Modular representations of exceptional supergroups
aa r X i v : . [ m a t h . R T ] M a y MODULAR REPRESENTATIONS OF EXCEPTIONAL SUPERGROUPS
SHUN-JEN CHENG, BIN SHU, AND WEIQIANG WANG
Abstract.
We classify the simple modules of the exceptional algebraic supergroups overan algebraically closed field of prime characteristic.
Introduction
Among the simple Lie superalgebras over the complex field C , the basic Lie superalgebrasdistinguish themselves by admitting a non-degenerate super-symmetric even bilinear form(see, e.g., [CW12]), and they include 3 exceptional Lie superalgebras: D (2 | ζ ) , G (3) and F (3 | C .There are algebraic supergroups associated to the basic Lie superalgebras, valid over analgebraically closed field k of prime characteristic p = 2. A general theory of Chevalleysupergroups was systematically developed by Fioresi and Gavarini [FG12] (also see [G14]).In representation theory of algebraic supergroups G over k , one of the basic questions is toclassify the simple G -modules. For type A , the answer is immediate as it is the same as forthe even subgroup G . For type Q such a classification was obtained in [BK03], and it hasapplications to classification of simple modules of spin symmetric groups over k . For type osp , the classification was obtained in [SW08] in terms of the Mullineux involution by usingodd reflections; also see Remark 1.3.The goal of this paper is to classify the simple G -modules, when G is a simply connectedsupergroup of exceptional type. We shall assume throughout the paper that p > D (2 | ζ ) and p > G (3) or F (3 |
1) (except in § G (3) for p = 3 in § G -modules to the classification of the highestweights of finite-dimensional simple modules L ( λ ) = L b ( λ ) over the distribution superalgebraDist( G ), where b is the standard Borel subalgebra. We then reduce the verification offinite-dimensionality of L ( λ ) to verifying that L ( λ ) is locally finite over its even distribution Mathematics Subject Classification.
Primary 20G05, 17B25.
Key words and phrases.
Exceptional supergroups, simple modules, odd reflections. subalgebra. The local finiteness criterion for L ( λ ) is finally established by means of oddreflections (see [LSS86]), and is based on the following observation which seems to be wellknown to experts (see [Se11]): For every positive even root α in the standard positive system, either α/ (if it is a root)or α appears as a simple root in some simple system Π ′ associated to some b ′ , where b ′ is aBorel subalgebra obtained via a sequence of odd reflections from b . For the exceptional Lie superalgebras, we make this observation explicit in this paper. Wecompute the highest weight L b ′ ( λ ′ ) for all possible Borel subalgebras b ′ as mentioned above.Requiring λ ′ to be dominant integral for all possible b ′ gives the local finiteness criterion for L ( λ ).Recently, an approach to obtain characters of projective and simple modules in the BGGcategory O for the exceptional Lie superalgebras over C has been systematically developed;see [CW17] for D (2 | ζ ). Building on this and the current work, one may hope to betterunderstand the characters of projective and simple modules of the exceptional supergroupsover a field of prime characteristic in the future.The organization of this paper is as follows. In Section 1, we review the equivalencebetween the category of finite-dimensional modules over a supergroup G and the categoryof finite-dimensional (Dist( G ) , T )-modules, where T is a maximal torus of G . We developa criterion for the finite-dimensionality of simple Dist( G )-modules L ( λ ) via odd reflections.We also review the formula for the Euler characteristic, which implies that a Dist( G )-module L ( λ ), with λ dominant integral and λ + ρ is regular, is always finite dimensional.In Section 2, we analyze the highest weight constraints given by odd reflections of asimple finite-dimensional Dist( G )-module when G is of type D (2 | ζ ). Here D (2 | ζ ) is afamily depending on a parameter ζ ∈ k \{ , − } . We then classify the simple G -modules inTheorem 2.1.In Section 3, we analyze the highest weight constraints given by odd reflections of a simplefinite-dimensional Dist( G )-module when G is of type G (3). We then classify the simple G -modules in Theorem 3.5.In Section 4, we study the supergroup G of type F (3 | λ = aω + bω + cω + dω with a, b, c ∈ N , d ≥ λ + ρ isregular and hence the Euler character formula implies that the Dist( G )-module L ( λ ) is finitedimensional. For d ≤
3, it is rather involved to analyze the highest weight changes undersequences of odd reflections and formulate sufficient and necessary conditions for L ( λ ) to befinite dimensional. We finally classify the simple G -modules in Theorem 4.8.Finally we remark that, although in this article we deal with an algebraically closed field ofpositive characteristic, the results also make sense in characteristic zero and give the knownclassification in this case; cf. [Kac77, Ma14]. Acknowledgment.
S.-J.C. is partially supported by a MoST and an Academia SinicaInvestigator grant; B.S. is partially supported by the National Natural Science Foundationof China (Grant Nos. 11671138, 11771279) and Shanghai Key Laboratory of PMMP (No.13dz2260400); W.W. is partially supported by an NSF grant DMS-1702254. We thank EastChina Normal University and Institute of Mathematics at Academia Sinica for hospitalityand support.
ODULAR REPRESENTATIONS OF EXCEPTIONAL SUPERGROUPS 3 Modular representations of algebraic supergroups
Algebraic supergroups and ( Dist ( G ) , T ) - mod . Throughout the paper, the groundfield k is assumed to be algebraically closed and of characteristic p > p > G is an affine superscheme whose coordinatering k [ G ] is a Hopf superalgebra that is finitely generated as a k -algebra, and gives rise toa functor from the category of commutative k -superalgebras to the the category of groups.The underlying purely even group G is a closed subgroup of G corresponding to the Hopfideal generated by k [ G ] , and it is an algebraic group in the usual sense. For an algebraicsupergroup G , the distribution superalgebra Dist( G ), which is by definition the restricteddual of the Hopf superalgebra k [ G ], is a cocommutative Hopf superalgebra.We denote by G - mod the category of rational G -modules with (not necessarily homoge-neous) G -homomorphisms. Note that a G -module is always locally finite , i.e., it is a sumof finite-dimensional G -modules. Given a closed subgroup T of G , a Dist( G )-module M is called a (Dist( G ) , T )-module if M has a structure of a T -module such that the Dist( T )-module structure on M induced from the actions of Dist( G ) and of T coincide. We denote by(Dist( G ) , T )- mod the category of locally finite (Dist( G ) , T )-modules, and denote by Dist( G )- mod the category of locally finite Dist( G )-modules. (We shall always take T to be a maximaltorus of G when G is of basic type.)1.2. Modules of basic algebraic supergroups.
Let g be a basic Lie superalgebra over k [CW12, FG12, G14], including the three exceptional types: D (2 | ζ ), G (3), and F (3 | · , · ) of g over k exists when the characteristic p of k satisfies p > gl , osp and D (2 | ζ ), and p > G (3) and F (3 | k associated with basic (including exceptional) Lie superalge-bras are constructed in analogy to Chevalley’s construction of semisimple algebraic groups(see [FG12] and [G14]); we shall use the same terminologies (such as basic type, excep-tional type) to refer to Lie superalgebras and corresponding supergroups. We shall call G simply connected if G is a simply connected algebraic group, taking advantage of [Mas12,Proposition 35]. Simple-connected supergroups of basic type exist, and we shall assume theexceptional supergroups in this paper to be simply connected.The assumption on Chevalley bases in [SW08, Theorem 2.8] is satisfied for all algebraicsupergroups of basic type, by the constructions in [FG12, G14]. Hence we have the following. Proposition 1.1. [SW08, Theorem 2.8] [MS17]
Let G be an algebraic supergroup of basictype. Then there is a natural equivalence of categories between G - mod and ( Dist ( G ) , T ) - mod . If we further assume G is simply connected, then (Dist( G ) , T )- mod in Proposition 1.1 abovecan be replaced by Dist( G )- mod ; cf. [Jan03, II.1.20].A supergroup G of basic type can be constructed as a Chevalley supergroup through aChevalley basis associated with a standard positive root system Φ + as described in [SW08, § § § B corresponding to Φ + , which contains a maximal torus T . The distribution superalgebraDist( G ) contains Dist( B ) as a subalgebra. Set Lie ( B ) = b . Let X ( T ) be the character group SHUN-JEN CHENG, BIN SHU, AND WEIQIANG WANG of T . For λ ∈ X ( T ), we denote the Verma module of Dist( G ) by M ( λ ) = Dist( G ) ⊗ Dist( B ) k λ , where k λ is the one-dimensional Dist( B )-module of weight λ . The Dist( G )-module M ( λ )has a unique simple quotient L ( λ ), and furthermore the Dist( G )-modules L ( λ ) are non-isomorphic for distinct λ ∈ X ( T ). By definition, L ( λ ) is X ( T )-graded and thus a T -module.Denote by X + ( T ) the set of G -dominant integral weights (with respect to Φ + ). Lemma 1.2. [SW08, Lemma 4.1]
Every simple module in the category ( Dist ( G ) , T ) - mod isisomorphic to a finite-dimensional highest weight module L ( λ ) for some λ ∈ X + ( T ) , andvice versa. By Proposition 1.1 and Lemma 1.2, the classification of simple G -modules can be refor-mulated as the determination of the following set:(1.1) X † ( T ) = (cid:8) λ ∈ X + ( T ) (cid:12)(cid:12) L ( λ ) is finite dimensional (cid:9) . For general supergroups of basic type, X † ( T ) turns out to be a nontrivial proper subset of X + ( T ). Remark 1.3.
