν 6 λ}. (4.7) w∈WJ Equipped with the Integrable Slice Decomposition, we provide a characterization of the weights of a parabolic Verma module that will be helpful in Section 5. Proposition 4.8. wt M(λ, J) = (λ + Zπ) ∩ conv M(λ, J). (4.9) Proof. The inclusion ⊂ is immediate. For the reverse ⊃, note that conv M(λ, J) lies in the J Tits cone and is WJ invariant. It then suﬃces to consider a point ν of the right-hand side which is J dominant. Write ν as a convex combination: n X k k ν = tk(λ − µI\J − µJ ), k=1 >0 P k >0 k >0 where tk ∈ R , k tk = 1, µI\J ∈ Z πI\J , µJ ∈ Z πJ , 1 6 k 6 n. By Propositions 4.3 P k and 4.1, it remains to observe that ν is non-degenerate with respect to λ − k tkµI\J , as a similar statement holds for each summand. 4.2. The Ray Decomposition. Using the Integrable Slice Decomposition, we obtain the following novel description of conv M(λ, J), which is of use in the proof of the main theorem and throughout the paper. Proposition 4.10 (Ray Decomposition).

[ >0 conv M(λ, J) = conv w(λ − Z αi). (4.11) w∈WJ , i∈I\J S When J = I, by the right-hand side we mean conv w∈W wλ.

Proof. The containment ⊃ is straightforward using the deﬁnition of integrability and WJ invariance. For the containment ⊂, it suﬃces to show every weight of M(λ, J) lies in the >0 right-hand side. Note that the right-hand side contains WJ (λ − Z πI\J ), so we are done by the Integrable Slice Decomposition 4.1. 4.3. Proof of the main result. With the Integrable Slice and Ray Decompositions in hand, we now turn to our main theorem. Theorem 4.12. Given a highest weight module V and a subset J ⊂ I, the following are equivalent:

(1) J = IV . (2) conv wt V = conv wt M(λ, J). (3) The stabilizer of conv V in W is WJ . 12 GURBIR DHILLON AND APOORVA KHARE

The connection to perverse sheaves promised in Section 1 is proven in Proposition 8.2.

>0 Proof. To show (1) implies (2), note that λ − Z αi ⊂ wt V, ∀i ∈ πI\IV . The implication now follows from the Ray Decomposition 4.10. The remaining implications follow from the assertion that the stabilizer of conv V in W is WIV . Thus, it remains to prove the assertion.

It is standard that WIV preserves conv V . For the reverse, since (1) implies (2), we may reduce to the case of V = M(λ, J). Suppose w ∈ W stabilizes conv M(λ, J). It is easy to see λ is a face of conv M(λ, J), hence so is wλ. However by the Ray Decomposition 4.10, it is clear that the only 0-faces of conv M(λ, J) are WJ (λ), so without loss of generality we may assume w stabilizes λ. + + Recalling that WJ is exactly the subgroup of W which preserves ∆ \∆J , it suﬃces to show + + + + >0 w preserves ∆ \ ∆J . Let α ∈ ∆ \ ∆J ; then by Proposition 3.7, λ − Z α ⊂ wt M(λ, J), 0 whence so is λ − Z> w(α) by Proposition 4.8. This shows that w(α) > 0; it remains to + show w(α) ∈/ ∆J . It suﬃces to check this for the simple roots αi, i ∈ I \ J. Suppose not, + + i.e. w(αi) ∈ ∆J for some i ∈ I \ J. In this case, note w(αi) must be a real root of ∆J , e.g. 0 by considering 2w(αi). Thus the w(αi) root string λ − Z> w(αi) is the set of weights of an integrable representation of g−w(αi) ⊕ [g−w(αi), gw(αi)] ⊕ gw(αi) ' sl2, which is absurd.

Corollary 4.13. The stabilizer of ch V in W is WIV .

Corollary 4.14. For any V , conv V is the Minkowski sum of conv WIV (λ) and the cone >0 R WIV (πI\IV ). Proof. By Theorem 4.12, we reduce to the case of V a parabolic Verma module. Now the result follows from the Ray Decomposition 4.10 and Equation (3.9). Remark 4.15. For g semisimple, the main theorem 4.12 and the Ray Decomposition 4.10

imply that the convex hull of weights of any highest weight module V is a WIV -invariant polyhedron. To our knowledge, this was known for many but not all highest weight modules by recent work [28], and prior to that only for parabolic Verma modules. We develop many other applications of the main theorem to the convex geometry of conv V , including the classiﬁcation of its faces and their inclusions, in the companion work [11].

5. A question of Bump and weight formulas for non-integrable simple modules We remind the three positive formulas for the weights of simple highest weight modules to be obtained in this section.

Proposition 5.1. Write l for the Levi subalgebra corresponding to IL(λ), and write Ll(ν) for the simple l module of highest weight ν ∈ h∗. Then: G wt L(λ) = wt Ll(λ − µ). (5.2) µ∈ >0π\π Z IL(λ) Proposition 5.3. ∗ wt L(λ) = conv L(λ) ∩ {µ ∈ h : µ 6 λ}. (5.4)

Proposition 5.5. Suppose λ has ﬁnite stabilizer in WIL(λ) . Then: + wt L(λ) = WI {µ ∈ P : µ λ}. (5.6) L(λ) IL(λ) 6 Remark 5.7. For an extension of Proposition 5.5 to arbitrary λ, see the companion work [11]. CHARACTERS OF HIGHEST WEIGHT MODULES AND INTEGRABILITY 13

Note that the three weight formulas above are immediate consequences of combining the following theorem with Propositions 4.1 and 4.8 and Corollary 4.6 respectively. Recall from Section 2.2 that for a highest weight module V , we deﬁned its potential integrability to be p IV := IL(λ) \ IV .

Theorem 5.8. Let V be a highest weight module, Vλ its highest weight line. Let l denote the p Levi subalgebra corresponding to IV . Then wt V = wt M(λ, IV ) if and only if wt U(l)Vλ = 0 wt U(l)M(λ, IV )λ, i.e. wt U(l)Vλ = λ − > π p . Z IV In particular, if V = L(λ), then wt L(λ) = wt M(λ, IL(λ)).

