Characters for Projective Modules in the BGG Category O for the Orthosymplectic Lie Superalgebra osp(3|4)
aa r X i v : . [ m a t h . R T ] J un CHARACTERS FOR PROJECTIVE MODULES IN THE BGGCATEGORY O FOR THE ORTHOSYMPLECTIC LIE SUPERALGEBRA osp (3 | ARUN S. KANNAN AND HONGLIN ZHU
Abstract.
We determine the Verma multiplicities of standard filtrations of projectivemodules for integral atypical blocks in the BGG category O for the orthosymplectic Liesuperalgebras osp (3 |
4) by way of translation functors. We then explicitly determine thecomposition factor multiplicities of Verma modules using BGG reciprocity.
Contents
1. Introduction 12. Preliminaries 33. Character Formulae for osp (3 |
4) 114. Jordan-H¨older Formulae for osp (3 |
4) 31References 331.
Introduction O of modules of semisimple Lie algebras. This category exhibits richand deep theory and a broad survey of results can be found in [Hum08]. A generalizationof semisimple Lie algebras are basic Lie superalgebras, which exhibit many of the samephenomena (for reference, see [CW12; Mus12]). The BGG category O can analogously bedefined for basic Lie superalgebras, and many of the results from the semisimple case extend.Among the most conceptual objects in this category are the Verma modules. In this paper,we determine Verma multiplicities of standard filtrations of projective modules of integralatypical highest weight in the BGG category O for the basic Lie superalgebra osp (3 | O whose degree of atypical-ity is greater than 0 to have infinitely many simple modules. The principal block in O for osp (2 m + 1 | n ), which contains the trivial module, always has nonzero degree of atypicalitywhen m, n ≥ osp (2 m + 1 | n ), the degree of atypicality is an integer in the range 0 to min( m, n ), inclusive.In the integral case, any typical (i.e. degree of atypicality 0) block can be reduced to thesemisimple Lie algebra case via an equivalence of categories (cf. [Gor02b]). Therefore, thenew cases arise primarily when the degree of atypicality is nonzero.1.3. The problem of determining the irreducible representations that appear in a Jordan-H¨older series of a Verma module of a semisimple Lie algebra has a detailed history. For adominant integral weight, Kazhdan and Lusztig conjectured that these multiplicities could bedetermined in terms of certain recursively defined polynomials generated from the Weyl groupof the semisimple Lie algebra (cf. [LK79]), and this can be extended to an arbitrary integralweight by Jantzen’s translation functors (cf. [Jan79]). The Kazhdan-Lusztig conjecture wasproven via geometric methods in the 1980s by Beilinson & Bernstein ([BB81]) and Brylinski& Kashiwara ([BK81]).Generalizing to the basic Lie superalgebra case has been difficult because the Weyl groupno longer solely dictates linkage, but some progress has been made. (cf. [Bru03; BLW16;CLW11; CLW15]). An entirely different approach (and therefore solving the problem forcertain semisimple Lie algebras in a novel way) was done for osp ( l | n ) by way of quantumsymmetric pairs by Bao and Wang (cf. [Bao17; BW18]).Nonetheless, these methods do not readily offer concrete multiplicities. By way of trans-lation functors, we explicitly compute standard filtration formulae for projectives. Thismethod is used to solve a similar problem for gl (3 |
1) and gl (2 |
2) in [Kan19], for G (3) in[CW18], and D (2 | ζ ) in [CW19].1.4. In this work, we use the tool of translation functors to determine the characters of pro-jective modules in the BGG category O for the orthosymplectic Lie superalgebras osp (3 | O . There are infinitely many inequivalent atypical blocks.Then, BGG reciprocity allows us to convert these formulae to formulae for compositionmultiplicities, which we also explicitly state.1.5. Our general approach of using translation functors is as follows. Given some projectivecover P λ for which we wish to deduce Verma multiplicities, we find some P µ with knownVerma multiplicities and some finite-dimensional representation N such that the Vermamodule M λ appears in a standard filtration of P µ ⊗ N . If λ is the lowest weight appearingamong all the weights linked to λ appearing in the Verma flag, then P λ is a direct summandfor the projection of P µ ⊗ N on to the block corresponding to λ . In most cases, it is the onlydirect summand.A particularly useful set of criteria for determining whether a summand is direct and forverifying indecomposability is stated in Proposition 2.7. These criteria follow from similarcriteria on tilting modules (cf. [CW18]) derived from the Super Jantzen sum formula (cf.[Mus12]). Verifying indecomposability is a non-trivial step, as it is not evident whether ornot translation functors yield an indecomposable projective. See § § O . HARACTERS FOR PROJECTIVE MODULES IN THE BGG CATEGORY O §
2, we recall basic structure theorems for osp (2 m + 1 | n ), fix a Cartan subalgebra, aroot system, a fundamental system, and define linkage. Also, we recall the BGG category O ,review relevant results in the super case, and offer conditions when Verma modules appearin the standard filtration of projective modules.The section § g = osp (3 | § osp (3 | Acknowledgements.
This paper is the result of MIT PRIMES, a program that provideshigh-school students an opportunity to engage in research-level mathematics and in whichthe first author mentored the second. The first author would like to thank the MassachusettsInstitute of Technology for financial support in the form of a MathWorks fellowship. Theauthors would also like to thank the MIT PRIMES organizers for providing this opportunity.Finally, the authors thank David Vogan and Kevin Coulembier for answering some questionsabout the project. 2.
Preliminaries
We shall recall elementary properties about the structure of the Lie superalgebra osp (2 m +1 | n ) and introduce some basic notations.2.1. Basic definitions.
Suppose V = C k | l = C k ⊕ C l . Let { , . . . , k } and { , , . . . , l } parametrize the standard bases for the even and odd subspaces of V , C k and C l , respectively.Denote(2.1) I ( k, l ) = { , , . . . , k ; 1 , , . . . , l } where we impose the total order(2.2) 1 < · · · < k < < < · · · < l. The Lie superalgebra gl ( k | l ) is the Lie superalgebra of k × l matrices over C with bracketto be defined. The basis I ( k, l ) for V induces a basis for gl ( k | l ) given by { E ij : i, j ∈ I ( k, l ) } , where E ij is the elementary matrix with a 0 in every entry except for a 1 in the i -th row and j -th column ( i, j ∈ I ( m, n )). The even subalgebra gl ( k | l ) of gl ( k | l ) has abasis { E ij : i, j < , i, j > , i, j ∈ I ( k, l ) } and the odd subspace gl ( k | l ) has a basis { E ij : i < < j, j < < i, i, j ∈ I ( k, l ) } . An element that is either purely even or purelyodd is said to be homogeneous, and its parity (denoted | · | ) is 0 or 1, respectively. The Liesuperbracket is defined on homogeneous elements x, y ∈ gl ( k | l )(2.3) [ x, y ] = xy − ( − | x || y | yx and extended by bilinearity. Define the supertranspose x st of an element x ∈ gl ( k | l ) in( k | l )-block form x = (cid:18) a bc d (cid:19) by x st = (cid:18) a t c t − b t d t (cid:19) , where t denotes the regular transpose.Then, we define the Lie superalgebra osp (2 m + 1 | n ) by stabilizing a non-degenerate evensupersymmetric bilinear form as follows: A.S. KANNAN AND H. ZHU (2.4) osp (2 m + 1 | n ) = { g ∈ gl (2 m + 1 | n ) | g st J m +1 , n + J m +1 , n g = 0 } , where if I m is the m × m identity matrix, J m +1 , n is the (2 m + 1 + 2 n ) × (2 m + 1 + 2 n )matrix in the (1 | m | m | n | n )-block form(2.5) J m +1 , n = I m I m I n − I n . The even subalgebra osp (2 m + 1 | n ) (resp. odd subspace osp (2 m + 1 | n ) ) consists of theelements in osp (2 m + 1 | n ) that are also in gl (2 m + 1 | n ) (resp. gl (2 m + 1 | n ) ). As asemisimple Lie algebra, osp (2 m + 1 | n ) is isomorphic to sp (2 n ) ⊕ so (2 m + 1).Let h j = E j,j − E n + j,n + j for 1 ≤ j ≤ n and let h ′ i = E i, i − E m + i for 1 ≤ i ≤ m , andlet h denote the Cartan subalgebra of osp (2 m + 1 | n ) given by the subalgebra with basisgiven by these diagonal matrices:(2.6) h = C { h j , h ′ i | ≤ j ≤ n, ≤ i ≤ m } . Then, consider the dual basis for h ∗ given by { δ j , ǫ i | ≤ j ≤ n, ≤ i ≤ m } , where(2.7) δ i ( h j ) = ǫ i ( h ′ j ) = δ ij , δ i ( h ′ j ) = ǫ i ( h j ) = 0for 1 ≤ j ≤ n and 1 ≤ i ≤ m . Define a bilinear form ( · , · ) : h ∗ × h ∗ → C given by(2.8) ( δ j , δ k ) = δ jk , ( ǫ i , ǫ l ) = − δ il ( δ j , ǫ i ) = ( ǫ i , δ j ) = 0 , where 1 ≤ i, l ≤ m and 1 ≤ j, k ≤ n and then we extend by bilinearity. Note that( δ i ± ǫ j , δ i ± ǫ j ) = 0for 1 ≤ i ≤ m and 1 ≤ j ≤ n . We can define the corresponding integral weight lattice X in h ∗ :(2.9) X := M ≤ j ≤ n Z δ j ⊕ M ≤ i ≤ m Z ǫ i . Furthermore, with this choice of h we have a triangular decomposition osp (2 m + 1 | n ) = n − ⊕ h ⊕ n + and root system Φ = Φ ∪ Φ where(2.10) Φ = {± δ ˜ j ; ± δ j ± δ k ; ± ǫ ˜ i ; ± ǫ i ± ǫ l } Φ = {± δ ˜ j ± ǫ ˜ i ; ± δ ˜ i } , is the even and odd root decomposition, where 1 ≤ ˜ j ≤ n, ≤ j < k ≤ n, ≤ ˜ i ≤ m, ≤ i < l ≤ m and signs are taken independently. Call a root α ∈ Φ isotropic if ( α, α ) = 0. Let
HARACTERS FOR PROJECTIVE MODULES IN THE BGG CATEGORY O (2.11) Π = { δ j − δ j +1 , ǫ i − ǫ i +1 } ∪ { δ n − ǫ } , Φ + = { δ ˜ j ; δ j ± δ k ; ǫ ˜ i ; ǫ i ± ǫ l } ∪ { δ ˜ j ± ǫ ˜ j ; δ ˜ i } be a fundamental system and positive system, respectively, with 1 ≤ ˜ j ≤ n, ≤ j < k ≤ n, ≤ ˜ i ≤ m, ≤ i < l ≤ m . Let Φ +0 = Φ + ∩ Φ denote the positive even roots andΦ +1 = Φ + ∩ Φ denote the positive odd roots. Lastly, let W = W sp (2 n ) × W so (2 m +1) ∼ =( Z n ⋊ S n ) × ( Z m ⋊ S m ) denote the Weyl group of osp (2 m + 1 | n ), which by definition is theWeyl group of the even subalgebra. The action on h ∗ given by signed permutations of the δ j ’s or of the ǫ i ’s. The dot-action ( · ) is given by ρ -shifting the regular action. Call a weight λ dot-regular if |W · λ | = |W| and dot-singular otherwise.Furthermore, we can define for any α ∈ Φ the corresponding coroot α ∨ ∈ h such that forany λ ∈ h ∗ (2.12) h λ, α ∨ i = 2( λ, α )( α, α ) . The associated reflection s α acts on h ∗ as expected: s α ( λ ) = λ − h λ, α ∨ i α . However, noticethat the Weyl group is not generated by the simple reflections.Define the Weyl vector ρ as follows:(2.13) ρ = n X j =1 ( n − m − j + 12 ) δ j − m X i =1 ( m − i + 12 ) ǫ i . A weight λ ∈ h ∗ is said to be antidominant if h λ + ρ, α ∨ i 6∈ Z > and dominant if h λ + ρ, α ∨ i 6∈ Z < for all α ∈ Φ +0 .2.2. Atypicality and linkage.