For a supergroup G of type spo (2 n | ℓ ), the subset X † ( T ) ⊂ X + ( T ) wasdetermined explicitly in [SW08]. Note the supergroup G therein has even subgroup G =Sp n × SO ℓ and hence is not simply connected. For a simply connected group of type spo (2 n | ℓ ), one would have additional simple modules L ( λ ), where λ ∈ X + ( T ) is of theform λ ∈ P i< Z δ i + P j> ( + Z ) δ j in the notation of [SW08, § L ′ ( λ ) and L ′′ ( λ ) the highest weight Dist( G )-modules with respect to positivesystems Φ ′ + and Φ ′′ + , respectively. Lemma 1.4. [BKu03, Lemma 4.2] [SW08, Lemma 5.7]
Let λ ∈ X ( T ) , and let β be anodd isotropic root for g . Suppose that Φ ′ + and Φ ′′ + are two positive systems of g such that Φ ′′ + = Φ ′ + ∪ {− β }\{ β } . Then, L ′′ ( λ ) ∼ = (cid:26) L ′ ( λ ) if ( λ, β ) ≡ p ) ,L ′ ( λ − β ) if ( λ, β ) p ) . We shall say Φ ′′ + is obtained from Φ ′ + by an odd reflection in the setup of Lemma 1.4.Often we shall abbreviate a ≡ b (mod p ) as a ≡ b later on. In the coming sections dealingwith exceptional supergroups, we shall be very explicit about the (positive) root systemsand odd reflections. Lemma 1.5.
Let L = L ( λ ) , for λ ∈ X + ( T ) . Suppose that L is isomorphic to L b ′ ( λ ′ ) with λ ′ ∈ X + ( T ) , for every Borel subalgebra b ′ that is obtained from b by a sequence of oddreflections. Then L is locally finite as a Dist ( G ¯0 ) -module, i.e., it is a rational G -module.Proof. We recall the following observation (cf., e.g., [Se11, Ma14]):
For every positive even root α in Φ +0 , either α/ (if it is a root) or α appears as a simpleroot in some simple system Π ′ associated to b ′ . Denote by SL ,α the root subgroup of G associated to α . Then by the assumption of thelemma, Dist( SL ,α ) acts on L locally finitely (i.e., L is a rational SL ,α -module). It follows ODULAR REPRESENTATIONS OF EXCEPTIONAL SUPERGROUPS 5 that L is a rational G -module, or equivalently, L is locally finite as a Dist( G ¯0 )-module byProposition 1.1. (cid:3) Lemma 1.6.
If a finitely generated Dist ( G ) -module M is locally finite as a Dist ( G ¯0 ) -module,then M is finite dimensional.Proof. Since Dist( G ) is finitely generated over the algebra Dist( G ¯0 ), as a Dist( G ¯0 )-module M is also finitely generated. Together with the locally finiteness assumption, this impliesthat M is finite dimensional. (cid:3) The combination of Proposition 1.1, Lemmas 1.4, 1.5 and 1.6 provides us with an effectiveapproach of classifying simple G -modules. Indeed, the problem of determining the finite-dimensional irreducible modules is thus reduced to determining the weights that remain tobe G ¯0 -dominant integral when transformed to highest weights with respect to any Borel(with fixed even part).1.3. Euler characteristic.
Let H be a closed subgroup of an algebraic supergroup G suchthat the quotient superscheme G/H is locally decomposable (cf. [B06, the paragraph aboveLemma 2.1]) and G /H is projective; that is, the superscheme X = G/H satisfies theassumptions (Q5)-(Q6) in [B06, § §
6] for the precise definitions for induction andrestriction functors below. Below, for a superspace M , we shall use S ( M ) to denote thecorresponding supersymmetric algebra. Lemma 1.7. ( [B06, Corollary 2.8] ) For any finite-dimensional H -module M , we have X i ≥ ( − i [ res GG R i ind GH M ] = X i ≥ ( − i [ R i ind G H S (cid:0) ( Lie G/ Lie H ) ∗ ¯1 (cid:1) ⊗ M ] , where the equality is understood in the Grothendieck group of G -modules. Now we take G to be an algebraic supergroup of basic type, H = B − to be the oppositeBorel subgroup. Since G /B − is projective and G/B − is locally decomposable (cf. [MZ17]and [Z18]), Lemma 1.7 is applicable. For M = k λ , we define H i ( λ ) := R i ind GB − ( k λ ) and thenthe Euler characteristic χ ( λ ) := X i ≥ ( − i ch H i ( λ ) . By Lemma 1.7 we have the following formula for the Euler characteristic χ ( λ ) = X i ≥ ( − i ch R i ind G B S (cid:0) (cid:0) g / b − (cid:1) ∗ ¯1 (cid:1) ⊗ k λ , (1.2)where b − is the opposite Borel subalgebra. Since the Euler characteristic is additive on shortexact sequences, it suffices to determine the Euler characteristic on the composition factorsof the B -module S (cid:0) ( g / b − ) ∗ ¯1 (cid:1) ⊗ k λ . Recall that the supersymmetric algebra of a purely oddspace is the exterior algebra in the usual sense. Let W be the Weyl group of g . Since SHUN-JEN CHENG, BIN SHU, AND WEIQIANG WANG Π β ∈ Φ +¯1 ( e β + e − β ) is W -invariant, it follows by (1.2) and Lemma 1.7 that χ ( λ ) = P w ∈ W ( − ℓ ( w ) w ( e ( λ + ρ ¯0 ) (Π β ∈ Φ +¯1 (1 + e − β )))Π α ∈ Φ +¯0 ( e α − e − α )= P w ∈ W ( − ℓ ( w ) w ( e ( λ + ρ ) (Π β ∈ Φ +¯1 ( e β + e − β )))Π α ∈ Φ +¯0 ( e α − e − α )= Π β ∈ Φ +¯1 ( e β + e − β )Π α ∈ Φ +¯0 ( e α − e − α ) X w ∈ W ( − ℓ ( w ) e w ( λ + ρ ) . Here as usual ℓ ( w ) denotes the length of w ∈ W , and ρ is the Weyl vector given by ρ = ρ ¯0 − ρ ¯1 , where ρ ¯0 = 12 X α ∈ Φ +¯0 α, ρ ¯1 = 12 X β ∈ Φ +¯1 β. Proposition 1.8.
Let λ ∈ X + ( T ) . The Euler characteristic is given by χ ( λ ) = Π β ∈ Φ +¯1 ( e β + e − β )Π α ∈ Φ +¯0 ( e α − e − α ) X w ∈ W ( − ℓ ( w ) e w ( λ + ρ ) . Proposition 1.9.
