Crystal structures arising from representations of $GL(m|n)$

This paper provides results on the modular representation theory of the supergroup $GL(m|n).$ Working over a field of arbitrary characteristic, we prove that the explicit combinatorics of certain crystal graphs describe the representation theory of a modular analogue of the Bernstein-Gelfand-Gelfand category $\mathcal{O}$. In particular, we obtain a linkage principle and describe the effect of certain translation functors on irreducible supermodules. Furthermore, our approach accounts for the fact that $GL(m|n)$ has non-conjugate Borel subgroups and we show how Serganova's odd reflections give rise to canonical crystal isomorphisms.


Introduction
In 1995 Serganova computed the characters of the finite dimensional irreducible representations of the Lie superalgebra gl(m|n, C) [17,18]. Recently, Brundan gave a more direct way to calculate these characters [1]. He also provides for the first time a conjectural formula for the characters of the irreducible representations belonging to the gl(m|n, C)-analogue of the Bernstein-Gelfand-Gelfand category O. Brundan's approach relates the Grothendieck group of this category O to a certain gl(∞, C)module. To be more precise, let V denote the natural gl(∞, C)-module and let V ∨ denote its dual. Brundan identifies the complexified Grothendieck group of category O with the gl(∞, C)-module V ∨ ⊗ · · · ⊗ V ∨ n times ⊗ V ⊗ · · · ⊗ V m times (1.1) so that the Verma supermodules in category O correspond to the natural monomial basis of (1.1). Then Brundan's conjecture is that the irreducible supermodules in category O correspond to Lusztig's dual canonical basis for (a certain completion of) the module (1.1). The subcategory F of O consisting of finite dimensional modules fits nicely into this picture: the Grothendieck group of the category F is identified with the submodule n V ∨ ⊗ m V of (1.1). This article is concerned instead with the crystal structures (in the sense of Kashiwara) which underlie Brundan's conjecture. Actually, we work throughout the article over an arbitrary field k of characteristic p, considering a modular analogue O p of the usual category O. Remarkably, all the results at the level of crystals remain true even if p > 0 provided one replaces the Lie algebra gl(∞, C) with the affine Kac-Moody algebra sl(p, C).
When p = 0 there are certain translation functors E r , F r (r ∈ Z) defined on category O which play a key role in [1]; actually they already appeared in [17] but in a slightly Date: November 16, 2021. 2000 Mathematics Subject Classification. Primary 20C20, 05E99; Secondary 17B10. different form. At the level of Grothendieck groups, these functors correspond to the usual Chevalley generators of gl(∞, C) acting on the module (1.1). When p > 0 one can define analogous functors E r , F r (r ∈ Z/pZ) on category O p . Let L(λ) be an irreducible module in category O or O p . In both cases we prove that the modules E r L(λ) and F r L(λ) are either zero or else are self-dual indecomposable modules with irreducible socle and cosocle isomorphic to L(ẽ * r (λ)) and L(f * r (λ)) respectively. This gives a representation theoretic definition of operatorsẽ * r ,f * r on the set of weights that parametrizes the irreducible supermodules. Our main result (Theorem 2.4) gives an explicit combinatorial description of these operators, allowing us to verify that they are dual to the crystal operatorsẽ r ,f r associated to Kashiwara's crystal basis of the module (1.1).
Let us remark that our main result contains as a special case branching rules for representations of the supergroup GL(m|n) in characteristic p. They are a natural extension of Kleshchev's modular branching rules in the case of GL(n); see e.g. [3]. Our proof is an adaptation of Kleshchev's original methods, based on some explicit computations with certain lowering operators in the universal enveloping algebra.
An important feature of the Lie superalgebra gl(m|n, C) is that it has various different conjugacy classes of Borel subalgebra. These may be parametrized by a sequence (v 1 , . . . , v m+n ) of parities v i ∈ Z 2 , m of which are equal to0 and n of which are equal tō 1. Brundan only considers the Borel subalgebra corresponding to the parity sequence (1, · · · ,1,0, · · · ,0); in the present article we consider all the conjugacy classes of Borel subalgebras. It turns out that in the general case one must replace the module (1.1) with the module V v1 ⊗ · · · ⊗ V vm+n (1.2) where V0 denotes V and V1 denotes V ∨ . (It appears that Brundan's conjecture itself also extends nicely to other Borel subalgebras, again replacing (1.1) with (1.2).) Odd reflections, introduced by Serganova [19] and, independently, by Dobrev and Petkova [6], give a simple way to translate between the parametrization of irreducible highest weight modules arising from different choices of Borel subalgebra. In the final section of the paper we explain how these odd reflections can also be interpreted as canonical crystal isomorphisms between the crystals associated to the modules (1.2) for different choices of parity sequences. commutative superalgebras to the category of groups: for a commutative superalgebra A let G(A) be the group of all invertible even (i.e. grading preserving) automorphisms of the A-supermodule V ⊗ A.
Once and for all we fix a ordered homogeneous basis v 1 , . . . , v m+n for V. Having made this choice, we can now introduce coordinates: given a commutative superalgebra A, we identify G(A) with the group of all invertible (m + n) × (m + n) matrices (g i,j ) 1≤i,j≤m+n (2.1) with g i,j ∈ A vi+vj for all 1 ≤ i, j ≤ m + n. Namely, if we identify elements of V ⊗ A with column vectors via Let T be the subgroup of G consisting of diagonal matrices. More precisely, given a commutative superalgebra A, T is the functor given by setting T (A) to be the subgroup of G(A) consisting of all diagonal matrices. Let X(T ) be the free abelian group We identify X(T ) with the character group of T by identifying ε i with the function which picks out the ith diagonal entry of a diagonal matrix. We put a symmetric bilinear form on X(T ) by declaring that (ε i , ε j ) = (−1) v i δ i,j . (2.2) Observe that we have an action by the symmetric group S m+n on X(T ) given by x · ε i = ε xi for all 1 ≤ i ≤ m + n and x ∈ S m+n . Given 1 ≤ t ≤ m + n, let V t denote the subspace of V generated by v 1 , . . . , v t . We fix a choice of Borel subgroup B of G so that, for a commutative superalgebra A, B(A) is the stabilizer of the full flag Note that B(A) equals the set of all upper triangular invertible matrices of the form (2.1).
The root system of G is the set Φ = {ε i − ε j : 1 ≤ i, j ≤ m + n, i = j}. There are even and odd roots, the parity of the root ε i − ε j being v i + v j . Our choice of Borel subgroup defines a set, of positive roots. The corresponding dominance order on X(T ) is denoted ≤ and is defined by λ ≤ µ if µ − λ ∈ Z ≥0 Φ + .