The notion of linkage in the super case is similar to thatof semisimple Lie algebras. However, the key distinction is that while blocks of modules inthe semisimple Lie algebra case contain finitely many simple modules, odd roots allow forblocks in the super case to have infinitely many simple modules. This arises because of anotion called atypicality.Let h be the Cartan subalgebra of osp (2 m + 1 | n ) and let Φ be the root system as above.The degree of atypicality of λ ∈ h ∗ , denoted λ , is the maximum number of mutuallyorthogonal positive odd roots α ∈ Φ +1 such that ( λ + ρ, α ) = 0. An element λ ∈ h ∗ is said tobe typical (relative to Φ + ) if λ = 0 and is atypical otherwise.A relation ∼ on h ∗ can be defined as following. We say λ ∼ µ λ, µ ∈ h ∗ if there existmutually orthogonal odd roots α , α , . . . , α l , complex numbers c , c , . . . , c l , and an element w ∈ W satisfying:(2.14) µ + ρ = w λ + ρ − l X i =1 c i α i ! , ( λ + ρ, α i ) = 0 , i = 1 . . . , l. The weights λ and µ are said to be linked if λ ∼ µ . It can be shown that linkage is anequivalence relation.Given a fundamental root system Π, we can establish the Bruhat order on h ∗ as follows.Let λ, µ ∈ h ∗ . We say λ ≥ µ if λ ∼ µ and λ − µ ∈ Z ≥ Π (i.e the nonnegative sum of simpleroots).
A.S. KANNAN AND H. ZHU
We introduce notation for both convenience and to make the degree of atypicality clear.If λ = P mj =1 q j δ j + P ni =1 r i ǫ i ∈ h ∗ , use the notation λ = ( q , q , . . . , q m | r , r , . . . r n ). Fur-thermore, the action of the Weyl group W is clear. We can permute with signs everythingto the left of the bar and to the right of the bar, but no coefficient may cross the bar.The degree of atypicality of the weight ( q , q , . . . , q m | r , r , . . . r n ) − ρ is read by countingthe number of pairs ( q i , r j ) such that | q i | = | r j | , with the important stipulation no q i or r j be reused. The corresponding set of mutually orthogonal roots are δ i − ǫ j if q i = − r j and δ i + ǫ j if q i = r j for each pair ( i, j ). The degree of the atypicality is also given by the size ofthe multiset {| q i |} mi =1 ∩ {| r j |} nj =1 . In particular, if none of the | q i | coincide with the | r j | , theweight is typical.2.3. The Lie superalgebra osp (3 | . Since the Lie superalgebra osp (3 |
4) is of primaryinterest, we explicitly restate some of the previous facts for this Lie superalgebra. The evensubalgebra is g = sp (4) ⊕ so (3). We write ǫ to abbreviate ǫ . The positive system is givenby Φ + = Φ +0 ∪ Φ +1 = { δ , δ , δ ± δ , ǫ } ∪ { δ , δ , δ ± ǫ, δ ± ǫ } . The integral weight lattice in h ∗ is given by X = Z δ ⊕ Z δ ⊕ Z ǫ . The Weyl vector is given by ρ = δ − δ + ǫ . Noticethat any vector in X + ρ is half-integer and therefore not orthogonal to the non-isotropicodd roots δ and δ .The weights in h ∗ of osp (3 |
4) are of the form ( a, b | c ) in our notation. We are mainlyinterested in modules of integral highest weight λ ∈ X , frequently written as λ = ( a, b | c ) − ρ where a, b, c ∈ Z + .The Weyl group is W = W sp (4) × W so (3) ∼ = ( Z ⋊ S ) × Z is the product of dihedralgroups. W sp (4) acts on a weight λ = ( a, b | c ) by signed permutations of a and b and W so (3) acts by sign changes of c . A weight λ = ( a, b | c ) − ρ is atypical (of degree one) if and onlyif c ∈ {± a, ± b } .Denote by r the reflection associated with δ − δ , by s the reflection associated with 2 δ ,and by t the reflection associated with ǫ . Then, the respective actions on h ∗ are given bypermuting δ and δ , negating δ , and negating ǫ . As a Coxeter group, the Weyl group hasa presentation W = h r, s, t | r , s , t , ( rs ) , ( rt ) , ( st ) i . The first two reflections r and s generate W sp (4) , and t generates W so (3) . We impose the Bruhat order on W , writing w ′ ≤ w if a reduced word for w ′ appears in some reduced word for w for w ′ , w ∈ W . By the BGGtheorem, this order is compatible with the partial order above on h ∗ in the sense that if λ − ρ is typical, dot-regular, and antidominant, then w ′ ≤ w if and only if w ′ λ ≤ wλ (cf. [Hum08]). W sp (4) is dihedral and therefore the restricted Bruhat order is determined by comparing thelengths of elements. The Bruhat graph of W sp (4) is given below:1 rs rssr rsrsrs rsrs = srsr Combined with the fact that t is central, this makes clear the Bruhat order on W . HARACTERS FOR PROJECTIVE MODULES IN THE BGG CATEGORY O The BGG category O . From now on, let g = osp (3 |
4) = g ⊕ g with the standardassociated bilinear form, root system, and triangular decomposition: g = n − ⊕ h ⊕ n + and b = h ⊕ n + . Recall that the BGG category O is the full subcategory of U ( g )-modules M subject to the following three conditions:(1) M is finitely generated.(2) M is h -semisimple: M = L λ ∈ X M λ , where M λ = { v ∈ M | h · v = λ ( h ) v for all h ∈ h } is a nonzero weight space.(3) M is locally n + -finite: U ( n + ) · v is finite dimensional for all v ∈ M .Observe that the abelian quotient algebra b / n + ∼ = h . Thus, any λ ∈ h ∗ naturally definesa one-dimensional b -module with trivial n + -action, which we denote as C λ . Specifically, if v ∈ C λ , then h · v = λ ( h ) v for all h ∈ h . Now, define(2.15) M λ := U ( g ) ⊗ U ( b ) C λ − ρ , where ρ is the Weyl vector. This is naturally a left U ( g )-module. This is called a Vermamodule of highest weight λ − ρ .We let L λ denote the unique simple quotient of M λ of highest weight λ − ρ , and use thenotation [ M µ : L λ ] to denote the multiplicity of L λ in a composition series of M µ . Such aseries exists for all M in O .In the notation introduced in § λ = ( a, b | c ), write M a,b | c to denote M λ and L a,b | c todenote L λ .2.5. Blocks in O . The integral blocks in O can be divided into typical and atypical blocks.By definition, any simple module in a typical block has typical highest weight. The typicalblocks in O are described by Gorelik (see section 8.5.1 in [Gor02a] and theorem 1.3.1 in[Gor02b]). Because any ρ -shifted integral weight is strongly typical in the sense of [Gor02b],we get Proposition 2.1 (Gorelik) . Any typical block in O is equivalent to a block in the BGGcategory O of g -modules of integral weights. For osp (3 | O are indexed by linkage classes. Inparticular, each a ∈ Z ≥ + 1 / B a , with a corresponding linkageclass representative given by ( a, b | b ) − ρ with b ∈ Z + 1 /
2. All integral atypical blocks aregiven this way. In particular, the principal block B / contains the trivial module.2.6. Key results in O . The primary means by which the goals of this paper are achievedare by using translation functors. We restate the necessary results to justify our steps. Thiscollection of results is justified in [Hum08, Chap. 1-3] for the BGG category O for semisimpleLie algebras; similar arguments extend them to the BGG category O of osp (3 | Theorem 2.2.