Proof. It is clear, e.g. by the Integrable Slice Decomposition 4.1 that if wt V = wt M(λ, IV ), 0 then wt p V = λ − > π p . For the reverse implication, by again using the Integrable Slice IV Z IV >0 Decomposition and the action of lIV , it suﬃces to show that λ − Z πI\IV ⊂ wt V . >0 p q Suppose not, i.e. there exists µ ∈ Z πI\IV with Vλ−µ = 0. We decompose I \IV = IV tI , 0 0 and write accordingly µ = µp + µq, where µp ∈ > π p and µq ∈ > πIq . By assumption Z IV Z − Vλ−µp is nonzero, so by the PBW theorem, there exists ∂ ∈ U(n p ) of weight −µp such that IV q P ∂ ·Vλ is nonzero. Now choose an enumeration of I = {ik : 1 6 k 6 n}, write µq = k mkαik , and consider the monomials E := Q emk ,F := Q f mk . By the vanishing of V , ∂F k ik k ik λ−µp−µq annihilates the highest weight line Vλ, whence so does E∂F . Since [ei, fj] = 0 ∀i 6= j, we may write this as ∂ Q emk f mk , and each factor emk f mk acts on V by a nonzero scalar, a k ik ik ik ik λ contradiction. Remark 5.9. Given the delicacy of Jantzen’s criterion for the simplicity of a parabolic Verma module (see [21, 22]), it is interesting that the equality on weights of L(λ) and M(λ, IL(λ)) always holds.

6. A question of Lepowsky on the weights of highest weight modules In Section 5, we applied our main result to obtain formulas for the weights of simple modules. We now explore the extent to which this can be done for arbitrary highest weight modules, thereby answering the question of Lepowsky discussed in Section 2.3. ∗ Theorem 6.1. Fix λ ∈ h and J ⊂ IL(λ). Every highest weight module V of highest weight p λ and integrability J has the same weights if and only if J := IL(λ) \ J is complete, i.e. 0 p (ˇαj, αj0 ) 6= 0, ∀j, j ∈ J . We ﬁrst sketch our approach. The subcategory of modules V with ﬁxed highest weight λ and integrability J is basically a poset modulo scaling. The source here is the parabolic Verma module M(λ, J), and we are concerned with the possible vanishing of weight spaces as we move away from the source. The most interesting ingredient of the proof is the observation that there is a sink, which we call L(λ, J), and hence the question reduces to whether M(λ, J) and L(λ, J) have the same weights.

Lemma 6.2. For J ⊂ IL(λ), there is a minimal quotient L(λ, J) of M(λ) satisfying the equivalent conditions of Theorem 4.12.

Proof. Consider all submodules N of M(λ) such that N 0 = 0, ∀i ∈ I \ J. There is a λ−Z> αi maximal such, namely their sum N 0, and it is clear that L(λ, J) = M(λ)/N 0 by construction. As the reader is no doubt aware, the inexplicit construction of L(λ, J) here is parallel to that of L(λ) – in fact, L(λ) = L(λ, IL(λ)). The existence of the objects L(λ, J) was noted by 14 GURBIR DHILLON AND APOORVA KHARE the second named author in [28], but we are unaware of other appearances in the literature. In [28], the character of L(λ, J) was not determined. However, it turns out to be no more difficult than ch L(λ) under the following sufficient condition, which as we explain below is often satisfied: ∗ 1 Proposition 6.3. Fix λ ∈ h ,J ⊂ IL(λ). Suppose that ExtO(L(λ),L(si · λ)) 6= 0, ∀i ∈ IL(λ) \ J. Then there is a short exact sequence: M 0 → L(si · λ) → L(λ, J) → L(λ) → 0. (6.4)

i∈IL(λ)\J 1 Proof. The choice of a nonzero class in each ExtO(L(λ),L(si · λ)), i ∈ IL(λ) \ J gives an extension E as in Equation (6.4). The space Eλ is a highest weight line, and we obtain an associated map M(λ) ' M(λ) ⊗ Eλ → E. This is surjective, as follows from an easy argument using Jordan–Höldercontent and the nontriviality of each extension. The consequent surjection E → L(λ, J) is an isomorphism, by considering Jordan–Höldercontent. 1 Remark 6.5. By identifying ExtO(L(λ),L(si · λ)) with Hom(N(λ),L(si · λ)), where N(λ) is the maximal submodule of M(λ), one sees it is at most one dimensional. It is nonzero in the following two situations: First, when g is symmetrizable and λ is dominant integral; here one uses the end of the BGG resolution. Second, when g is of finite type and λ is arbitrary with si · λ 6 λ. In this case, by work of Soergel [41] (see also [20, §3.4]), it suffices to consider λ integral, in which case this is a standard consequence of Kazhdan–Lusztig theory. See [20, Theorem 8.15(c)] for λ regular and Irving [19, Corollary 1.3.5] for λ singular. We sketch a proof in the regular case, by considering the corresponding ICs on G/B. By parity vanishing, the Cousin spectral sequence for RHom associated to the Schubert stratification collapses on E1, cf. [5, §3.4]. This reduces the question to only the Schubert strata indexed by si · λ, λ where it is clear. We expect the hypotheses in Proposition 6.3 to hold in most cases of interest, e.g. for λ in the dot orbit of a dominant integral weight. Returning to Lepowsky’s question for g possibly non-symmetrizable, we first prove the following weaker form of Equation (6.4). Proposition 6.6. Let g be of arbitrary type and λ ∈ P +. Then: [ wt L(λ, J) = wt L(λ) ∪ wt L(si · λ). (6.7) i∈I\J 0 Proof. The restriction of ch M(λ) to the root string λ−Z> αi, i ∈ I \J is entirely determined by the Jordan–Höldercontent [M(λ): L(λ)] = [M(λ): L(si · λ)] = 1. Using this, it suffices to exhibit a module with weights given by the right-hand side of (6.7). Consider the end of the BGG resolution: M max M(si · λ) → M(λ) → L (λ) → 0. (6.8) i∈I

For each i ∈ I, let N(si ·λ) denote the maximal submodule of M(si ·λ). Consider the quotient L L Q of M(λ) by the image of j∈J M(sj · λ) ⊕ i∈I\J N(si · λ), under (6.8). We deduce: max X ch Q = ch L (λ) + ch L(si · λ), i∈I\J which implies that wt Q, whence wt L(λ, J), coincides with the right-hand side of (6.7). CHARACTERS OF HIGHEST WEIGHT MODULES AND INTEGRABILITY 15

We now answer Lepowsky’s question. By the above results, it suﬃces to determine when wt L(λ, J) = wt M(λ, J).

Proof of Theorem 6.1. It will be clarifying, though not strictly necessary for the argument, to observe the following compatibility of the construction of L(λ, J) and restriction to a Levi.