The Superalgebra of Distributions.
There is an abstract notion of the superalgebra of distributions for a supergroup; see, for example, [4, §3]. In this case however we can realize the superalgebra of distributions for G, Dist(G), explicitly as the reduction modulo p from an analogue of Kostant's Z-form for the Lie superalgebra over C corresponding to G. This Lie superalgebra consists of the set of (m + n) × (m + n) matrices over C with homogeneous basis given by the matrix units e i,j (1 ≤ i, j ≤ m + n) and with the parity of e i,j equal to v i + v j . The superbracket of this Lie superalgebra is given by Let U C denote the universal enveloping superalgebra of this Lie superalgebra. By the PBW theorem for Lie superalgebras [10], U C has basis consisting of all monomials , and the product is taken in any fixed order. We shall write h i = e i,i for short. Define the Kostant Z-form U Z to be the Z-subalgebra of U C generated by ele- i,j := e r i,j /(r!) and hi for all a i,j , r i ∈ Z ≥0 and d i,j ∈ {0, 1}, where the product is taken in any fixed order.
The enveloping superalgebra U C is a Hopf superalgebra in the canonical way and, furthermore, this structure restricts to make U Z a Hopf superalgebra over Z. Finally, set Dist(G) = k ⊗ Z U Z , naturally a Hopf superalgebra over k. We will abuse notation by using the same symbols e (r) i,j , hi r etc... for the canonical images of these elements of U Z in Dist(G). It is also easy to describe the superalgebras of distributions of our various natural subgroups of G as subalgebras of Dist(G). For example, Dist(T ) is the subalgebra generated by all hi r (1 ≤ i ≤ m + n, r ≥ 1), and Dist(B) is the subalgebra generated by Dist(T ) and all e (a) Let us describe the category of Dist(G)-supermodules. The objects are all left Dist(G)-modules which are Z 2 -graded; i.e., k-superspaces, M, satisfying Dist(G) r M s ⊆ M r+s for all r, s ∈ Z 2 . A morphism of Dist(G)-supermodules is a linear map f : M → M ′ satisfying f (xm) = (−1) f x xf (m) for all m ∈ M and all x ∈ Dist(G). Note that this definition makes sense as stated only for homogeneous elements; it should be interpreted via linearity in the general case. We emphasize that we allow all morphisms and not just graded (i.e. even) morphisms. However, we note that for superspaces M and M ′ the space s+r for all r, s ∈ Z 2 . This gives a Z 2 -grading on Hom Dist(G) (M, M ′ ) ⊆ Hom k (M, M ′ ). The category of Dist(G)-supermodules is not an abelian category. However, the underlying even category, consisting of the same objects but only the even morphisms, is an abelian category. This, along with the parity change functor, Π, which simply interchanges the Z 2 -grading of a supermodule, allows us to make use of the tools of homological algebra. For We call a Dist(G)-supermodule M integrable if it is locally finite over Dist(G) and satisfies M = λ∈X(T ) M λ . The category of G-supermodules can naturally be identified with the category of integrable Dist(G)-supermodules [4, Corollary 3.5].
2.3. Highest Weight Theory. Given λ ∈ X(T ) let Let O p denote the the full subcategory of the category of all Dist(G)-supermodules for any µ ∈ X(T ). Note that any supermodule in O p is locally finite over Dist(B). We also remark that in the case p = 0 Brundan's category O discussed in the introduction is a full subcategory of O 0 . From now on we will assume all Dist(G)-supermodules under discussion are objects in O p .
For λ ∈ X(T ), we have the Verma module where k λ denotes k viewed as a Dist(B)-supermodule of weight λ concentrated in degree0. Note that by Lemma 2.1 it follows that M (λ) is an object in O p . We say that a homogeneous vector v in a Dist(G)-supermodule M is a primitive vector of weight λ (or simply a primitive vector ) if Dist(B)v ∼ = k λ as a Dist(B)-supermodule. Familiar arguments exactly as for semisimple Lie algebras over C show: In this way, we get a parametrization of the irreducible objects in O p by their highest weights with respect to the ordering ≤.
Given a Dist(G)-supermodule, M, we can consider its graded dual with the usual Z 2 -grading and Dist(G) action. We have an automorphism, τ, of Dist(G) induced by e i,j → −(−1) v i (vi+vj ) e j,i (the negative of the supertranspose). Twisting M * by τ yields a new Dist(G)-supermodule, which we call the contravariant dual and denote by M τ . In particular, for λ ∈ X(T ) we define the co-Verma supermodule of highest weight λ. We remark that for a module M in category O p we have (M τ ) τ ∼ = M and the characters of M and M τ coincide. In particular, we have L(λ) τ ∼ = L(λ) for any λ ∈ X(T ).

