Borel-Weil Theorem for Algebraic Supergroups 3

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[27] gave a systematic construction of simple supermodules for quasireductive supergroups over an algebraically closed ﬁeld of characteristic zero, in terms of their Lie superalgebras. On the other hand, the study of quasireductive supergroups over a ﬁeld of positive characteristic has just started. A. Zubkov and F. Marko [14, 15, 32,

33] have obtained may results for representations of GÄ(m n). J. Brundan and A. Kleshchev [3] studied representations of Q(n) which| have a close relationship to modular representations of spin symmetric groups. By using

representations of GÄ(m n), Mullineux conjecture (now is Mullineux theorem) was re-proven by J.| Brundan and J. Kujawa [4]. B. Shu and Wang

[29] classiﬁed simple supermodules of ËÔÇ(m n) whose parameter set is described by combinatorics terms, which is related| to Mullineux bijection. Recently, S.-J. Cheng, B. Shu and W. Wang [5] explicitly classiﬁed all simple supermodules of (simply connected) Chevalley groups of type D(2 1; ζ), G(3) and F (3 1). | This paper| proposes a framework of characteristic-free study of represen-

tation theory of quasireductive supergroups G. The ﬁrst purpose of the pa-

per is to establish the Borel-Weil theorem for G. Namely, we give a system-

atic construction of all simple G-supermodules which extends Serganova’s construction to arbitrary characteristic. In the analytic situation, the Borel- Weil-Bott theorem for classical Lie supergroups over C, was established by

I. Penkov [25] in 1990. Recently, A. Zubkov [33] proved the Borel-Weil- GÄ Bott theorem for G = (m n) over a ﬁeld. In the following, we ex- plain the details of our approach.| First, using a concept of the theory

of Harish-Chandra pairs due to A. Masuoka [17], we are able to ﬁnd a

B G super-torus Ì and a Borel super-subgroup of . Next, we construct all

simple Ì-supermodules u(λ) by the theory of Cliﬀord algebras, where

{ }λ∈Λ

G B Λ is the character group of Ìev. For the quotient superscheme / (due to A. Masuoka and A. Zubkov [21]), we may consider cohomology groups

n n B H (λ) := H (G/ , L(u(λ))), where λ Λ, n 0 and L(u(λ)) is the asso-

♭ ∈ ≥ 0 B ciated sheaf to u(λ) on G/ . Set Λ := λ Λ H (λ) = 0 . Our main result is the following. { ∈ | 6 } ♭ 0 Theorem (Theorem 4.12). For λ Λ , the G-socle L(λ) of H (λ) is simple. The map λ L(λ) gives a∈ one-to-one correspondence between ♭ 7→ Λ and the isomorphism classes of all simple G-supermodules up to parity change.

The second purpose of the paper is to study quasireductive supergroups È G which include a distinguished “parabolic” super-subgroup such that

G È B Èev = ev and Lie( )1¯ = Lie( )1¯. Then we can describe H (λ) in terms

n n λ

B Ì of the non-super cohomology group Hev(λ) := H (Gev/ ev, L(k )) as ev- λ supermodules, where k is the one-dimensional Ìev-module of weight λ. This

is a generalization of Zubkov’s result [32, Proposition 5.2 and Lemma 5.1] GÄ for G = (m n), see also Marko [14, 1.2]. As a corollary, we have the

following results.| § È Theorem (Section 5.2). If G has such a parabolic super-subgroup , then ♭ + (1) Λ coincides with the set of all dominant weights Λ for Gev, (2) Hn(λ) = 0 for λ Λ+, n 1, and ∈ ≥ BOREL-WEIL THEOREM FOR ALGEBRAIC SUPERGROUPS 3

0 0 −δ (3) the formal character of H (λ) is given by ch(Hev(λ)) + (1+e ) δ∈∆1¯ + + for λ Λ , where ∆ is the set of all positive odd roots of Lie(G). ∈ 1¯ Q The above (2) is a super-analogue of the Kempf vanishing theorem. The paper is organized as follows. In Section 2, we review some basic def- initions and results for algebraic supergroups over a principal ideal domain (PID). As a generalization of [18, Definition 6.3], we re-define the notion of sub-pairs (Definition 2.4). In Section 3, we give a definition of quasireductive supergroups over Z (Definition 3.1) as a generalization of Serganova’s one.

We study structure of quasireductive supergroups G over a PID; super-tori

+ +

Í Í B B Ì, unipotent super-subgroups , and Borel super-subgroups , are given in 3.2 and 3.3. In particular, we give a PBW basis of Kostant’s § § Z-form for G (Theorem 3.11). As a consequence, we prove an algebraic in-

B G terpretation of the “density” of the big cell B in (Corollary 3.13). × This is needed to prove the Borel-Weil theorem for our G. In the non-super situation, see B. Parshall and J. Wang [24], and J. Bichon and S. Riche [1]. In Section 4, we work over a ﬁeld k of characteristic not equal to 2. First, in 4.1, we construct all simples u(λ) λ∈Λ of a super-torus Ì, using the

theory§ of Clifford algebras (Appendix{ } B). Here, Λ is the character group Ì X(Ìev) of ev. When k is algebraically closed, we show that our simples are the same as Serganova’s (Remark 4.7). Next two subsections are devoted to n cohomology groups H (λ) and determination of all simple B-supermodules. 0 In 4.4, we prove our first main result, that is, the G-socle L(λ) of H (λ) = 0 § 6 is simple, and this gives a one-to-one correspondence between the set Λ♭ := 0 λ Λ H (λ) = 0 and the set of all isomorphism classes Simple (G) { ∈ | 6 } Π of simple G-supermodules up to parity change. We also give a necessary Q and sufficient condition for L(λ) to be of type , that is, L(λ) ∼= ΠL(λ) as G-supermodules, using the language of Clifford algebras (Theorem 4.12). Here, Π is the parity change functor. Some further properties of the induced supermodule H0(λ) are discussed in 4.5. In the final Section 5, we still work over§ a field k of characteristic not equal

to 2. A. Zubkov [32] establish various results on GÄ (m n), using its “para-

bolic” super-subgroups. In this section, we assume that| our quasireductive

È È G

supergroup G having a parabolic super-subgroup such that ev = ev B and Lie(È)¯ = Lie( )¯. In 5.1, we generalize some of results stated in [32, 1 1 § 5]. In 5.2, we prove that for all λ Λ♭ and n 0, Hn(λ) is isomorphic

§ n § + ∗ ∈ ≥ n Ì to Hev(λ) Lie(Í )1¯ as ev-supermodules (Theorem 5.5), where Hev(λ)

⊗∧ + ∗ Í is the cohomology group for Gev and Lie( )1¯ is the dual space of the odd + part of Lie(Í ). As a consequence, we get the following three results: (1) ♭ + Λ coincides with the set of all dominant weights Λ for Gev, (2) the Kempf vanishing theorem Hn(λ)=0for λ Λ+, n 1 (Corollary 5.6), and (3) the Weyl character formula, that is, an∈ explicit≥ description of the formal character of H0(λ) for λ Λ+ (Corollary 5.8). In 5.3, we see some examples. ∈ § 2. Preliminaries In this section, the base ring k supposed to be a principal ideal domain (PID) of characteristic different from 2. The unadorned denoted the tensor product over k. Note that, a k-module is projective if⊗ and only if it 4 T. SHIBATA is free. A k-module is said to be finite (resp. k-free/k-flat) if it is finitely generated (resp. free/flat). In particular, we say that a k-module is finite free if it is finite and k-free. All k-modules form a symmetric tensor category Mod with the trivial symmetry V W W V ; v w w v, where V,W Mod. ⊗ → ⊗ ⊗ 7→ ⊗ ∈ 2.1. Superspaces. The group algebra of the group Z = 0¯, 1¯ of order 2 { } two over k is denoted by kZ2 which has a unique Hopf algebra structure over k. We let := (ModkZ2 , , k) denote the tensor category of right kZ - C ⊗ 2 comodules, that is, Z2-graded modules over k. Here, k is regarded as a purely even object; k = k 0. In what follows, for V = V0¯ V1¯ , an element v in V is always regarded⊕ as a homogeneous element⊕ of ∈V . C We denote the parity of v by v , that is, v = ǫ if v Vǫ (ǫ Z2). For each V,W , the following is| the| so-called| supersymmetry| ∈ . ∈ ∈C c = c : V W W V ; v w ( 1)|v||w|w v. V,W ⊗ −→ ⊗ ⊗ 7−→ − ⊗ In this way, (ModkZ2 , , k, c) forms a symmetric tensor category which we denote by SMod. An object⊗ in SMod is called a superspace. We let SMod(V,W ) denote the set of all morphisms from V to W in SMod. For V SMod, we define an object ΠV of SMod so that (ΠV ) = V ¯ for ∈ ǫ ǫ+1 each ǫ Z2. For a morphism f : V W in SMod, we define a morphism Πf : Π∈V ΠW in SMod so that Π→f = f. In this way, we get a functor Π : SMod → SMod, called the parity change functor on SMod. For V,W SMod, we→ define an object SMod(V,W ) of SMod as follows. ∈

SMod(V,W )0¯ := SMod(V,W ), SMod(V,W )1¯ := SMod(ΠV,W ). Given an object V of SMod, we define a superspace V ∗ := SMod(V, k), called the dual of V . A superalgebra (resp. supercoalgebra/Hopf superalgebra/Lie superalgebra) is defined to be an algebra (resp. coalgebra/Hopf algebra/Lie algebra) object in SMod. A superalgebra A is said to be commutative if ab = ( 1)|a||b|ba for all a, b A. For a supercoalgebra C with the comultiplication− : ∈ △C C C C, we use the Hyneman-Sweedler notation C (c)= c c(1) c(2), where→ c⊗ C. △ ⊗ ∈ P For a superalgebra A, we let ASMod denote the category of all left A- supermodules. Similarly, we let SModC denote the category of all right C-supercomodules for a supercoalgebra C. We also define a superspace SModC (V,W ) := SModC (V,W ) SModC (ΠV,W ) for V,W SModC . The definition of SMod (M,N) for M,N⊕ SMod will be clear.∈ A ∈ A 2.2. Algebraic Supergroups. An affine supergroup scheme (supergroup,

for short) over k is a representable functor G from the category of commu-

tative superalgebras to the category of groups. We denote the representing G object of G by ( ) which forms a commutative Hopf superalgebra. A

O G supergroup G is said to be algebraic (resp. flat) if A := ( ) is finitely generated as a superalgebra (resp. k-flat). A closed super-subgroupO of a

supergroup G is a supergroup which is represented by a quotient Hopf su- G peralgebra of A. For a supergroup G, we deﬁne its even part ev as the

restricted functor of G form the category of commutative algebras to the BOREL-WEIL THEOREM FOR ALGEBRAIC SUPERGROUPS 5

category of groups. This Gev is an ordinary aﬃne group scheme represented by the quotient (ordinary) Hopf algebra A := A/IA, where IA is the super-

ideal of A generated by the odd part A1¯ of A. We denote the quotient map G by A ։ A; a a. If G is algebraic, then so is ev. An algebraic supergroup is said to be connected7→ if its even part is connected, see [17, Deﬁnition 8].

Example 2.1. For a superspace V over k, we deﬁne a group functor GÄ(V ) so that

GÄ(V )(R) := Aut (V R), R ⊗ where R is a commutative superalgebra and AutR(V R) is the set of all bijective maps in SMod (V R,V R). Suppose⊗ that V is ﬁnite free R ⊗ ⊗ with m = rank(V0¯) and n = rank(V1¯). Then GÄ(V ) becomes an algebraic

supergroup, called a general linear supergroup, and is denoted by GÄ(m n). | The even part GÄ(m n)ev is isomorphic to GL GL . For the explicit form | m × n of the Hopf algebra structure of (GÄ(m n)), see [4, 32] for example.

O | G Let G be a supergroup with A := ( ). A superspace M is said to

O G

be a left G-supermodule if there is a group functor morphism from to

GÄ G

(M). We let G SMod denote the tensor category of left -supermodules. One easily sees that M forms a right A-supercomodule. Conversely, any

right A-supercomodule can be regard as a left G-supermodule. In this way,

À G

we may identify G SMod and SMod . Let be a closed super-subgroup of

G À with B := (À). For a left -supermodule M and a left -supermodule V ,

O G we may consider its restricted À-supermodule and induced -supermodule,

respectively; G

res G (M) := resA(M) and ind (V ) := indA(V ), À À B B

via the quotient Hopf superalgebra map A ։ B, see Appendix A.3. Suppose À that G and are ﬂat. We also obtain the following Frobenius reciprocity, see (A.4). ∼ (2.1) SModB(resA(M),V ) = SModA(M, indA(V )).

B −→ B À

Zubkov [32, Propositon 3.1] showed that the category À SMod of all left -

supermodules has enough injectives. Since the induction functor indG ( ) À

n G −

is left exact, we get its right derived functor R ind ( ) form À SMod to À −

G SMod (n Z = 0, 1, 2,... ). ∈ ≥0 { }

A non-zero left G-supermodule L is said to be simple (or irreducible) if L is simple as a right A-supercomodule, that is, L has no non-trivial A- super-subcomodule. If L is simple, then so is ΠL. As in [3], we shall use the following terminology.