Let λ ∈ X + ( T ) be such that λ + ρ is G -dominant and regular. Then L ( λ ) is finite dimensional.Proof. By the same arguments as in [B06, Corollary 2.8, Lemma 4.2], all H i ( λ ) are finite-dimensional G -modules. By assumption λ + ρ is G -dominant and regular, and hence, thehighest weight of the Euler characteristic in Proposition 1.8 equals λ + ρ + ( ρ − ρ ) = λ .The proposition now follows from Proposition 1.1 and Lemma 1.2. (cid:3) Modular representations of the supergroup of type D (2 | ζ )2.1. Weights and roots for D (2 | ζ ) . The Lie superalgebra g = D (2 | ζ ) is a family ofsimple Lie superalgebras of basic type, which depends on a parameter ζ ∈ k \ { , − } . Thereare isomorphisms of Lie superalgebras with different parameters(2.1) D (2 | ζ ) ∼ = D (2 | − − ζ − ) ∼ = D (2 | ζ − ) . Then g = g ⊕ g , where g ∼ = sl ⊕ sl ⊕ sl and, as a g ¯0 -module, g ∼ = k ⊠ k ⊠ k . Here k is the natural representation of sl .Let h ∗ be the dual of the Cartan subalgebra with basis { δ, ǫ , ǫ } . We equip h ∗ with a k -valued bilinear form ( · , · ) such that { δ, ǫ , ǫ } are orthogonal and( δ, δ ) = − (1 + ζ ) , ( ǫ , ǫ ) = 1 , ( ǫ , ǫ ) = ζ . (2.2)The root system for g = g ⊕ g is denoted by Φ = Φ ¯0 ∪ Φ ¯1 . The set of simple roots of thestandard simple system in h ∗ of D (2 | ζ ) is chosen to beΠ = { α = δ − ǫ − ǫ , α = 2 ǫ , α = 2 ǫ } . The Dynkin diagram associated to Π is depicted as follows:
ODULAR REPRESENTATIONS OF EXCEPTIONAL SUPERGROUPS 7 N (cid:13)(cid:13) δ − ǫ − ǫ ǫ ǫ Π:The set of positive roots is Φ + = Φ +0 ∪ Φ +1 , whereΦ +¯0 = { δ, ǫ , ǫ } , Φ +¯1 = { δ − ǫ − ǫ , δ + ǫ − ǫ , δ − ǫ + ǫ , δ + ǫ + ǫ } . One computes the Weyl vector ρ = − δ + ǫ + ǫ (= − α ) . Let X = Z δ + Z ǫ + Z ǫ denote the weight lattice of g .We denote the positive odd roots by(2.3) β = δ − ǫ − ǫ , β = δ + ǫ − ǫ , β = δ − ǫ + ǫ , β = δ + ǫ + ǫ . There are 4 conjugate classes of positive systems under the Weyl group action. The 4positive systems containing Φ +0 admit the following simple systems Π i (0 ≤ i ≤ § := Π = { δ − ǫ − ǫ , ǫ , ǫ } , Π := r β (Π) = {− δ + ǫ + ǫ , δ + ǫ − ǫ , δ − ǫ + ǫ } , Π := r β (Π ) = { ǫ , − δ − ǫ + ǫ , δ } , Π := r β (Π ) = { ǫ , δ, − δ + ǫ − ǫ } . The Dynkin diagrams of Π , Π , and Π are respectively as follows: N NN − δ + ǫ + ǫ δ + ǫ − ǫ δ − ǫ + ǫ Π N (cid:13)(cid:13) − δ − ǫ + ǫ δ ǫ Π N (cid:13)(cid:13) − δ + ǫ − ǫ δ ǫ Π The corresponding positive systems are denoted by Φ i + , for 0 ≤ i ≤
3, with Φ = Φ + , andthe corresponding Borel subalgebras of g are denoted by b i .2.2. Highest weight computations.
The simply connected algebraic supergroup G oftype D (2 | ζ ) was constructed in [G14]. With respect to the standard Borel subalgebra b (associated to Φ + ), we have X + ( T ) = { λ = dδ + aǫ + bǫ ∈ X | a, b, d ∈ N } . SHUN-JEN CHENG, BIN SHU, AND WEIQIANG WANG
Denote the simple Dist( G )-module of highest weight λ by L ( λ ), where λ = dδ + aǫ + bǫ ∈ X + ( T ). We denote by λ i the highest weight of L ( λ ) with respect to Π i , for 0 ≤ i ≤
3. So λ = λ . We shall apply (2.2) and Lemma 1.4 repeatedly to compute λ i , for 1 ≤ i ≤ λ, β ) = − d (1 + ζ ) − a − bζ = − ( a + d ) − ( b + d ) ζ . We now divide into 2 cases (1)-(2).(1) Assume x := ( a + d ) + ( b + d ) ζ p ). Then λ = λ − β = ( d − δ + ( a + 1) ǫ + ( b + 1) ǫ . First we compute ( λ , β ) = − ( d − ζ )+( a +1) − ( b +1) ζ = ( a − d +2) − ( b + d ) ζ ,and then further divide into 2 subcases (a)-(b).(a) If y := ( a − d + 2) − ( b + d ) ζ p ), then λ = λ − β = ( d − δ + aǫ + ( b + 2) ǫ . (b) If y ≡ p ), then λ = λ = ( d − δ + ( a + 1) ǫ + ( b + 1) ǫ . We also have ( λ , β ) = − ( d − ζ ) − ( a + 1) + ( b + 1) ζ = − ( a + d ) + ( b − d + 2) ζ ,and then divide into 2 subcases (a ′ )-(b ′ ).(a ′ ) If z := − ( a + d ) + ( b − d + 2) ζ p ), then λ = λ − β = ( d − δ + ( a + 2) ǫ + bǫ . (b ′ ) If z ≡ p ), then λ = λ = ( d − δ + ( a + 1) ǫ + ( b + 1) ǫ . (2) Assume x ≡ p ). Then λ = λ = dδ + aǫ + bǫ . First we compute ( λ , β ) = − d (1 + ζ ) + a − bζ = ( a − d ) − ( b + d ) ζ , and thendivide into 2 subcases (a)-(b).(a) If y := ( a − d ) − ( b + d ) ζ p ), then λ = λ − β = ( d − δ + ( a − ǫ + ( b + 1) ǫ . (b) If y ≡ p ), then λ = λ = dδ + aǫ + bǫ . We also have ( λ , β ) = − d (1 + ζ ) − a + bζ = − ( a + d ) + ( b − d ) ζ , and then divideinto 2 subcases (a ′ )-(b ′ ).(a ′ ) If z := − ( a + d ) + ( b − d ) ζ p ), then λ = λ − β = ( d − δ + ( a + 1) ǫ + ( b − ǫ . (b ′ ) If z ≡ p ), then λ = λ = dδ + aǫ + bǫ . Simple modules for the supergroup D (2 | ζ ) .Theorem 2.1. Let p > . Let G be the supergroup of type D (2 | ζ ) . A complete list ofinequivalent simple G -modules consists of L ( λ ) , where λ = dδ + aǫ + bǫ , with d, a, b ∈ N ,such that one of the following conditions is satisfied:(1) d = 0 , and a ≡ b ≡ p ) ;(2) d = 1 , and ( a + 1) − ( b + 1) ζ ≡ p ) ;(3) d = 1 , and ( a + 1) + ( b + 1) ζ ≡ p ) ;(4) d ≥ , (and a, b ∈ N are arbitrary). ODULAR REPRESENTATIONS OF EXCEPTIONAL SUPERGROUPS 9
Proof.
From the computations in § λ i (1 ≤ i ≤
3) and theirassociated conditions, we obtain the following (mutually exclusive) sufficient and necessaryconditions for L ( λ ) to be finite dimensional:(i) d = 0, ( a + d ) + ( b + d ) ζ ≡
0, ( a − d ) − ( b + d ) ζ ≡ − ( a + d ) + ( b − d ) ζ ≡ d = 1, ( a + d ) + ( b + d ) ζ
0, ( a − d + 2) − ( b + d ) ζ ≡ − ( a + d ) + ( b − d + 2) ζ ≡ d = 1, ( a + d ) + ( b + d ) ζ ≡
0, ( a − d ) − ( b + d ) ζ a ≥ − ( a + d ) + ( b − d ) ζ b ≥ d = 1, ( a + d ) + ( b + d ) ζ ≡
0, ( a − d ) − ( b + d ) ζ ≡ − ( a + d ) + ( b − d ) ζ b ≥ d = 1, ( a + d ) + ( b + d ) ζ ≡
0, ( a − d ) − ( b + d ) ζ a ≥ − ( a + d ) + ( b − d ) ζ ≡ d = 1, ( a + d ) + ( b + d ) ζ ≡
0, ( a − d ) − ( b + d ) ζ ≡ − ( a + d ) + ( b − d ) ζ ≡ d ≥
2, (and a, b ∈ N are arbitrary).In Case (i), we obtain d = 0 and a ≡ b ≡
0, that is, Condition (1) in the theorem. Case (v)is the same as Condition (4).Condition (ii) with the help of Condition (iii-a) simplifies to Condition (2).We note that the seemingly additional constraints a ≥ b ≥ (cid:3) Remark 2.2.