0 Lemma 6.9. Let I ⊂ I, l = lI0 the corresponding Levi subalgebra, and for J ⊂ I write J 0 = J ∩ I0. Then: 0 Ll(λ, J ) ' U(l)L(λ, J)λ, 0 0 where Ll(λ, J ) is the smallest l-module with integrability J , constructed as in Lemma 6.2. 0 Proof. Since the integrability of U(l)L(λ, J)λ is J , we have a surjection: 0 U(l)L(λ, J)λ → Ll(λ, J ) → 0. (6.10) 0 To see that Equation (6.10) is an isomorphism, we need to show the kernel K of Ml(λ, J ) → 0 0 Ll(λ, J ) is sent into the kernel of M(λ, J) → L(λ, J) under the inclusion Ml(λ, J ) → M(λ, J). To do so, choose a ﬁltration of K:

0 ⊆ K1 ⊆ K2 ⊆ · · · ,K = ∪m>0Km,

where Km/Km−1 is generated as a U(l)-module by a highest weight vector vm. It suﬃces to prove by induction on m that the U(g)-module generated by Km lies in the kernel of M(λ, J) → L(λ, J), ∀m > 0. For the inductive step, suppose that M(λ, J)/U(g)Km−1 has integrability J. Then choosing a liftv ˜m of vm to Km, observe thatv ˜m is a highest weight vector in M(λ, J)/U(g)Km−1 for the action of all of U(g), by the linear independence of sim- 0 ple roots. It is clear that U(g)˜vm has trivial intersection with the (λ−Z> αi)-weight spaces of M(λ, J)/U(g)Km−1, for all i ∈ I \ J, hence M(λ, J)/U(g)Km again has integrability J. We are now ready to prove Theorem 6.1. By Theorem 5.8, we have wt L(λ, J) = wt M(λ, J) if and only if wtJp L(λ, J) = wtJp M(λ, J). By Lemma 6.9, we may therefore assume that J = ∅, and λ ∈ P +, and hence J p = I. By Proposition 6.6, we have: [ wt L(λ, ∅) = wt L(λ) ∪ wt L(si · λ), i∈I wt L(λ, I \ i) = wt L(λ) ∪ wt L(si · λ), ∀i ∈ I. Combining the two above equalities, we have: [ wt L(λ, ∅) = wt L(λ, I \ i). i∈I For L(λ, I \ i) the set of potentially integrable directions is {i}, so we may apply Theorem 5.8 to conclude: [ wt L(λ, ∅) = wt M(λ, I \ i). (6.11) i∈I S Thus, it remains to show that for λ dominant integral, wt M(λ) = i∈I wt M(λ, I \ i) if and only if the Dynkin diagram of g is complete. 0 Suppose ﬁrst that the Dynkin diagram of g is not complete. Pick i, i ∈ I with (ˇαi, αi0 ) = 0. Then M(λ) and M(λ)/sisi0 · M(λ)λ have distinct weights, as can be seen by a calculation in type A1 × A1. 16 GURBIR DHILLON AND APOORVA KHARE

Suppose now that the Dynkin diagram of g is complete. By (6.11) it suﬃces to show:

X >0 wt M(λ, I \ i) ⊃ {λ − kjαj : kj ∈ Z , ki > kj, ∀j ∈ I}. j∈I But this follows using the assumption that every node is connected to i, and the representation theory of sl2. Remark 6.12. Theorem 6.1 can be proved without working directly with the objects L(λ, J), but instead using only the lower bound (6.7) for the weights: [ wt V ⊃ wt L(λ, IV ) = wt L(λ) ∪ wt L(si · λ).

i∈IL(λ)\IV 7. The Weyl–Kac formula for the weights of simple modules The main goal of this section is to prove the following: ∗ Theorem 7.1. Let λ ∈ h be such that the stabilizer of λ in WIL(λ) is finite. Then: X eλ wt L(λ) = w . (7.2) Q (1 − e−α) w∈W α∈π IL(λ) We remind that on the left hand side of (7.2) we mean the ‘multiplicity-free’ character P µ ∗ e . On the right hand side of (7.2), in each summand we take the ‘highest µ∈h :L(λ)µ6=0 weight’ expansion of w(1 − e−αi )−1, i.e.: ( −wα −2wα 1 1 + e i + e i + ··· , wαi > 0, w −α := wα 2wα 3wα (7.3) 1 − e i −e i − e i − e i − · · · , wαi < 0. By Theorem 5.8, we will deduce Theorem 7.1 from the following: ∗ Proposition 7.4. Fix λ ∈ h , J ⊂ IL(λ) such that the stabilizer of λ in WJ is finite. Then: X eλ wt M(λ, J) = w Q −α . α∈π(1 − e ) w∈WJ Before proving Proposition 7.4, we note the similarity between the result and the following formulation of the Weyl–Kac character formula. This is due to Atiyah and Bott for finite dimensional simple modules in finite type, but appears to be new in this generality: ∗ Proposition 7.5. Fix λ ∈ h , J ⊂ IL(λ). Then: X eλ ch M(λ, J) = w . (7.6) Q −α dim gα α>0(1 − e ) w∈WJ The right hand side of Equation (7.6) is expanded in the manner explained above. We now prove Proposition 7.4, and then turn to Proposition 7.5. Proof of Proposition 7.4. By expanding w Q (1 − e−α)−1, note the right hand side α∈πI\J equals: X X eλ−µ w . Q (1 − e−α) 0 α∈πJ w∈WJ µ∈Z> πI\J Therefore we are done by the Integrable Slice Decomposition 4.1 and the existing result for integrable highest weight modules with finite stabilizer [26]. CHARACTERS OF HIGHEST WEIGHT MODULES AND INTEGRABILITY 17

Specialized to the trivial module in finite type, we obtain the following curious consequence, which roughly looks like a Weyl denominator formula without a choice of positive roots: Corollary 7.7. Let g be of finite type with root system ∆, and let Π denote the set of all π, where π is a simple system of roots for ∆. Then we have: Y X Y (1 − eα) = (1 − eβ). (7.8) α∈∆ π∈Π β∈ /π Remark 7.9. When g is of finite type, Theorem 7.1 follows from Proposition 2.4 and general formulae for exponential sums over polyhedra due to Brion [8] for rational polytopes and Lawrence [34] and Khovanskii–Pukhlikov [29] in general, cf. [3, Chapter 13]. For regular weights, one uses that the tangent cones are unimodular and hence the associated polyhedra are Delzant. For general highest weights one may apply a deformation argument due to Postnikov [39]. We thank Michel Brion for sharing this observation with us. Moreover, it is likely that a similar approach works in infinite type, though a cutoff argument needs to be made owing to the fact that conv V is in general locally, but not globally, polyhedral, cf. our companion work [11].