Crystals.
Let us recall the general definition of a crystal in the sense of Kashiwara [11, 7.2]. Assume we have the following data: P = a free Z-module (called the weight lattice) I = an index set α i ∈ P for all i ∈ I (called a simple root) h i ∈ P * = Hom Z (P, Z) for all i ∈ I (called a simple coroot) −, − : P × P → Q a symmetric bilinear form.
Additionally, we assume the data satisfies the following axioms: With this fixed data, we define a crystal B as a set along with maps wt : B → P subject to the following axioms: We also remind the reader of the notion of the tensor product of two crystals. If B 1 and B 2 are crystals, then set If M 1 and M 2 are Lie algebra modules with associated crystals B 1 and B 2 , respectively, then by [11,Thm. 4.1] the crystal associated to 2.5. Affine Lie Algebras. Recall that we have a fixed ground field k of characteristic p. There are two cases to consider: when p = 0 and when p > 0. In each case we define the requisite Cartan datum in the notation of subsection 2.4 and use this data to define an affine Lie algebra over C, g, in the manner of [9]. We first consider the case when p = 0. Let P = r∈Z Zγ r . The index set is Z/pZ = Z. Define the simple roots by α r = γ r − γ r+1 for r ∈ Z. We define a nondegenerate symmetric bilinear form −, − : P × P → Q by setting γ r , γ s = δ r,s for r, s ∈ Z. Observe that otherwise; for all r, s ∈ Z. Using the form we identify P and P * via x ↔ x, − . Under this identification the simple coroot h r is α r for all r ∈ Z. Now we consider the case when p > 0. Then we let P = Zδ ⊕ r∈Z/pZ ZΛ r . Again, the index set is Z/pZ. The simple roots are defined by α r = 2Λ r − Λ r−1 − Λ r+1 + δ r,0 δ for r ∈ Z/pZ. Let −, − : P × P → Q be the nondegenerate bilinear form determined by requiring δ, Λ 0 , . . . , Λ p−1 and Λ 0 , α 0 , . . . , α p−1 to be dual bases with respect to the form. Observe that if p > 2, then and if p = 2, then α r , α s = 2, if r ≡ s (mod 2); −2, if r ≡ s + 1 (mod 2); for all r, s ∈ Z/pZ. In particular this implies the form is symmetric. Using the form we identify Q := ZΛ 0 ⊕ r∈Z/pZ Zα r ⊆ P with P * via x ↔ x, − . Under this identification, the simple coroot h r is α r for all r ∈ Z/pZ. Finally, given a ∈ Z we write a = pd + s with d ∈ Z and s = 1, . . . , p and define γ a ∈ P by Observe that if a = pd + s, then γ a − γ a+1 = α s .
In both cases we define a Lie algebra over C, g, generated by h := P ⊗ Z C and {E r , F r : r ∈ Z/pZ} subject to the relations for all r, s ∈ Z/pZ, all H, H ′ ∈ h, and where −, − denotes the bilinear form on P extended to h. Note that if p = 0, then g = gl ∞ (C). If p > 0, then g = sl p (C).
2.6. The Crystal B. We are now prepared to describe the crystal which plays a central role in this paper. Let V denote the natural "evaluation" g-module with basis {x b : b ∈ Z} and action given by where g is the affine Lie algebra defined in subsection 2.5. We say a vector x ∈ V is of weight λ ∈ P if Hx = H, λ x for all H ∈ h.
There is a crystal (B0,ẽ r ,f r , ε r , ϕ r , wt) associated to the module V (for both p = 0 and p > 0) where the underlying set B0 is {x b : b ∈ Z} (the given basis). The crystal operators are defined byẽ r = E r andf r = F r for all r ∈ Z/pZ and, given b ∈ Z, ε r (x b ) = 1, if r + 1 ≡ b (mod p) and is zero otherwise; and ϕ r (x b ) = 1, if r ≡ b (mod p) and is zero otherwise. Finally, wt is the usual weight function on V, hence wt(x b ) = γ b for all b ∈ Z. We leave it to the reader to verify the crystal axioms.
We have an automorphism of g given by for all r ∈ Z/pZ and all H ∈ h. We can twist V by this automorphism and obtain a new g-module, V ∨ . This module also has an associated crystal (B1,ẽ r ,f r , ε r , ϕ r , wt) which is, roughly speaking, B0 with the roles ofẽ r andf r interchanged. Namely, the crystal B1 is the set {x , for all r ∈ Z/pZ and all b ∈ Z. The tensor product of crystals B v1 ⊗ · · · ⊗ B vm+n ,ẽ r ,f r , ε r , ϕ r , wt is then the crystal associated to the g-module V v1 ⊗ · · · ⊗ V vm+n where V0 := V and V1 := V ∨ . The key combinatorial object in this article is a different crystal structure (B,ẽ * r ,f * r , ε * r , ϕ * r , wt) on the same underlying set, We call this the dual crystal structure following Brundan [1]. The dual crystal operators are defined bỹ The weight function is given by See [1] for a discussion of the sense in which these crystal structures are dual to one another. In the next subsection we will give a more explicit combinatorial description of the dual crystal B.