Theorem 2.1 makes sense over C , providing an odd reflection approach tothe classification of finite-dimensional simple modules over C (due to [Kac77]). Indeed thisclassification can be read off from Theorem 2.1 by regarding p = ∞ .3. Modular representations of the supergroup of type G (3)3.1. Weights and roots for the supergroup G (3) . Let g = g ⊕ g be the exceptionalsimple Lie superalgebra G (3). We assume ǫ , ǫ , ǫ satisfy the linear relation ǫ + ǫ + ǫ = 0 . The root system is Φ = Φ ∪ Φ . We choose the standard simple system Π = { α , α , α } ,where α = ǫ − ǫ , α = ǫ , α = δ + ǫ . The Dynkin diagram associated to Π is depicted as follows: (cid:13) (cid:13) > N α = ǫ − ǫ α = ǫ α = δ + ǫ Π:Then the standard positive roots are Φ + = Φ +0 ∪ Φ +1 , whereΦ +0 = { δ, ǫ , ǫ , − ǫ , ǫ − ǫ , ǫ − ǫ , ǫ − ǫ } , Φ +1 = { δ, δ ± ǫ i | ≤ i ≤ } . The Weyl vector for g is(3.1) ρ = − δ + 2 ǫ + 3 ǫ , ρ ¯1 = 72 δ. We have g ∼ = G ⊕ sl and g ∼ = k ⊠ k as an adjoint g -module, where k denotes denotesthe 7-dimensional simple G -module and, as before, k the natural sl -module. Note that { α , α } forms a simple system of G , and we denote by ω , ω the corresponding fundamentalweights of G . We have ω = ǫ + 2 ǫ , ω = ǫ + ǫ ; ǫ = 2 ω − ω , ǫ = ω − ω . We can rewrite the formulae for ρ in (3.2) as(3.2) ρ = − δ + ω + ω . Denote the weight lattice of g by X = Z δ ⊕ X , where X = Z ω ⊕ Z ω = Z ǫ ⊕ Z ǫ is the weight lattice of G .The bilinear form ( · , · ) on X is given by( δ, δ ) = − , ( δ, ǫ i ) = 0 , ( ǫ i , ǫ i ) = 2 , ( ǫ i , ǫ j ) = − , for 1 ≤ i = j ≤ . It follows that ( ω , ǫ ) = 0 , ( ω , ǫ ) = 3 , ( ω , ǫ ) = − , ( ω , ǫ ) = 1 , ( ω , ǫ ) = 1 , ( ω , ǫ ) = − . (3.3)Denote the following positive odd roots of G (3) by(3.4) β = δ + ǫ , β = δ − ǫ , β = δ − ǫ . There are 4 conjugate classes of positive systems under the Weyl group action. The 4positive systems containing Φ +0 admit the following simple systems Π i (0 ≤ i ≤ § := Π = { ε − ε , ε , δ + ε } , Π := r β (Π) = { ǫ − ǫ , δ − ǫ , − δ − ǫ } , Π := r β (Π ) = { δ − ǫ , − δ + ǫ , ǫ } , Π := r β (Π ) = {− δ + ǫ , ǫ − ǫ , δ } . The Dynkin diagrams of Π , Π , and Π are respectively as follows: (cid:13) N > N ǫ − ǫ δ − ǫ − δ − ǫ Π N N (cid:13) δ − ǫ − δ + ǫ ǫ Π (cid:13) N > ǫ − ǫ − δ + ǫ δ Π The corresponding positive systems are denoted by Φ i + , for 0 ≤ i ≤
3, with Φ = Φ + ,and the corresponding Borel subalgebras of g are denoted by b i . ODULAR REPRESENTATIONS OF EXCEPTIONAL SUPERGROUPS 11
Highest weight computations.
The (simply connected) algebraic supergroup G oftype G (3) was constructed in [FG12]. With respect to the standard Bore subalgebra b (associated to Φ + ), we have X + ( T ) = { λ = nδ + rω + sω ∈ X | n, r, s ∈ N } . Denote by L ( λ ) = L b ( λ ) the irreducible Dist( G )-module of highest weight λ with respect tothe standard Borel subalgebra b , where λ = dδ + rω + sω ∈ X + ( T ) . Assume the simple module L ( λ ) = L b ( λ ) has b i -highest weight λ i , for i = 1 , ,
3. We shallapply (3.3) and Lemma 1.4 repeatedly to compute λ i , for 1 ≤ i ≤
3. We have ( λ, β ) = − d − r − s. We now divide into 2 cases (1)–(2).(1) Assume x := + + p ). Then λ = λ − β = ( d − δ + rω + ( s + 1) ω . We obtain ( λ , β ) = − d − r − s + 1 . We then divide into 2 subcases (a)-(b).(a) Assume y := + + s − p ). Then λ = λ − β = ( d − δ + ( r + 1) ω + sω . We have ( λ , β ) = − d − s + 4 . (i) If z := + s − p ), then λ = λ − β = ( d − δ + rω + ( s + 2) ω . (ii) If z ≡ p ), then λ = λ = ( d − δ + ( r + 1) ω + sω . (b) Assume y = 2 d + 3 r + s − ≡ p ). Then λ = λ = ( d − δ + rω + ( s + 1) ω . We have ( λ , β ) = − d − s + 1 . (i) If z := + s − p ), then λ = λ − β = ( d − δ + ( r − ω + ( s + 3) ω . (ii) If z ≡ p ), then λ = λ = λ = ( d − δ + rω + ( s + 1) ω . (2) Assume x = 2 d + 3 r + 2 s ≡ p ). Then λ = λ = dδ + rω + sω . We have( λ , β ) = − d − r − s. We then divide into 2 subcases (a)-(b).(a) Assume y := + + s p ). Then λ = λ − β = λ − β = ( d − δ + ( r + 1) ω + ( s − ω . We have ( λ , β ) = − d − s + 3 . (i) If z := + s − p ), then λ = λ − β = ( d − δ + rω + ( s + 1) ω . (ii) If z ≡ p ), then λ = λ = ( d − δ + ( r + 1) ω + ( s − ω . (b) Assume y = 2 d + 3 r + s ≡ p ). Then λ = λ = λ = dδ + rω + sω . We have ( λ , β ) = − d − s. (i) If z := + s p ), then λ = λ − β = ( d − δ + ( r − ω + ( s + 2) ω . (ii) If z ≡ p ), then λ = λ = λ = dδ + rω + sω . Proposition 3.1.
Assume λ = dδ + rω + sω , for d, r, s ∈ N . Then L ( λ ) is finite dimensionalif only if one of the following conditions holds:(1) (a) (i) d ≥ , d + 3 r + 2 s , d + 3 r + s − , d + s − ;(ii) d ≥ , d + s − ≡ , r + 1) , d + 3 r + 2 s ;(b) (i) d ≥ , r , d + 3 r + s − ≡ , d + 3 r + 2 s ;(ii) d ≥ , r ≡ , d + s − ≡ , d + 3 r + 2 s ;(2) (a) (i) d ≥ , s , d + 3 r + 2 s ≡ , r + s + 3 ;(ii) d ≥ , s , d + 3 r + 2 s ≡ , r + s + 3 ≡ ;(b) (i) s ≡ , d , d + 3 r ≡ ;(ii) d ≡ r ≡ s ≡ .Proof. The conditions in the proposition are summary of the dominant conditions for thenew highest weights after odd reflections, which were computed in § d from the summary in § d ≥ d = 1” is quickly ruled out by the other conditions 3 r ≡ , d + s − ≡ , d + 3 r + 2 s L ( λ ) to befinite dimensional. (cid:3) Note the conditions in Proposition 3.1 are obtained without using any division on theconditions arising from odd reflections; some scalars 2 , p >
3, and they are kept for the case when p = 3 below.3.3. Simple modules for the supergroup G (3) for p > . We assume the characteristicof the ground field k is p > d ≥ Proposition 3.2.
For λ = dδ + rω + sω ∈ X + ( T ) with d ≥ , the module L ( λ ) is alwaysfinite dimensional.Proof. Recall ρ from (3.2). The proposition now follows by Proposition 1.9 since λ + ρ =( d − ) δ + ( r + 1) ω + ( s + 1) ω is in X + ( T ). (Alternatively, the proposition also follows fromthe analysis in § (cid:3) We then analyze the case when d = 2. Proposition 3.3.
Let p > . The module L ( λ ) is finite dimensional, for λ = 2 δ + rω + sω ∈ X + ( T ) , if and only if one of the following 3 conditions are satisfied:(i) s ≡ p ) ;(ii) r + s + 3 ≡ p ) ;(iii) r + 2 s + 4 ≡ p ) . ODULAR REPRESENTATIONS OF EXCEPTIONAL SUPERGROUPS 13
The three conditions (i)-(iii) in Proposition 3.3 are not mutually exclusive. Three mutuallyexclusive conditions are given in (3.5)-(3.7) below.
Proof.
Let us set d = 2 in Proposition 3.1.Condition (1a)(ii) becomes s ≡ , r + 1 , r + 4
0, while Condition (2b)(i) becomes s ≡ , r +4 ≡ r +1 s ≡ , r + 1 . Condition (1b)(i) becomes r , r + s + 3 ≡ , r + 2 s + 4
0, while Condition (1b)(ii)becomes r ≡ , r + s +3 ≡ , r +2 s +4
0. Hence the combination of Conditions (1b)(i)-(ii)gives us the following conditions:(3.6) 3 r + s + 3 ≡ , r + 2 s + 4 . Condition (2a)(i) becomes s , r + 2 s + 4 ≡ , r + s + 3
0, while Condition (2a)(ii)becomes s , r +2 s +4 ≡ , r + s +3 ≡
0. Hence the combination of Conditions (2a)(i)-(ii)gives us the following conditions:(3.7) 3 r + 2 s + 4 ≡ , s . So by Proposition 3.1, L ( λ ) is finite dimensional, for λ = 2 δ + rω + sω ∈ X + ( T ), if andonly if one of the 3 (mutually exclusive) conditions (3.5), (3.6), (3.7) holds.Let us show that Conditions (3.5)-(3.7) are equivalent to Conditions (i)-(iii) in the propo-sition. Clearly if r, s satisfy one of Conditions (3.5)-(3.7), then they satisfy one of Con-ditions (i)-(iii). On the other hand, if r, s satisfy Condition (i) but not (3.5), that is, s ≡ r + 1 ≡
0, then (3.6) is satisfied. If r, s satisfy Condition (ii) but not (3.6), thatis, 3 r + s + 3 ≡ r + 2 s + 4 ≡
0, then (3.7) is satisfied. Finally, if r, s satisfy Condition (iii)but not (3.7), that is, s ≡ r + 2 s + 4 ≡
0, then (3.5) is satisfied.The proof of Proposition 3.3 is completed. (cid:3)
We finally analyze the case when d = 1. Proposition 3.4.