Let N be a finite dimensional U ( g ) -module. For any λ ∈ h ∗ , the tensormodule T := M λ ⊗ N has a finite filtration with quotients isomorphic to Verma modules ofthe form M λ + µ , where µ ranges over the weights of N , each occurring dim N µ times in thefiltration. A module N ∈ O has a standard filtration or a Verma flag if there is a sequence ofsubmodules 0 = N ⊂ N ⊂ N ⊂ · · · N k = N such that each N i /N i − ≤ i ≤ k is A.S. KANNAN AND H. ZHU isomorphic to a Verma module. The number of times the Verma module M λ appears in astandard filtration of N is denoted by ( N : M λ ).It can be shown that the length and the Verma multiplicities in a standard filtration areindependent of choice of a standard filtration. Therefore, the following informal notationto indicate a standard filtration of a module is useful. If M λ i , λ i ∈ h ∗ , ≤ i ≤ k are theVerma modules appearing with multiplicity c i ∈ Z > in a standard filtration of a module N ,we write:(2.16) N = c M λ + c M λ + · · · + c k M λ k Similarly, if L µ i , µ i ∈ h ∗ , ≤ i ≤ k are the irreducibles appearing with multiplicity d i ∈ Z > in a composition series of a module N , we write(2.17) N = d L µ + d L µ + · · · + d k L µ k We let P λ denote the (unique) projective cover for L λ for all λ ∈ h ∗ , that is the indecom-posable projective such that P λ ։ L λ →
0. We recall the following facts about projectives.(1) All projectives have a standard filtration.(2) The category O has enough projectives.(3) If P = Q ⊕ R with P, Q, R ∈ O , P is projective if and only if Q and R are projective.(4) If P ∈ O is projective and indecomposable, then P ∼ = P λ for some λ ∈ h ∗ .(5) The Verma modules M µ which appear in a standard filtration of P λ satisfy µ ≥ λ inthe Bruhat ordering, and M λ appears with multiplicity 1.These facts yield the following lemma. Lemma 2.3. If λ − ρ is the lowest weight in a standard filtration of a projective object P ,then P λ is a direct summand of P . The following proposition, which follows from Theorem 2.2, is a critical part of our trans-lation functor arguments.
Proposition 2.4.
If a projective P has a standard filtration given by P λ = P ν M ν , the ν not necessarily distinct, then for any finite-dimensional representation N with weights µ , thestandard filtration for P ⊗ N is given by P ν P µ M ν + µ , where µ appears in the sum withmultiplicity given by dim N µ . Knowing the Verma flag structure of typical projectives will be key in determining thoseof atypical projectives. We have the following lemmas.
Lemma 2.5. If λ ∈ X + ρ is such that λ − ρ is typical and dot-regular, then the Vermamodules that appear in a standard filtration of P λ are of the form M wλ , where w ∈ W suchthat wλ ≥ λ , and each Verma module appears with multiplicity .Proof. By Proposition 2.1, we have an equivalence of categories to the Lie algebra g = sp (4) ⊕ so (3). Since the Weyl group W is the product of dihedral groups, it is well known inthis case that the Kazhdan-Lusztig polynomials are all 1 (and for our particular case it can bedirectly verified by computation). The result follows by the Kazhdan-Lusztig conjecture. (cid:3) The lemma also extends to typical and dot-singular weights.
HARACTERS FOR PROJECTIVE MODULES IN THE BGG CATEGORY O Lemma 2.6.
Let λ ∈ X + ρ be such that λ − ρ is a typical anti-dominant dot-singular weight.Let W λ be a minimal set of left-coset representatives of W / W λ , where W λ = { w ∈ W | wλ = λ } . Then, if σ ∈ W λ , (2.18) P σλ = X τ ≥ σ,τ ∈W λ M τλ . Proof.
The proof is analogous to that of Lemma 3.5 in [CW18]. Since λ = aδ + bδ + cǫ with a, b, c ∈ Z + is singular and in particular c = 0, changing the sign of c does not stabilize λ . Hence, the action of W so (3) is always regular. Therefore, { e } 6 = W λ ⊆ W sp (4) . The centralcharacter corresponding to the integral weight λ − ρ is strongly typical in the sense of Gorelik(cf. [Gor02a]) and by Proposition 2.1 we have an equivalence of categories between the blockcontaining the irreducible module L λ and a singular integral block of sp (4) ⊕ so (3)-modules.Since the action of W so (3) is regular, it suffices to check the analog of (2.18) for a singularintegral block of W sp (4) modules. Since the corresponding Weyl group is dihedral and theKazhdan-Lusztig polynomials are 1, the lemma follows by Theorem 3.11.4 in [BGS96]. (cid:3) Lastly, we recall BGG reciprocity.(2.19) ( P λ : M µ ) = [ M µ : L λ ] , λ, µ ∈ h ∗ . Some representations of osp (3 | . The strategy of using translation functors in-volves choosing appropriate representations to tensor with projective modules to producenew modules.The simplest module we use is the seven-dimensional natural representation V = C | of osp (3 | g ) of osp (3 | V V of the natural representation (call it the wedge-squaredof the natural). In general, the k -th exterior power of a vector superspace W = W ⊕ W isdefined as:(2.20) ^ k ( W ) := M i + j = k (cid:0) Λ i ( W ) ⊗ S j ( W ) (cid:1) where Λ i and S j acting on vector spaces are the i -th exterior power and j -th symmetricpower in the traditional sense, respectively.In the case of osp (3 | Conditions for nonzero Verma flag multiplicities in projective modules.
Wehave the following proposition, which uses BGG reciprocity to reformulate the conditions fortilting modules in [CW12, Proposition 2.2] as conditions for projective modules.
Proposition 2.7.
Suppose that λ ∈ X, α i ∈ Φ +¯0 , ≤ i ≤ k, and β, γ ∈ Φ +¯1 . Let w = s α k s α k − · · · s α ∈ W .(1) Suppose that h λ, α ∨ i < . Then ( P λ : M s α λ ) > .(2) Suppose that h s α i − · · · s α λ, α ∨ i i < for all i ∈ , , . . . , k . then ( P λ : M wλ ) > .(3) Suppose that ( λ, β ) = 0 . Then ( P λ : M λ + β ) > .(4) Suppose that ( λ, β ) = 0 and h s α i − · · · s α ( λ + β ) , α ∨ i i < for all i ∈ , , . . . , k . Then ( P λ : M w ( λ + β ) ) > . (5) Suppose that ( λ, β ) = ( λ + β, γ ) = 0 and ht( β ) < ht( γ ) . Then ( P λ : M λ + β + γ ) > .(6) Suppose that ( λ, β ) = ( λ + β, γ ) = 0 , ht( β ) < ht( γ ) , and h s α i − · · · s α ( λ + β + γ ) , α ∨ i i < for all i ∈ , , . . . , k . Then ( P λ : M w ( λ + β + γ ) ) > .Proof. The proposition is originally derived using the Super Jantzen sum formula (cf. [Gor02a;Mus12]), giving conditions for composition factors. BGG Reciprocity (2.19) immediatelytranslates the conditions from those on tilting modules to those on projective modules. (cid:3)
We now rederive a well-known but useful corollary, which goes back to a fundamentallemma of Penkov and Serganova.
Corollary 2.8.
Suppose λ − ρ ∈ h ∗ is atypical. Then P λ must have a Verma flag of lengthgreater than .Proof. M λ appears in the standard filtration. Furthermore, because λ − ρ is atypical, thereexists β such that β ∈ Φ +1 and ( λ, β ) = 0. Therefore, apply Proposition 2.7(4) to see that M λ + β also appears in the standard filtration. (cid:3) Strategy.
Given an atypical λ − ρ ∈ h ∗ , we seek to deduce the standard filtrationformula of P λ . To do so, we choose a µ ∈ h ∗ such that we know a standard filtration for P µ . This is often accomplished by letting µ := λ − ν , where ν is a weight (often the lowest)in some finite-dimensional representation W such that µ − ρ is typical; Lemma 2.5 andLemma 2.6 tell us the structure of P µ . Proposition 2.4 can be used to deduce the Vermamodules which appear in a standard filtration of the projective P µ ⊗ W , which must include M λ . Our next step is to project P µ ⊗ W onto the block corresponding to the linkage classof λ − ρ . We denote the resulting projection as pr λ ( P µ ⊗ W ). By Lemma 2.3, if M λ has thelowest weight of all the Verma modules in the standard filtration of the projection, P λ mustappear in that projection as a direct summand. The projection itself is done by collectingall Verma modules in the standard filtration whose weights are linked to λ − ρ .In this projection, we apply Proposition 2.7 to see which Verma modules appear in thestandard filtration of P λ . These necessarily appear in the projection because P λ is a directsummand. Then, we generally try to argue that there is no other direct summand (i.e. P λ is the projection). This is often done by showing that no other indecomposable projectivecan appear in the projection, since there are not enough terms. In certain special cases, thismethod fails, and we get two possible standard filtrations of P λ . To determine which one iscorrect, we generally show that one of them is not a projective.For convenience, we introduce the following notation which we use extensively in thepresentation of our results and proofs to save space and improve clarity. Let λ ∈ X + ρ besuch that λ − ρ is anti-dominant. Let W λ be a minimal set of left-coset representatives of W / W λ , where W λ = { w ∈ W | wλ = λ } . Then, if σ ∈ W λ , we denote X τ ≥ σ,τ ∈W λ M τλ by X M σλ . For example, we may write M , − | + M , | + M , − | + M , | HARACTERS FOR PROJECTIVE MODULES IN THE BGG CATEGORY O as X M , − | . Character Formulae for osp (3 | osp (3 |
4) with integral, atypical highest weight.3.1.