Deﬁnition 2.2. A simple left G-supermodule L is said to be of type Q if L

is isomorphic to ΠL as left G-supermodules, and type M otherwise. G Let Simple(G) denote the set of isomorphism classes of simple left -

supermodules. The functor Π naturally acts on Simple(G) as a permutation G of order 2. Let SimpleΠ(G) denote the set of Π -orbits in Simple( ). In

′ h i G other words, two elements L,L in Simple(G) coincides in SimpleΠ( ) if and ′ ′ only if L ∼= L or ΠL as left G-supermodules. 6 T. SHIBATA

2.3. Lie superalgebras and Super-hyperalgebras. In the following, we

G G let be an algebraic supergroup with A := ( ). Set A := Ker(εG ), O ∗

where εG : A k is the counit of A. As a k-super-submodule of A , we let → + + 2 ∗

Lie(G) := (A /(A ) ) .

This naturally forms a Lie superalgebra (see [18, Proposition 4.2]), which

G G

we call the Lie superalgebra of G. Note that, Lie( ) is ﬁnite free and Lie( )0¯ G can be identiﬁed with the (ordinary) Lie algebra Lie(Gev) of ev. For any n 1, we may regard (A/(A+)n)∗ as a k-super-submodule of A∗ through the dual≥ of the canonical quotient map A ։ A/(A+)n. Set + n ∗

hy(G) := (A/(A ) ) . n[≥1 ∗

This hy(G) forms a super-subalgebra of A . We call it the super-hyperalgebra

G G of G. It is sometimes called the super-distribution algebra Dist( ) of . + n Suppose that G is infinitesimally flat, that is, A/(A ) is finitely presented and flat as k-module (or equivalently, finite free) for any n 1. Then one ≥ sees that hy(G) has a structure of a cocommutative Hopf superalgebra such that the restriction

(2.2) , : hy(G) A k h i × −→ of the canonical pairing A∗ A k is a Hopf pairing, see [18, Lemma 5.1].

× → G

One sees that Lie(G) coincides with the set of all primitive elements P (hy( )) := G u hy(G) u u = u 1 + 1 u in hy( ). { ∈ | u (1) ⊗ (2) ⊗ ⊗ } Remark 2.3P(See [17, Remark 2]). Suppose that k is a ﬁeld of characteristic

zero. Then it is known that hy(G) coincides with the universal enveloping

G G superalgebra (Lie(G)) of the Lie superalgebra Lie( ) of .

U G For a left G-supermodule V , we regard V as a left hy( )-supermodule by letting u.v := ( 1)|v(0)||u| v u, v , − (0) h (1)i v X where u hy(G), v V and v v v(0) v(1) is the right A-supercomodule structure∈ of V . Suppose∈ that V7→is ﬁnite⊗ free. Then the dual superspace V ∗ of V forms a right A-supercomoduleP by using the antipode S : A A of → A. The induced left hy(G)-supermodule structure, which we shall denote by ⇀, satisﬁes the following equation. u ⇀ f, v = ( 1)|u||f| f, S∗(u).v , h i − h i

∗ ∗ G where v V , f V , u hy(G) and S is the antipode of hy( ). ∈ ∈ ∈

2.4. Sub-pairs. Let G be an algebraic supergroup. It is known that the

G G Lie superalgebra Lie(G) of is 2-divisible, that is, for any v Lie( )¯, ∈ 1 there exists x Lie(G)0¯ such that [v, v] = 2x, see [18, Proposition 4.2]. We ∈ 1 sometimes denote x by [v, v]. Set A := (G) and 2 O A + + 2 W := (A /(A ) )1¯. It is easy to see that the right coaction on A (2.3) coad : A A A; a ( 1)|a(1)||a(2)|a S(a )a −→ ⊗ 7−→ − (2) ⊗ (1) (3) a X BOREL-WEIL THEOREM FOR ALGEBRAIC SUPERGROUPS 7

A induces on W a right A-comodule (or equivalently, left Gev-module) struc- A ture, where S is the antipode of A. Thus, if W is ﬁnite free, then Lie(G)1¯ can be regarded as a left Gev-module.

Suppose that G satisﬁes the following three conditions.

(E0) Gev is flat, (E1) W A is finite free, and + + (E2) A /(A )2 is finite free. The labels (E0)–(E2) above are taken from [19, 5.2]. In other words, we § ′ suppose that G is an object of the category (gss-fsgroups)k, see [19, Re-

mark 5.8] for the notation. In this case, G is -split, in the sense that there exists a counit-preserving isomorphism ⊗ (2.4) A A (W A) −→ ⊗∧ of left A-comodule superalgebras, where (W A) is the exterior superalgebra ∧ on W A. Here, we naturally regard A as a left A-comodule superalgebra via A A A; a a a . → ⊗ 7→ a (1) ⊗ (2) Note that, a k-(super-)submodule of a finite free k-(super)module is again finite free, since kPis a PID. The following is a generalization of [19, Defini- tion 6.3].

Deﬁnition 2.4. Let G be an algebraic supergroup. For a closed subgroup G

K of Gev and a Lie super-subalgebra k of Lie( ) with Lie(K) = k0¯, we say G that the pair (K, k) is a sub-pair of the pair (Gev, Lie( )), if (S0) K is ﬂat,

(S1) k1¯ is K-stable in Lie(G)1¯, and (S2) k is 2-divisible. Remark 2.5. In general, a Lie super-subalgebra k of a 2-divisible Lie superalgebra g is not always 2-divisible. For example, as super-subspaces of Mat1|1(Z) (see Example A.1 for the notation), we take 2a 3b a b k = a, b, c Z g = a, b, c Z . { 3b 2a | ∈ } ⊆ { c a | ∈ } The following calculation shows that this k is not 2-divisible. 0 3 0 3 9 0 [ , ] = 2 . 3 0 3 0 0 9 Working over a PID, it is easy to see that a Lie super-subalgebra of an admissible Lie superalgebra (for the deﬁnition, see [18, Deﬁnition 3.1]) is admissible if and only is it is 2-divisible.

Proposition 2.6. Let G be an algebraic supergroup satisfying (E0)–(E2). G

For a sub-pair (K, k) of the pair (Gev, Lie( )), there exists a closed subgroup

G Ã Ã Ã Ã of which satisﬁes ev = K and Lie( )= k. Moreover, is -split. ⊗ Proof. We freely use the notations in [19, 5]. To construct the desired § supergroup Ã, it is enough to show that the pair (K, k) is object of the ′ ′ category (sHCP)k ( (gss-fsgroups)k), that is, K and k satisfy the conditions (F0)–(F5) deﬁned in≈ [19, 5.1, 5.2]. § The condition (F0) is just our (S0). By the assumption Lie(K) = k0¯, the condition (F1) is obvious. Since (K) is a quotient Hopf algebra of O

8 T. SHIBATA G (Gev) and ev satisﬁes the condition (E2), one sees that (K) satisﬁes

O ′ O

G G the condition (F2). Since G (gss-fsgroups)k, the pair ( ev, Lie( )) is an ∈′ object of the category (sHCP)k. Then by the assumption (S1), we see that (K, k) satisfies the conditions (F3) and (F4). By the assumption (S2), we h2i h2i 1 can define a 2-operation ( ) : k1¯ k0¯ by letting v := 2 [v, v]. This is obviously K-stable, which− shows that→ the condition (F5) is satisfied.

3. Structures of Quasireductive Supergroups over a PID It is known that the category of all representations of a split reductive groups over a ﬁeld of characteristic zero is semisimple, that is, any object is decomposed into a direct sum of simple modules (i.e., irreducible represen-

tations). However, algebraic supergroups G whose representation category is semisimple are rather restricted. Over an algebraically closed ﬁeld of

characteristic zero, Weissauer showed that such G (which are not ordinary algebraic groups) are essentially exhausted by ortho-symplectic supergroups

ËÔÇ(2n 1), see [31, Theorem 6]. On the other hand, over a ﬁeld of positive |

characteristic, Masuoka showed that such G are necessarily purely even, that G is, G = ev, see [17, Theorem 45]. Hence, it is natural to ask what is a nice/good generalization of the notion of split reductive groups. In 2011, Serganova [27] answered this problem. She introduced the notion of “quasireductive supergroups” over a field. In this section, we still suppose that k is a PID of characteristic not equal to 2, and we generalize her definition to our k and study its structure. 3.1. Quasireductive Supergroups. A split and connected reductive Z- group GZ is a certain connected algebraic group over Z having a split maximal torus TZ such that the pair (GZ, TZ) corresponds to some root datum, see [10, Part II, Chap. 1] (see also [18, 5.2]). It is known that (G ) is free § O Z as a Z-module and GZ is infinitesimally flat.

Deﬁnition 3.1. An algebraic supergroup GZ over Z is said to be quasireductive if its even part (GZ)ev is a split and connected reductive group over Z and W O(GZ) is ﬁnitely generated and free as a Z-module. There are plenty of examples of quasireductive supergroups. Examples 3.2. The following are quasireductive supergroups over Z.

(1) General linear supergroups GÄ(m n). (2) Queer supergroups Q(n): | X Y

Q(n)(R) := GÄ(n n)(R) , { Y X ∈ | } − where R is a commutative superalgebra over Z. The even part is

Q(n)ev = GÄn. For the Hopf superalgebra structures of Q(n), see Brundan and Kleshchev [3]. The Lie superalgebra q(n) of Q(n) is the so-called queer Lie superalgebra. (3) Chevalley supergroups of classical type deﬁned by Fioresi and Gavarini [8, 9], see also [18, 6]. For example, special linear supergroups

§ GÄ ËÄ(m n) (consists of elements in (m n) whose Berezinian deter- | | minants are trivial), ortho-symplectic supergroups ËÔÇ(m n) (see Shu and Wang [29] for details), etc. | BOREL-WEIL THEOREM FOR ALGEBRAIC SUPERGROUPS 9

(4) Periplectic supergroups P(n), see Example 5.11.

In what follows, we ﬁx a quasireductive algebraic supergroup GZ over Z G and a split maximal torus TZ of (GZ)ev. Set GZ := ( Z)ev and Λ := X(TZ). Note that, the character group X(TZ) of TZ can be identiﬁed with the set

of all group-like elements of the Hopf algebra (T ). Let G (resp. G, T ) O Z

denote the base change of GZ (resp. GZ, TZ) to our base ring k, that is, G (G) := ( Z) Z k. Since k is an integral domain, we can identify Λ with O Oℓ ⊗ X(T ) ∼= Z , where ℓ is the rank of G.

By deﬁnition, we see that the algebraic supergroup G satisﬁes the condi-

tions (E0)–(E2). Therefore, G is -split: ⊗ ∼ = O(G) (3.1) ψ : (G) (G) (W ). O −→ O ⊗∧

In particular, we have the following. G Proposition 3.3. (G) is k-free and is inﬁnitesimally ﬂat.

O G Let g = g¯ g¯ be the Lie superalgebra Lie(G) of . The left adjoint 0 ⊕ 1 action of T on G induces an action of T on g which preserves the parity of g. Since T is a diagonalizable group scheme, the left T -supermodule g decomposes into weight superspaces as follows. g = gα = ( gα) ( gδ), 0¯ ⊕ 1¯ αM∈Λ αM∈Λ Mδ∈Λ where gα is the α-weight super-subspace of g. It is known that gα = X g u ⇀ X = u, α X for all u hy(T ) , { ∈ | h i ∈ } see [10, Part I, 7.14]. Here, we regard Λ as a subset of (T ). Let h := gα=0 be the α = 0 weight super-subspace of g which is a LieO super-subalgebra of g. Note that, the even part g0¯ of g coincides with the Lie algebra Lie(G) of

G = Gev. By deﬁnition, we see that h¯ = Lie(T ). For ǫ Z , we set 0 ∈ 2 ∆ := α Λ gα = 0 0 ǫ { ∈ | ǫ 6 } \ { } and ∆¯ ∆¯ if h¯ = 0, ∆ := 0 ∪ 1 1 ∆¯ ∆¯ 0 otherwise. ( 0 ∪ 1 ∪ { }

We shall call ∆ the root system of G with respect to T . Note that, ∆0¯ is

the root system of G with respect to T in the usual sense. Suppose that

G GÄ G = Q(n) with the standard maximal torus T of ev = n. Then one sees that h¯ = 0, hence 0 ∆. 1 6 ∈

By taking the linear dual of (3.1), we see that hy(G) is -split, that is, there exists a (unique) unit-preserving isomorphism ⊗

=∼

(3.2) φ : hy(G) (g¯) hy(G) ⊗∧ 1 −→ of left hy(G)-module supercoalgebras satisfying

(3.3) φ(z), a = z, ψ(a) , a (G), z hy(G) (g¯). h i h i ∈ O ∈ ⊗∧ 1 For details, see [18, Lemma 5.2]. 10 T. SHIBATA

∗ G The dual hy(G) of the cocommutative Hopf superalgebra hy( ) naturally has a structure of a commutative superalgebra. Since our quasireduc-

tive supergroup G is connected, the canonical Hopf paring (2.2) induces an

∗ G .[injection (G) ֒ hy( ) of superalgebras, see [18, Lemma 5.3

O → G A left hy(G)-supermodule M is called a left hy( )-T -supermodule if the restricted left hy(T )-module structure on M arises from some left T -module structure on M. In this case, M is said to be locally ﬁnite if M is so as a

left hy(G)-supermodule. Then by [18, Theorem 5.8], we have the following. G Theorem 3.4. For a left G-supermodule, the induced left hy( )-supermodule

is a locally ﬁnite left hy(G)-T -supermodule. This gives an equivalence of

categories between k-free, left G-supermodules and k-free, locally ﬁnite left

hy(G)-T -supermodules. 3.2. Super-tori and Unipotent Super-subgroups. Recall that, G =

Gev, Lie(T )= h¯ and (T ) is a free Z-module. 0 O Lemma 3.5. The pair (T, h) forms a sub-pair of (G, g). Proof. The condition (S0) is obvious. By deﬁnition, we see that h is hy(T )- stable, and hence (S1) is clear. To see (S2), we take K h1¯ and X g so that [K, K] = 2X. For all u hy(T ) with u k, we see∈ that 2(u ⇀∈ X) = ∈ 6∈ u⇀ [K, K] = 0. Since char(k) = 2, we conclude that X h¯.