Let p > . The module L ( λ ) is finite dimensional, for λ = δ + rω + sω ∈ X + ( T ) , if and only if one of the following 2 conditions are satisfied:(i) s − ≡ r + 4 ≡ p ) ;(ii) s ≡ r + 2 ≡ p ) .Proof. Let us set d = 1 in Proposition 3.1. The case d = 1 only occurs in Cases (2a)(ii) and(2b)(i). Condition (2a)(ii) reads s , r + 2 s + 2 ≡ , r + s + 3 ≡
0, which is clearlyequivalent to (i) in the proposition. Condition (2b)(i) is the same as (ii) above. (cid:3)
Summarizing Propositions 3.1, 3.2, 3.3 and 3.4 (and recalling Proposition 1.1, Lemma 1.2),we have established the following.
Theorem 3.5.
Let p > . Let G be the supergroup of type G (3) . A complete list of inequiv-alent simple G -modules consists of L ( λ ) , where λ = dδ + rω + sω , with d, r, s ∈ N , suchthat one of the following conditions is satisfied:(1) d = 0 , and r ≡ s ≡ p ) .(2) d = 1 , and r, s satisfy either of (i)-(ii) below:(i) s − ≡ r + 4 ≡ p ) ; (ii) s ≡ r + 2 ≡ p ) .(3) d = 2 , and r, s satisfy either of (i)-(iii) below:(i) s ≡ p ) ;(ii) r + s + 3 ≡ p ) ;(iii) r + 2 s + 4 ≡ p ) .(4) d ≥ , (and r, s ∈ N are arbitrary). Remark 3.6.
Theorem 3.5 makes sense over C , providing an odd reflection approach to theclassification of finite-dimensional simple modules over C (due to [Kac77]; also cf. [Ma14]).Indeed this classification can be read off from Theorem 3.5 (by regarding p = ∞ ) as follows. The g -modules L ( λ ) over C is finite dimensional if and only λ = dδ + rω + sω , for d, r, s ∈ N ,satisfies one of the 3 conditions: (1) d = r = s = 0; (2) d = 2 , s = 0; (3) d ≥ . Simple modules for the supergroup G (3) for p = 3 . The assumption p > G and classification of simple G -modules. The (lesspolished) conditions in Proposition 3.1 remain valid for p = 3. When one works it through,it turns out to be the same as setting p = 3 in Theorem 3.5; note the scalar 3 in (1) therein.We summarize this in the following. Theorem 3.7.
Let p = 3 . Let G be the supergroup of type G (3) . A complete list of inequiv-alent simple G -modules consists of L ( λ ) , where λ = dδ + rω + sω , with d, r, s ∈ N , suchthat one of the following conditions is satisfied:(1) d = 0 , and s ≡ ;(2) d = 2 , s ≡ or ;(3) d ≥ , (and r, s ∈ N are arbitrary). Modular representations of the supergroup of type F (3 | k is p > Weights and roots for F (3 | . Let g = g ⊕ g be the exceptional simple Lie superal-gebra F (3 |
1) (which is sometimes denoted by F (4) in the literature). We have g ∼ = sl ⊕ so and g ∼ = k ⊠ k as g -module, where k here is the 8-dimensional spin representation of so . The root system of g can be described via the basis { ǫ , ǫ , ǫ , δ } in h ∗ ∼ = C with anon-degenerate bilinear form ( · , · ) as follows:( δ, δ ) = − , ( δ, ǫ i ) = 0 , ( ǫ i , ǫ i ) = 1 , ( ǫ i , ǫ j ) = 0 , i, j = 1 , , , i = j. (4.1)The root system Φ = Φ ∪ Φ is as below:Φ = {± δ ; ± ǫ i ± ǫ j ; ± ǫ i | i, j = 1 , , , i = j } ; Φ = {
12 ( ± δ ± ǫ ± ǫ ± ǫ ) } The standard Borel subalgebra b corresponds to the simple root systemΠ = (cid:8) α := ǫ − ǫ , α := ǫ − ǫ , α := ǫ , α := 12 ( δ − ǫ − ǫ − ǫ ) (cid:9) . The fundamental weights of g associated with the g -simple roots α , α , α , δ are: ω := ǫ , ω := ǫ + ǫ , ω := 12 ( ǫ + ǫ + ǫ ) , ω := 12 δ. ODULAR REPRESENTATIONS OF EXCEPTIONAL SUPERGROUPS 15
Denote the weight lattice by X = { λ = aω + bω + cω + dω | a, b, c, d ∈ Z } . Sometimes we simply denote λ = aω + bω + cω + dω ∈ X as λ = ( a, b, c, d ) ∈ Z . With respect to b , the Weyl vector ρ can be expressed in terms of the fundamental weightsas ρ = ω + ω + ω − ω . The Dynkin diagram associated to Π is depicted as follows: (cid:13) (cid:13) > (cid:13) N ǫ − ǫ ǫ − ǫ ǫ ( δ − ǫ − ǫ − ǫ ) Π:From (4.1), we have( ω , ǫ ) = 1 , ( ω , ǫ ) = 0 , ( ω , ǫ ) = 0 , ( ω , δ ) = 0 , ( ω , ǫ ) = 1 , ( ω , ǫ ) = 1 , ( ω , ǫ ) = 0 , ( ω , δ ) = 0 , ( ω , ǫ ) = 12 , ( ω , ǫ ) = 12 , ( ω , ǫ ) = 12 , ( ω , δ ) = 0 , ( ω , ǫ ) = 0 , ( ω , ǫ ) = 0 , ( ω , ǫ ) = 0 , ( ω , δ ) = − . (4.2)Denote the positive odd roots for F (3 |
1) by γ = 12 ( δ − ǫ − ǫ − ǫ ) , γ = 12 ( δ − ǫ − ǫ + ǫ ) , γ = 12 ( δ − ǫ + ǫ − ǫ ) ,γ = 12 ( δ − ǫ + ǫ + ǫ ) , γ = 12 ( δ + ǫ − ǫ − ǫ ) . In terms of the fundamental weights, we can reexpress the odd roots γ i as follows: γ = 12 δ − ω , γ = 12 δ − ω + ω , γ = 12 δ − ω + ω − ω ,γ = 12 δ − ω + ω , γ = 12 δ + ω − ω . Besides the conjugate class of the standard simple system Π := Π = { ǫ − ǫ , ǫ − ǫ , ǫ , γ } there are five other conjugate classes of simple systems under the Weyl group action as listedbelow. They all are obtained via sequences of odd reflections from Π (cf. [CW12, § = r γ (Π ) = { ǫ − ǫ , ǫ − ǫ , γ , − γ } , Π = r γ (Π ) = { ǫ − ǫ , γ , − γ , ǫ } , Π = r γ (Π ) = { γ , − γ , ǫ − ǫ , γ } , Π = r γ (Π ) = { δ, ǫ , ǫ − ǫ , − γ } , Π = r γ (Π ) = {− γ , ǫ − ǫ , ǫ − ǫ , δ } . (4.3)Their corresponding Dynkin diagrams are listed as follows: (cid:13) (cid:13) N N ǫ − ǫ ǫ − ǫ γ − γ Π (cid:13) N (cid:13) N ǫ − ǫ γ − γ ǫ Π (cid:13) N NN ǫ − ǫ − γ γ γ Π (cid:13) N (cid:13) (cid:13) δ − γ ǫ ǫ − ǫ < Π (cid:13) N (cid:13) (cid:13) δ − γ ǫ − ǫ ǫ − ǫ Π The corresponding positive systems are denoted by Φ i + , for 0 ≤ i ≤
5, with Φ = Φ + , andthe corresponding Borel subalgebras of g are denoted by b i .4.2. Constraints on highest weights.