Results.
Let g = osp (3 |
4) have the standard choices of Cartan subalgebra, bilinearform, root system, positive, and fundamental system as described in §
2. Recall the notationdescribed in § h ∗ . We have the following Theorems 3.1 to 3.4 thatdescribe standard filtrations of projectives in these blocks. Theorem 3.1.
Let λ − ρ = ( a, b | c ) − ρ be an atypical weight with a, b, c ∈ + Z , a, b > ,and c ∈ {± a, ± b } . The projective modules P λ of highest weight λ − ρ have the followingVerma flag formulae.(1) Suppose that a > b > .(1.1) When c = a , we have P a,b | a = M a,b | a + M a +1 ,b | a +1 . (1.2) When c = − a , we have P a,b |− a = M a,b |− a + M a,b | a + M a +1 ,b |− a − + M a +1 ,b | a +1 . (1.3) When c = b , we have P a,b | b = M a,b | b + M a,b +1 | b +1 for b < a − , and P a,a − | a − = M a,a − | a − + M a,a | a + M a +1 ,a | a +1 . (1.4) When c = − b , we have P a,b |− b = M a,b |− b + M a,b | b + M a,b +1 |− b − + M a,b +1 | b +1 for b < a − , and P a,a − |− a +1 = M a,a − |− a +1 + M a,a − | a − + M a,a |− a + M a,a | a + M a +1 ,a |− a − + M a +1 ,a | a +1 . (2) Suppose that b > a > .(2.1) When c = a , we have P a,b | a = M a,b | a + M b,a | a + M a +1 ,b | a +1 + M b,a +1 | a +1 for b > a + 1 , and P a,a +1 | a = M a,a +1 | a + M a +1 ,a | a + M a +1 ,a +1 | a +1 . (2.2) When c = − a , we have P a,b |− a = M a,b |− a + M a,b | a + M b,a |− a + M b,a | a + M a +1 ,b |− a − + M a +1 ,b | a +1 + M b,a +1 |− a − + M b,a +1 | a +1 for b > a + 1 , and P a,a +1 |− a = M a,a +1 |− a + M a,a +1 | a + M a +1 ,a |− a + M a +1 ,a | a + M a +1 ,a +1 |− a − + M a +1 ,a +1 | a +1 . (2.3) When c = b , we have P a,b | b = M a,b | b + M b,a | b + M a,b +1 | b +1 + M b +1 ,a | b +1 . (2.4) When c = − b , we have P a,b |− b = M a,b |− b + M a,b | b + M b,a |− b + M b,a | b + M a,b +1 |− b − + M a,b +1 | b +1 + M b +1 ,a |− b − + M b +1 ,a | b +1 = X M a,b |− b + X M a,b +1 |− b − . (3) Suppose that a = b > .(3.1) When c = a , we have P a,a | a = M a,a | a + M a,a +1 | a +1 + M a +1 ,a | a +1 . (3.2) When c = − a , we have P a,a |− a = M a,a |− a + M a,a | a + M a,a +1 |− a − + M a,a +1 | a +1 + M a +1 ,a |− a − + M a +1 ,a | a +1 . Theorem 3.2.
Let λ − ρ = ( a, b | c ) − ρ be an atypical weight with a, b, c ∈ + Z , a > > b ,and c ∈ {± a, ± b } . The projective modules P λ of highest weight λ − ρ have the followingVerma flag formulae.(1) Suppose that a > − b > (1.1) When c = a , P a,b | a = M a,b | a + M a, − b | a + M a +1 ,b | a +1 + M a +1 , − b | a +1 . (1.2) When c = − a , P a,b |− a = M a,b |− a + M a,b | a + M a, − b |− a + M a, − b | a + M a +1 ,b |− a − + M a +1 ,b | a +1 + M a +1 , − b |− a − + M a +1 , − b | a +1 . (1.3) When c = − b , P a,b |− b = M a,b |− b + M a, − b |− b + M a,b +1 |− b − + M a, − b − |− b − for b < − , and P a, − | = M a, − | + M a, | + M a, |− + M a, | for a > , and P , − | = M , − | + M , | + M , |− + M , | + M , | . HARACTERS FOR PROJECTIVE MODULES IN THE BGG CATEGORY O (1.4) When c = b , P a,b | b = M a,b | b + M a,b |− b + M a, − b | b + M a, − b |− b + M a,b +1 | b +1 + M a,b +1 |− b − + M a, − b − | b +1 + M a, − b − |− b − for b < − , and P a, − |− = M a, − |− + M a, − | + M a, |− + M a, | = X M a, − |− . (2) Suppose that − b > a > .(2.1) When c = a , we have P a,b | a = X M a,b | a + X M a +1 ,b | a +1 . (2.2) When c = − a , we have P a,b |− a = X M a,b |− a + X M a +1 ,b |− a − . (2.3) When c = − b , we have P a,b |− b = X M a,b |− b + X M a,b +1 |− b − . (2.4) When c = b , we have P a,b | b = X M a,b | b + X M a,b +1 | b +1 . (3) Suppose that a = − b > .(3.1) When c = a , we have P a, − a | a = M a, − a | a + M a,a | a + M a, − a +1 | a − + M a,a − | a − + M a +1 , − a | a +1 + M a +1 ,a | a +1 for a > , and P , − | = M , − | + M , | + M , |− + M , | + M , − | + M , |− + 2 M , | + M , | . (3.2) When c = − a , we have P a, − a |− a = X M a, − a |− a + X M a, − a +1 |− a +1 + X M a +1 , − a |− a − for a > , and P , − |− = X M , − |− + X M , − |− . Theorem 3.3.
Let λ − ρ = ( a, b | c ) − ρ be an atypical weight with a, b, c ∈ + Z , b > > a ,and c ∈ {± a, ± b } . The projective modules P λ of highest weight λ − ρ have the followingVerma flag formulae.(1) Suppose that a < − b < .(1.1) When c = − a , we have P a,b |− a = X M a,b |− a + X M a +1 ,b |− a − . (1.2) When c = a , we have P a,b | a = X M a,b | a + X M a +1 ,b | a +1 . (1.3) When c = b , we have P a,b | b = X M a,b | b + X M a,b +1 | b +1 . (1.4) When c = − b , we have P a,b |− b = X M a,b |− b + X M a,b +1 |− b − . (2) Suppose that − b < a < .(2.1) When c = − a , we have P a,b |− a = X M a,b |− a + X M a +1 ,b |− a − for a < − , and P − ,b | = X M − ,b | + M ,b |− + M b, |− + M ,b | + M b, | for b > , and P − , | = X M − , | + M , |− + M , |− + M , | . (2.2) When c = a , we have P a,b | a = X M a,b | a + X M a +1 ,b | a +1 for a < − , and P − ,b |− = X M − ,b |− . (2.3) When c = b , we have P a,b | b = X M a,b | b + X M a,b +1 | b +1 . (2.4) When c = − b , we have P a,b |− b = X M a,b |− b + X M a,b +1 |− b − . (3) Suppose that a = − b < .(3.1) When c = − a , we have P a, − a |− a = M a, − a |− a + M − a,a |− a + 2 M − a, − a |− a + M a, − a +1 |− a +1 + M − a, − a +1 |− a +1 + M − a +1 ,a |− a +1 + M − a +1 , − a |− a +1 + M a +1 , − a |− a − + M − a − , − a |− a − + M − a,a +1 |− a − + M − a, − a − |− a − = X M a, − a |− a + M − a, − a |− a + X M a, − a +1 |− a +1 + X M a +1 , − a |− a − HARACTERS FOR PROJECTIVE MODULES IN THE BGG CATEGORY O for a < − , and P − , | = M − , | + M , − | + M , | + M , |− + M − , | + M , | + M , − | + M , | = X M − , | + M , |− + X M − , | . (3.2) When c = a , we have P a, − a | a = X M a, − a | a + M − a, − a | a + M − a, − a |− a + X M a, − a +1 | a − + X M a +1 , − a | a +1 for a < − , and P − , |− = X M − , |− + M , |− + M , | + X M − , |− . Theorem 3.4.
Let λ − ρ = ( a, b | c ) − ρ be an atypical weight with a, b, c ∈ + Z , a, b < ,and c ∈ {± a, ± b } . The projective modules P λ of highest weight λ − ρ have the followingVerma flag formulae.(1) Suppose that a < b < .(1.1) When c = − a , we have P a,b |− a = X M a,b |− a + X M a +1 ,b |− a − . (1.2) When c = a , we have P a,b | a = X M a,b | a + X M a +1 ,b | a +1 . (1.3) When c = − b , we have P a,b |− b = X M a,b |− b + X M a,b +1 |− b − for b < − , and P a, − | = X M a, − | + X M − , − a | + M a, |− + M − , − a |− + M ,a |− + M , − a |− + M − a, − |− + M − a, |− + X M a, | . (1.4) When c = b , we have P a,b | b = X M a,b | b + X M a,b +1 | b +1 for b < − , and P a, − |− = X M a, − |− . (2) Suppose that b < a < . (2.1) When c = − a , we have P a,b |− a = X M a,b |− a + X M a +1 ,b |− a − for a < − , and P − ,b | = X M − ,b | + M − b, − | + M − b, | + M ,b |− + M , − b |− + M − b, − |− + M − b, |− + X M ,b | . (2.2) When c = a , we have P a,b | a = X M a,b | a + X M a +1 ,b | a +1 for a < − , and P − ,b |− = X M − ,b |− . (2.3) When c = − b , we have P a,b |− b = X M a,b |− b + X M a,b +1 |− b − for b < a − , and P a,a − |− a +1 = X M a,a − |− a +1 + X M a +1 ,a |− a − + X M a,a |− a + X M − a,a |− a for a < − , and P − , − | = X M − , − | + M , − | + M , | + X M − , − | + X M , − |− . (2.4) When c = b , we have P a,b | b = X M a,b | b + X M a,b +1 | b +1 for b < a − , and P a,a − | a − = X M a,a − | a − + X M a +1 ,a | a +1 + X M a,a | a + X M − a,a | a for a < − , and P − , − |− = X M − , − |− + X M − , − |− + X M , − |− . (3) Suppose that a = b < . HARACTERS FOR PROJECTIVE MODULES IN THE BGG CATEGORY O (3.1) When c = − a , we have P a,a |− a = X M a,a |− a + X M a,a +1 |− a − for a < − , and P − , − | = X M − , − | + M , | + M − , |− + M , − |− + M , |− + X M − , | . (3.2) When c = a , we have P a,a | a = X M a,a | a + X M a,a +1 | a +1 for a < − , and P − , − |− = X M − , − |− . Proof.