2.7.
The Crystal Structure on X(T ). We now lift the dual crystal structure on B to X(T ). To do so we require some additional notation. Let ρ ∈ X(T ) denote the unique element which satisfies the following conditions: and let Then observe that we have As an example, say our fixed homogeneous basis for V satisfies v i =1 if 1 ≤ i ≤ n and v i =0 if n + 1 ≤ i ≤ m + n (the parity choice in [1]), then we have We define a bijection X(T ) → B by Using this bijection we can lift the dual crystal structure on B to the set X(T ). Let us describe the combinatorics of this crystal structure explicitly. It is convenient to use a combinatorial description of the crystal tensor product rule which uses certain sequences commonly called signatures. See, for example, [7,Sec. 4.4].
For λ ∈ X(T ) and 1 ≤ j ≤ m + n, define the j-residue of λ to be Given the r-signature of λ we form the reduced r-signature by successively replacing −+ pairs with 00 (where the − and + may be separated by zeros, which are ignored) until no − appears to the left of a +. Given an r-signature σ = (σ 1 , . . . , σ m+n ), we writẽ σ = (σ 1 , . . . ,σ m+n ) for the reduced r-signature. In particular, given λ ∈ X(T ) we writẽ σ r (λ) for the reduced r-signature of λ. We then defineẽ * r ,f * r : We also define . The weight function is given by (2.13) Taken together the datum is the crystal of interest in the present work. We emphasize that this crystal structure on X(T ) depends on (but only on) the sequence of parities v 1 , . . . , v m+n which we fixed at the beginning.
2.8. Main Results. We can now summarize our main results. Namely, that the crystal on X(T ) given in (2.14) describes aspects of the category O p .
In section 3 we prove that the function wt given as part of the crystal structure on X(T ) partitions the central characters of Dist(G) arising from the irreducible supermodules of category O p . In particular if L(λ) and L(µ) have the same central character, then wt(λ) = wt(µ). As a consequence we obtain the following linkage principle.
To continue we need to define certain translation functors on category O p . These functors should be compared with the translation functors defined by Jantzen [8, II.7], Brundan and Kleshchev [3], and Brundan [1]. For ν ∈ P, define O ν p to be the full subcategory of O p of all modules with all their irreducible subquotients isomorphic to However, by our above remarks we can rewrite the decomposition as where the sum runs over all ν ∈ P and M ν is a Dist(G)-supermodule lying in category where V denotes the natural G-supermodule and V * denotes its dual. On a morphism ϕ : M → N , E r ϕ and F r ϕ are the restriction of the map ϕ ⊗ 1.
In section 5 we prove that the action of these translation functors on irreducible supermodules is regulated by the crystal structure on X(T ). Namely, we have the following theorem: As a corollary to the previous theorem we also obtain an explicit description of the socle of L(λ) ⊗ V * and L(λ) ⊗ V and combinatorial criterion for when these supermodules are semisimple.

Central Characters
3.1. Some Central Elements. Following Sergeev [20] (where the characteristic zero case was considered), we define certain central elements of Dist(G). Recall from subsection 2.2 that {e i,j : 1 ≤ i, j ≤ m + n} is the usual homogeneous basis for the Lie superalgebra associated to G. Define x [r] k,l ∈ Dist(G) inductively as follows: x Lemma 3.1. Let 1 ≤ i, j, k, l ≤ m + n and r ∈ Z, r ≥ 1. Then, k,k for r ≥ 1 are central. Proof. Each of the statements is a straightforward induction on r.
where the sum runs over all 1 ≤ k 1 < · · · < k r ≤ m + n and ϑ i is as in (2.7). Since Z r differs fromZ r by a scalar, these elements are still central and generate the same subalgebra of Dist(G). We remark that if char k = 0 one can use the results of [21] to prove that these elements in fact generate Z(Dist(G)).

3.2.
The Linkage Principle. Multiplication by an element of Z(Dist(G)) defines an endomorphism of M (λ) (λ ∈ X(T )) so takes the canonical generator v λ ∈ M (λ) λ to scalar multiple of itself. From this we conclude that the elements of Z(Dist(G))0 must act by scalars on M (λ) and any of its subquotients. We now prove that the even central element Z r defined in the previous subsection acts on M (λ) by the scalar Z r (λ) given below.
where ϑ i is as defined in (2.7). Observe that if M is a Dist(G)-supermodule and v ∈ M λ , then r i v = r i (λ)v, the ith residue of λ (2.11). Given λ ∈ X(T ) define the integer where the unmarked sum runs over all 1 ≤ k 1 < · · · < k s ≤ m + n and nonnegative integers a 1 , . . . , a s such that a 1 + · · · + a s = r − s + 1.
Proof. Given our assumption that v is annihilated by all e i,j when i < j, it suffices to work work modulo the left ideal generated by these elements. We shall write ≡ for congruence modulo this ideal. We shall prove the statement via several intermediate claims: k,l ≡ 0 for each 1 ≤ k < l ≤ m + n, and r ≥ 1. We prove this by inducting on r with the base case being clear. For r > 1 by the induction hypothesis, Lemma 3.1(ii), and (3.1) we have: s,s = (−1) vs ). We do a direct calculation using (3.1), Claim 1, and Lemma 3.1 (ii): s,s .
Claim 3: for r ≥ 1; where the second sum is over all k = k 1 < k 2 < · · · < k s and a 1 , . . . , a s ∈ Z ≥0 such that a 1 + · · · + a s = r − s. We induct on r ≥ 1 with the case r = 1 being clear. Let r > 1, then by Claim 2 and the induction hypothesis we have Which, one observes, is equal to the double sum given in (3.6). Consequently, we havẽ where the second sum is over all k 1 < k 2 < · · · < k s and a 1 , . . . , a s ∈ Z ≥0 such that a 1 + · · · + a s = r − s.