Let G be the simply connected algebraic super-group of type F (3 |
1) whose even subgroup is SL ( k ) × Spin ( k ). With respect to the standardBorel subalgebra b (associated to Φ + ), we have X + ( T ) = { λ = aω + bω + cω + dω ∈ X | a, b, c, d ∈ N } . Denote the simple Dist( G )-module of highest weight λ by L ( λ ), where λ ∈ X + ( T ). Assumethat the simple module L ( λ ) = L b ( λ ) has b i -highest weight λ i , for 0 ≤ i ≤
5, where we haveset λ = λ, b = b .4.2.1. The cases of d ≥ and d = 0 . Lemma 4.1.
For any fixed ≤ i ≤ , assume the module L b i ( λ i ) is finite dimensional and λ i is of the form ( x, y, z, . Let j = i + 1 if i ≤ , and let j = 4 or if i = 3 . Then ( λ i , γ j ) ≡ p ) , and λ j = λ i . Proof.
The second equality is an immediate consequence of the first one by Lemma 1.4.Assume that ( λ i , γ j ) . Then, by applying the odd reflection r γ j and Lemma 1.4, wehave L b i ( λ i ) = L b j ( λ j ), where λ j = λ i − γ j is of the form ( ∗ , ∗ , ∗ , − L b j ( λ j )cannot be finite dimensional due to the fact λ j X + ( T ), which is a contradiction. (cid:3) Proposition 4.2.
Let λ = aω + bω + cω + dω ∈ X + ( T ) .(1) If d ≥ , then L ( λ ) is finite dimensional for arbitrary a, b, c ∈ N .(2) If d = 0 , then L ( λ ) is finite dimensional if and only if a ≡ b ≡ c ≡ p ) .Proof. (1) Let d ≥
4. Then λ + ρ = ( a + 1 , b + 1 , c + 1 , d − ∈ X + ( T ) and it is regular.Hence L ( λ ) is finite dimensional by Proposition 1.9.(2) Assume L ( λ ) is finite dimensional, for λ = ( a, b, c, λ, γ ) ≡ ( λ, γ ) ≡ ( λ, γ ) ≡ p ). A direct computation shows( λ, γ ) = − a − b − c, ( λ, γ ) = − a − b − c, ( λ, γ ) = − a − c. From these we conclude that a ≡ b ≡ c ≡ p ). In this case we have λ = λ = λ = λ = λ = λ .By Lemma 1.5, we see the condition a ≡ b ≡ c ≡ p ) is also sufficient for L ( λ ) tobe finite dimensional (this also follows easily by Steinberg tensor product theorem). (cid:3) ODULAR REPRESENTATIONS OF EXCEPTIONAL SUPERGROUPS 17
The case of d = 1 . Proposition 4.3.
Let λ = aω + bω + cω + dω ∈ X + ( T ) , with d = 1 . Then L ( λ ) is finitedimensional if only if one of the following conditions holds.(i) a ≡ b + 3 ≡ c − ≡ p ) ;(ii) a + 1 ≡ b + 1 ≡ c ≡ p ) ;(iii) a + 3 ≡ b ≡ c ≡ p ) .Proof. Assume L ( λ ) is finite dimensional, for λ = ( a, b, c, ∈ X + ( T ). We compute( λ, γ ) = − a − b −
34 ( c + 1) . For now let us assume − a − b − ( c + 1) p ). Then λ = λ − γ = ( a, b, c + 1 , . Hence Lemma 4.1 is applicable and gives us ( λ , γ ) ≡ ( λ , γ ) ≡ ( λ , γ ) ≡
0. A directcomputation shows( λ , γ ) = − a − b −
14 ( c + 1) , ( λ , γ ) = − a −
14 ( c + 1) , ( λ , γ ) = − a + 14 ( c + 1) . From these we conclude that a ≡ b ≡ c + 1 ≡
0. This contradicts the assumption − a − b − ( c + 1) − a − b −
34 ( c + 1) ≡ p ) , and λ = λ = ( a, b, c, . (4.4)Using the above equations, we compute( λ , γ ) = − a − b − c − ≡ c (mod p ) . We now divide into 2 cases (1)-(2).(1) Assume c p ). Then λ = λ − γ = ( a, b + 1 , c − , c ≥
1. Hence Lemma 4.1 is applicable and gives us that ( λ , γ ) ≡ ( λ , γ ) ≡
0. A directcomputation shows( λ , γ ) = − a −
14 ( c − , ( λ , γ ) = − a + 14 ( c − . From these we conclude a ≡ c − ≡
0; a revisit of (4.4) then gives us b ≡ − . This givesus Condition (i) in the proposition. (Note the conditions c ≥ λ = λ = λ = λ = ( a, b + 1 , c − ,
0) and λ = λ .(2) Assume c ≡ p ). So λ = λ = λ = ( a, b, c, λ , γ ) = − a −
14 ( c + 3) ≡ − a −
34 (mod p ) . Now we divide (2) into two subcases (2a)-(2b).(2a) Assume − a − p ). Then λ = λ − γ = ( a + 1 , b − , c + 1 , b ≥
1. Hence Lemma 4.1 is applicable and gives us that ( λ , γ ) ≡ λ , γ ) = − a + c − . Recalling c ≡
0, we concludethat a + ≡
0. A revisit of (4.4) then gives us b ≡ − . This gives us Condition (ii) in the proposition. (Note the conditions b ≥ λ = λ = λ = ( a + 1 , b − , c + 1 ,
0) and λ = λ = λ .(2b) Assume − a − ≡ p ). Then a ≡ − , and it follows by c ≡ b ≡
0. This gives us Condition (iii). In this case, we have λ = λ = λ = λ = λ ,and λ = ( a − , b, c + 1 , λ i lie in X + ( T ) for all i in all casesabove, we see the conditions (i)-(iii) are sufficient for L ( λ ) to be finite dimensional. Theproposition is proved. (cid:3) The case of d = 2 . Proposition 4.4.
Assume λ = aω + bω + cω + dω ∈ X + ( T ) with d = 2 . Then L ( λ ) isfinite dimensional if only if one of the following conditions hold:1. 1.1. a ≡ c ≡ , b
6≡ − and b
6≡ − ;1.2. a
6≡ − , b ≡ − a − , c ≡ a , and b ≥ ;1.3. a
6≡ − , b ≡ and c ≡ − a − .2. 2.1. c ≡ a + 2 , b ≡ − a − , c ≥ , and a
6≡ − ;2.2. (i) c ≡ − a − , b ≡ a , a
6≡ − , a
6≡ − , c ≥ , and a ≥ ;(ii) a ≡ − , b ≡ − , c ≡ ;2.3. c ≡ , b ≡ − a − , and a
6≡ − ;2.4. c ≡ b ≡ and a ≡ − .Proof. Assume L ( λ ) is finite dimensional, for λ = ( a, b, c, ∈ X + ( T ). We compute ( λ, γ ) = − a − b − c − , and then divide into two cases (1)-(2) below.(1) Assume − a − b − c − p ). Then λ = λ − γ = ( a, b, c + 1 , λ , γ ) = − a − b − c −
1, and then divide into 2 subcases (1a)-(1b).(1a) Assume − a − b − c − p ). Then λ = λ − γ = ( a, b + 1 , c, λ , γ ) ≡ ( λ , γ ) ≡ ( λ , γ ) ≡
0. From these anda direct computation of ( λ , γ ) = − a − c , ( λ , γ ) = − a + c , and ( λ , γ ) = a − c ,we conclude that a ≡ c ≡ , b
6≡ − , b
6≡ − , whence Condition 1.1. In this case, we have λ = λ = λ = λ = ( a, b + 1 , c, − a − b − c − ≡ p ). Then λ = λ = ( a, b, c + 1 , λ , γ ) = − a − c −
1, and then again divide into 2 subcases (1b-1)-(1b-2):(1b-1) Assume − a − c −
0. Then λ = λ − γ = ( a + 1 , b − , c + 2 , λ , γ ) ≡ ( λ , γ ) ≡