In this subsection, we prove Theorems 3.1 through 3.4. We use the methodof translation functors by effecting certain finite-dimensional representations. These repre-sentations, which are all irreducible, highest-weight, and self-dual (cf. [CW12]), and theirweights are given below. All weights, except the zero weight, appear with multiplicity 1.The zero weight is stated with its total multiplicity (i.e. 3 · V ±{ δ , δ , ǫ } ∪ { } δ V V ±{ δ ± δ , δ ± ǫ, δ , δ ± ǫ, δ , ǫ, ǫ } δ + δ ∪{ · } g ±{ δ , δ ± δ , δ ± ǫ, δ , δ , δ ± ǫ, δ , ǫ }
25 2 δ ∪{ · } In particular, we have as osp (3 | V ∼ = L / , − / | / = L δ + ρ , V V ∼ = L / , / | / = L δ + δ + ρ , and g ∼ = L / , − / | / = L δ + ρ .We now offer justification for the formulae above, separated into cases that have differentformulae, based on the strategy in § P µ and representation fortranslation functor. Proof of Theorem 3.1.
Let λ − ρ = ( a, b | c ) − ρ be an atypical weight with a, b, c ∈ + Z and a, b > a > b > λ = ( a, b | a ),pr λ (cid:0) P a +1 ,b | a ⊗ V (cid:1) = M a,b | a + M a +1 ,b | a +1 . By Lemma 2.3 and Proposition 2.7, the projection is equal to P λ . (1.2) When λ = ( a, b | − a ),pr λ (cid:0) P a +1 ,b |− a ⊗ V (cid:1) = M a,b |− a + M a,b | a + M a +1 ,b |− a − M a +1 ,b | a +1 . By Lemma 2.3 and Proposition 2.7, the projection is equal to P λ .(1.3) When λ = ( a, b | b ):(i) If b < a − λ (cid:0) P a,b +1 | b ⊗ V (cid:1) = M a,b | b + M a,b +1 | b +1 . By Lemma 2.3 and Proposition 2.7, the projection is equal to P λ .(ii) If b = a − λ (cid:0) P a +1 ,a − | a − ⊗ V (cid:1) = M a,a − | a − + M a,a | a + M a +1 ,a | a +1 By Lemma 2.3 and Proposition 2.7, the projection is equal to P λ .(1.4) When λ = ( a, b | − b ):(i) If b < a − λ (cid:0) P a,b +1 |− b ⊗ V (cid:1) = M a,b |− b + M a,b | b + M a,b +1 |− b − + M a,b +1 | b +1 . By Lemma 2.3 and Proposition 2.7, the projection is equal to P λ .(ii) If b = a − λ (cid:0) P a +1 ,a − |− a +1 ⊗ V (cid:1) = M a,a − |− a +1 + M a,a − | a − + M a,a |− a + M a,a | a + M a +1 ,a |− a − + M a +1 ,a | a +1 . By Lemma 2.3 and Proposition 2.7, the projection is equal to P λ .(2) Suppose that b > a > λ = ( a, b | a ):(i) If b > a + 1,pr λ (cid:0) P a +1 ,b | a ⊗ V (cid:1) = M a,b | a + M b,a | a + M a +1 ,b | a +1 + M b,a +1 | a +1 . By Lemma 2.3 and Proposition 2.7, the projection is equal to P λ .(ii) If b = a + 1,pr λ (cid:0) P a +1 ,a +1 | a ⊗ V (cid:1) = M a,a +1 | a + M a +1 ,a | a + M a +1 ,a +1 | a +1 . By Lemma 2.3 and Proposition 2.7, the projection is equal to P λ .(2.2) When λ = ( a, b | − a ):(i) If b > a + 1,pr λ (cid:0) P a +1 ,b |− a ⊗ V (cid:1) = M a,b |− a + M a,b | a + M b,a |− a + M b,a | a + M a +1 ,b |− a − + M a +1 ,b | a +1 + M b,a +1 |− a − + M b,a +1 | a +1 . By Lemma 2.3 and Proposition 2.7, the projection is equal to P λ .(ii) If b = a + 1,pr λ (cid:0) P a +1 ,a +1 |− a ⊗ V (cid:1) = M a,a +1 |− a + M a,a +1 | a + M a +1 ,a |− a + M a +1 ,a | a + M a +1 ,a +1 |− a − + M a +1 ,a +1 | a +1 . By Lemma 2.3 and Proposition 2.7, the projection is equal to P λ . HARACTERS FOR PROJECTIVE MODULES IN THE BGG CATEGORY O (2.3) When λ = ( a, b | b ),pr λ (cid:0) P a,b +1 | b ⊗ V (cid:1) = M a,b | b + M b,a | b + M a,b +1 | b +1 + M b +1 ,a | b +1 . By Lemma 2.3 and Proposition 2.7, the projection is equal to P λ .(2.4) When λ = ( a, b | − b ),pr λ (cid:0) P a,b +1 |− b ⊗ V (cid:1) = M a,b |− b + M a,b | b + M b,a |− b + M b,a | b + M a,b +1 |− b − + M a,b +1 | b +1 + M b +1 ,a |− b − + M b +1 ,a | b +1 = X M a,b |− b + X M a,b +1 |− b − . By Lemma 2.3 and Proposition 2.7, the projection is equal to P λ .(3) Suppose that a = b > λ = ( a, a | a ),pr λ (cid:0) P a,a | a +1 ⊗ V (cid:1) = M a,a | a + M a,a +1 | a +1 + M a +1 ,a | a +1 . By Lemma 2.3 and Proposition 2.7, the projection is equal to P λ .(3.2) When λ = ( a, a | − a ),pr λ (cid:0) P a,a |− a − ⊗ V (cid:1) = M a,a |− a + M a,a | a + M a,a +1 |− a − + M a,a +1 | a +1 + M a +1 ,a |− a − + M a +1 ,a | a +1 . By Lemma 2.3 and Proposition 2.7, the projection is equal to P λ . (cid:3) Proof of Theorem 3.2.