7.1. Some related results. We conclude this section with several related observations which are not needed in the remainder of the paper, beginning with Proposition 7.5.

7.1.1. Parabolic Atiyah–Bott formula and the Bernstein–Gelfand–Gelfand–Lepowsky resolution. Proposition 7.5 may be deduced from the usual presentation of the parabolic Weyl–Kac character formula:

∗ Proposition 7.10. Fix λ ∈ h ,J ⊂ IL(λ). Then:

X ew(λ+ρ)−ρ ch M(λ, J) = . (7.11) Q −α dim gα α>0(1 − e ) w∈WJ To our knowledge, even Proposition 7.10 does not appear in the literature in this generality. However, via Levi induction one may reduce to the case of λ dominant integral, J = I, where it is a famed result of Kumar [31, 32]. Along these lines, we also record the BGGL resolution for arbitrary parabolic Verma modules, which may be of independent interest: Proposition 7.12. Fix M(λ, J) and J 0 ⊂ J. Then M(λ, J) admits a BGGL resolution, i.e., there is an exact sequence: M M · · · → M(w · λ, J 0) → M(w · λ, J 0) → M(λ, J 0) → M(λ, J) → 0. J0 J0 w∈WJ :`(w)=2 w∈WJ :`(w)=1 (7.13) J0 Here WJ runs over the minimal length coset representatives of WJ0 \WJ . In particular, we have the Weyl–Kac character formula: X ch M(λ, J) = (−1)`(w) ch M(w · λ, J 0). (7.14) J0 w∈WJ This can be deduced by Levi induction from the result of Heckenberger and Kolb [17], which builds on [16, 32, 38]. 18 GURBIR DHILLON AND APOORVA KHARE

7.1.2. Generalization of a result of Kass. In the paper of Kass [26] that proves Theorem 7.1 for integrable modules, the main result is a recursive formula for the character of an integrable module. After establishing notation, we will state the analogous result for parabolic Verma modules. ∗ For J ⊂ I, if ν ∈ h lies in the J dot Tits cone, write ν = wν · ν for the unique J dot + dominant weight in its WJ orbit. For λ ∈ PJ , we write χλ := ch M(λ, J), and for ν lying in the dot J Tits cone, we set:

( `(wν ) (−1) χν if ν is in the J dominant chamber, χν := 0 otherwise. This makes sense because if ν lies in the J dominant chamber, it is regular dominant under the J dot action, so there is a unique wν ∈ WJ such that wν · ν = ν. >0 Deﬁne the elements hΦi ∈ Z π and the integers chΦi via the expansion,

Y −α dim gα X −hΦi (1 − e ) =: chΦie . α∈∆+\π hΦi Informally, this expansion counts the number of ways to write hΦi as sums of non-simple positive roots, with consideration of the multiplicity of the root and the parity of the sum. For a parabolic Verma module M(λ, J), by abuse of notation identify its weights with the corresponding multiplicity-free character: X wt M(λ, J) = eµ.

µ:M(λ,J)µ6=0 With this notation, we are ready to state the following: Proposition 7.15. Let M(λ, J) be a parabolic Verma module such that the stabilizer of λ in WJ is finite. Then: X wt M(λ, J) = chΦiχλ−hΦi, (7.16) hΦi Furthermore, for all hΦi, we have (i) λ − hΦi 6 λ, (ii) with equality if and only if Φ is empty. In Equation (7.16), we expand denominators as ‘highest weight’ power series as explained in Equation (7.3). The arguments of [26] or [40] apply with suitable modification to prove Proposition 7.15. Remark 7.17. As the remainder of the paper concerns only symmetrizable g, we now explain the validity of earlier results for g (cf. Section 3.2), should g and g differ. The proofs in Sections 4–7 prior to Remark 7.9 apply verbatim for g. Alternatively, recalling that the roots of g and g coincide [25, §5.12], it follows from Equation (3.9) and the remarks following Proposition 4.3 that the weights of parabolic Verma modules for g, g coincide, as do the weights of simple highest weight modules. Hence many of the previous results can be deduced directly for g from the case of g. These arguments for g apply for any intermediate Lie algebra between g and g.

8. A question of Brion on localization of D-modules and a geometric interpretation of integrability Throughout this section, g is of ﬁnite type. We will answer the question of Brion [9] discussed in Section 2.5. We ﬁrst remind notation. For λ a dominant integral weight, consider CHARACTERS OF HIGHEST WEIGHT MODULES AND INTEGRABILITY 19

0 Lλ, the line bundle on G/B with H (Lλ) ' L(λ) as g-modules. For w ∈ W , write Cw := BwB/B for the Schubert cell, and write Xw for its closure, the Schubert variety. To affirmatively answer Brion’s question for any regular integral infinitesimal character, it suffices by translation to consider the regular block O0. Let V be a highest weight module with highest weight w · (−2ρ), w ∈ W . Write V for the corresponding regular holonomic D-module on G/B, and Vrh for the corresponding perverse sheaf under the Riemann–Hilbert correspondence. Then V, Vrh are supported on Xw. Note that for i ∈ IV , Xsiw is a Schubert divisor of Xw. We will use D to denote the standard dualities on Category O, regular holonomic D-modules, and perverse sheaves, which are intertwined by Beilinson–Bernstein localization and the Riemann–Hilbert correspondence.

Theorem 8.1. For V, V as above, and J ⊂ IV , consider the union of Schubert divisors ˜ 0 ˜ Z := ∪i∈IV \J Xsiw. Write U := G/B \ Z. Then DH (U, DV) is of highest weight w · (−2ρ) and has integrability J. To prove Theorem 8.1, we will use the following geometric characterization of integrability, which was promised in Theorem 2.1.

Proposition 8.2. For J ⊂ IL(λ), consider the smooth open subvariety of Xw given on complex points by: G UJ := Cw t Csiw. (8.3) i∈J

Let V, Vrh be as above, and set U := UIL(λ) . Upon restricting to U we have: V | ' j [`(w)]. (8.4) rh U !CUIV

In particular, IV = {i ∈ I : the stalks of Vrh along Csiw are nonzero}.