6 ∈ 0 G

Then by Proposition 2.6, we get a closed super-subgroup Ì of satisfying Ì

(3.4) Ìev = T, Lie( )= h. G We shall call Ì a super-torus of . Let Z∆ be the abelian subgroup of X(T ) generated by ∆. In what follows, we ﬁx a homomorphism γ : Z∆ R of abelian groups such that γ(α) = 0 for any 0 = α ∆. As in [27, 9.3],→ we set 6 6 ∈ § ∆± := α ∆ 0 γ(α) > 0 , ∆± := ∆ ∆± { ∈ \ { }|± } ǫ ǫ ∩ for ǫ Z . Set ∈ 2 n± := gα, b± := n± h. ± ⊕ αM∈∆ One easily sees that, these are Lie super-subalgebras of g. As in [10, Part II, 1.8], there is a connected and unipotent subgroup U of G such that U G , Lie(U) = n−, ∼= add 0¯ α∈∆− Y0¯ where Gadd is the one-dimensional additive group scheme over k. In the − product above, we have ﬁxed an (arbitrary) ordering of ∆0¯ . Lemma 3.6. The pair (U, n−) forms a sub-pair of (G, g). Proof. The condition (S0) is clear. Take X gδ (δ ∆−) and Y g so ∈ ∈ 1¯ ∈ that [X, X] = 2Y . For all u hy(T ), we see that 2(u⇀Y ) = 2 u, δ [X, X]. − char ∈ − h i − Since 2δ ∆0¯ and (k) = 2, we conclude that Y n0¯ , and hence n ∈ 6 ∈− is 2-divisible (S2). It remains to prove (S1), that is, n1¯ is U-stable in g1¯. − Since n1¯ is obviously G-stable in g1¯, we see that it is a right hy(G)-module via the induced action. BOREL-WEIL THEOREM FOR ALGEBRAIC SUPERGROUPS 11

− First, we show that n1¯ is hy(U)-stable in g1¯. It is known that we can choose an element X gα for each α ∆− so that the set α ∈ 0¯ ∈ 0¯ X(nα) n = 0, 1, 2, 3,... { α | α } α∈∆− Y0¯

(nα) forms a k-basis of hy(U). Here, Xα denotes the divided power, see [10, − (nα) hy Part II, 1.12] (see also 3.4 below). We take Xβ n1¯ and Xα (U). § (nα) ∈ ∈ It is easy to see that the T -weight of Xα ⇀ X ( g) is given by β + n α. β ∈ α Since β ∆− and α ∆−, we see γ(β + n α) = γ(β)+ n γ(α) 0, and ∈ 1¯ ∈ 0¯ α α ≤ − (nα) − hence β + nαα is in ∆1¯ . Thus, Xα ⇀ Xβ is an element of n1¯ . This − hy implies that n1¯ is (U)-stable in g1¯. − Next, we show that n1¯ is indeed U-stable. By construction, the corresponding Hopf algebra (U) of U is isomorphic to the polynomial algebra − O− k[Tα; α ∆0¯ ] in #(∆0¯ )-variables Tα over k with each Tα is primitive. −∈ − Since n1¯ is ﬁnite free, the right hy(U)-module structure on n1¯ induces the following (well-deﬁned) left (U)-comodule structure on n−. O 1¯ n− (U) n−; Y T nα (X(nα) ⇀Y ), 1¯ −→ O ⊗ 1¯ 7−→ α ⊗ α α∈∆− nα≥0 X0¯ X −

see [10, Part II, 1.20] for detail. Thus, we conclude that n1¯ is U-stable. G By Proposition 2.6, we obtain a closed super-subgroup Í of which satisﬁes

− Í (3.5) Íev = U, Lie( ) = n .

The supergroup Í is connected, and hence the canonical Hopf paring in-

∗ Í ,duces an injection (Í) ֒ hy( ) of superalgebras, as before. By [17 Theorem 41], we getO the following.→

Proposition 3.7. Suppose that k is a ﬁeld. Then the supergroup Í is

unipotent, that is, the corresponding Hopf superalgebra (Í) is irreducible. O

+ + +

G Í Similarly, we get a closed super-subgroup Í of such that ev = U , + + + + Lie Í ( )= n , where U is an unipotent subgroup of G corresponds to ∆0¯ .

3.3. Borel Super-subgroups. Our next aim is to construct “Borel super-

subgroups” of G. Í

Lemma 3.8. Ì normalizes . Ì Proof. First, we work over Q := Frac(k) and show that Ì := Q is

Q ×k

Í Í G included in the normalizer G ( ) of in . In the following, for N Q Q Q Q

simplicity, we shall drop the subscript Q. By [19, Thorem 6.6(1)], to show

Ì Í

G ( ), it is enough to check the following conditions (i) and (ii): ⊆N Stab − (i) T is contained in G(U) G(n1¯ ); N ∩ − − − (ii) h¯ is contained in Inv (g¯/n ) (n : n ). 1 U 1 1¯ ∩ 0¯ 1¯ 12 T. SHIBATA

− − − Here, StabG(n1 ) is the stabilizer of n1¯ in G, InvU (g1¯/n1¯ ) is the largest U- − − submodule of g1¯ including n1¯ whose quotient U-module by n1¯ is trivial, and − − − − (n0¯ : n1¯ ) := X g1¯ [X, n1¯ ] n0¯ . { ∈ | ⊆ −} By the deﬁnition of root spaces, n1¯ is trivially T -stable. Since T G(U) ⊆N− − is obvious, the condition (i) is clear. It is easy to see that [h1¯, n1¯ ] n0¯ . − − ⊆ Thus, we have h1¯ (n0¯ : n1¯ ). By using the Hopf pairing hy(U) (U) Q, one sees that ⊆ ⊗O → − ∗ − Inv (g¯/n ) = X g¯ (u ⇀ X) ε (u)X n for all u hy(U) , U 1 1¯ { ∈ 1 | − U ∈ 1¯ ∈ } ∗ (n) where εU is the counit of hy(U). For each X h1¯ and u = Xα hy(U) − ∈ − ∗ ∈ (α ∆0¯ , n Z≥1), the weight of u ⇀ X is nα ∆ . Since εU (u) = 0, we ∈ ∈ ∗ − ∈ ∗ have (u ⇀ X) εU (u)X n1¯ . If u Q, then (u ⇀ X) εU (u)X = 0. This − − ∈ ∈ − shows h¯ Inv (g¯/n ), and hence the condition (ii) follows. Thus, we get

1 ⊂ U 1 1¯

Ì Í

G ( ) over Q.

⊆N

Ì Í Ì

Next, we prove that G ( ) over k. Since ( ) is k-free, the canoni-

⊆N O Ì cal map (Ì) ( ) is injective. Then one easily sees that the canonical

O → O Q Ì Í ։ quotient map ( G ( )) ( ) induces the following diagonal arrow:

O N Q Q O Q Í

G / /

( ) ( G ( )) r O rO N r r yr

(Ì) O Here, the horizontal and vertical arrows are the canonical quotient maps.

Thus, we are done.

Í Ì Í

Let Ì ⋉ be the crossed product supergroup scheme of and . Namely,

Í Ì Í for any superalgebra R, the group (Ì ⋉ )(R) is just (R) (R) as a set, and the group multiplication is given by ×

−1 Í (t, u) (s, v) = (ts, s usv), s,t Ì(R), u, v (R).

∈ ∈ G We deﬁne a closed super-subgroup B of as the image (in the sense of [10,

−

Í G Part I, 5.5]) of the multiplication map Ì ⋉ . Since T U and h n

→ ∩ ∩ Í are trivial, the intersection Ì is also trivial. Thus, the multiplication

∩

Ì Í B map of G induces an isomorphism ⋉ ∼= of algebraic supergroups. By dualizing, we have an isomorphism of Hopf superalgebras ∼

= ◮

Ì Í Ì Í (3.6) (B) ( ⋉ ) = ( ) < ( ), O −→ O O O

◮

Í Í Ì where (Ì) < ( ) denotes the Hopf-crossed product of ( ) and ( )

O O ◮ O O

Í Ì Í in SMod. Note that, (Ì) < ( ) is just ( ) ( ) as a superalgebra.

O O O ⊗ O

Ì Í The identiﬁcation B = ⋉ implies that

− B (3.7) B := Bev = T ⋉ U, Lie( )= b .

Thus, B is a Borel subgroup of G. In this sense, we shall call B a Borel B super-subgroup of G (with respect to γ). The supergroup is connected,

∗ B ( )and hence the canonical Hopf paring induces an injection (B) ֒ hy

+ O +→

Ì Í G of superalgebras, as before. Similarly, we deﬁne B := Im( ⋉ ) =

+ → ∼ Í Ì ⋉ . BOREL-WEIL THEOREM FOR ALGEBRAIC SUPERGROUPS 13

+ +

Í B B Remark 3.9. Note that, the groups Í, , and are depending on the choice of the homomorphism γ : Z∆ R, deﬁned in 3.2. In addition, not all (possible) Borel super-subgroups are→ conjugate. § − Since T B and h b , we see that Ì is a closed super-subgroup of

⊆ ⊆

Ì B B. Then the inclusion map ֒ induces a Hopf superalgebra surjection

→ Ì (B) ։ ( ).

O O Ì Proposition 3.10. The morphism (B) ։ ( ) is split epic. O O Proof. We consider the following Hopf superalgebra map id ∼

1⊗ ◮ =

Ì Í B

spl Ì < (3.8) B : ( ) ( ) ( ) ( ), O −→ O O −→ O where the ﬁrst arrow is given by x 1 x and the second arrow is the

7→ ⊗ spl

inverse map of (3.6). Then one easily sees that this B gives a section of Ì (B) ։ ( ). O O 3.4. Kostant’s Z-Form and PBW Theorem. As a superspace, we have α δ g = (h¯ g ) (h¯ g ). 0 ⊕ 0¯ ⊕ 1 ⊕ 1¯ αM∈∆0¯ δM∈∆1¯ α Note that, for α ∆¯, the rank of g is always 1. However, for δ ∆¯, the ∈ 0 0¯ ∈ 1 rank of gδ may be greater than 1, see [27, Lemma 9.6]. For ǫ Z , we set 1¯ ∈ 2 ℓǫ := rank(hǫ).

Set gZ := Lie(GZ) and hZ := h gZ. For X (gZ)0¯ (hZ)0¯ and H (hZ)0¯, as elements in hy(G ) Q, we set∩ ∈ \ ∈ Z ⊗Z m−1 1 1 X(n) := Xn ,H(m) := ( (H j)) , ⊗Z n! − ⊗Z m! jY=0 where n,m Z . Here, we set H(0) := 1. As in [10, Part II, 1.11], we ∈ ≥0 choose a Z-free basis of (gZ)0¯ α ¯ := X (g ) α ∆¯ H (h )¯ 1 i ℓ¯ B0 { α ∈ Z 0¯ | ∈ 0} ∪ { i ∈ Z 0 | ≤ ≤ 0} (nα) (mi) so that the set of all products of factors of type Xα and H (n ,m i α i ∈ Z , α ∆¯, 1 i ℓ¯), taken in hy(G ) with respect to any order on ¯ ≥0 ∈ 0 ≤ ≤ 0 Q B0 forms a Z-basis of hy(GZ). Since (gZ)1¯ is also Z-free, we take a Z-free basis of gZ as follows. δ δ ¯ := Y (g ) δ ∆¯, 1 p rank(g ) K (h )¯ 1 t ℓ¯ .