Let λ − ρ = ( a, b | c ) − ρ be an atypical weight with a, b, c ∈ + Z and a > > b .(1) Suppose that a > − b > λ = ( a, b | a ),pr λ (cid:0) P a +1 ,b | a ⊗ V (cid:1) = M a,b | a + M a, − b | a + M a +1 ,b | a +1 + M a +1 , − b | a +1 . By Lemma 2.3 and Proposition 2.7, the projection is equal to P λ .(1.2) When λ = ( a, b | − a ),pr λ (cid:0) P a +1 ,b |− a ⊗ V (cid:1) = M a,b |− a + M a,b | a + M a, − b |− a + M a, − b | a + M a +1 ,b |− a − + M a +1 ,b | a +1 + M a +1 , − b |− a − + M a +1 , − b | a +1 . By Lemma 2.3 and Proposition 2.7, the projection is equal to P λ .(1.3) When λ = ( a, b | − b ):(i) If b < − ,pr λ (cid:0) P a,b +1 |− b ⊗ V (cid:1) = M a,b |− b + M a, − b |− b + M a,b +1 |− b − + M a, − b − |− b − . By Lemma 2.3 and Proposition 2.7, the projection is equal to P λ .(ii) If b = − and a > ,pr λ (cid:16) P a, | ⊗ V (cid:17) = M a, − | + M a, | + M a, |− + M a, | . By Lemma 2.3, P λ must appear in the projection as a direct summand, andProposition 2.7 ensures that the first three terms appear in P λ . However, since M a, | does not form a projective on its own, it must also belong to P λ .(iii) if b = − and a = ,pr λ (cid:16) P , | ⊗ V (cid:17) = M , − | + M , | + M , |− + M , | + M , | . By Lemma 2.3, P λ must appear in the projection as a direct summand,and Proposition 2.7 ensures that the first three terms appear in P λ . Sincethe standard filtration of P , | does not appear in the projection, M , | must belong to P λ . Similarly, M , | must belong to P λ .(1.4) When λ = ( a, b | b ):(i) If b < − ,pr λ (cid:0) P a,b +1 | b ⊗ V (cid:1) = M a,b | b + M a,b |− b + M a, − b | b + M a, − b |− b + M a,b +1 | b +1 + M a,b +1 |− b − + M a, − b − | b +1 + M a, − b − |− b − By Lemma 2.3 and Proposition 2.7, the projection is equal to P λ .(ii) If b = − and a > ,pr λ (cid:16) P a, − | ⊗ V (cid:17) = 2 M a, − |− + 2 M a, − | + 2 M a, |− + 3 M a, | + M a, | . By Lemma 2.3, P λ must appear twice in the projection as a direct sum-mand, and By Proposition 2.7, one copy of each of the first four termsmust be in P λ . Now, one copy of the fourth term and the last term re-main. However, since only one copy of these two terms remains, theycannot appear in P λ . Thus, we get thatpr λ (cid:16) P a, − | ⊗ V (cid:17) = 2 P a, − |− + P a, | , and P a, − |− = M a, − |− + M a, − | + M a, |− + M a, | . (iii) If b = − and a = , we get a formula consistent with the previous caseby applying the same method.(2) Suppose that − b > a > λ = ( a, b | a ),pr λ (cid:0) P a +1 ,b | a ⊗ V (cid:1) = X M a,b | a + X M a +1 ,b | a +1 . By Lemma 2.3 and Proposition 2.7, the projection is equal to P λ . Note thatwhen b = − a − P M a +1 ,b | a +1 has two instead of four terms.(2.2) When λ = ( a, b | − a ),pr λ (cid:0) P a +1 ,b |− a ⊗ V (cid:1) = X M a,b |− a + X M a +1 ,b |− a − . By Lemma 2.3 and Proposition 2.7, the projection is equal to P λ . Note thatwhen b = − a − P M a +1 ,b |− a − has four instead of eight terms. HARACTERS FOR PROJECTIVE MODULES IN THE BGG CATEGORY O (2.3) When λ = ( a, b | − b ),pr λ (cid:0) P a,b +1 |− b ⊗ V (cid:1) = X M a,b |− b + X M a,b +1 |− b − . By Lemma 2.3 and Proposition 2.7, the projection is equal to P λ . Note thatwhen b = − a − P M a,b +1 |− b − has two instead of four terms.(2.4) When λ = ( a, b | b ),pr λ (cid:0) P a,b +1 | b ⊗ V (cid:1) = X M a,b | b + X M a,b +1 | b +1 . By Lemma 2.3 and Proposition 2.7, the projection is equal to P λ . Note thatwhen b = − a − P M a,b +1 | b +1 has four instead of eight terms.(3) Suppose that a = − b > λ = ( a, − a | a ):(i) If a > ,pr λ (cid:0) P a +1 , − a | a ⊗ V (cid:1) = M a, − a | a + M a,a | a + M a, − a +1 | a − + M a,a − | a − + M a +1 , − a | a +1 + M a +1 ,a | a +1 . By Lemma 2.3 and Proposition 2.7, the projection is equal to P λ .(ii) If a = ,pr λ (cid:16) P , − | ⊗ V (cid:17) = M , − | + M , | + M , |− + M , | + M , − | + M , |− + 2 M , | + M , | . By Lemma 2.3, P λ must appear in the projection. By Proposition 2.7, thefirst six term and one copy of M , | must appear in P λ . However, werun into some trouble here, since the two remaining terms, M , | , M , | could actually form the projective P , | , which means we have to devisesome different method to show they are also included in P λ .We have two possible standard filtrations of P λ . Call the shorter one,which do not include the two unexplained terms, Q , and call P , | = M , | + M , | R. We shall show that P λ has the longer standard filtration,which we shall denote by abuse of notation Q + R , by proving that Q is nota projective. We calculate the projections pr µ ( Q ⊗ g ) and pr µ ( R ⊗ g ).Projective Terms pr µ ( − ⊗ g ) Q M , − | M , − | M , − | M , | M , | M , | M , |− M , |− M , |− M , | M , | M , | M , | M , − | M , − | M , − | M , − | M , |− M , |− M , |− M , |− M , | M , | M , | M , | R M , | M , | M , | M , | M , | M , | M , | If Q were a projective, then pr µ ( Q ⊗ g ) is again a projective and byLemma 2.3 must split into indecomposable projectives. We see that thelowest weight appearing is (cid:0) , − | (cid:1) , so P , − | must appear, and itsterms are colored red. Next, we must have P , |− appear, whose termsare colored blue. Then, as (cid:0) , − | (cid:1) is the next lowest weight, P , − | must appear (colored violet). However, we see that there are not enoughterms left in pr µ ( Q ⊗ g ). Thus, Q is not a projective and we must have P λ = Q + R . It turns out thatpr µ (( Q + R ) ⊗ g ) = P , − | + P , |− + P , − | + P , | . (3.2) When λ = ( a, − a | − a ):(i) If a > ,pr λ (cid:0) P a +1 , − a |− a ⊗ V (cid:1) = X M a, − a |− a + X M a, − a +1 |− a +1 + X M a +1 , − a |− a − . By Lemma 2.3 and Proposition 2.7, the projection is equal to P λ .(ii) If a = ,pr λ (cid:16) P , − |− ⊗ V (cid:17) = X M , − |− + X M , − |− . By Lemma 2.3 and Proposition 2.7, the projection is equal to P λ . (cid:3) Proof of Theorem 3.3.
Let λ − ρ = ( a, b | c ) − ρ be an atypical weight with a, b, c ∈ + Z and b > > a .(1) Suppose that a < − b < λ = ( a, b | − a ),pr λ (cid:0) P a +1 ,b |− a ⊗ V (cid:1) = X M a,b |− a + X M a +1 ,b |− a − . By Lemma 2.3 and Proposition 2.7, the projection is equal to P λ . Notice thatwhen b = − a − P M a +1 ,b |− a − has three instead of six terms.(1.2) When λ = ( a, b | a ),pr λ (cid:0) P a +1 ,b | a ⊗ V (cid:1) = X M a,b | a + X M a +1 ,b | a +1 . By Lemma 2.3 and Proposition 2.7, the projection is equal to P λ . Notice thatwhen b = − a − P M a +1 ,b | a +1 has six instead of twelve terms.(1.3) When λ = ( a, b | b ),pr λ (cid:0) P a,b +1 | b ⊗ V (cid:1) = X M a,b | b + X M a,b +1 | b +1 . By Lemma 2.3 and Proposition 2.7, the projection is equal to P λ . Notice thatwhen b = − a − P M a,b +1 | b +1 has three instead of six terms.(1.4) When λ = ( a, b | − b ),pr λ (cid:0) P a,b +1 |− b ⊗ V (cid:1) = X M a,b |− b + X M a,b +1 |− b − . By Lemma 2.3 and Proposition 2.7, the projection is equal to P λ . Notice thatwhen b = − a − P M a,b +1 |− b − has six instead of twelve terms. HARACTERS FOR PROJECTIVE MODULES IN THE BGG CATEGORY O (2) Suppose that − b < a < λ = ( a, b | − a ):(i) If a < − ,pr λ (cid:0) P a +1 ,b |− a ⊗ V (cid:1) = X M a,b |− a + X M a +1 ,b |− a − . By Lemma 2.3 and Proposition 2.7, the projection is equal to P λ .(ii) If a = − and b > ,pr λ (cid:16) P ,b |− ⊗ V (cid:17) = X M − ,b | + M ,b |− + M b, |− + M ,b | + M b, | . By Lemma 2.3 and Proposition 2.7, P M − ,b | must belong to P λ . Thelowest remaining term is M ,b |− , and since the remaining terms do notcontain the standard filtration of P ,b |− , it must also belong to P λ . Bythe same argument, each of the remaining terms belongs to P λ .(iii) If a = − and b = ,pr λ (cid:16) P , |− ⊗ V (cid:17) = X M − , | + M , |− + M , |− + M , | . By Lemma 2.3 and Proposition 2.7, P M − , | must belong to P λ . Thelowest remaining term is M , |− , and since the remaining terms do notcontain the standard filtration of P , |− , it must also belong to P λ . Bythe same argument, each of the remaining terms belongs to P λ .(2.2) When λ = ( a, b | a ):(i) If a < − ,pr λ (cid:0) P a +1 ,b | a ⊗ V (cid:1) = X M a,b | a + X M a +1 ,b | a +1 . By Lemma 2.3 and Proposition 2.7, the projection is equal to P λ .(ii) If a = − and b > ,pr λ (cid:16) P − ,b | ⊗ V (cid:17) = 2 X M − ,b |− + M ,b | + M b, | + M ,b | + M b, | . By Lemma 2.3, two copies of P λ must appear in the projection. By Propo-sition 2.7, P M − ,b |− belongs to P λ . Now, since the remaining four termseach only appear with multiplicity one, they cannot belong to P λ . Theyform P ,b | .(iii) If a = − and b = , we get the same result with the same projectionas above, except that the projection has three (instead of four) remainingterms after subtracting two copies of P M − ,b |− . The three terms stillform P ,b | , which has three instead of four terms when b = . (2.3) When λ = ( a, b | b ),pr λ (cid:0) P a,b +1 | b ⊗ V (cid:1) = X M a,b | b + X M a,b +1 | b +1 . By Lemma 2.3 and Proposition 2.7, the projection is equal to P λ .(2.4) When λ = ( a, b | − b ),pr λ (cid:0) P a,b +1 |− b ⊗ V (cid:1) = X M a,b |− b + X M a,b +1 |− b − . By Lemma 2.3 and Proposition 2.7, the projection is equal to P λ .(3) Suppose that a = − b < λ = ( a, − a | − a ):(i) If a < − ,pr λ (cid:0) P a +1 , − a |− a ⊗ V (cid:1) = X M a, − a |− a + M − a, − a |− a + X M a, − a +1 |− a +1 + X M a +1 , − a |− a − . By Lemma 2.3 and Proposition 2.7, one copy of each term must appear in P λ . Now, there remains only the second copy of the term M − a, − a |− a , andas it clearly cannot form a projective, it must also belong to P λ .(ii) If a = − ,pr λ (cid:16) P , | ⊗ V (cid:17) = X M − , | + M , |− + X M − , | . By Lemma 2.3 and Proposition 2.7, every term except for M , |− mustappear in P λ . As the one remaining term cannot form a projective, it mustalso belong to P λ . Notice that unlike the previous term, each term hereonly appears with multiplicity one.(3.2) When λ = ( a, − a | a ):(i) If a < − ,pr λ (cid:0) P a +1 , − a | a ⊗ V (cid:1) = X M a, − a | a + M − a, − a | a + M − a, − a |− a + X M a, − a +1 | a − + X M a +1 , − a | a +1 . By Lemma 2.3 and Proposition 2.7, one copy of each term must appear in P λ . Now, there remain only the second copies of the terms M − a, − a | a and M − a, − a |− a , and as they cannot form a projective, they must also belong to P λ .(ii) If a = − ,pr λ (cid:16) P , |− ⊗ V (cid:17) = X M − , |− + M , |− + M , | + X M − , |− . By Lemma 2.3 and Proposition 2.7, one copy of each term must appear in P λ . Now, there remain only the second copies of the terms M , |− and M , | , and as they cannot form a projective, they must also belong to P λ . (cid:3) HARACTERS FOR PROJECTIVE MODULES IN THE BGG CATEGORY O Proof of Theorem 3.4.