0. From these and adirect computation of ( λ , γ ) = − ( a +1)+ ( c +2) and ( λ , γ ) = ( a +1) − ( c +2),we conclude that c ≡ a . Combining with the conditions on (1), (1b) and (1b-1),this gives us b ≡ − a − a
6≡ −
1, whence Condition 1.2. In this case we have λ = λ = ( a, b, c + 1 , λ = λ = λ = ( a + 1 , b − , c + 2 , − a − c − ≡
0. Then λ = λ = λ = ( a, b, c + 1 , b ≡ , c ≡ − a − , a
6≡ − , whenceCondition 1.3. (cid:0) We then compute ( λ , γ ) = − a + c − ≡ − a −
0. Thus, λ = λ − γ = ( a + 1 , b, c, λ , γ ) = a − c − ≡ a ). Hence λ = λ = ( a, b, c + 1 ,
1) if a ≡ λ = ( a − , b, c,
0) if a (cid:1) ODULAR REPRESENTATIONS OF EXCEPTIONAL SUPERGROUPS 19
Case (1b) and hence Case (1) are completed.(2) Assume − a − b − c − ≡ p ). We have λ = λ = ( a, b, c, λ , γ ) = − a − b − c − , and divide into 2 subcases (2a)-(2b).(2a) Assume − a − b − c −
0. Then we compute λ = λ − γ = ( a, b + 1 , c − , c ≥
1. (Note the combination of the condition c ≥ λ , γ ) = − a − c − , and thendivide into 2 subcases (2a-1)-(2a-2).(2a-1) Assume − a − c −
0. Then λ = λ − γ = ( a + 1 , b, c, λ , γ ) ≡ ( λ , γ ) ≡
0. Combining with the computa-tions of ( λ , γ ) = − ( a + 1) + c and ( λ , γ ) = ( a + 1) − c , this implies c ≡ a + 2and b ≡ − a −
3; moreover the condition on (2a-1) becomes a
6≡ −
1. Thus, we haveobtained Condition 2.1. In this case, we have λ = λ , λ = ( a, b + 1 , c − , λ = λ = λ = ( a + 1 , b, c, − a − c − ≡
0. The conditions on (2), (2a) and (2a-2) can be rephrasedas c ≡ − a − b ≡ a and a
6≡ −
1. We have λ = λ = ( a, b + 1 , c − , c ≥
1. We further compute ( λ , γ ) = − a + c − ≡ − a − , andagain divide into 2 subcases:(i) Assume a
6≡ − . Then we have λ = λ − γ = ( a +1 , b +1 , c − , c ≥
2. Moreover, if ( λ , γ ) = a
0, then λ = λ − γ = ( a − , b + 1 , c, a ≥
1; otherwise, λ = λ . This gives us Condition 2.2(i).(ii) Assume a ≡ − . Then we have b ≡ − and c ≡
1, whence Condition 2.2(ii). Inthis case, we have λ = λ , λ = λ = λ = ( a, b + 1 , c − , λ = λ − γ =( a − , b + 1 , c, − a − b − c − ≡ p ). Then λ = λ = ( a, b, c, λ , γ ) = − a − c − , and divide into 2 subcases (2b-1)-(2b-2).(2b-1) Assume − a − c − p ). Then we have c ≡ b ≡ − a − , and a
6≡ −
3, whence Condition 2.3. In this case, we have λ = λ = λ = ( a, b, c, λ = λ − γ = ( a + 1 , b − , c + 1 , λ , γ ) = − a + c − ≡ − a − λ , γ ) ≡ a − . So λ = λ − γ = ( a + 2 , b − , c,
0) if a
6≡ −
2, and λ = λ otherwise; moreover, if a λ = λ − γ = ( a, b − , c + 2 , λ = λ .(2b-2) Assume − a − c − ≡ p ). Then we have a ≡ − b ≡ c ≡
0, whenceCondition 2.4. In this case, we have λ i = λ for 1 ≤ i ≤ L ( λ ) to be finite dimensional.By inspection, we have all weights λ i ∈ X + ( T ) for all i in every case above. Hence byLemma 1.5 we conclude that the conditions as listed in the proposition are also sufficient for L ( λ ) to be finite dimensional. (cid:3) Now we simplify the above conditions by removing all inequalities. We caution that theresulting conditions are no longer mutually exclusive.
Proposition 4.5.
Set d = 2 . Assume λ = aω + bω + cω + dω ∈ X + ( T ) . Then L ( λ ) isfinite dimensional if only if one of the following conditions (i)–(vi) holds:(i) a ≡ c ≡ p ) ;(ii) a − c ≡ a + b + 1 ≡ p ) ;(iii) b ≡ a + c + 4 ≡ p ) ;(iv) a − c + 2 ≡ a + b + 3 ≡ p ) , and c ≥ ;(v) a + c + 2 ≡ a − b ≡ p ) , and a ≥ ;(vi) a + 2 b + 3 ≡ c ≡ p ) .Proof. One first observes that all conditions listed in Proposition 4.4 are part of conditionslisted above in this proposition. Indeed the conditions above are basically obtained byremoving the inequalities in the conditions in Proposition 4.4; the cases (1.4) and (2.4) withno inequalities in Proposition 4.4 are part of (iii) and (vi) above, respectively.It remains to show that all conditions above in this proposition are included in the list ofconditions (1.1)–(1.3) and (2.1)–(2.4) in Proposition 4.4.If Condition (i) is satisfied but (1.1) in Proposition 4.4 is not, then either (A) b ≡ −
1, inwhich case a ≡ c ≡
0, and so (1.2) is satisfied, or (B) b ≡ − , in which case a ≡ c ≡
0, andso (2.3) is satisfied.If Condition (ii) is satisfied but (1.2) in Proposition 4.4 is not, then either (A) a ≡ − b ≡ c ≡ −
2, and hence (1.3) is satisfied, or (B) b = 0, in which case, a ≡ − , c ≡ −
2, and so (1.3) is satisfied.If Condition (iii) is satisfied but (1.3) in Proposition 4.4 is not, then a ≡ − , in whichcase b ≡ , c ≡ −
1, and so (2.1) is satisfied.If Condition (iv) is satisfied but (2.1) in Proposition 4.4 is not, then a ≡ −
1, in whichcase b ≡ − , c ≡
0, and so (2.3) is satisfied.If Condition (v) is satisfied but (2.2)(i) in Proposition 4.4 is not, then either (A) a ≡ − ,in which case b ≡ − , c ≡
1, and so (2.2)(ii) is satisfied; or (B) a ≡ −
1, in which case b ≡ − , c ≡
0, and so (2.3) is satisfied; or (C) c = 0, in which case a ≡ b ≡ −
1, and so (2.3)is satisfied.If Condition (vi) is satisfied but (2.3) in Proposition 4.4 is not, then a ≡ − b ≡ c ≡ (cid:3) The case of d = 3 . Proposition 4.6.
Assume λ = aω + bω + cω + dω ∈ X + ( T ) , with d = 3 . Then L ( λ ) isfinite dimensional if only if one of the following conditions holds:1. 1.1. c ≡ a + 1 , and b
6≡ − a − , b
6≡ − a − , a
6≡ − ;1.2. c ≡ − a − , and b
6≡ − , b a ;1.3. b ≡ − a − c − , and b , c
6≡ − ;1.4. b ≡ , c ≡ − a − , and a
6≡ − .2. 2.1. b ≡ − a − c − , and c , c
6≡ − a − ;2.2. b ≡ a + , c ≡ − a − , and c ;2.3. b ≡ − a − , c ≡ , and b ;2.4. a ≡ − , b ≡ c ≡ . ODULAR REPRESENTATIONS OF EXCEPTIONAL SUPERGROUPS 21
Proof.