Using (3.3) and Claim 2 we have
however r ks = (−1) ks (h ks + ϑ ks ) so a substitution yields: We now observe that the first sum is (3.7) in all cases when a s ≥ 1, the second sum (after reindexing) is (3.7) for a s = 0 and s = 2, . . . , r plus one additional term which exactly cancels the third sum. It remains to note that the final possible case (when s = 1 and a s = 0) in fact does not occur in (3.7). Finally, since r t v = r t (λ)v, Claim 4 exactly implies the statement given in the lemma.
To prove (ii) ⇔ (iii), we observe that Comparing the multiplicity of zeros and poles we see To prove (iii) ⇔ (iv), we need to analyze the weight function more closely. In characteristic zero, we have otherwise.
We can now deduce the main results of this section.  Let v ′ λ ∈ N be a homogeneous preimage of v λ , the canonical generator of M (λ). By the weight assumption we have that v ′ λ is a primitive vector of weight λ. By the universal property of M (λ) there is a homomorphism which maps v λ → v ′ λ , providing a splitting of the sequence. Hence Ext 1 Dist(G) (M (λ), D) = 0. Now if λ, µ ∈ X(T ) are arbitrary, by contravariant duality we have , W (λ)), so we can assume without loss that λ < µ. The result then immediately follows.

Lowering operators
In a series of papers ( [12], [13], [14]) Kleshchev gave certain elements of the algebra of distributions of SL(n) called lowering operators which allowed him to prove modular branching rules for SL(n), and via Schur functor arguments, the symmetric group. Brundan generalized Kleshchev's construction to the case of quantum GL(n). In this section we develop the appropriate theory for the supergroup G = GL(V ) and use them to prove a representation theoretic interpretation of the crystal operatorsẽ * r and f * r .

A Combinatorial Interlude.
We now introduce the super analogue of Kleshchev's combinatorial notions of normal, conormal, good, and cogood. We first define them in terms of the crystal structure on X(T ) and then relate them to the appropriate generalization of Kleshchev's original definitions. Let r ∈ Z/pZ and λ ∈ X(T ).
Definition 4.1. We define 1 ≤ i ≤ m + n to be r-normal for λ if one of the following equivalent conditions hold:  (1)ẽ * r (λ) = λ − ε i ; (2) i is the position of the leftmost − in the reduced r-signature of λ.
We also have the analogous definitions forf * r : Definition 4.3. We define 1 ≤ i ≤ m+n to be r-conormal for λ if one of the following equivalent conditions hold: (1) (f * r ) a+1 (λ) = (f * r ) a (λ) + ε i for some a ∈ Z ≥0 ; (2) i is the position of a + in the reduced r-signature of λ.
Definition 4.4. We define 1 ≤ i ≤ m + n to be r-cogood for λ if one of the following equivalent conditions hold: (1)f * r (λ) = λ + ε i ; (2) i is the position of the rightmost + in the reduced r-signature of λ.
We now relate these crystal theoretic definitions to the super analogue of Kleshchev's original definitions. To do so we require additional notation. Given 1 ≤ i < j ≤ m + n, let (i. For 1 ≤ i, j ≤ m + n, 1 ≤ k ≤ m + n − 1, and λ ∈ X(T ), let Let 1 ≤ i < j ≤ m + n. Define the following subsets of {1, . . . , m + n}: Let λ ∈ X(T ) and let 1 ≤ i, h ≤ m + n. The following remarks are immediate from the definitions.
Using the above remarks and the definitions of normal and good, a straightforward combinatorial argument proves the following result. We remark that in the purely even case these are the definitions of normal and good used by Brundan. They are transpose to those given by Kleshchev (see [2]).
We end this subsection with two combinatorial results which will be useful later. A straightforward combinatorial argument proves the following lemma.
Lemma 4.8. Given 1 ≤ i ≤ m + n, r ∈ Z/pZ, and λ ∈ X(T ), i is r-good for λ if and only if i is r-normal for λ and i is r-conormal for λ − ε i .
The following lemma links the notions of normal and good with conormal and cogood, respectively. Let us fix some notation which we will use again in subsection 4.5. Let w 0 denote the longest element of S m+n . Let V denote a superspace with dim k V0 = n and dim k V1 = m and let G denote GL( V ). Fix a homogeneous basis v 1 , . . . ., v m+n satisfying v i = v w0i +1 for all 1 ≤ i ≤ m + n, where v 1 , . . . , v m+n is our usual fixed basis for V. Let us write T ∼ = T for the subgroup of all diagonal matrices of G. Following the procedure discussed in subsection 2.7, we put a crystal structure on X( T ) = m+n i=1 Zε i lifted from the dual crystal B = B v 1 ⊗ · · · ⊗ B v m+n ,ẽ * r ,f * r , ε * r , ε * r , wt . Lets : X(T ) → X( T ) be given by Proof. Let r ∈ Z/pZ. Recall our notation σ r (λ) = (σ r (λ) 1 , . . . , σ r (λ) m+n ) for the rsignature of λ. Given 1 ≤ i < j ≤ m + n, we call (i, j) an r-canceling pair for λ if σ r (λ) i = −, σ r (λ) j = +, and σ k (λ) t = 0 for i < k < j.
Given λ ∈ X(T ) and 1 ≤ t ≤ m + n, let ξ denote m+n k=1 (−1) v k , r = r t (λ), and r ′ = r − ξ. It is straightforward to verify that (i, j) is an r-canceling pair for λ if and only if (w 0 j, w 0 i) is an r ′ -canceling pair fors(λ). This along with our combinatorial description of the crystal structure on X(T ) immediately implies t is normal for λ if and only if w 0 t is conormal fors(λ). It is clear from the definitions that i is good for λ if and only if w 0 i is cogood fors(λ).