B1 { (δ,p) ∈ Z 1¯ | ∈ 1 ≤ ≤ 1¯ }∪{ t ∈ Z 1 | ≤ ≤ 1}

G G We see that hy(G) can be identiﬁed with hy( ) k, since is inﬁnites-

Z ⊗Z Z G imally ﬂat (Proposition 3.3). As elements of hy(G)= hy( ) k, we set Z ⊗Z X := X(n) 1,H := H(m) 1, α,n α ⊗Z i,m i ⊗Z Y := Y ǫ 1, K := Kǫ 1,

(δ,p),ǫ (δ,p) ⊗Z t,ǫ t ⊗Z G where ǫ = 0 or 1. Then the supercoalgebra isomorphism hy(G) = hy( ev) ∼ ⊗ (g¯) given in (3.2) implies the following Poincar´e-Birkhoﬀ-Witt (PBW) ∧ 1 theorem for hy(G). 14 T. SHIBATA

Theorem 3.11. For any total order on the set 0¯ 1¯, the set of all products of factors of type B ∪B

Hi,m(i), Xα,n(α), Kt,ǫ(t), Y(δ,p),ǫ(δ,p) δ (n(α),m(i) Z , α ∆¯, 1 i ℓ¯, δ ∆¯, 1 p rank(g ), ∈ ≥0 ∈ 0 ≤ ≤ 0 ∈ 1 ≤ ≤ 1¯

1 t ℓ¯ and ǫ(t),ǫ(δ, p) 0, 1 ), taken in hy(G) with respect to the ≤ ≤ 1 ∈ { } order, forms a k-basis of hy(G). Remark 3.12. To construct Chevalley supergroups over Z, it is necessary to prove that (1) any ﬁnite-dimensional simple Lie superalgebra s over C has a Chevalley basis; (2) the Kostant’s Z-form of (s) has a PBW basis,

like above. These were done by Fioresi and GavariniU [8, Theorems 3.7, 4.7].

G G Since the k-valued points of G is just (k)= ev(k), it is hard to consider

B G a geometrical “denseness” of B in (or, a “big cell” as in [10, Part II, 1.9]) directly. The following is an× algebraic interpretation of the denseness, see also [24] and [1]. Corollary 3.13. The following superalgebra map is injective.

G G B B (3.9) (G) ( ) ( ) ( ) ( ), O −→ O ⊗ O −→ O ⊗ O

where the ﬁrst arrow is the comultiplication map of (G) and the second

O B arrow is the tensor product of the canonical quotient maps (G) ։ ( )

+ O O B and (G) ։ ( ). O O

֒ (Proof. As in the proof Proposition 3.10, we also get an injection (Í

O → G (B). Thus, to prove the claim, it is enough to see that the map ( )

O + O → B (Í) ( ) is injective.

O ⊗ O +

B G The multiplication map µ : Í induces a morphism hy(µ) :

+ × →

B G hy(Í) hy( ) hy( ) of supercoalgebras which is indeed bijective, by n− b⊗+ = g and→ Theorem 3.11. Then the k-linear dual hy(µ)∗ is an iso-

⊕ + B morphism of superalgebras. Since hy(Í) and hy( ) are both k-free, the

∗ + ∗ + ∗

B Í B canonical map hy(Í) hy( ) hy( ) hy( ) is injective. There- fore, we get the following⊗ commutative→ diagram⊗ of superalgebras: hy(µ)∗ ∗ + ∗ ? _ ∗ + ∗

G / o

B Í B hy( ) hy(Í) hy( ) hy( ) hy( ) O =∼ ⊗ ⊗O +

G / B ( ) (Í) ( ). O O ⊗ O

Í B The vertical canonical maps are injection, since G, and are connected. Therefore, the lower horizontal arrow is also injective.

4. Borel-Weil Theorem for G

Throughout the rest of the paper, we assume that k is a ﬁeld of character-

B Í Ì istic diﬀerent from 2. All supergroups G, , , , etc. are deﬁned over the

ﬁeld k. In this section, we will construct all simple G-supermodules, which extends Serganova’s construction [27, 9] to arbitrary characteristic. The main idea is based on Brundan and Kleshchev’s§ argument [3, 6], see also Parshall and Wang [24], Bichon and Riche [1] for the non-super§ situation. BOREL-WEIL THEOREM FOR ALGEBRAIC SUPERGROUPS 15

4.1. Simple Ì-Supermodules. We will construct all simple supermodules G of the super-torus Ì of . In the following, we freely use the notations used in Appendix B. Recall that, ℓ denotes the dimension of h for ǫ Z . By Theorem 3.11,

ǫ ǫ ∈ 2 Ì the super-hyperalgebra hy(Ì) of has a k-basis

ℓ0¯ ℓ1¯ (4.1) H K m(i) Z , ǫ(t) 0, 1 . { i,m(i) t,ǫ(t) | ∈ ≥0 ∈ { }} t=1 Yi=1 Y In the following, we ﬁx λ Λ and write λ(H) := H, λ for H h0¯ ( ∈ h i λ ∈ ⊆ hy(T )). Since H (h )¯, we see that λ(H ) Z. We let hy(Ì) denote the

i ∈ Z 0 i ∈ Ì quotient superalgebra of the super-hyperalgebra hy(Ì) of by the two-sided λ(Hi) super-ideal of hy(Ì) generated by all Hi,m m , where 1 i ℓ0¯ and m 0. Here, we used − ≤ ≤ ≥ m−1 c 1 c := (c j), := 1 m m! − 0 jY=0 for m 1 and c Z. Then we get the following. ≥ ∈ λ ℓ1¯ dim hy(Ì) = 2 .

For x,y h1¯, we see that [x,y] h0¯, and hence we can deﬁne a bilinear λ ∈ ∈ map b : h¯ h¯ k as follows. 1 × 1 → bλ(x,y) := λ([x,y]), λ where x,y h1¯. It is easy to see that (h1¯, b ) forms a quadratic space over ∈ λ λ k. Thus, we get the Cliﬀord superalgebra Cl(h1¯, b ) := T (h1¯)/I(h1¯, b ) over k. We may regard h1¯ as a subspace of hy(Ì), since h coincides with the set

of all primitive elements in hy(Ì). Then we have the following map.

λ Ì (4.2) T (h¯) hy(Ì) hy( ) , 1 −→ −→ where T (h1¯) is the tensor algebra of h1¯. λ λ Lemma 4.1. The map above induces an isomorphism Cl(h1¯, b ) ∼= hy(Ì) of superalgebras. Proof. It is easy to see that (4.2) is surjective and the kernel contains the λ λ λ ideal I(h1¯, b ) of T (h1¯). Thus, (4.2) induces a surjection Cl(h1¯, b ) hy(Ì) of superalgebras. On the other hand, it is known that the dime→nsion of λ ℓ¯ Cl(h1¯, b ) is 2 1 . Thus, we are done. λ Take a non-degenerate subspace (h1¯)s of h1¯ so that h1¯ = rad(b ) (h1¯)s, and set ⊥ λ (4.3) d := ℓ¯ dim rad(b ) . λ 1 − We choose an orthogonal basis x1,...,xd of (h¯)s and we let { λ } 1 (4.4) δ := ( 1)dλ(dλ+1)/2 λ([x ,x ]) λ([x ,x ]), λ − 1 1 · · · dλ dλ the signed determinant of (h1¯)s, see (B.4). For simplicity, we let δλ = 0 if bλ = 0. By Lemma 4.1 and Proposition B.3, we have the following. 16 T. SHIBATA

λ Proposition 4.2. The superalgebra hy(Ì) has a unique simple supermodule u(λ) up to isomorphism and parity change. If (1) δλ = 0 or (2) dλ is even and δ (k×)2, then Πu(λ) = u(λ). Otherwise, Πu(λ)= u(λ). λ ∈ 6

λ Ì For a left hy(Ì) -supermodule M, we may regard M as a left hy( )-

λ Ì supermodule via the canonical quotient map hy(Ì) hy( ) . It is easy to → see that M is a left hy(Ì)-T -supermodule. Since u(λ) is ﬁnite-dimensional,

it is obviously locally ﬁnite as a left hy(Ì)-T -supermodule.

Lemma 4.3. Let L be a locally ﬁnite left hy(Ì)-T -supermodule. If L is simple, then there exists λ Λ such that L is isomorphic to u(λ) or Πu(λ). ∈ Proof. Since L is non-zero, there exists λ Λ such that the λ-weight space λ λ ∈ λ L of L is non-zero. By deﬁnition, L is a hy(Ì) -supermodule. Hence by Proposition 4.2, Lλ contains u(λ) or Πu(λ), say u(λ). The simplicity

assumption on L implies L = u(λ). Ì By Theorem 3.4 for Ì, we see that the category of locally ﬁnite left hy( )-

T -supermodules is equivalent to the category of Ì-supermodules. Thus, we

may regard u(λ) as a Ì-supermodule, and hence we get the following map.

Λ= X(T ) Simple (Ì); λ u(λ). −→ Π 7−→ Proposition 4.4. The above map is bijective. Moreover, u(λ) is of type M if and only if (1) δ = 0 or (2) d is even and δ (k×)2. λ λ λ ∈

Remark 4.5. As a Ì-supermodule, this u(λ) is isomorphic to some copies of kλ up to parity change, where kλ is the one-dimensional purely even left λ hy(T )-supermodule of weight λ. If h = h0¯, then we have hy(Ì) = k, and hence u(λ) is just kλ or Π kλ. By Proposition B.4, we have the following. Proposition 4.6. If k is algebraically closed, then the dimension of u(λ) is given by 2⌊(dλ+1)/2⌋. Remark 4.7. In [27, 9.2], Serganova constructed simple h-supermodules § Cλ for each λ Λ, when the base ﬁeld is algebraically closed of characteristic zero. Assume∈ that our base ﬁeld k is algebraically closed (not necessarily char(k) = 0). In the following, we shall generalize her construction (in the supergroup level) to our k, and show that our u(λ) is isomorphic to her Cλ up to parity change, when char(k) = 0. † λ Fix a maximal totally isotropic subspace h1¯ of (h1¯, b ). We set † † h := h¯ h . 0 ⊕ 1¯ This forms a Lie super-subalgebra of h. It is obvious that the pair (T, h†) is † a sub-pair of the pair (T, h), and hence we get a closed super-subgroup Ì of

† † Ì Ì. In particular, hy( ) is isomorphic to hy(T ) (h ) as supercoalgebras. ⊗∧ 1¯ We regard the one-dimensional purely even left hy(T )-supermodule kλ as † λ † a left hy(Ì )-supermodule by letting K.k = 0 for any K h . Set ∈ 1¯

Ì λ

coind hy Ì † (4.5) † (λ) := ( ) Ì k . Ì ⊗hy( ) BOREL-WEIL THEOREM FOR ALGEBRAIC SUPERGROUPS 17

† Ì ℓ1¯−dim(h¯) λ

By (4.1), we see that dim(coind † (λ)) = 2 1 . Since the radical rad(b ) Ì

† Ì is contained in h , one sees that coind † (λ) is a hy(Ì)-T -supermodule, and

1¯ Ì Ì

hence it is a Ì-supermodule. Thus, coind † (λ) must contain u(λ) or Πu(λ),

Ì Ì

by Proposition 4.4. Then by (B.6), we conclude that coind † (λ) is iso-

Ì Ì

morphic to u(λ) up to parity change. If char(k) = 0, then our coind † (λ) Ì coincides with Serganova’s Cλ by Remark 2.3. 4.2. Cohomology Groups and Induced Supermodules. In this sec-

tion, we shall prepare some notations. Let À be a closed super-subgroup of

G À G À G. We consider the functor ( / )(n) : R (R)/ (R) from the category

of superalgebras to the category of sets, which7→ is called the naive quotient of

À G À À G G over . Here, (R)/ (R) is the set of right cosets of (R) in (R), as

usual. Then by Masuoka and Zubkov [21, Theorem 0.1], the sheaﬁﬁcation

À G À

G/ of the native quotient ( / )(n) becomes a noetherian superscheme

G G À À endowed with a morphism πG : / satisfying the conditions (Q1)– / → (Q5) described in Brundan’s article [2, 2]. See also Masuoka and Takahashi

§ L L À [20, 4.4]. As in [2, 2] (see also [10, Part I, 5.8]), we let = G be a § § § /

O G

À SMod À functor from to the category of quasi-coherent G/ -supermodules satisfying L U −1 U

(V )( ) = V À (π ( )) À O( ) O G/

−1 À U G U

for V À SMod and an open super-subfunctor / such that π ( ) À

∈ ⊆ G/ À

is aﬃne. This L(V ) is the so-called associated sheaf to V on G/ . B

Now, let us return to our situation. Recall that, G = Gev and B = ev. B It is easy to see that the quotient G/ satisﬁes the condition (Q6) in [2, 2],

B G B that is, the even part G/B = (G/ )ev of the quotient / is projective.