Proof. For J ⊂ IL(λ), let PJ denote the corresponding parabolic subgroup of G, i.e. with Lie + algebra lJ + n . Then it is well known that the perverse sheaf corresponding to M(λ, J) is

j!CPJ wB/B[`(w)]. The map M(λ, IV ) → V → 0 yields a surjection on the corresponding perverse sheaves. By considering Jordan–H¨oldercontent, it follows this map is an isomorphism when restricted

to UIL(λ) , as for y 6 w the only intersection cohomology sheaves ICy := ICXy which do not

vanish upon restriction are ICw, ICsiw, i ∈ IL(λ). We ﬁnish by observing: ∗ jU j!CUI [`(w)] ' j!CU ∩P wB/B[`(w)] = j!CUI [`(w)]. IL(λ) V IL(λ) IV V We deduce the following D-module interpretation of integrability:

Corollary 8.5. Let V, U be as above. The restriction of V to U is j∗O . If we define D UIV IV := {i ∈ I : the fibers of DV are nonzero along Csiw}, then IV = IV . Here we mean fibers in the sense of the underlying quasi-coherent sheaf of DV. Before proving Theorem 8.1, we first informally explain the idea. Proposition 8.2 says that for i ∈ IL(w·−2ρ), the action of the corresponding sl2 → g on V is not integrable if and

only if Vrh has a ‘pole’ on the Schubert divisor Xsiw. Therefore to modify V so that it loses

integrability along Xsiw, we will restrict Vrh to the complement and then extend by zero.

Proof of Theorem 8.1. For ease of notation, write X := Xw. Write X = Z t U, where Z is ˜ ˜ ˜ as above, and G/B = Z t U. Note U ∩ Xw = U. Write jU : U → Xw, jU˜ : U → G/B for 0 the open embeddings, and iZ : Z → Xw, iZ : Z → G/B, iXw : Xw → G/B for the closed embeddings. 20 GURBIR DHILLON AND APOORVA KHARE

0 We will study H (U,˜ V) by studying the behavior of V on U, i.e. before pushing off of Xw. ∗ Formally, write Vrh = iX ∗iX Vrh =: iX ∗P, where P is perverse. The distinguished triangle: ∗ ∗ +1 jU !jU → id → iZ ∗iZ −−→ (8.6) gives the following exact sequence in perverse cohomology: p −1 ∗ p 0 ∗ p 0 ∗ 0 → H iZ ∗iZ P → H jU !jU P → P → H iZ ∗iZ P → 0. p 0 ∗ We now show in several steps that H jU !jU P is a highest weight module with the desired integrability. p 0 ∗ Step 1: H jU !jU P is a highest weight module of highest weight w · −2ρ. By definition, we have a surjection jCw !CCw [`(w)] → P → 0. Since jCw !CCw is supported ∗ off of the Schubert divisors, we have jU !jU CCw ' CCw . By right exactness, we obtain p 0 ∗ p 0 ∗ jCw !CCw [`(w)] → H jU !jU P → 0, as desired. We remark that similarly, H iZ ∗iZ P = 0. p 0 ∗ Step 2: The integrability of H jU !jU P is J. In ‘ground to earth’ terms, by Proposition 8.2 we need to look at this sheaf on U, where by design it has the correct behavior. More carefully, we have: ∗ p 0 ∗ p 0 ∗ ∗ jU H jU !jU P' H jU jU !jU P p 0 ∗ ' H jU∩U !jU∩U P p 0 ∗ ' H jU∩U j jU [`(w)] ! U∩U IV !C p 0 ' H jU ∩U [`(w)] = jU ∩U [`(w)]. IV !C IV !C

As UIV ∩ U = UJ , we are done by Proposition 8.2. We now push our analysis oﬀ of Xw. To do so, we use the isomorphism of distinguished triangles: ∗ 0 0 ∗ +1 jU˜ !jU˜ iX ∗ −−−−→ iX ∗ −−−−→ iZ !iZ iX ∗ −−−−→       (8.7) y y y ∗ ∗ +1 iX ∗jU !jU −−−−→ iX ∗ −−−−→ iX ∗iZ ∗iZ −−−−→ Here the middle vertical map is the identity, and the left and right vertical isomorphisms are induced by adjunction. Plainly, the isomorphism (8.7) comes from an isomorphism of short exact sequences of functors for the abelian categories of sheaves of abelian groups. p 0 ∗ Translating our analysis of H jU !jU P into the corresponding statement for g-modules and using Equation (8.7), we obtain a surjection of highest weight modules: 0 ∗ Γ(G/B, H jU˜ !jU˜ V) → V → 0, where the former module has integrability J. To ﬁnish the proof of Theorem 8.1, it remains 0 ∗ ˜ only to identify the dual of Γ(G/B, H jU˜ !jU˜ V) with sections of DV on U. But using standard compatibilities of D, and of composition of derived functors, we obtain: 0 ∗ 0 ∗ DΓ(G/B, H jU˜ !jU˜ V) ' Γ(G/B, DH jU˜ !jU˜ V) ' Γ(G/B, H0 j j∗ V) D U˜ ! U˜ 0 ∗ ' Γ(G/B, H RjU˜ ∗jU˜ DV) ' R0Γ(G/B, Rj j∗ V) U˜ ∗ U˜ D ˜ ' Γ(U, DV). CHARACTERS OF HIGHEST WEIGHT MODULES AND INTEGRABILITY 21

If one thinks about the above proof, in fact all we used about U (and U˜) was the intersection of U with U. The following proposition shows our U has the correct components in codimension > 2 to mimic another feature of Brion’s example: Proposition 8.8. The construction of Theorem 8.1 sends parabolic Verma modules to parabolic Verma modules.

Proof. For J ⊂ K ⊂ IL(w·−2ρ), we know that M(w·−2ρ, K) corresponds to j!CPK wB/B, where PK is the parabolic subgroup corresponding to K ⊂ I. Applying the construction (before ∗ taking perverse cohomology), we obtain jU !jU j!CPK wB/B ' j!CU∩PK wB/B. The claim follows from the identity PK wB/B \ Z = PJ wB/B, i.e. the identity

WK w \ ∪k∈K\J {y ∈ W : y 6 skw} = WJ w. To see this identity, recall by [7, Exercise 2.26 and proof of Proposition 2.4.4] that the assignment wkw 7→ wkw◦ is an isomorphism of posets WK w ' WK , where w◦ is the longest −1 element of WK . Multiplying the claimed identity on the right by w , it is therefore equivalent to: WK \ ∪k∈K\J {y ∈ WK : y > sk} = WJ , which is clear. While the results in this section concern g of ﬁnite type, we expect and would be interested to see that similar results hold for g symmetrizable.