Lowering Operators.
In what follows for 1 ≤ i < j ≤ m + n and 1 ≤ k ≤ m + n we write E i,j = e i,j , F i,j = e j,i , and H k = e k,k . For 1 ≤ j ≤ m + n, let (4.6) A straightforward calculation shows that the L j 's commute with one another and, since they are of weight 0, with any element of Dist(T ). For 1 ≤ i, j ≤ m + n and 1 ≤ k ≤ m + n − 1, let and let (4.9) Note that this notation is compatible with that of subsection 4.1 in the sense that if M is a Dist(G)-supermodule with v ∈ M λ then the elements c i,j , b i,k ∈ Dist(G) act on v by the scalars c i,j (λ) and b i,k (λ), respectively. For 1 ≤ i < j ≤ m + n and A ⊆ (i..j), define Note that our observations on the commutativity of the L j 's imply that the order of the product does not need to be specified. Let B − denote the opposite Borel subgroup of G consisting of all lower triangular invertible matrices of the form (2.1). Let U denote the unipotent radical of B given by letting U (A) ⊆ B(A) be the set of upper triangular unipotent matrices for any commutative superalgebra A. Fix an ordering of the PBW basis (see Lemma 2.1) of Dist(G) so that each PBW monomial is of the form XY where X ∈ Dist(B − ) and Y ∈ Dist(U ). We define the lowering operators S i,j (A) by expanding S i,j (A) in the PBW basis given by this ordering and taking S i,j (A) to be the sum of those terms lying in Dist(B − ). The idea is that we are interested applying the lowering operators to primitive vectors and the nonconstant elements of Dist(U ) annihilate all primitive vectors. Consequently, in many of our calculations we will work modulo a left ideal of Dist(G) which annihilates any primitive vector. The following lemma illustrates this point of view. Lemma 4.10. Let J be the left ideal of Dist(G) generated by the nonconstant elements of Dist(U ). Let 1 ≤ i < t < j ≤ m + n. Then: Proof. Throughout we write ≡ for congruent modulo J.
To prove (i) we simply calculate the left hand side working modulo J: To prove (ii) one uses (i) and calculates that On the other hand, one calculates that ≡ −a r<t (−1) vr e t,r e r,t e t,i e j,t .
However, e t,r e r,t e t,i e j,t ≡      0 if r < i; e i,i − (−1) vi+vt e t,t + (−1) vi+vt + 1 e t,i e j,t if r = i; e t,i e j,t + (−1) (vt+vr )(vr+vi) e r,i e t,r e j,t if r > i.
Together, these imply Finally, through the miracle of A useful fact about the lowering operators, both for calculating them and for proving results, is the recurrence relation given in the following theorem.
Proof. Throughout we write ≡ for equivalence modulo the left ideal J generated by the nonconstant elements of Dist(U ). If h = i, theñ Using this equation, we replace the kth term in the product in the definition of S i,j (A). Distributing yields S i,j (A) ≡ S i,j (A) ≡ P 1 + P 2 + P 3 + P 4 where since, as i < h, k < j, the c h,k commutes with all the terms; by Lemma 4.10(ii); and, by Lemma 4.10(i) and a careful calculation. Together these imply the result. The case when h = i is argued similarly using the equatioñ

Technical Lemmas.
We now prove several technical lemmas about the lowering operators which will be useful in what follows. The proof of the lemmas in this subsection follow the arguments of the analogous results of Brundan in [2]. The exception is the proof of Lemma 4.13. This proof is new and is simpler than Brundan's proof of the analogous result, which requires the introduction of certain formal polynomials.
The following lemma records how the lowering operators commute with E l = E l,l+1 .
Lemma 4.12. Let 1 ≤ i < j ≤ m + n and let A ⊆ (i..j). Let 1 ≤ l ≤ m + n − 1 and let J l be the left ideal of Dist(G) generated by E l . Then, where b h,j−1 is as defined in (4.9).
A calculation using this and condition (b) shows E l S i,j (A) ≡ 0. If A = {i + 1} and condition (a) holds, then l = i and again a calculation shows E l S i,j (A) ≡ 0. Now suppose ht(A) > ht({i + 1}) and the result holds for all sets of smaller height. If A = {l + 1}, then by our inductive assumption l = i and by applying Theorem 4.11 twice we obtain One proves (iii) by using Theorem 4.11 to induct on ht(A). One proves (iv) by applying Theorem 4.11 and the previous parts of the lemma.
Proof. We prove the statement by induction on d(i, j) := |(i..j)|. The base case is d(i, j) = 0. Then A = B = ∅, j = i + 1, and the left hand side of (4.13) is which is the right hand side of (4.13). In this case, incidentally, ε = (−1) v i . Now we assume d(i, j) > 0 and that (4.13) holds for all smaller d(i, j). We proceed by considering cases: Observe that this along with A ↓ B implies j − 1 / ∈ B. Then the left hand side of (4.13) is by Lemma 4.12(iii) and the inductive assumption.
.l)\A). Then by Lemma 4.12 (iv) Then necessarily either . If the former occurs, then by induction we have B i,j−1 (λ) ↓ C i..j−1 . As in Case 2, one concludes that B i,j (λ) ↓ C. If the latter occurs, then h = i and by induction B i,j−1 (λ) ↓ C\{h}. Again one can extend any weakly increasing injective map to prove that B i,j (λ) ↓ C.
All other possibilities are eliminated by Lemma 4.12 and our assumption that E l S i,j (A).v λ ∈ Rad M (λ). This proves the desired result in all possible situations.