Recall that, u(λ) is a simple Ì-supermodule for λ Λ. For simplicity,

n Ì∈ n B we shall denote the cohomology group H G/ , L(res u(λ)) by H (λ) for n B λ Λ and n Z≥0. Note that, H (λ) does depend on the choice of the morphism∈ γ : ∈Z∆ R deﬁned in 3.2. Brundan [2, Corollary 2.4] showed that for each n 0,→ we have an isomorphism§

≥ Ì

Hn(λ) RnindG res (u(λ)) B ∼= B of G-supermodules. This is a super-analogue result of [10, Part I, Proposi- tion 5.12]. In the following, we shall use this identiﬁcation. By construction,

Ì λ

res (u(λ)) is isomorphic to some (ﬁnite) copies of k as B-supermodules. B

Thus, for c a u(λ) (G), we have ⊗ ∈ ⊗ O c a H0(λ) c λ a = c a B a , ⊗ ∈ ⇐⇒ ⊗ ⊗ ⊗ (1) ⊗ (2) a

B B where a is the canonical image of a (G) in ( ).

∈ O O

Í B B 4.3. Simple B-Supermodules. The inclusion makes ( ) into

⊆ O

Ì Í Í a right (Í)-supercomodule. We regard ( ) ( ) as a right ( )- O O ⊗ O O

id B supercomodule via Í . Then one sees that the isomorphism ( ) =

⊗△O( ) O ∼

Í Í (Ì) ( ) given in (3.6) is right ( )-colinear. By taking the functor O ⊗ O O

( ) Í k to both sides, we get

− O( )

Í Ì (4.6) (B) = ( ), O ∼ O

18 T. SHIBATA

Í B where (B) is the -invariant super-subspace of ( ), in other words, the

O coO(Í) O

B B (Í)-coinvariant super-subspace ( ) of ( ), see Appendix A.2. It

O O O Ì is easy to see that this isomorphism is left (B)- right ( )-colinear.

O O B

In the following, for a left B-supermodule V , we consider res (V ) and

Ì B Í

ind (V ) via the quotient maps (B) ։ ( ) and ( ) ։ ( ) respec- B

tively. O O O O

B Í

Lemma 4.8. For a left B-supermodule V , res (V ) is isomorphic to ind (V )

B Í

as left Ì-supermodules.

Proof. It is easy to see that the canonical isomorphism M B ( ) = V O( ) O ∼

Í is left ( )-colinear. Then by taking the functor ( ) O(Í) k to both sides,

we getO −

B Í

B res Í V B ( ) k (V ) . = Í

O( ) O O( ) ∼ Í

The left hand side is isomorphic to V B ( ( ) ). Thus, by combining O( ) O

this with (4.6), we are done. Ì

For a left Ì-supermodule N, we consider res (N) via the Hopf superalge-

B B spl Ì

bra splitting B : ( ) ֒ ( ) given in (3.8). Since the composition the Ì

O → O B Ì

spl B ։ res res

splitting B with the quotient ( ) ( ) is identity, ( (N)) is

Ì B

just the original N. O O Ì

Proposition 4.9. For λ Λ, the left B-supermodule res (u(λ)) is simple. ∈ B Moreover, this gives a one-to-one correspondence between Λ and SimpleΠ(B).

Proof. It is enough to show that for any simple B-supermodule L, there Ì

exists λ Λ such that L = res Ì (u(λ)) or Πres (u(λ)). Since L = 0, we

B B

∈Í 6 see that L = 0, by Proposition 3.7 and Lemma A.4. Then by Lemma 4.8,

there exists λ6 Λ such that indÌ (L) contains u(λ) or Πu(λ). For simplicity, B

∈ Ì

we shall concentrate on the case when u(λ) ind (L). Then by Frobenius B

reciprocity (2.1), we get ⊆

Ì Ì Ì

B SMod res u(λ) , L SMod u(λ), ind (L) = 0. B B =

∼ 6 Ì

Hence, we get a B-supermodule surjection res (u(λ)) ։ L. By applying the

B B

functor res B ( ) to both sides, we have u(λ) ։ res (L). Since this is indeed

Ì Ì a bijection, we− are done.

4.4. Simple G-Supermodules. Set ♭ 0 0

Λ := λ Λ H (λ) = 0 , L(λ) := soc G (H (λ)). { ∈ | 6 } ♭ Note that, for λ Λ , the left G-supermodule L(λ) is non zero by Lemma A.3.

∈ Ì

Ì res Let N be a left -supermodule. By taking the functor (N) B ( ) B O( )

+ −

B B to the superalgebra inclusion (G) ֒ ( ) ( ) given in (3.9), we get

O → O ⊗ O Ì

Ì +

B B

res G res B

.(( ) ( ) ) (N) B ( ) ֒ (N) B

B O( ) O → O( ) O ⊗ O G + Ì

֒ ((Since the right hand side is equal to N (B ), we have ind (res (N B + ⊗ O B → N (B ). It is easy to see that the image of the above map lies in ⊗ O +

N Ì ( ). Thus, we get an inclusion O( ) O

G G Ì B

(res + (ind (res (N))) ֒ ind (N (4.7) Ì B B B →

+ + B of right (B )-supercomodules, or equivalently left -supermodules. O BOREL-WEIL THEOREM FOR ALGEBRAIC SUPERGROUPS 19

G 0 0

For simplicity, we write res + (H (λ)) as H (λ) for λ Λ. Then by (4.7), B ∈+ 0 + B

H (λ) can be regarded as a B -super-submodule of ind (u(λ)). Ì

♭ 0 Ì soc + res Lemma 4.10. For λ Λ , we have B (H (λ)) = + (u(λ)). ∈ B + 0 B

soc + soc + ind B

Proof. We see that B (H (λ)) is contained in (u(λ)) . To Ì conclude the proof, it is enough to show that

B Ì soc + ind res

(4.8) B (u(λ)) = + (u(λ)). Ì B + + B By Proposition 4.9, any simple B -super-submodule of ind (u(λ)) is ether

Ì Ì

(i) res + (u(µ)) or (ii) Πres + (u(µ)) for some µ Λ. First, we consider the

B B case (i). In this case, we have ∈

Ì B + SMod res ind Ì SMod

0 = B + u(µ) , (u(λ)) u(µ), u(λ) , Ì = 6 B ∼

Ì B by Frobenius reciprocity (2.1) and ind + (ind (u(λ))) = u(λ). Thus, we

Ì B conclude that λ = µ, and hence the equation (4.8) holds. Next, we consider the case (ii). Similarly, by Frobenius reciprocity, we have Πu(µ) ∼= u(λ). Thus, we conclude that µ = λ and u(λ) is of type Q, and hence the equation (4.8) also holds. ♭ Proposition 4.11. For λ Λ , the left G-supermodule L(λ) is a unique simple super-submodule of H∈0(λ). ′ 0

Proof. Suppose that L,L are two simple G-super-submodules of H (λ). G G ′

soc + res soc + res B

Then by Lemma 4.10, B ( + (L)) and ( + (L )) should coin-

B B

Ì ′ ′ cide with res + (u(λ)). Thus, u(λ) is included in L L , and hence L = L . B ∩ By Proposition 4.11, we get the following map. ♭

Λ Simple (G); λ L(λ). −→ Π 7−→ The next is our main theorem, which is a generalization of Serganova’s result [27, Theorem 9.9].

Theorem 4.12. The map above is bijective. Moreover, if (1) δλ = 0 or (2) × 2 dλ is even and δλ (k ) , then L(λ) is of type M. Otherwise, L(λ) is of type Q. ∈

G ∗ G

Proof. Let L be a simple -supermodule. We see that socB (res (L )) = 0

Ì Ì G 6

by Lemma A.3. Then res (L∗) includes res (u(µ)) or Πres (u(µ)) for some

B B B

µ Λ. Taking the linear dual ( )∗, we get a morphism from resG (L) to

B Ì ∈Ì −

res (u(µ))∗ or Πres (u(µ))∗. Here, we used L = L∗∗. Since u(µ)∗ is simple,

B B

there exists λ Λ such that u(µ)∗ = u(λ) by Proposition 4.9. Thus, we get

Ì Ì

a non trivial morphism∈ from resG (L) to (i) res (u(λ)) or (ii) Πres (u(λ)).

B B B

First, we consider the case (i). By Frobenius reciprocity (2.1), we get Ì

G 0 Ì

0 = B SMod res (L), res (u(λ)) SMod L, H (λ) . B B = 6 ∼ 0 Thus, we conclude that L ֒ H (λ), and hence L = L(λ). Next, we consider ,(the case (ii). In this case,→ a similar argument ensures that L ֒ ΠH0(λ and hence L = ΠL(λ). → 0 Since ΠL(λ) = socG(ΠH (λ)), we conclude that L(λ) is of type M (resp. Q) if and only if u(λ) is of type M (resp. Q). Thus, the last statement directly follows from Proposition 4.4. 20 T. SHIBATA

λ If h1¯ = 0, then b = 0 by deﬁnition. Hence, in this case, L(λ) is always

of type M, that is, L(λ) is not isomorphic to ΠL(λ) as G-supermodules.

Remark 4.13. For the queer supergroup G = Q(n), the result above is a part of Brundan and Kleshchev’s result [3, Theorem 6.11] (where the base ﬁeld k is assumed to be algebraically closed). Indeed, one easily sees that their hp′ (λ) coincides with our dλ, where p := char(k) and hp′ (λ) :=# i n { ∈ 1,...,n p ∤ d for λ = d λ + + d λ Λ = Zλ . { } | i} 1 1 · · · n n ∈ ∼ i=1 i 4.5. Some Properties of Induced Supermodules.L In this section, we study some properties of H0(λ) for λ Λ♭. ∈ + ♭ 0 Í Lemma 4.14. For λ Λ , we have H (λ) = u(λ) as left Ì-supermodules. ∈ ∼

+ B Ì 0

,((Proof. By taking the functor ind + ( ) to the inclusion H (λ) ֒ ind (u(λ

Ì B

Ì 0 − → we get an inclusion ind + (H (λ)) ֒ u(λ) of Ì-supermodules, and hence this B + → is bijective. Then by a B -version of Lemma 4.8, we are done. For λ Λ♭, we regard H0(λ) as a left T -supermodule via the left adjoint action, as∈ usual. Then for µ Λ ( (T )), the µ-weight superspace of H0(λ) is described as follows. ∈ ⊆ O H0(λ)µ = ξ H0(λ) u⇀ξ = u, µ ξ for u hy(T ) . { ∈ | h i ∈ } We define a partial order on Λ as follows. ≤ (4.9) µ λ : λ µ = nαα for some nα Z≥0. ≤ ⇐⇒ − + ∈ αX∈∆ The following proof is based on the proof of [10, Part II, Proposition 2.2(a)]. Proposition 4.15. For λ Λ♭, λ is a maximal T -weight of H0(λ) with 0 λ ∈ respect to and H (λ) = u(λ) as left Ì-supermodules. ≤ ∼ Proof. Suppose that µ Λ is a maximal T -weight of H0(λ). Then by ∈ 0 µ + definition, the µ weight superspace H (λ) is included in the Í -invariant + super-subspace H0(λ)Í of H0(λ). On the other hand, since u(λ) is isomorphic to some copies of kλ as T - modules, we see that H0(λ)λ = 0 by Lemma 4.14. We fix a non-zero element x H0(λ)λ and consider the6 following map. ∈ 0 0 λ

H (λ) H (λ) ; c a c εG (a)x,

−→ ⊗ 7−→ ⊗

G G G where c a u(λ) ( ) and εG : ( ) k is the counit of ( ). We ⊗ ∈ ⊗ O will show that this is injective. For simplicity,O → we assume that uO(λ) = kλ. + 0 Í

For c a H (λ) with c = 0 and εG (a) = 0, it is easy to see that the

⊗ ∈ 6 +

Í B canonical images of a (G) in ( ) and ( ) are both zero. As in ∈ O O O the proof of Corollary 3.13, we can also prove that the composition (G)

+ O →

G Í B (G) ( ) ( ) ( ) is injective. Thus, we can conclude that a = + O ⊗O → O ⊗O 0 Í 0 λ 0, and hence we may regard H (λ) as a Ì-super-submodule of H (λ) . + Combined with the above, we get µ = λ and H0(λ)Í = H0(λ)λ.

Set A := (G). Recall that, IA is the super-ideal of A generated by the O 0 odd part A¯, see 2.2. Then the (ﬁnite) descending chain I := A I 1 § A ⊇ A ⊇ BOREL-WEIL THEOREM FOR ALGEBRAIC SUPERGROUPS 21

I2 deﬁnes the graded (ordinary) algebra A ⊇··· n n+1 gr(A) := IA/IA Mn≥0 of A. On the other hand, we regard A as a right A-comodule via the right coadjoint action (2.3). This coaction makes (W A) into a right A-comodule ∧ Hopf algebra, and hence we can construct the cosmash product A ◮< (W A). ◮ A ∧ Then we have an isomorphism gr(A) ∼= A < (W ) of graded (ordinary) Hopf algebras, see [16, Proposition 4.9(2)]. Recall∧ that, T is a split maximal torus of G. Since (T ) is a cosemisimple Hopf algebra, we see that the graded algebra gr(AO) is isomorphic to A as right (T )-comodules via the adjoint action of T . Hence, we get an isomorphismO =∼ (4.10) A A ◮< (W A) −→ ∧ of left A- right (T )-comodules (or equivalently, right G- left T -modules). O Here, the left A-comodule structure of right hand side is id.