9. Highest weight modules over symmetrizable quantum groups We now extend many results of the previous sections to highest weight modules over quantum groups Uq(g), for g a Kac–Moody 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, one uses the symmetrizability of A. When g is non-symmetrizable, even the formulation of Uq(g) is subtle and is the subject of recent research [13]. In light of this, we restrict to Uq(g) where g is symmetrizable. 9.1. Notation and preliminaries. We begin by reminding standard definitions and notation. Fix g = g(A) for A a symmetrizable generalized Cartan matrix. Fix a diagonal >0 matrix D = diag(di)i∈I such that DA is symmetric and di ∈ Z , ∀i ∈ I. Let (h, π, πˇ) be ∨ ˇ a realization of A as before; further fix a lattice P ⊂ h, with Z-basisα ˇi, βl, i ∈ I, 1 6 ∨ ˇ l 6 |I| − rk(A), such that P ⊗Z C ' h and (βl, αi) ∈ Z, ∀i ∈ I, 1 6 l 6 |I| − rk(A). ∗ ∨ Set P := {λ ∈ h :(P , λ) ⊂ Z} to be the weight lattice. We further retain the notations + ρ, b, h, lJ ,PJ ,M(λ, J),IL(λ) from previous sections; note we may and do choose ρ ∈ P . We ∗ normalize the Killing form (·, ·) on h to satisfy: (αi, αj) = diaij for all i, j ∈ I. Let q be an indeterminate. Then the corresponding quantum Kac–Moody algebra Uq(g) is h ∨ a C(q)-algebra, generated by elements fi, q , ei, i ∈ I, h ∈ P , with relations given in e.g. [18, d αˇ P ∨ Definition 3.1.1]. Among these generators are distinguished elements Ki = q i i ∈ q . Also ± define Uq to be the subalgebras generated by the ei and the fi, respectively. P ∨ A weight of the quantum torus Tq := C(q)[q ] is a C(q)-algebra homomorphism χ : × 2|I|−rk(A) Tq → C(q), which we identify with an element µq ∈ (C(q) ) given an enumeration h h ofα ˇi, i ∈ I. We will abuse notation and write µq(q ) for χµq (q ). There is a partial ordering −ν 0 on the set of weights, given by: q µq 6 µq, for all weights ν ∈ Z> π. We will mostly be µ µ h (h,µ) concerned with integral weights µq = q for µ ∈ P , which are defined via: q (q ) = q , h ∈ P ∨. 22 GURBIR DHILLON AND APOORVA KHARE

Given a Uq(g)-module V and a weight µq, the corresponding weight space of V is: h h ∨ Vµq := {v ∈ V : q v = µq(q )v ∀h ∈ P }.

Denote by wt V the set of weights {µq : Vµq 6= 0}.A Uq(g)-module is highest weight if there exists a nonzero weight vector which generates V and is killed by ei, ∀i ∈ I. For a weight µq, let M(µq),L(µq) denote the Verma and simple Uq(g)-modules of highest weight µq, respectively. Writing λq for the highest weight of V , the integrability of V equals:

IV := {i ∈ I : dim C(q)[fi]Vλq < ∞}. (9.1)

In this case, the parabolic subgroup WIV acts on wt V by

h αˇi −αi(h) h ∨ si(λq)(q ) := λq(q ) λq(q ), h ∈ P . The braid relations can be checked using by specializing q to 1, cf. [36]. In particular, w(qλ) = qwλ for w ∈ W and λ ∈ P .

Given a weight λq and J ⊂ IL(λq), the parabolic Verma module M(λq,J) co-represents the functor:

M {m ∈ Mλq : eim = 0 ∀i ∈ I, fj acts nilpotently on m, ∀j ∈ J}. (9.2) αˇ n Note λq(q j ) = ±q j , nj > 0, ∀j ∈ J cf. [23, Proposition 2.3]. Therefore: nj +1 M(λq,J) ' M(λq)/(fj M(λq)λq , ∀j ∈ J). (9.3) In what follows, we will specialize highest weight modules at q = 1, as pioneered by Lusztig [36]. Let A1 denote the local ring of rational functions f ∈ C(q) that are regular at q = 1. Fix λ P an integral weight λq = q ∈ q and a highest weight module V with highest weight vector 1 vλq ∈ Vλq . Then the classical limit of V at q = 1, defined to be V := A1 · vλq /(q − 1)A1 · vλq , is a highest weight module over U(g) with highest weight λ. Moreover the characters of V and V 1 are “equal”, i.e. upon identifying qP with P . 9.2. A classification of highest weight modules to first order. We begin by extending Theorem 2.1 to quantum groups.

Theorem 9.4. Let V be a highest weight module with integral highest weight λq. The following data are equivalent:

(1) IV , the integrability of V . (2) conv V 1, the convex hull of the specialization of V . (3) The stabilizer of ch V in W . Proof. The equivalence of the three statements follows from their classical counterparts, using the equality of characters under specialization. 9.3. A question of Bump and weight formulas for non-integrable simple modules. We now extend the results of Section 5 to quantum groups.

Theorem 9.5. Let λq be integral. The following are equivalent: (1) wt V = wt M(λq,IV ). >0 p −Z πI p (2) wt p V = q V λq, where I = I \ IV . IV V L(λq) p In particular, if V is simple, or more generally |IV | 6 1, then wt V = wt M(λq,IV ). To prove Theorem 9.5, we will need the Integrable Slice Decomposition for quantum parabolic Verma modules. CHARACTERS OF HIGHEST WEIGHT MODULES AND INTEGRABILITY 23

Proposition 9.6. Let λq be an integral weight and M(λq,J) =: V a parabolic Verma module. Then: G −µ wt M(λq,J) = wt LlJ (q λq), (9.7) 0 µ∈Z> (π\πJ ) 1 where LlJ (ν) denotes the simple Uq(lJ )-module of highest weight ν. In particular, wt V = wt M(λ1,J). Proof. It suffices to check the formula after specialization. Since the characters of V and V 1 1 are “equal”, IV = IV 1 . It therefore suffices to check the surjection M(λ1,J) → V induces an equality of weights. By the Integrable Slice Decomposition 4.1, it suffices to show that 0 −Z> πI\J q λq ⊂ wt M(λq,J). But this is clear from (9.3) by considering weights and using nj +1 that fj M(λq)λq is a highest weight line, for all j ∈ J.