Let λ − ρ = ( a, b | c ) − ρ be an atypical weight with a, b, c ∈ + Z and a, b < a < b < λ = ( a, b | − a ),pr λ (cid:0) P a +1 ,b |− a ⊗ V (cid:1) = X M a,b |− a + X M a +1 ,b |− a − . By Lemma 2.3 and Proposition 2.7, the projection is equal to P λ . Note thatwhen b = a + 1, P M a +1 ,b |− a − has four instead of eight terms.(1.2) When λ = ( a, b | a ),pr λ (cid:0) P a +1 ,b | a ⊗ V (cid:1) = X M a,b | a + X M a +1 ,b | a +1 . By Lemma 2.3 and Proposition 2.7, the projection is equal to P λ . Note thatwhen b = a + 1, P M a +1 ,b | a +1 has eight instead of sixteen terms.(1.3) When λ = ( a, b | − b ):(i) If b < − ,pr λ (cid:0) P a,b +1 |− b ⊗ V (cid:1) = X M a,b |− b + X M a,b +1 |− b − . By Lemma 2.3 and Proposition 2.7, the projection is equal to P λ .(ii) If b = − , we start with the case λ = (cid:0) − , − | (cid:1) . We project P − , | ⊗ V onto the λ linkage block. Notice that in the table below, to save space,we use the ± sign to combine two terms into one. For example, M , ± | represents the two terms M , | and M , − | . P − , | pr λ (cid:16) P − , | ⊗ V (cid:17) M − , | M − , − | M − , |− M − , | M − , | M − , |− M − , | M , | M , ± | M − , ± | M , ± |− M , ± | M , ± | M , ± | M , | M , − | M , ± |− M − , | M − , | M , − | M , − | M , | M , | We start by finding the terms that must appear in P λ . By Proposition 2.7,the terms colored red belong to P λ . Now, the lowest remaining term is M − , | , and since P − , | does not appear in the projection, it mustbelong to P λ . By the same reasoning, the second copy of M − , | must alsoappear in P λ . Now, the next lowest term is M , | . However, the threeterms in the standard filtration of P , | all remain in the projection,colored blue. The two remaining unsorted terms must belong to P λ asno more projective can form among them. Again, we face two possiblestandard filtrations of P λ . Denote the one containing all red and blackterms Q , and denote the three blue terms R . We shall show that Q is nota projective and thereby prove that P λ = Q + R . Let µ = (cid:0) − , − | (cid:1) . We project Q ⊗ V and R ⊗ V onto the µ linkageblock. In the table we use our notation of P M λ to combine terms.Projective Terms pr µ ( − ⊗ V ) Q P M − , − | P M − , − | P M − , − | P M − , − | M − , |− M − , |− M − , |− M − , |− M − , |− M − , |− M , ± |− M , ± |− M , ± |− M , ± |− M , ± |− M , ± |− M − , | M − , | M − , | M , − | M , − | M , − | M − , | M − , | M − , | M , − | M , − | M , − | R M , | M , | M , | M , | M , | M , | M , | M , | M , | We find indecomposable projectives in this projection starting with thelowest term. First, P µ appears, colored red. The next lowest is M − , |− ,so P − , |− must appear, colored blue. Since Q does not have enough termsfor this, it cannot be a projective and we must have that P λ = Q + R .(iii) Here we calculate the projective P ν where ν = (cid:0) − , − | (cid:1) . If we keepisolating projectives from the projection above, we find that the next pro-jective that must appear is P − , − | . By Proposition 2.7, the terms coloredviolet must appear. Of the five remaining terms, the two in the projection(colored brown) of Q must also belong to P − , − | as they cannot belongto another projective, while the three in the projection of R could formthe projective T = P , | . Thus, we have two possible standard filtrationsof P − , − | , which we denote by S and S + T . Now, we project S ⊗ V and T ⊗ V back onto the linkage block of λ = (cid:0) − , − | (cid:1) . HARACTERS FOR PROJECTIVE MODULES IN THE BGG CATEGORY O Projective Terms pr λ ( − ⊗ V ) S P M − , − | P M − , − | M − , |− M − , |− M − , |− M , ± |− M , ± |− M , ± |− M − , | M − , | M − , | M , − | M , − | M , − | T M , | M , | M , | M , | M , | M , | M , | M , | M , | By Lemma 2.3, P λ must appear in the projection, and we color its termsred. We see that S does not have enough terms and thus cannot be aprojective. Thus, P ν = S + T , andpr λ (( S + T ) ⊗ V ) = P − , − | + P , | . (iv) If b = − and a < − , we project P a, | ⊗ V onto the λ linkage block.Similar to the previous case, we obtain two possible standard filtrations,denoted Q ( a ) and ( Q + R )( a ). Now, we consider the specific case of a = − ,and we project the corresponding Q ( − ) ⊗ V and R ( − ) ⊗ V onto the (cid:0) − , − | (cid:1) block. It turns out that Q ( − ) is not a projective andpr − , − , (( Q + R ) ⊗ V ) = P − , | + P − , − | . Thus, P − , | = ( Q + R )( − ). Then, we proceed by induction, projecting Q ( a − ⊗ V and R ( a − ⊗ V onto the (cid:0) a, | (cid:1) block. Since we findthat the projection of ( Q + R )( a − ⊗ V is equal to P a, | = ( Q + R )( a ), Q ( a −
1) does not have enough terms and thus is not a projective. Thisway, we show that P a, | = ( Q + R )( a ) for all a < − .(1.4) When λ = ( a, b | b ):(i) If b < − ,pr λ (cid:0) P a,b +1 | b ⊗ V (cid:1) = X M a,b | b + X M a,b +1 | b +1 . By Lemma 2.3 and Proposition 2.7, the projection is equal to P λ .(ii) We defer the case of λ = (cid:0) a, − | − (cid:1) to the next part, since the methodrequires a projective we have not yet calculated.(2) Suppose that a = b <
0. (Note that we prove part 3 of Theorem 3.4 before part 2)(2.1) When λ = ( a, a | − a ),(i) If a < − ,pr λ (cid:16) P a +1 ,a +1 |− a ⊗ ^ V (cid:17) = X M a,a |− a + X M a,a +1 |− a − . By Lemma 2.3 and Proposition 2.7, the projection is equal to P λ .(ii) The case λ = (cid:0) − , − | (cid:1) was resolved in case 1. (2.2) When λ = ( a, a | a ),(i) If a < − ,pr λ (cid:16) P a +1 ,a +1 | a ⊗ ^ V (cid:17) = X M a,a | a + X M a,a +1 | a +1 . By Lemma 2.3 and Proposition 2.7, the projection is equal to P λ .(ii) If a = − ,pr λ (cid:16) P − , − | ⊗ V (cid:17) = 3 X M − , − |− + 2 (cid:16)X M − , | + M , |− + X M − , | (cid:17) . Since the lowest term is M − , − |− and it appears with multiplicity 3, byLemme 2.3, P λ must appear three times in the projection. By Proposi-tion 2.7, P M − , − |− belongs to P λ . Since these are also the only termswith multiplicity at least 3, P λ = X M − , − |− . It turns out thatpr λ (cid:16) P − , − | ⊗ V (cid:17) = 3 P − , − |− + 2 P − , | . (iii) Here we calculate the projective P ν where ν = (cid:0) a, − | − (cid:1) , which wedeferred from case 1. First, we consider the specific case where a = − .We have thatpr( − , − |− ) (cid:16) P − , − |− ⊗ V (cid:17) = X M − , − |− . By Lemma 2.3 and Proposition 2.7, the projection is equal to P − , − |− .Now, we proceed by induction,pr( a, − |− ) (cid:16) P a +1 , − |− ⊗ V (cid:17) = X M a, − |− , and get By Proposition 2.7 that P a, − |− = X M a, − |− for all a < − (in fact, for a = − as well, as shown in the previoussubcase).(3) Suppose that b < a < λ = ( a, b | − a ):(i) If a < − ,pr λ (cid:0) P a +1 ,b |− a ⊗ V (cid:1) = X M a,b |− a + X M a +1 ,b |− a − . By Lemma 2.3 and Proposition 2.7, the projection is equal to P λ . HARACTERS FOR PROJECTIVE MODULES IN THE BGG CATEGORY O (ii) If a = − , we start with the case λ = (cid:0) − , − | (cid:1) . We have thatpr λ (cid:16) P , − | ⊗ V (cid:17) = X M − , − | + M , − | + M , | + M , − |− + M , |− + M , − |− + M , |− + M , − | + M , | . By Proposition 2.7, all terms except for the four colored red must appearin P λ . Proceeding from the lowest remaining term, we can show that eachof the remaining term must appear in P λ since no other projective canform in the projection.(iii) If a = − and b < − ,pr λ (cid:16) P ,b | ⊗ V (cid:17) = X M − ,b | + M − b, − | + M − b, | + M ,b |− + M , − b |− + M − b, − |− + M − b, |− + M ,b | + M , − b | + M − b, − | + M − b, | . By a similar argument as above, we can show that all terms except forthe four terms colored red, which can form the projective P − b, − | , mustappear in P λ . We now have two possible standard filtrations for P λ . Asusual, call them Q ( b ) and ( Q + R )( b ). First we consider the case b = − .We project Q ( − ) ⊗ V and R ( − ) ⊗ V back to the (cid:0) − , − | (cid:1) block.Projective Terms pr − , − | ( − ⊗ V ) Q ( − ) P M − , − | P M − , − | M , ± |− M , ± |− M , ± |− M , ± |− M , ± | M , ± | M , ± | R ( − ) M , ± | M , ± | M , ± | M , ± | M , ± | Since the lowest term appearing in the projection is M − , − | , the projec-tive P − , − | must appear in the projection, with its terms colored red.We see that Q again does not have enough terms and is therefore not aprojective. Thus, P − , − | = ( Q + R )( − ). As we have the base case now,we may proceed by induction. By projecting Q ( b − ⊗ V and R ( b − ⊗ V onto the (cid:0) − , b | (cid:1) block, we see that Q ( b −