△A ⊗ B Recall that G = Gev and B = ev. Then we have the following lemma.

Lemma 4.16. For a left B-supermodule V , there is an inclusion

B G (( )indG (V ) ֒ indG(res (V )) (W O B → B B ⊗∧ of left T -modules. G Proof. Taking the functor V O(B) resB( ) to both sides in (4.10), we get ∼ − = A V O(B) A (V O(B) A) (W ). By the deﬁnition of the cotensor, we → ⊗∧ see that V O(B) A is contained in V O(B) A. This completes the proof.

Proposition 4.17. For a ﬁnite-dimensional left B-supermodule V , the in-

duced supermodule indG (V ) is ﬁnite-dimensional. In particular, H0(λ) is B ﬁnite for any λ Λ. ∈

O(G) Proof. Since (G) is a finitely generated superalgebra, W is finite- dimensional byO [16, Proposition 4.4]. On the other hand, it is known that G indB(V ) is finite dimensional, see [10, Part I, 5.12(c)] for example. This claim follows immediately from Lemma 4.16.

0 B Recall that, G = Gev and B = ev. As a non-super version of H (λ), we set H0 (λ) := indG(kλ) and Λ+ := λ Λ H0 (λ) = 0 . ev B { ∈ | ev 6 } for λ Λ. It is known that Λ+ consists of all dominant weights of T with respect∈ to ∆+, that is, Λ+ = λ Λ λ, α∨ 0 for all α ∆+ , see 0¯ { ∈ | h i ≥ ∈ 0¯ } [10, Part II, 2.6] for example. Then by Lemma 4.16, we get the following Proposition. Proposition 4.18. For λ Λ, there is an inclusion ∈ ((H0(λ) ֒ H0 (λ)⊕nλ (W O(G → ev ⊗∧ of left T -modules, where nλ := dim(u(λ)). In particular, we see that the parameter set Λ♭ is included in Λ+. 22 T. SHIBATA

Remark 4.19. Assume that k is algebraically closed. Brundan and Kleshchev [3, Theorem 6.11] determined Λ♭ explicitly for queer supergroups Q(n); ♭ + ♭ Λ = Xp (n) in their notation. Shu and Wang [29, Theorem 5.3] describe Λ ♭ † combinatorially for ortho-symplectic supergroups ËÔÇ(m n); Λ = X (T ) in their notation. Recently, Cheng, Shu and Wang [5]| explicitly determined Λ♭ for simply connected Chevalley groups of type D(2 1; ζ), G(3) and F (3 1). | | A uniﬁed description of the parameter set Λ♭ for all quasireductive supergroups, as in the case of (non-super) split reductive groups, is not known. This is an open problem.

5. Quasireductive Supergroups admitting Distinguished Parabolic Super-subgroups

Let G be a quasireductive supergroup over a ﬁeld k, as before. Recall

G B G that, G is the even part Gev of and is the Borel super-subgroup of

− G satisfying b = Lie(B). In this section, we assume that contains a closed

super-subgroup È such that

− È È Lie (5.1) ev = G and ( )1¯ = b1¯ ,

which we shall call a maximal parabolic super-subgroup of G. Note that, the assumption does depend on the choice of the homomorphism γ : Z∆ R, deﬁned in 3.2. → § Examples 5.1. The following, with a suitable choice of γ, satisfy the as- sumptions above.

(1) General linear supergroups GÄ(m n), see Example 5.9. |

(2) Chevalley supergroups G of classical type such that the Lie superal- G gebra g = Lie(G) of is a ﬁnite-dimensional simple Lie superalgebra of type I. (3) Periplectic supergroups P(n), see Example 5.11. However, queer supergroups Q(n) do not satisfy the assumption.

For simplicity, we also assume that h1¯ = 0, or equivalently, Ì = T .

5.1. A -Splitting Property. First, we ﬁx a totally ordered k-basis G = ⊗ ((X ) , ) of g¯ and deﬁne the following unit-preserving supercoalgebra map i i ≤ 1

ιG : (g¯) hy(G); X X X X X X , ∧ 1 −→ i1 ∧ i2 ∧···∧ ir 7−→ i1 i2 · · · ir where Xi1 < Xi2 < < Xir in G. The left T -supermodule structure on g1¯ naturally induces a· left · · T -supermodule structure on the exterior superalgebra (g1¯). It is easy to see that ιG is left T -linear. Then one can retake the

isomorphism∧ (3.2) so that G

φG G : hy(G) (g¯) hy( ); u ξ u ιG(ξ). , ⊗∧ 1 −→ ⊗ 7−→ · Moreover, by dualizing ιG, we get a counit-preserving left T -module super-

O(G) algebra map πG : (G) (W ) satisfying O →∧

O(G) ιG(ξ), η = ξ, πG(η) , ξ (g¯), η (W ) h i h i ∈∧ 1 ∈∧ BOREL-WEIL THEOREM FOR ALGEBRAIC SUPERGROUPS 23 and the isomorphism (3.1) can be chosen as follows.

O(G) G

ψG G : ( ) (G) (W ); a a πG(a ), , O −→ O ⊗∧ 7−→ (1) ⊗ (2) a X

see [17, Proposition 22]. It is easy to see that this ψG,G is also left T -linear. Take B, U+ G so that G = B U+ (disjoint union) and B (resp. U+)

⊆ − + ⊔ O(B) forms a k-basis of b1¯ (resp. n1¯ ). Then we also get πB : (B) (W ) + O →∧ + O(Í ) − + and πU+ : (Í ) (W ). The canonical identiﬁcation g = b u

O →∧ ∼ + ⊕

B Í induces an isomorphism (W O(G)) = (W O( )) (W O( )). One sees that the following diagram∧ commutes.→ ∧ ⊗∧

G / Í ( ) (B) ( ) O O ⊗ O π (5.2) G πB⊗πU+

G Í (W O( )) / (W O(B)) (W O( )), ∧ =∼ ∧ ⊗∧

where the upper arrow is given by (3.9). È In the following, we regard (G) as a right -supermodule via the canon- +

O O(Í )

È È G ։ ical quotient map ̟È : ( ) ( ), and regard ( ) and (W ) as left T -supermodules, asO before. ThenO we have theO following. ∧

O(Í )

È È Proposition 5.2. (G) is isomorphic to ( ) (W ) as right - left T -supermodules.O O ⊗∧

Proof. By the deﬁnition of È, we also have a counit-preserving left (G)-

O È right (T )-comodule superalgebra isomorphism ψÈ,B : ( ) (G)

OB O → O ⊗ (W O( )) which satisﬁes

∧

È B ψÈ B(̟ (a)) = a πB(̟ (a )), a ( ), , (1) ⊗ (2) ∈ O a

G B where ̟B : ( ) ( ) is the canonical quotient map. Then by (5.2), we get the followingO commutative→ O diagram in the category of left T -supermodules.

=∼ O(G) (G) / (G) (W )

O ψG,G O ⊗∧ id ( ⊗πG)◦△O(G)

O(G) (G) (W ) =∼ O ⊗∧

̟È ⊗∧f

+ =∼ +

B Í O(Í ) / O( ) O( ) (È) (W ) (G) (W ) (W ). O ⊗∧ ψÈ,B⊗id O ⊗∧ ⊗∧

+ Í Here, f : W O(G) W O( ) is the canonical projection induced by the inclusion U+ G.→ Thus, we are done.

⊂

B B Recall that, G = Gev and B = ev. For a left -supermodule V , it is

È È easy to see that the map id ε : V B ( ) V is left B-linear, where

⊗ O( ) O →

È È

εÈ : ( ) k is the counit of ( ). Then by Frobenius reciprocity (2.1), we haveO the→ following left G-moduleO map

: V B ( ) V (G); v p v p, NV O( ) O −→ O(B) O ⊗ 7−→ ⊗ where p is the canonical image of p (È) in (G). ∈ O O

24 T. SHIBATA

È B Lemma 5.3. The above : res È ind ( ) indGres ( ) is a natural iso- G B B B morphism. N − → −

Proof. The proof is essentially the same as Zubkov’s [32, Proposition 5.2]. È For simplicity, we write R := (B), R := (B), P := ( ). Note that, O O O A = (G) = (Èev). First, we prove that is an isomorphism. By the O O NR right version of (3.1), we see that R is isomorphic to (W R) R as a right ∧ ⊗ R-supercomodule. Thus, we have : R P R A = coRR A. NR R −→ R ∼ ⊗ coR coR Here, R is the right R-coinvariant subspace of R, that is, R = k R R, where k is regarded as the trivial one-dimensional right R-comodule. It is known that coRR is isomorphic to (W R), see [16, Theorem 4.5]. Then one ∧ sees that this map coincides with the tensor decomposition P = (W R) A ∼ ∧ ⊗ of P , which is the right version of ψÈ,B, and hence B is an isomorphism. We see that is also an isomorphism. N NΠB Next, we prove that V is an isomorphism for all injective object V in SModB. By [32, PropositionN 3.1], it is shown that V is a direct summand of a direct sum of some copies of B and ΠB. Thus, V is an isomorphism. Then by [24, Lemma 8.4.5], we are done. N The following is a generalization of Zubkov’s result [32, Proposition 5.2].

Theorem 5.4. For a left B-supermodule V , there is an isomorphism

B Í RnindG (V ) RnindG(res (V )) (W O( )) B = ∼ B B ⊗∧

of left T -supermodules.

È È

Proof. Since resÈ ( ) is exact, we get res Rnind ( ) Rn(res ind ( )), B

G G B ∼= G È

− ◦ − È − see [10, Part I, 4.1(2)]. Then by Lemma 5.3, we see that res Rnind ( ) G B ∼=

n G B ◦ − R (indBresB( )). By Proposition 5.2, we see that (G) is an injective object

− O G

in the category of left (È)-supercomodules, and hence the functor ind ( ) È È O G −

is exact. Thus, again by [10, Part I, 4.1(2)], we get ind Rn(ind ( )) B

È ∼=

G G È G ◦ −

Rnind ( ), since ind ( )= ind ind ( ). On the other hand, for any right

B È B B − − − (È)-supercomodule Y , we get an isomorphism O

+ Í G O( )

ind (Y ) = Y È ( ) Y (W ) È = O( ) O ∼ ⊗∧ of left (T )-supercomodules by Proposition 5.2. Thus, we are done. O n n G T λ 5.2. Some Applications. As before, we set Hev(λ) := R indB(resB(k )) for λ Λ and n Z . By Theorem 5.4, we get the following theorem. ∈ ∈ ≥0 Theorem 5.5. For λ Λ and n 0, there is an isomorphism ∈ ≥ + Hn(λ) = Hn (λ) (W O(Í )) ∼ ev ⊗∧ of left T -supermodules. In particular, we have Λ♭ = Λ+. This seems to be a certain dual version of the notion of Kac modules (cf. [12, 2.2 (b)]). As a corollary, we get a super-analogue of the Kempf vanishing§ theorem [10, Part II, Proposition 4.5]. Corollary 5.6. For λ Λ+ and n 1, we have Hn(λ) = 0. ∈ ≥ BOREL-WEIL THEOREM FOR ALGEBRAIC SUPERGROUPS 25

The result above is ﬁrst proved by Zubkov [32, Theorem 5.1] for G =

GÄ(m n) with the standard Borel super-subgroup. Later he also gave a

| GÄ partial generalization of the Kempf vanishing theorem for G = (m n) with all Borel super-subgroups [33, Proposition 13.3] (i.e., for all homomorphism| γ : Z∆ R). → 1 Set ρ0¯ := 2 α∈∆+ α in Λ Z Q, and 0¯ ⊗ P + + Λ if p = 0, Λp := + ∨ + λ Λ 0 λ + ρ¯, β p, β ∆ if p> 2. ({ ∈ | ≤ h 0 i≤ ∀ ∈ 0¯ } + Here, we put p := char(k). Then for λ Λp , it is known that the induced 0 0∈ 0 G-module Hev(λ) is simple, that is, Hev(λ) = Lev(λ) := socG(Hev(λ)). See [10, Part II, 5.5]. Thus, in this case, we see that H0(λ) is isomorphic to the dim g¯

direct sum of 2 1 -copies of Lev(λ) as T -modules. GÄ Remark 5.7. For the case of G = (m n), Marko [14] gave a necessary | 0 and suﬃcient condition for the induced G-supermodule H (λ) to be simple, see also Zubkov [33, Proposition 12.10].