1 Remark 9.8. If g is of finite type, then in fact V ' M(λ1,J). The proof uses [6, Chapter 2] and the PBW theorem, as well as arguments similar to the construction of Lusztig’s canonical − basis [37] to define Uq(uJ ). It would be interesting to know if the parabolic Verma module M(λq,J) specializes to M(λ1,J) for all symmetrizable g. Proof of Theorem 9.5. By Proposition 9.6, (1) implies (2). For the converse, by the Integrable 0 −Z> (π\πI ) Slice Decomposition (9.7), it suffices to show that q V λq ⊂ wt V . This follows by mimicking the proof of Theorem 5.8.

As an application of Theorem 9.5, we obtain the following positive formulas for weights of simple modules wt L(λq).

Proposition 9.9. Suppose λq is integral. Write l for the Levi subalgebra corresponding to

IL(λq), and write Ll(νq) for the simple Uq(l) module with highest weight νq. Then:

G −µ wt L(λq) = wt Ll(q λq). (9.10) 0 µ∈ > π\πI Z L(λq)

Proposition 9.11. Let λq be integral and V := L(λq). Then:

1 1 wt V = (λ + Zπ) ∩ conv V . (9.12)

By specialization, this determines the weights of L(λq). Proposition 9.13. Suppose λ is integral and has ﬁnite W -isotropy. Then: q IL(λq)

[ ν + ν wt L(λq) = w{q : ν ∈ P , q λq}. (9.14) IL(λq) 6 w∈WI L(λq) 9.4. A question of Lepowsky on the weights of highest weight modules. The result and arguments of Section 6 apply without change to quantum groups.

p Theorem 9.15. Fix J ⊂ IL(λq) and deﬁne Jq := IL(λq) \ J. The following are equivalent:

(1) wt V = wt M(λq,J) for every V with IV = J. (2) The Dynkin diagram for g p is complete. Jq 24 GURBIR DHILLON AND APOORVA KHARE

9.5. The Weyl–Kac formula for the weights of simple modules. The main result of Section 7 follows from combining Theorems 7.1, 9.5, Proposition 9.6, and specializing. Theorem 9.16. For all integral λ with ﬁnite stabilizer in W , we have: q IL(λq)

X λq wt L(λq) = w Q −α . (9.17) α∈π(1 − q ) w∈WI L(λq) 9.6. The case of non-integral weights. We now explain how to extend the above results in this section to other highest weights. We do so in two ways. First, we observe that with h some modiﬁcations, Lusztig’s specialization method applies to any weight λq such that λq(q ) is regular at q = 1 with value 1, ∀h ∈ P ∨. Calling such weights specializable, one can show that as before, a highest weight module with specializable highest weight λq specializes to a ∗ highest weight U(g)-module with highest weight λ1 ∈ h , given by: h λq(q ) − 1 ∨ (h, λ1) := , h ∈ P . q − 1 q=1 Moreover, the following holds:

Theorem 9.18. The above results in this section all extend to λq specializable. Second, we extend many of the above results to generic highest weights, i.e. with finite integrable stabilizer. In particular, this covers all cases in finite and affine type, the remaining cases in affine type being trivial modules. As before, we obtain:

Theorem 9.19. Fix a weight λq and a highest weight module V such that the stabilizer of its highest weight λq in WIV is ﬁnite. The following are equivalent: (1) wt V = wt M(λq,IV ). >0 p −Z πI p (2) wt p V = q V λq, where I = I \ IV . IV V L(λq) p In particular, if V is simple, or more generally |IV | 6 1, then wt V = wt M(λq,IV ). To prove Theorem 9.19, we will need the Integrable Slice Decomposition for quantum parabolic Verma modules.

Proposition 9.20. Let M(λq,J) be a parabolic Verma module such that the stabilizer of λq in WJ is ﬁnite. Then: G −µ wt M(λq,J) = wt LlJ (q λq), (9.21) 0 µ∈Z> (π\πJ )

where LlJ (ν) denotes the simple Uq(lJ )-module of highest weight ν. Proof. The inclusion ⊃ follows from considering weights as in Proposition 4.1. The inclusion ⊂ follows by using Proposition 4.3(2), which holds for quantum groups by specialization to q = 1. Proof of Theorem 9.19. By Proposition 9.20, (1) implies (2). For the converse, by the In- 0 −Z> (π\πI ) tegrable Slice Decomposition (9.21), it suﬃces to show that q V λq ⊂ wt V . This follows by mimicking the proof of Theorem 5.8.

As an application of Theorem 9.19, Equation (9.10) holds on the nose for all λq with ﬁnite stabilizer in W , and Equation (9.14) holds, rephrased as follows: IL(λq) CHARACTERS OF HIGHEST WEIGHT MODULES AND INTEGRABILITY 25

Proposition 9.22. Suppose λ has ﬁnite W -isotropy. Then: q IL(λq)

[ αˇi ni >0 wt L(λq) = w{νq : νq 6 λq, νq(q ) = ±q , ni ∈ Z , ∀i ∈ IL(λq)}. (9.23) w∈WI L(λq) Finally, we extend the results of Section 6 to quantum groups, as the same proof applies.

Theorem 9.24. Fix λq and J ⊂ IL(λq) such that the stabilizer of λq in WJ is ﬁnite, and p deﬁne Jq := IL(λq)\J . The following are equivalent: (1) wt V = wt M(λq,J) for every V with IV = J. (2) The Dynkin diagram for g p is complete. Jq