4.4.
Filtrations and Hom-spaces. By weights and Frobenious reciprocity, we have the following lemma.
Proof. Let k λ be the Dist(B)-supermodule of highest weight λ ∈ X(T ), and let v λ denote its canonical generator. We have a filtration of k λ ⊗ V * as a Dist(B)-supermodule given by Proof. It follows from Lemma 4.17 and taking contravariant duals that we have the filtration 4.5. Return to Normal, Good, Conormal, and Cogood. We are now able to provide representation theoretic interpretations of the combinatorial notions of normal, good, conormal, and cogood. Note that the arguments used here are an adaptation of those used in [2].
Let V ′ denote the subspace of V spanned by the vectors v 1 , . . . , v m+n−1 . We consider G ′ = GL(V ′ ) as a subgroup of G = GL(V ) in the natural way and make the corresponding identification of Dist(G ′ ) as a subalgebra of Dist(G). A direct calculation verifies the following lemma.
The map e is a homomorphism of Dist(G ′ )-supermodules.
Theorem 4.20. Let λ ∈ X(T ) and let 1 ≤ i ≤ m + n. Let v λ ∈ M (λ) λ denote the canonical generator of M (λ). The following are equivalent: Proof. Say i = m + n. A consideration of weights shows that the image of v λ ⊗ w m+n ∈ L(λ) ⊗ V * is a primitive vector. Consequently, by Frobenious reciprocity we have Hom Dist(G) (M (λ − ε m+n ), L(λ) ⊗ V * ) = 0. When i = m + n it is straightforward to verify that (ii) and (iii) also always hold. Therefore, for the rest of the proof we assume 1 ≤ i ≤ m + n − 1. .v λ / ∈ Ker ϕ. As a vector space, we have the direct sum decomposition From the definition of S i,m+n (A), one can verify that However, by the definition of B, the right hand side is a nonzero scalar multiple of v λ . Therefore, z := S i,m+n (A).v λ / ∈ Rad M (λ) and the image of z in L(λ) is nonzero. Consequently z ⊗ w m+n ∈ L(λ) ⊗ V * is nonzero and, hence, is nonzero.
The preceding paragraph proves z ∈ L(λ) is a Dist(G ′ ) primitive vector hence, by Lemma 4.19, v λ (i) is a Dist(G ′ ) primitive vector. Furthermore, observe that a weight argument along with a direct calculation verifies E  Proof. By Lemma 4.9, i is conormal for λ ∈ X(T ) if and only if w 0 i is normal for s(λ) ∈ X(T ). By the above remarks we have the following isomorphisms of Homspaces: The result then follows immediately from Theorem 4.20.
The Hom-space given in (4.17) is naturally a subspace of the Hom-spaces in (4.18 Consequently, the map f factors through to give a nonzerof : This implies the desired result.
Arguing as in the proof of Corollary 4.21 using the mapS one obtains the following corollary of the previous result.

Translation Functors and Irreducible Supermodules.
Recall from (2.16) the definition of the translation functors E r and F r for r ∈ Z/pZ. We can now describe how these functors act on irreducible Dist(G)-supermodules.
Proof. First, let µ ∈ X(T ). We observe that by Theorem 4.20 if and only if µ = λ − ε i for some 1 ≤ i ≤ m + n which is normal for λ and wt(λ From this we conclude that (5.1) holds if and only if µ = λ − ε i for some 1 ≤ i ≤ m + n which is r-normal for λ.
Recall from (4.14) the filtration of W (λ) ⊗ V * by co-Verma supermodules. Intersecting with L(λ) ⊗ V * and projecting onto E r L(λ), we obtain a filtration 0 = Q 0 ⊆ · · · ⊆ Q m+n = E r L(λ) of E r L(λ) where each Q k /Q k−1 is a (possibly zero) submodule of a co-Verma module. Refining this filtration by requiring strict inclusions, we obtain 0 = Q 0 ⊆ · · · ⊆ Q l = E r L(λ) with Q k /Q k−1 a nonzero submodule of W (λ − ε i k ) and i 1 < · · · < i l . Now for any µ ∈ X(T ) we have, If 1 ≤ t ≤ m + n is r-normal for λ, then the left hand side is nonzero for µ = λ − ε t . However, by Lemma 4.15, the right hand side is one if µ = λ − ε i k for some k = 1, . . . , l and zero otherwise. Consequently, t = i k for some 1 ≤ k ≤ l. On the other hand, assume Q k /Q k−1 is nonzero. Applying the functor Hom . . By weights we have Hom Dist(G) (M (λ − ε i k ), Q k−1 ) = 0 and by Lemma 3.5 we have hence i k is r-normal for λ.
Theorem 5.2. Let λ ∈ X(T ) and r ∈ Z/pZ. holds if and only if µ = λ − ε i for some 1 ≤ i ≤ m + n which is r-good for λ. Also we note that if i is r-good for λ, then it is necessarily unique by the definition of r-good.
From this we see that if E r L(λ) = 0, then the socle of E r (λ) is precisely L(λ − ε i ) ∼ = L(ẽ * r (λ)) where i is r-good for λ. Therefore, E r L(λ) = 0 if and only if ε * r (λ) = 0 and, if it is nonzero, it has irreducible socle and is indecomposable. The self-duality statement and, hence, the description of the head follows from the fact that L(λ) is self-dual with respect to contravariant duality and E r commutes with this duality.
Finally, we prove the statement about the simplicity of E r L(λ). If E r L(λ) is simple, it is clear that i is r-normal if and only if i is r-good. Consequently, ε * r (λ) = 1. One the other hand, assume ε * r (λ) = 1 but E r L(λ) is not irreducible. Fix i to be the position of the unique − in the reduced r-signature for λ. From Lemma 5.1 we deduce that E r L(λ) is a submodule of W (λ − ε i ). This implies by the reducibility assumption that Hom Dist(G) (E r L(λ), L(λ − ε i )) = 0. However, taking contravariant duals and using that i is r-good for λ, we see that Hom Dist(G) (E r L(λ), L(λ − ε i )) ∼ = Hom Dist(G) (L(λ − ε i ), E r L(λ)) = 0.
This gives the desired contradiction. Therefore E r L(λ) is irreducible.
One deduces (ii) from (i) using the mapS defined in (4.16) or can be proven directly with an argument similar to the one used above.
In particular, in both cases the socle is multiplicity free and contains no more than p irreducible summands.