Let ZΛ be the group algebra of Λ over Z, and let eλ λ Λ be the standard basis of ZΛ. For a ﬁnite-dimensional T -supermodule{ | ∈M,} we let ch(M) λ λ denote the formal character of M, that is, ch(M)= λ∈Λ dim(M )e . Set

A(µ) := ( 1)ℓ(w) ewµP, − w∈WX(G,T )

where µ Λ Z Q and W(G, T ) is the Weyl group of G = Gev with respect to T . As a∈ super-analogue⊗ of the Weyl character formula, we get the following description of the Euler characteristic for λ Λ+. ∈ Corollary 5.8. For λ Λ+, we have ∈ A(λ + ρ¯) ch(H0(λ)) = 0 (1 + e−δ). A(ρ0¯) · δ∈∆+ Y1¯ Proof. By the Weyl character formula for G, it is known that the formal 0 character of Hev(λ) is given by A(λ + ρ0¯)/A(ρ0¯), see [10, Part II, 5.10]. On + O(Í ) + ∗ + the other hand, by deﬁnition, W coincides with the dual (n1¯ ) of n1¯ as + ∗ −δ + e right T -modules. The formal character of (n1¯ ) is given by δ∈∆ (1+ ). ∧ 1¯ Thus, we are done. Q 1 Set ρ1¯ := 2 δ∈∆+ δ and ρ := ρ0¯ ρ1¯ in Λ Z Q. Assume that wρ1¯ = ρ1¯ 1¯ − ⊗ for all w W(G, T ). Then for each λ Λ+, we have ∈ P ∈ δ/2 −δ/2 δ∈∆+ (e + e ) (5.3) ch(H0(λ)) = A(λ + ρ) 1¯ . · Q A(ρ0¯)

This assumption is satisﬁed if g = Lie(G) is a simple Lie superalgebra of GÄ type I or gl(m n). For the case of G = (m n), see [33, Corollary 13.4]. | | 26 T. SHIBATA

5.3. Some Examples. In this section, we shall see some examples.

Example 5.9. Let us consider the case of GÄ(m n). In our setting, we shall see Corollary 5.4 in [32]. As usual, we choose a| split maximal torus T of

GÄ(m n)ev = GL GL so that | ∼ m × n T (R) = diag(x ,...,x ), diag(x ,...,x ) GL (R) GL (R) , { 1 m m+1 m+n ∈ m × n } m+n

where R is a commutative algebra. We may identify Λ = i=1 Zλi. Then GÄ we have Lie(GÄ (m n)) = gl(m n) = Matm|n(k) and W( (m n)ev, T ) ∼= S S , where S| is the symmetric| group on m letters.L The root| system m × n m of GÄ(m n) with respect to T is ∆= λ λ Λ 1 i = j m + n and | { i − j ∈ | ≤ 6 ≤ } ∆¯ = λ λ 1 i = j m or m + 1 i = j m + n . 0 { i − j | ≤ 6 ≤ ≤ 6 ≤ } Note that, ∆ = ∆0¯ ∆1¯ (disjoint union). Deﬁne γ : Z∆ R so that ⊔ ± → γ(λi) := i for each 1 i m + n. Then one sees that ∆ = (λi λj)

− ≤ ≤ + {± − |

B GÄ 1 i < j m + n and the Borel super-subgroup B (resp. ) of (m n) consists≤ of≤ lower (resp.} upper) triangular matrices. In this case, the maximal|

parabolic super-subgroup of GÄ(m n) is given as |

X 0 GÄ È(R) = (m n)(R) { Z W ∈ | } where R is a commutative superalgebra. By Theorem 5.5, we see that m+n Λ♭ = Λ+ = d λ Λ d d and d d . { i i ∈ | 1 ≥···≥ m m+1 ≥···≥ m+n} Xi=1 For λ = m+n d λ Λ♭, we shall write λ := m d λ and λ := i=1 i i ∈ + i=1 i i − m+n d λ . Then we have H0 (λ) = H0 (λ ) H0 (λ ), see [10, i=m+1 iPi ev ∼ GLm + P GLn − 0 ⊗ λ+ Part I, Lemma 3.8]. Here, HGLm (λ+) is the induced GLm-module for k . P ± ± Let t1,...,tm+n be indeterminates. As an element of Z[t1 ,...,tm+n], s (t ,...,t ) := det(tdj +m−j) (t t ) λ+ 1 m i 1≤i,j≤m i − j . 1≤i

λi Here, we have replaced each e with ti.

Remark 5.10. We consider the case of GÄ(2 1). Let T be the standard | torus of Gev as before. If we choose γ : Z∆ R so that γ(λ1) > γ(λ3) > γ(λ ), then ∆− = (λ λ ), λ λ , (λ → λ ) and the corresponding

2 {− 1 − 2 2 − 3 − 1 − 3 } GÄ Borel super-subgroup B of (2 1) is given by | 0 0

∗ GÄ B(R) = (2 1)(R) , ∗ ∗ ∗ { 0  ∈ | } ∗ ∗   BOREL-WEIL THEOREM FOR ALGEBRAIC SUPERGROUPS 27 where R is a commutative superalgebra. In this case, one easily sees that

GÄ(2 1) does not have a maximal parabolic super-subgroup, that is, there

| GÄ is no such a super-subgroup È of (2 1) satisfying the condition (5.1). Recently, Zubkov [33] studied general| properties of the cohomology Hn(λ)=

n λ

B B GÄ H (GÄ(m n)/ , (k )) for all Borel super-subgroups of (m n). | L | For a superalgebra R, the super-transpose of A Mat (R) is given as ∈ m|n tX tZ X Y stA := , where A = . tY tW Z W − Here, tX denotes the ordinary transpose of the matrix X. Example 5.11. For n 2, we consider the following closed super-subgroup ≥

P(n) of GÄ (n n). For a commutative superalgebra R, | st

P(n)(R) := g GÄ(n n)(R) g J g = J , { ∈ | | n n} 0 I where J := n and I is the identity matrix of size n. This is a n I 0 n n quasireductive supergroup whose even part is P(n)ev ∼= GLn. We ﬁx a split maximal torus T of P(n)ev = GLn as the subgroup of all n ∼ diagonal matrices. We may identify Λ = i=1 Zλi and W(P(n)ev, T )= Sn. It is easy to see that the Lie superalgebra of P(n) is given as L Lie(P(n)) = A Mat (k) stAJ + J A = 0 . { ∈ n|n | n n } This is the so-called periplectic Lie superalgebra, see [11, 2.1.3] and [6, 1.1.5]. For this reason, we shall call P(n) the periplectic supergroup§ . § Then the root system of P(n) with respect to T is given as follows. ∆ = (λ λ ), (λ + λ ), 2λ 1 i < j n, 1 p n . {± i − j ± i j p | ≤ ≤ ≤ ≤ } ∈ ∆0¯ ∈ ∆1¯

Deﬁne γ : Z∆| R{zso that} | γ(λi){z := i for} each 1 i n. Then we see that + → − ≤− ≤ ∆ = λi λj, (λi +λj) 1 i < j n and ∆ = (λi λj), λp +λq 1 i <{ j− n, −1 p q | ≤n . In particular,≤ } one sees{− that− ∆+ = ∆−|. ≤ ≤ ≤ ≤ ≤ } + − 6 In this case, the Borel subgroup B (resp. B ) of P(n)ev ∼= GLn consists of all lower (resp. upper) triangular matrices. As a closed super-subgroup of P(n)(R) (where R is a commutative superalgebra), one sees that

X 0 t t B(R) = X B(R¯), ZX = XZ , { Z tX−1 | ∈ 0 − } + X Y + t −1 t −1 B (R) = X B (R¯), X Y = Y X , { 0 tX−1 | ∈ 0 } X 0 t t È(R) = X GL (R¯), ZX = XZ . { Z tX−1 | ∈ n 0 − } By Theorem 5.5, we see that Λ♭ = Λ+ = n d λ Λ d d . { i=1 i i ∈ | 1 ≥···≥ n} For λ Λ♭, the formal character of H0(λ) is given as follows. ∈ P 1 sλ(t1,...,tn) (1 + ). · titj 1≤i

Remark 5.12. As we mentioned before, the Borel super-subgroup B, the induced supermodule H0(λ), the order on Λ given in (4.9), and the notion of maximal T -weight (see Proposition 4.15),≤ and the existence of a maximal

parabolic super-subgroup È do depend on the choice of the homomorphism γ : Z∆ R, defined in 3.2. → § Appendix A. Some General Properties of Superspaces In this appendix, k is a field of characteristic not equal to 2. A.1. Supermodules. Let A be a superalgebra. We consider the (ordinary) crossed product algebra A ⋊ kZ2 via the non-trivial action of the algebra kZ2 on A. By definition, the multiplication of A ⋊ kZ2 is given as follows. (a ⋊ ǫ)(b ⋊ η) = a(b + ( 1)ǫb ) ⋊ (ǫ + η), 0 − 1 where ǫ, η Z , a, b = b + b A with b A¯, b A¯. ∈ 2 0 1 ∈ 0 ∈ 0 1 ∈ 1 The category of all left A-supermodules ASMod is, by definition, the cat- kZ2 egory of left A- right kZ2-Hopf modules AMod . Since the characteristic ∗ of k is not 2, there is a unique Hopf algebra isomorphism kZ2 ∼= (kZ2) . By using the isomorphism, we get the following equivalence of tensor categories. (A.1) SMod ≈ Mod; M M. A −→ A⋊kZ2 7−→ Here, the left A ⋊ kZ2-module structure on M is given by (a ⋊ ǫ).m = a.(m + ( 1)ǫm ), 0 − 1 where a A, ǫ Z2 and m = m0 + m1 M with m0 M0¯, m1 M1¯. A non-zero∈ superalgebra∈ is said to be∈simple if it has∈ no non-trivial∈ two- sided super-ideal. Here, the notion of a two-sided super-ideal is defined in an obvious way. Example A.1. For natural numbers m,n and an algebra R (not necessarily commutative), we let Matm|n(R) denote the set of all (m + n) (m + n) matrices over R. This naturally forms a superalgebra whose grading× is given as follows. X 0 Mat (R)¯ = X Mat (R), W Mat (R) , m|n 0 { 0 W | ∈ m ∈ n } 0 Y Mat (R)¯ = Y Mat (R),Z Mat (R) , m|n 1 { Z 0 | ∈ m,n ∈ n,m } where Mat (R) (resp. Mat (R)) is the set of all m m (resp. m n) m m,n × × matrices over R. For simplicity, we let Matm|0(R) denote the ordinary matrix algebra Matm(R).

Proposition A.2. Let A be a simple superalgebra. If A¯ = 0, then the 1 6 algebra A0¯ is Morita equivalent to A ⋊ kZ2.

Proof. Since A¯A¯ A¯ is a non-zero super-ideal of A, the Z -grading of A 1 1 ⊕ 1 2 is strongly graded, that is, A1¯A1¯ = A0¯. This means that A over A0¯ is a right kZ2-Galois, see [22, 8]. Then by Hopf-Galois theory, we get the following equivalence. § ≈ (A.2) Mod ModkZ2 (= SMod); V A V. A0¯ −→ A A −→ ⊗A0¯ By combining the equivalence above with (A.1), the claim follows. BOREL-WEIL THEOREM FOR ALGEBRAIC SUPERGROUPS 29

In the situation as above, the parity change functor Π : ASMod ASMod can be translated into the following functor. →

Mod Mod; V A¯ V. A0¯ −→ A0¯ 7−→ 1 ⊗A0¯ A.2. Supercomodules. Let C be a supercoalgebra. As a dual of A ⋊kZ2, we can consider the (ordinary) cosmash product coalgebra kZ2 ◮< C of kZ2 and C which is kZ C as a vector space. The comultiplication and the 2 ⊗ couinit of kZ2 ◮< C are respectively given as follows.