1) does not have enough termsand pr − ,b | (( Q + R )( b − ⊗ V ) = ( Q + R )( b ) . Thus, for all b < − , P − ,b | = ( Q + R )( b ).(3.2) When λ = ( a, b | a ): (i) If a < − ,pr λ (cid:0) P a +1 ,b | a ⊗ V (cid:1) = X M a,b | a + X M a +1 ,b | a +1 . By Lemma 2.3 and Proposition 2.7, the projection is equal to P λ .(ii) If a = − , we start with the case λ = (cid:0) − , − | − (cid:1) . We project P − , − | ⊗ V onto the λ block. We get thatpr λ (cid:16) P − , − | ⊗ V (cid:17) = 2 X M − , − |− + 2 X M , − |− + X M , − | + M , − | + M , | . First, by Lemma 2.3 and Proposition 2.7, two copies of P λ must appearin the projection and the terms in P M − , − |− belong to P λ . Now, thelowest remaining term is M , − | , and since only one copy of it remains, itcannot belong to P λ , which means P , − | must appear in the projectionas a separate projective. Now, the only remaining terms are the two copiesof R ( − ) = P M , − |− , which could form P , − |− . Thus, we again facetwo possibilities for P λ , namely, Q ( − ) = P M − , − |− and ( Q + R )( − ).Now, by projecting Q ( − ) ⊗ V and R ( − ) ⊗ V onto the (cid:0) − , − | − (cid:1) block, we see that P − , − | also has two possible standard filtrations Q ( − ) and ( Q + R )( − ), defined similarly. In addition, P − , − | = Q ( − )if and only if P − , − | = Q ( − ), as otherwise when we project the shorterprojective onto the block of the longer, there would not be enough terms.The same argument carries as we induct on b . Thus, it remains to find thecorrect filtration for any specific value of b .We examine µ = (cid:0) − , − | − (cid:1) and show that P µ = Q ( − ). Considerthe projection pr µ (cid:16) P − , − |− ⊗ V V (cid:17) , which has 180 terms. By applyingLemma 2.3, we find that P − , − |− and four copies of P − , − |− mustappear in the projection. Now, 60 terms remain, and the lowest term is M − , − |− , which appear 4 times, which means that P µ must appear fourtimes. However, ( Q + R )( − ) has 16 terms and thus does not fit. Thus, P − ,b |− = Q ( b ) = X M − ,b |− for all b < − . It turns out thatpr µ (cid:16) P − , − |− ⊗ ^ V (cid:17) = P − , − |− + 4 P − , − |− + 4 P − , − |− + P , − |− . (3.3) When λ = ( a, b | − b ):(i) If b < a − λ (cid:0) P a,b +1 |− b ⊗ V (cid:1) = X M a,b |− b + X M a,b +1 |− b − . By Lemma 2.3 and Proposition 2.7, the projection is equal to P λ . HARACTERS FOR PROJECTIVE MODULES IN THE BGG CATEGORY O (ii) If b = a − a < − ,pr λ (cid:0) P a +1 ,a − |− a +1 ⊗ V (cid:1) = X M a,a − |− a +1 + X M a +1 ,a |− a − + X M a,a |− a + X M − a,a |− a . By Proposition 2.7, all terms except for P M − a,a |− a must belong to P λ .As no projective can form among the remaining two terms, the projectionis equal to P λ .(iii) If b = a − a = − ,pr λ (cid:16) P , − | ⊗ V (cid:17) = X M − , − | + M , − | + M , | + X M − , − | + X M , − |− . By Proposition 2.7, P M − , − | and P M − , − | must belong to P λ . Asno projective can form among the remaining six terms, the projection isequal to P λ .(3.4) When λ = ( a, b | b ):(i) If b < a − λ (cid:0) P a,b +1 | b ⊗ V (cid:1) = X M a,b | b + X M a,b +1 | b +1 . By Lemma 2.3 and Proposition 2.7, the projection is equal to P λ .(ii) If b = a − a < − ,pr λ (cid:0) P a +1 ,a − | a − ⊗ V (cid:1) = X M a,a − | a − + X M a +1 ,a | a +1 + X M a,a | a + X M − a,a | a . By Proposition 2.7, all terms except for P M − a,a |− a must belong to P λ .As no projective can form among the remaining four terms, the projectionis equal to P λ .(iii) If b = a − a = − ,pr λ (cid:16) P − , − |− ⊗ V (cid:17) = X M − , − |− + X M − , − |− + X M , − |− . By Proposition 2.7, P M − , − |− and P M − , − |− must belong to P λ .As no projective can form among the remaining four terms, the projectionis equal to P λ . (cid:3) Jordan-H¨older Formulae for osp (3 | Let λ ∈ X + ρ such that λ − ρ is atypical, integral, and antidominant. Let W λ be aminimal set of left-coset representatives of W / W λ . Then, by applying the BGG reciprocityto Proposition 2.7, we immediately get that the composition series of M σλ ( σ ∈ W λ ) mustinclude X τ ≤ σ,τ ∈W λ ( L τλ + L τλ − α + L τλ − α − β ) , where each term in the sum appears with multiplicity one only if it is linked to λ , and α, β ∈ Φ and ht( α ) > ht( β ). For convenience, we denote this summation by X L σλ . Theorem 4.1.
Let λ − ρ = ( a, b | c ) − ρ be an atypical weight with a, b, c ∈ + Z and c ∈ {± a, ± b } . The Verma modules M λ of highest weight λ − ρ have Jordan-H¨older formulae M λ = X L λ except in the following cases.(1) Suppose that λ − ρ = (cid:0) a ′ , b ′ | (cid:1) − ρ is atypical, and at least one of a ′ , b ′ is positive.Since at least one of | a ′ | , | b ′ | is equal to , suppose that {| a ′ | , | b ′ |} = { a, } . Then,unless specified otherwise in cases below, M λ has the following composition series: M λ = X L λ + X ∗ L a, − | , where P ∗ L a, − | denotes the the sum of the terms in the set { L − a, − | , L a, − | , L − , − a | , L − ,a | } that are lower than L λ . The following subcases are exceptions to this case, whichcontain some additional terms than those given above.(i) When λ = (cid:0) , − | (cid:1) . We have M λ = X L λ + L − , − | + L − , − | , where we use red to emphasize terms with multiplicity two (its first copy appearsin P L λ ).(ii) When λ = (cid:0) , | (cid:1) . We have M λ = X L λ + L − , − | + L − , − | + L , − | . (iii) When λ = (cid:0) , − | (cid:1) . We have M λ = X L λ + L − , − | + L − , − | + L − , − | + L − , − |− . (iv) When λ = (cid:0) , | (cid:1) . We have M λ = X L λ + X ∗ L , − | + L − , | + L − , |− + L − , − | + L − , − |− . EFERENCES 33 (2) Suppose that λ − ρ = (cid:0) a ′ , b ′ | (cid:1) − ρ is atypical, and at least one of a ′ , b ′ is greater than .(i) When λ = (cid:0) − , b | (cid:1) or λ = (cid:0) , b | (cid:1) with b > . We have M λ = X L λ + L − b, − | . (ii) Suppose that λ = (cid:0) a, − | (cid:1) or λ = (cid:0) a, | (cid:1) with a > . We have M λ = X L λ + L − , − a | + L − a, − | . (3) Suppose that λ − ρ = ( a, b | c ) − ρ is atypical, and a = | b | = | c | .(i) When λ = ( a, − a | − a ) , we have M λ = X L λ + L − a, − a − |− a − . (ii) When λ = ( a, − a | a ) and a = , we have M λ = X L λ + L − a, − a − |− a − + L − a, − a − | a +1 . (iii) The case λ = (cid:0) , − | (cid:1) is given above.(iv) When λ = ( a, a | − a ) , we have M λ = X L λ + L − a,a |− a + L − a, − a − |− a − . (v) When λ = ( a, − a | a ) and a > , we have M λ = X L λ + L − a,a |− a + L − a,a | a + L − a, − a − |− a − + L − a, − a − | a +1 . (vi) The case λ = (cid:0) , | (cid:1) is given above.(vii) When λ = (cid:0) , | (cid:1) , we have M λ = X L λ + L − , |− + L − , − | + L − , − |− + L − , − | . (4) When λ = (cid:0) , | (cid:1) , we have M λ = X L λ + L , − | . (5) When λ = (cid:0) , | (cid:1) , we have M λ = X L λ + L , − | . References [BW18] H. Bao and W. Wang.
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EFERENCES 35
Department of Mathematics, Massachusetts Institute of Technology, Cambridge, MA02139
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