Odd Reflections
The results of this article depend on our choice of a homogeneous basis for V. More precisely, the crystal structure on X(T ) depends on the sequence of parities, v 1 , . . . , v m+n , of the vectors we have chosen. In this section, we discuss how to translate from one choice to another using Serganova's odd reflections. Note that it suffices to determine how to translate from the fixed homogeneous basis v 1 , . . . , v m+n to the homogeneous basis v 1 , . . . , v i−1 , v i+1 , v i , v i+2 , . . . , v m+n when v i + v i+1 =1 for some 1 ≤ i ≤ m + n − 1. Let 1 ≤ i ≤ m + n − 1 and let σ i : V → V denote the linear map defined by sending the first basis to the second. That is, for all 1 ≤ j ≤ m + n and where (i i + 1) ∈ S m+n . We then have an automorphism on GL(V ) induced by σ i which in turn induces an automorphism s i : Dist(G) → Dist(G). Explicitly, s i : e k,l → e (i i+1)k,(i i+1)l (6.1) for all 1 ≤ k, l ≤ m + n. We can twist a Dist(G)-supermodule M by s i in the usual way, where M si = M as a superspace and a.m = s i (a)m for all a ∈ Dist(G) and all m ∈ M. In particular, L(λ) si is again an irreducible Dist(G)-supermodule of some highest weight. The following lemma allows us to determine the highest weight of L(λ) si .
Recall that S m+n acts on X(T ) via x · ε j = ε xj for x ∈ S m+n and 1 ≤ j ≤ m + n. For 1 ≤ i ≤ m + n − 1, let s i : X(T ) → X(T ) be the involution given by where (−, −) is the bilinear form on X(T ) defined in (2.2).
Lemma 6.1. Let λ ∈ X(T ) and let 1 ≤ i ≤ m + n − 1 satisfying v i + v i+1 =1. Then, Proof. We observe that by the action of s i on Dist(T ), if v ∈ L(λ) µ , then v ∈ (L(λ) si ) (i i+1)·µ . Now let v λ be a Dist(G) primitive vector in L(λ) of weight λ, c.f. Lemma 2.2. We observe that the vector e i,i+1 .v λ = e i+1,i v λ ∈ L(λ) si is a Dist(G) primitive vector. For if 1 ≤ r < s ≤ m + n, then, unless r = i and s = i + 1, it is straightforward to verify that e r,s .e i+1,i v λ = 0 by the action of s i and by (2.4).
Thus, to complete the proof of the lemma, it suffices to show that e i+1,i v λ = 0 if and only if (λ, ε i − ε i+1 ) ≡ 0 (mod p). But e i+1,i v λ = 0 if and only if there is some element x ∈ Dist(B) such that xe i+1,i v λ is a non-zero multiple of v λ . By weights, the only x that needs to be considered is e i,i+1 . Finally, e i,i+1 e i+1,i v λ = (−1) v i (λ, ε i −ε i+1 )v λ .
This result is closely related to the fact that G has non-conjugate Borel subgroups and, hence, different labelings of the irreducible Dist(G)-supermodules by highest weight. The problem of translating between labelings was first solved by Serganova [19] and is also discussed in [4]. Corollary 6.2. Let 1 ≤ i ≤ m + n with v i + v i+1 =1. Let X(T ) 1 denote X(T ) with the crystal structure lifted (as in subsection 2.7) from the crystal B v1 ⊗ · · · ⊗ B vi ⊗ B vi+1 ⊗ · · · ⊗ B vm+n ,ẽ * r ,f * r , ε * r , ϕ * r , wt , and let X(T ) 2 denote X(T ) with the crystal structure lifted from the crystal B v1 ⊗ · · · ⊗ B vi+1 ⊗ B vi ⊗ · · · ⊗ B vm+n ,ẽ * r ,f * r , ε * r , ϕ * r , wt . The map s i : X(T ) 1 → X(T ) 2 defined in (6.1) gives an isomorphism of crystals.