Assume L ( λ ) is finite dimensional, for λ = ( a, b, c, ∈ X + ( T ). We compute ( λ, γ ) = − a − b − c − , and divide into 2 cases (1)-(2).(1) Assume − a − b − c − p ). We have λ = λ − γ = ( a, b, c + 1 , λ , γ ) = − a − b − ( c + 1) − , and then divide into 2 cases (1a)-(1b).(1a) Assume − a − b − ( c + 1) − p ). Then λ = λ − γ = ( a, b + 1 , c, λ , γ ) = − a − c − , and again divide into two subcases (1a-i)-(1a-ii):(1a-i) Assume − a − c −
0. Then λ = λ − γ = ( a +1 , b, c +1 , λ , γ ) ≡ ( λ , γ ) ≡
0. Combining with the computationof ( λ , γ ) = − ( a + 1) + ( c + 1) and ( λ , γ ) = ( a + 1) − ( c + 1), this implies c ≡ a + 1. The conditions on (1), (1a) and (1a-i) can be simplified to a
6≡ − b
6≡ − a − b
6≡ − a −
3, whence Condition 1.1. In this case, we have λ = ( a, b, c + 1 , λ = ( a, b + 1 , c, λ = λ = λ = ( a + 1 , b, c + 1 , − a − c − ≡
0. Then λ = λ = ( a, b + 1 , c, c ≡ − a − b
6≡ − b a , whenceCondition 1.2. In this case, we have λ = ( a, b, c + 1 , λ = λ = ( a, b + 1 , c, a
6≡ − , then λ = λ − γ = ( a + 1 , b + 1 , c − , λ = λ . If a λ = λ − γ = ( a − , b + 1 , c + 1 , λ = λ .This completes Subcase (1a).(1b) Assume − a − b − ( c + 1) − ≡ p ). Then λ = λ = ( a, b, c + 1 , λ , γ ) = − a − c − , and again divide into two subcases (1b-i)-(1b-ii):(1b-i) Assume − a − c −
0. We compute λ = λ − γ = ( a + 1 , b − , c + 2 , b ≡ − a − c − , b
0, and c
6≡ − λ = λ = ( a, b, c + 1 , λ =( a +1 , b − , c +2 , c a +3, then λ = λ − γ = ( a +2 , b − , c +1 , λ = λ . If c a −
3, then λ = λ − γ = ( a, b − , c + 3 , λ = λ .(1b-ii) Assume − a − c − ≡
0. Then λ = λ = ( a, b, c + 1 , a
6≡ − b ≡ c ≡ − a −
7, whence Condition 1.4.In this case, we have λ = λ = λ = ( a, b, c + 1 , a
6≡ −
3, then λ = λ − γ = ( a + 1 , b, c, λ = λ . If a
0, then λ = λ − γ =( a − , b, c + 2 , λ = λ .This completes Subcase (1b) and then Case (1).(2) Assume − a − b − c − ≡
0. Then λ = λ = ( a, b, c, λ , γ ) = − a − b − c − , and divide into 2 subcases (2a)-(2b).(2a) Assume − a − b − c −
0. Then λ = λ − γ = ( a, b + 1 , c − , λ , γ ) = − a − c − , and again divide into 2 subcases (2a-i)-(2a-ii):(2a-i) Assume − a − c −
0. Then the conditions on (2), (2a) and (2a-i) become b ≡ − a − c − , c c
6≡ − a −
5, whence Condition 2.1. In this case,we have λ = λ , λ = ( a, , b + 1 , c − , λ = ( a + 1 , b, c, c a + 5, then λ = ( a + 2 , b, c − , λ = λ . If c a −
1, then λ = ( a, b, c + 1 , λ = λ . (2a-ii) Assume − a − c − ≡
0. Then the conditions on (2), (2a) and (2a-ii) become c ≡ − a − b ≡ a + , and c
0, whence Condition 2.2. In this case, we have λ = λ , λ = λ = ( a, b + 1 , c − , a
6≡ −
3, then λ = ( a + 1 , b + 1 , c − , c ≥ λ = λ . If a
0, then λ = ( a − , b + 1 , c, λ = λ .This completes Subcase (2a).(2b) Assume − a − b − c − ≡
0. Then λ = λ = λ . We compute ( λ , γ ) = − a − c − ,and divide into 2 subcases (2b-i)-(2b-ii):(2b-i) Assume − a − c −
0. Then λ = λ − γ = ( a + 1 , b − , c + 1 , c ≡ b ≡ − a − , and b
0, whence Condition 2.3.In this case, we have λ = λ = λ , and λ = ( a + 1 , b − , c + 1 , a
6≡ − , then λ = ( a + 2 , b − , c, λ = λ . If a , then λ = ( a, b − , c + 2 , λ = λ .(2b-ii) Assume − a − c − ≡
0. The conditions on (2), (2b) and (2b-ii) become b ≡ c ≡ a ≡ − , whence Condition 2.4. In this case, we have λ i = λ for 1 ≤ i ≤ λ = λ − γ = ( a − , b, c + 1 , L ( λ ) to be finite dimensional.By inspection, we see that λ i ∈ X + ( T ) for all i in every case above. By Lemma 1.5, theconditions listed in the proposition are also sufficient for L ( λ ) to be finite dimensional. (cid:3) Now we simplify the conditions in Proposition 4.6 by removing all inequalities.
Proposition 4.7.
Set d = 3 . Assume λ = aω + bω + cω + dω ∈ X + ( T ) . Then L ( λ ) isfinite dimensional if only if one of the following conditions (i)–(v) holds:(i) a − c + 1 ≡ p ) ;(ii) a + c + 3 ≡ p ) ;(iii) a + 4 b + c + 7 ≡ p ) ;(iv) a + c + 7 ≡ b ≡ p ) ;(v) a + 4 b + 3 c + 9 ≡ p ) .Proof. One first observes that all conditions listed in Proposition 4.6 are part of conditionslisted above in this proposition. Indeed the conditions above are basically obtained by re-moving the inequalities in the conditions in Proposition 4.6, and the case (2.4) with equalitiesonly is included in (v).It remains to show that all conditions above in this proposition are included in the list ofconditions (1.1)–(1.4) and (2.1)–(2.4) in Proposition 4.6.We first check that the 4 subcases (2.1)–(2.4) of Proposition 4.6 are equivalent to Condi-tion (v). If Condition (v) is satisfied but (2.1) of Proposition 4.6 is not, then we have 2 cases(A)-(B):(A) c c ≡ − a −
5, in which case b ≡ a + , a
6≡ − , and so (2.2) is satisfied;(B) c ≡
0. Then b ≡ − a − . We further divide into 2 subcases:(B1) b
0, in which case (2.3) is satisfied,(B2) b ≡
0, in which case a ≡ − , and so (2.4) is satisfied. ODULAR REPRESENTATIONS OF EXCEPTIONAL SUPERGROUPS 23
If Condition (i) is satisfied but (1.1) of Proposition 4.6 is not, then we have the following3 cases (A)-(B)-(C):(A) b ≡ − a −
3, in which case c ≡ a + 1, and so (v) is satisfied;(B) a ≡ − b
6≡ − a −
3, in which case c ≡ − b
6≡ −
1, and so (1.2) is satisfied;(C) b ≡ − a − a
6≡ −
1. Hence c
6≡ − c ≡ a + 1. We further divide into2 subcases below:(C1) b
0, in which case c ≡ a + 1, and so (1.3) is satisfied,(C2) b ≡
0, in which case a ≡ − , c ≡ −
3, and so (1.4) is satisfied.If Condition (ii) is satisfied but (1.2) of Proposition 4.6 is not, then either (A) b ≡ a , inwhich case c ≡ − a −
3, and so (v) is satisfied, or (B) b ≡ − b a , in which case c ≡ − a − c
6≡ −
1, and so (1.3) is satisfied.If Condition (iii) is satisfied but (1.3) of Proposition 4.6 is not, then either (A) c ≡ − b ≡ − a − c − implies that (v) is satisfied; or (B) c
6≡ − b ≡
0, in which case a
6≡ −
3, and so (1.4) is satisfied.If Condition (iv) is satisfied but (1.4) of Proposition 4.6 is not, then a ≡ −
3, in whichcase b ≡ , c ≡ −
1, and so (v) is satisfied.The proposition is proved. (cid:3)
Simple modules of the supergroup F (3 | . Summarizing Propositions 4.2, 4.3, 4.5,and 4.7, we have established the following classification of simple modules for type F (3 | Theorem 4.8.
Let p > . Let G be the simply connected supergroup of type F (3 | . Acomplete list of inequivalent simple G -modules consists of L ( λ ) , where λ = aω + bω + cω + d δ , with a, b, c, d ∈ N , such that one of the following conditions is satisfied:(1) d = 0 , and a ≡ b ≡ c ≡ p ) .(2) d = 1 , and a, b, c satisfy either of (i)-(iii) below:(i) a ≡ b + 3 ≡ c − ≡ p ) ;(ii) a + 1 ≡ b + 1 ≡ c ≡ p ) ;(iii) a + 3 ≡ b ≡ c ≡ p ) .(3) d = 2 , and a, b, c satisfy either of (i)-(vi) below:(i) a ≡ c ≡ p ) ;(ii) a − c ≡ a + b + 1 ≡ p ) ;(iii) b ≡ a + c + 4 ≡ p ) ;(iv) a − c + 2 ≡ a + b + 3 ≡ p ) and c ≥ ;(v) a + c + 2 ≡ a − b ≡ p ) and a ≥ ;(vi) a + 2 b + 3 ≡ c ≡ p ) .(4) d = 3 , and a, b, c satisfy either of (i)-(v) below:(i) a − c + 1 ≡ p ) ;(ii) a + c + 3 ≡ p ) ;(iii) a + 4 b + c + 7 ≡ p ) ;(iv) a + c + 7 ≡ b ≡ p ) ;(v) a + 4 b + 3 c + 9 ≡ p ) .(5) d ≥ , (and a, b, c ∈ N are arbitrary). We do no attempt the classification of simple G -modules for p = 3 in this paper, and leaveit to the reader. Remark 4.9.
Theorem 4.8 makes sense over C , providing an odd reflection approach to theclassification of finite-dimensional simple modules over C (due to [Kac77]; also cf. [Ma14]).Indeed this classification can be read off from Theorem 4.8 (by regarding p = ∞ ) as follows. The simple g -modules L ( λ ) over C are finite dimensional if and only if λ = aω + bω + cω + d δ , for a, b, c, d ∈ N , satisfies one of the 3 conditions: (1) a = b = c = d = 0 ; (2) d = 2 and a = c = 0 ; (3) d ≥ . References [B06] J. Brundan,
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Institute of Mathematics, Academia Sinica, Taipei, Taiwan 10617
E-mail address : [email protected] Department of mathematics, East China Normal University, Shanghai, China 200241
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