ǫ ◮< c (ǫ ◮< c ) (ǫ + c ) ◮< c , ǫ ◮< c ε (c), 7−→ (1) ⊗ | (1)| (2) 7−→ C c X where ǫ Z2 and c C0¯ C1¯. Here, εC : C k is the counit of C. Then as a dual∈ of (A.1), we∈ get∪ → ≈ ◮ (A.3) SModC ModkZ2

v v ( v ◮< v ), 7−→ (0) ⊗ | (0)| (1) v X where v V and v v v(0) v(1) is the given right C-supercomodule structure∈ on V . 7→ ⊗ P For a right C-supercomodule V , we let socC (V ) denote the sum of all simple C-super-subcomodules of V . In particular, corad(C) := socC (C) is called the coradical of C. The identification (A.3) ensures that any simple supercomodules are finite dimensional, for example. Moreover, we get the following lemma. Lemma A.3. Suppose that C is a Hopf superalgebra. Then for a right C-supercomodule V , V = 0 if and only if soc (V ) = 0. 6 C 6 A Hopf superalgebra U is said to be irreducible if corad(U) is trivial, or equivalently, the simple U-supercomodules are exhausted by the purely even or odd trivial U-comodule k, see [17, Definition 2]. For a right U- supercomodule V , V coU := v V v v = v 1 { ∈ | (0) ⊗ (1) ⊗ } v X is called the U-coinvariant super-subspace of V . coU Lemma A.4. Let V be a right U-supercomodule. Then V = socU (V ). In particular, V = 0 if and only if V coU = 0. 6 6 A.3. Cotensors. Let C, D be supercoalgebras and let f : C D be a supercoalgebra map. For a right C-supercomodule V with ρ : V→ V C ρ id⊗f → ⊗ its structure map, the map ρ D : V V C V D makes V into a | −→ ⊗ −→C ⊗ right D-supercomodule. We shall denote it by resD(V ), called the restriction C C D functor. Then resD( ) becomes a functor from SMod to SMod . For a left C-supercomodule V−′ with ρ′ : V ′ C V ′ its structure map, the cotensor ′ ′ → ⊗ product V C V of V and V is given by ρ⊗id−id⊗ρ′ V V ′ := Ker(V V ′ V C V ′). C ⊗ −→ ⊗ ⊗ 30 T. SHIBATA

′ This is a super-subspace of V V . It is easy to see that V C C = V and ′ ′ ⊗ ∼ C C V = V as superspaces. For a Hopf superalgebra H, the H-coinvariant ∼ coH superspace V of a right supercomodule V is nothing but V H k, where k is regarded as the trivial right H-supercomodule. We regard C as a left D-supercomodule whose structure map is given by (f id) C , where C : C C C is the comultiplication of C. For ⊗ ◦△ △ → ⊗ C a right D-supercomodule W , one easily sees indD(W ) := W D C forms a right C-supercomodule whose structure map is given by id C . In this C D ⊗△ C way, we get a (left exact) functor indD( ) from SMod to SMod , called the induction functor. − In the super-situation, the following Frobenius reciprocity also holds: For a right D-supercomodule W and a left C-supercomodule V , we have an isomorphism ∼ (A.4) SModD(resC (V ),W ) = SModC (V, indC (W )) D −→ D of superspaces, which is given by f (f id ) ρ, where ρ : V V C 7→ ⊗ C ◦ → ⊗ is the C-supercomodule structure of V . Since indC ( ) is right adjoint to D − resC ( ), we see that the functor indC ( ) preserves injective objects. D − D − Appendix B. Clifford Superalgebras In this Appendix, we review some basic facts and notations of Clifford superalgebras over a field k of characteristic not equal to 2. Set k× := k 0 . For details, see Wall [30], Musson [23, A.3.2] and Lam [13]. \{ } B.1. Structures of Clifford Superalgebras. Let V be a finite-dimensional vector space. Set r := dim V . For a symmetric bilinear form b : V V k on V , the pair (V, b) is called a quadratic space. Let I(V, b) be the two-sided× → ideal of the tensor algebra T (V ) generated by all xy + yx b(x,y), where x,y V . Set − ∈ Cl(V, b) := T (V )/I(V, b). For simplicity, we sometimes write Cl(V ) instead of Cl(V, b). This forms a superalgebra over k with each element in V regards as odd, and is called the Clifford superalgebra for (V, b). Since the characteristic of k is not 2, we can choose an orthogonal basis x1,...,xr of V with respect to b, that is, a basis of V satisfying b(x ,x )=0if{ i = j}. For 1 i r, set i j 6 ≤ ≤ (B.1) δi := b(xi,xi). Then one sees that the superalgebra Cl(V, b) is generated by the odd elements x1,...,xr with the relations x2 δ ; x x + x x , i − i j k k j r−1 where 1 i r and 1 j < k r. One sees that Cl(V, b)ǫ is a 2 - ≤ ≤ ≤ ≤ r dimensional space for each ǫ Z2, and hence Cl(V, b) is 2 -dimensional. If b is trivial (i.e., b = 0), then∈Cl(V, b) is the so-called exterior superalgebra (V ) on V . ∧ The orthogonal sum (V, b) (V ′, b′) (or V V ′, for short) of two quadratic spaces (V, b) and (V ′, b′⊥) is a quadratic space⊥ whose underlying space is the direct sum V V ′ and the bilinear form is given as follows. ⊕ (v, v′), (w, w′) b(v, w)+ b′(v′, w′), 7−→ BOREL-WEIL THEOREM FOR ALGEBRAIC SUPERGROUPS 31 where v, w V and v′, w′ V ′. There is an isomorphism of superalgebras ∈ ∈ ∼ (B.2) Cl(V V ′) = Cl(V ) Cl(V ′) ⊥ −→ ⊗ given by (v, v′) v 1 + 1 v′, where v V , v′ V ′. The radical of7→ the⊗ quadratic⊗ space (V, b)∈ is defined∈ as follows. (B.3) rad(b) := v V b(v, w) = 0 for all w V . { ∈ | ∈ } We say that a quadratic space (V, b) is non-degenerate if rad(b) = 0. Suppose that (V, b) is non-degenerate. We define the signed determinant of V as follows. (B.4) δ := ( 1)r(r−1)/2 δ δ δ . − 1 2 · · · r It is known that the Clifford superalgebra Cl(V, b) is central simple over k in the category of superalgebras. In addition, if r = dim V is even, then Cl(V, b) is still central simple over k in the category of algebras. Moreover, in this case (i.e., r is even), the following structure theorem is known. For more details, see [13, Chapter V, 2]. × 2 § If δ (k ) , then the algebra Cl(V, b)0¯ is central simple over k(√δ). • 6∈ × 2 If δ (k ) , then the superalgebra Cl(V, b) is isomorphic to Matn|n(D), • where∈ n N and D is a central division algebra over k. ∈ Here, k(√δ) is the quadratic field extension of k, and (B.5) (k×)2 := a k a = b2 for some b k× . { ∈ | ∈ } B.2. Simples of Clifford Superalgebras. In the following, we let (V, b) be a quadratic space (not necessarily non-degenerate). Then there is a non- degenerate subspace V of V such that V = rad(b) V . Let δ be the s ⊥ s sign determinant of Vs. For simplicity, if b = 0, then we let δ := 0. The following fact is well-known, see Chevalley [7, Chapter II, 2.7] for example. For convenience of the reader, we shall give a proof here. Lemma B.1. The (ordinary) Jacobson radical J of the algebra Cl(V ) coincides with the two-sided ideal of Cl(V ) generated by rad(b), and the quotient (ordinary) algebra Cl(V )/J is isomorphic to Cl(Vs). Proof. For simplicity, we let r be the two-sided ideal of Cl(V ) generated by rad(b). We shall identify Cl(Vs) as a subalgebra of Cl(V ) via the isomorphism Cl(V ) ∼= Cl(rad(b)) Cl(Vs). Then r + Cl(Vs) forms a subalgebra of Cl(V ). Since V = rad(b) ⊗V , we see that r + Cl(V )= Cl(V ), and hence we have ⊥ s s Cl(V )/r = Cl(V )/(r Cl(V )). ∼ s ∩ s Fix x rad(b). For each u = u0 + u1 Cl(V ) with u0 Cl(V )0¯ and ∈ 2 ∈2 ∈ 2 u1 Cl(V )1¯, we see that (xu) = xuxu = x (u0 u1)u = 0, since x = 0. Thus,∈ we get that 1 xu is invertible in Cl(V ),− and hence rad(b) J. In − ⊆ particular, we have r J. Thus, r Cl(Vs) is a nil-ideal of Cl(Vs). Since the ⊆ ∩ Jacobson radical of Cl(Vs) is trivial, we get Cl(V )/r ∼= Cl(Vs) and r = J. To find simple supermodules, the following lemma is useful. Lemma B.2. There exists a (1+dim V )-dimensional quadratic space (V ′, b′) such that the category of Cl(V )-supermodules is equivalent to the category of Cl(V ′)-modules. 32 T. SHIBATA

Proof. For simplicity, we set A := Cl(V, b). Let kx be a one-dimensional vector space over k with a base x, and let b : kx kx k be a sym- x × → metric bilinear form defined by bx(x,x) = 1. Then we have the following isomorphism of (ordinary) algebras. ∼ A ⋊ kZ = A Cl(kx, b ); a ⋊ ǫ a xǫ. 2 −→ ⊗ x 7−→ ⊗ where a A and ǫ Z2. Here, the tensor product A Cl(kx, bx) is Z2- ∈ ∈ ′ ′ ⊗ graded, as before. Set (V , b ) := (V, b) (kx, bx). Then we have an iso- ⊥ ′ ′ morphism of (ordinary) algebras A ⋊ kZ2 ∼= Cl(V , b ). Hence, by the equivalence (A.1), the category of all left A-supermodules can be identified with the category of all left Cl(V ′, b′)-modules. Proposition B.3. There is a unique simple left Cl(V, b)-supermodule u up to isomorphism and parity change. If (1) δ = 0 or (2) dim V is even and δ (k×)2, then u = Πu. Otherwise, u = Πu. ∈ 6 Proof. First, suppose that δ = 0. Then Cl(V, b) coincides with the exterior superalgebra (V ) on V . It is easy to see that (V ) has a unique one- ∧ ∧ dimensional simple supermodule u up to isomorphism which satisfies u ∼= Πu. Next, suppose that δ k×. For simplicity, we set A := Cl(V, b) and6 r := dim V . By Lemma B.1, it∈ is enough to consider the case when (V, b) is non- degenerate. (i) Assume that r is odd. We use the notation in Lemma B.2. Note that, V ′ is also non-degenerate. Since dim V ′ = r + 1 is even, we see that Cl(V ′) is (Artinian) simple as an ordinary algebra. Thus, A has a unique simple supermodule u up to isomorphism which satisfies Πu = u. × 2 (ii) Assume that r is even and δ (k ) . In this case, A¯ is (Artinian) 6∈ 0 simple. By Proposition A.2, A ⋊ kZ2 is Morita equivalent to A0¯. Thus, A has a unique simple supermodule u up to isomorphism which satisfies Πu = u, by using the equivalence (A.1). (iii) Assume that r is even and × 2 δ (k ) . Then the superalgebra A can be identified with Matn|n(D) for some∈ n N and a central division algebra D over k. Thus, in the following, ∈ we identify A0¯ with X 0 Mat (D)¯ = X,W Mat (D) . n|n 0 { 0 W | ∈ n } As column vectors of length 2n with entries in D, we set e := t( 1,..., 1, 0,..., 0 ), e′ := t( 0,..., 0, 1,..., 1 ). n-times n-times ′ ′ Then M := A0¯ e |and{z M} := A0¯ e are the distinct| {z simple} left A0¯-modules. By Proposition A.2, the corresponding simple left A-supermodules are given ′ ′ ′ by u := A A0¯ M and u := A A0¯ M . One easily sees that Πu = u , and hence u = Π⊗u. ⊗ 6 B.3. The Algebraically Closed Case. Assume that the base field k is algebraically closed. In this section, we shall calculate the dimensions of simple supermodules of Clifford superalgebras. Let (V, b) be a quadratic space over k, in general. The quotient space V := V/rad(b) is non-degenerate. Set d := dim V . By Proposition B.3, Cl(V, b) has a unique simple supermodule u up to isomorphism and parity change. BOREL-WEIL THEOREM FOR ALGEBRAIC SUPERGROUPS 33

Proposition B.4. The dimension of the vector space u is given by 2⌊(d+1)/2⌋, where (d + 1)/2 is the largest integer less than or equal to (d + 1)/2. ⌊ ⌋ Proof. First, we suppose that the signed determinant δ of V , deﬁned in (B.4), is zero. In this case, dim u = 1 and d = 0, and hence the claim follows. Next, we suppose that δ = 0 and d is odd. In this case, the 6 algebra Cl(V , b) ⋊ kZ2 is central simple over k. Since k is algebraically closed, one sees that the algebra Cl(V , b) ⋊ kZ2 is isomorphic to Matm(k), where m = 2(d+1)/2. Hence, we are done. Finally, we suppose that δ = 0 and d is even. Note that, (d + 1)/2 = d/2. Since k is algebraically6 closed, the element δ (= 0)⌊ is always in⌋ (k×)2 = k×. Thus, the algebra 6 d/2 Cl(V , b)¯ is isomorphic to Mat (k) Mat (k), where n = 2 . Hence, we 0 n × n have dim u = n = 2d/2. A maximal totally isotropic subspace V † of (V, b) is a maximal subspace W of V satisfying b(W, W ) = 0. It is known that the Witt index Ind(V ) := d/2 of (V, b) coincides with the dimension of the quotient space V †/rad(b⌊), see⌋ [13, Chapter I, Corollary 4.4] for example. Therefore, we obtain dim V dim V † = (d + 1)/2 . Here, we used d = d/2 + (d + 1)/2 . Then by− Proposition⌊ B.4, we get⌋ ⌊ ⌋ ⌊ ⌋ † (B.6) dim u = 2dim V −dim V . This is another description of the dimension of u.

Acknowledgements Almost all of the manuscript was prepared while the author was a Paciﬁc Institute for the Mathematical Sciences (PIMS) Postdoctoral Fellow position at University of Alberta. The author would like to thank PIMS for their hospitality and ﬁnancial support.

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(T. Shibata) Department of Applied Mathematics, Okayama University of Science, 1-1 Ridai-cho Kita-ku Okayama-shi, Okayama 700-0005, Japan E-mail address: [email protected]