References [1] Tomoyuki Arakawa and Peter Fiebig. The linkage principle for restricted critical level representations of affine Kac–Moody algebras. Compos. Math., 148(6):1787–1810, 2012. [2] Michael F. Atiyah and Raoul Bott. A Lefschetz fixed point formula for elliptic complexes. II. Applications. Ann. of Math. (2), 88:451–491, 1968. [3] Alexander Barvinok. Integer points in polyhedra. Zurich Lectures in Advanced Mathematics. European Mathematical Society (EMS), Zürich, 2008. [4] Alexander Beilinson and Joseph Bernstein. Localisation de g-modules. C. R. Acad. Sci. Paris, 292(1):15– 18, 1981. [5] Alexander Beilinson, Victor Ginzburg, and Wolfgang Soergel. Koszul duality patterns in representation theory. J. Amer. Math. Soc., 9(2):473–527, 1996. [6] Christopher P. Bendel, Daniel K. Nakano, Brian J. Parshall, and Cornelius Pillen. Cohomology for quantum groups via the geometry of the nullcone. Mem. Amer. Math. Soc., 229(1077):x+93, 2014. [7] Anders Björnerand Francesco Brenti. Combinatorics of Coxeter groups, volume 231 of Graduate Texts in Mathematics. Springer, New York, 2005. [8] Michel Brion. Polyèdreset réseaux. Enseign. Math. (2), 38(1-2):71–88, 1992. [9] Michel Brion. Personal communication. 2013. [10] Jean-Luc Brylinski and Masaki Kashiwara. Kazhdan–Lusztig conjecture and holonomic systems. In- vent. Math., 64(3):387–410, 1981. [11] Gurbir Dhillon and Apoorva Khare. Faces of highest weight modules and the universal Weyl polyhedron. Adv. Math., 319:111–152, 2017. [12] Gurbir Dhillon and Apoorva Khare. The Weyl–Kac weight formula. Sém.Lothar. Combin., Vol. 78B, Article #77, 12 pages, 2017. [13] Xin Fang. Non-symmetrizable quantum groups: defining ideals and specialization. Publ. Res. Inst. Math. Sci., 50(4):663–694, 2014. [14] Edward Frenkel. Langlands correspondence for loop groups, volume 103 of Cambridge Studies in Advanced Mathematics. Cambridge University Press, Cambridge, 2007. [15] Edward Frenkel and Dennis Gaitsgory. Localization of g-modules on the affine Grassmannian. Ann. of Math. (2), 170(3):1339–1381, 2009. [16] Howard Garland and James Lepowsky. Lie algebra homology and the Macdonald–Kac formulas. In- vent. Math., 34(1):37–76, 1976. [17] Istvan Heckenberger and Stefan Kolb. On the Bernstein–Gelfand–Gelfand resolution for Kac–Moody algebras and quantized enveloping algebras. Transform. Groups, 12(4):647–655, 2007. [18] 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. [19] Ronald S. Irving. Singular blocks of the category O. Math. Z., 204(2):209–224, 1990. [20] James E. Humphreys. Representations of semisimple Lie algebras in the BGG category O, volume 94 of Graduate Studies in Mathematics. American Mathematical Society, Providence, RI, 2008. [21] Jens C. Jantzen. Kontravariante Formen auf induzierten Darstellungen halbeinfacher Lie-Algebren. Math. Ann., 226(1):53–65, 1977. [22] Jens C. Jantzen. Moduln mit einem höchsten Gewicht, volume 750 of Lecture Notes in Mathematics. Springer, Berlin, 1979. 26 GURBIR DHILLON AND APOORVA KHARE

[23] Jens C. Jantzen. Lectures on quantum groups, volume 6 of Graduate Studies in Mathematics. American Mathematical Society, Providence, RI, 1996. [24] Victor G. Kac. Infinite-dimensional Lie algebras and the Dedekind η-function (Russian). Funkt. analys y ego prilozh., 8(1):77–78, 1974. (English translation in: Funct. Anal. Appl., 8(1):68–70, 1974.) [25] Victor G. Kac. Infinite-dimensional Lie algebras. Cambridge University Press, Cambridge, third edition, 1990. [26] Steven N. Kass. A recursive formula for characters of simple Lie algebras. J. Algebra, 137(1):126–144, 1991. [27] David A. Kazhdan and George Lusztig. Representations of Coxeter groups and Hecke algebras. In- vent. Math., 53(2):165–184, 1979. [28] Apoorva Khare. Faces and maximizer subsets of highest weight modules. J. Algebra, 455:32–76, 2016. [29] Askold G. Khovanskii and Alexander V. Pukhlikov. Finitely additive measures of virtual polyhedra. In: St. Petersburg Mathematical Journal, 4(2):337–356, 1993. [30] Bertram Kostant. Lie algebra cohomology and the generalized Borel–Weil theorem. Ann. of Math. (2), 74:329–387, 1961. [31] Shrawan Kumar. Demazure character formula in arbitrary Kac–Moody setting. Invent. Math., 89(2):395– 423, 1987. [32] Shrawan Kumar. Bernstein–Gel0fand–Gel0fand resolution for arbitrary Kac–Moody algebras. Math. Ann., 286(4):709–729, 1990. [33] Shrawan Kumar. Kac–Moody groups, their flag varieties and representation theory, volume 204 of Progress in Mathematics. BirkhäuserBoston, Inc., Boston, MA, 2002. [34] Jim Lawrence. Rational-function-valued valuations on polyhedra. Discrete and computational geometry (New Brunswick, NJ, 1989/1990), pp. 199–208. DIMACS Series in Discrete Mathematics and Theoretical Computer Science, 6, American Mathematical Society, Providence, 1991. [35] James Lepowsky. A generalization of the Bernstein–Gelfand–Gelfand resolution. J. Algebra, 49(2):496– 511, 1977. [36] George Lusztig. Quantum deformations of certain simple modules over enveloping algebras. Adv. Math., 70(2):237–249, 1988. [37] George Lusztig. Canonical bases arising from quantized enveloping algebras. J. Amer. Math. Soc., 3(2):447–498, 1990. [38] Michael Murray and John Rice. A geometric realisation of the Lepowsky Bernstein Gel0fand Gel0fand resolution. Proc. Amer. Math. Soc., 114(2):553–559, 1992. [39] Alexander Postnikov. Permutohedra, associahedra, and beyond. Int. Math. Res. Not. IMRN, (6):1026– 1106, 2009. [40] Waldeck Schützer.A new character formula for Lie algebras and Lie groups. J. Lie Theory, 22(3):817–838, 2012. [41] Wolfgang Soergel. Kategorie O, perverse Garben und Moduln über den Koinvarianten zur Weylgruppe. J. Amer. Math. Soc., 3(2):421–445, 1990. [42] Mark A. Walton. Polytope sums and Lie characters. In Symmetry in physics, volume 34 of CRM Proc. Lecture Notes, pages 203–214. Amer. Math. Soc., Providence, RI, 2004. [43] Hermann Weyl. “Theorie der Darstellung kontinuierlicher halbeinfacher Gruppen durch lineare Transfor- mationen. I, II, III”. Math. Z., 23: pp. 271–309, 24: pp. 328–376, 24: pp. 377–395, 1925–1926.

(G. Dhillon) Department of Mathematics, Stanford University, Stanford, CA 94305, USA E-mail address: [email protected]

(A. Khare) Department of Mathematics, Indian Institute of Science, Bangalore 560012, India E-mail address: [email protected]