COHOMOLOGY AND SUPPORT VARIETIES FOR LIE SUPERALGEBRAS II

BRIAN D. BOE, JONATHAN R. KUJAWA, AND DANIEL K. NAKANO

Abstract. In [BKN] the authors initiated a study of the representation theory of classical Lie superalgebras via a cohomological approach. Detecting subalgebras were constructed and a theory of support varieties was developed. The dimension of a detecting subalgebra coincides with the defect of the Lie superalgebra and the dimension of the support variety for a simple supermodule was conjectured to equal the atypicality of the supermodule. In this paper the authors compute the support varieties for Kac supermodules for Type I Lie superalgebras and the simple supermodules for 𝔤𝔩(m|n). The latter result verifies our earlier conjecture for 𝔤𝔩(m|n). In our investigation we also delineate several of the major differences between Type I versus Type II classical Lie superalgebras. Finally, the connection between atypicality, defect and superdimension is made more precise by using the theory of support varieties and representations of Clifford superalgebras.

1. Introduction

1.1. Let 𝔤 = 𝔤¯0 ⊕ 𝔤¯1 be a classical Lie superalgebra over the complex numbers. For any classical Lie superalgebra there exists by definition a connected reductive algebraic group G¯0 such that Lie (G¯0) = 𝔤¯0. The simple classical Lie superalgebras were classified by Kac [Kac1]. In [BKN] the authors used relative cohomology for the pair (𝔤, 𝔤¯0) to investigate the combinatorics and representation theory of the blocks in the category of finite dimensional representations of the Lie superalgebra 𝔤. In this situation the cohomology ring R = H^{•}(𝔤, 𝔤¯0; ℂ) is finitely generated because G¯0 is reductive. By using invariant theoretic results due to Luna and Richardson [LR] and Dadok and Kac [DK], the authors were able to construct natural “detecting” subalgebras 𝔢 = 𝔢¯0 ⊕ 𝔢¯1 of 𝔤 such that the restriction map in cohomology induces an isomorphism

R = H^{•}(𝔤, 𝔤¯0; ℂ) ≅ H^{•}(𝔢, 𝔢¯0; ℂ)^{W} ,

where W is a finite pseudoreflection group. One striking outcome of our construction was that the dimension of the odd part of the detecting subalgebras and the Krull dimension of R both coincide with the combinatorially defined defect of 𝔤 introduced earlier by Kac and Wakimoto [KW].

1.2. Given a finite dimensional 𝔤-supermodule, M, one can use the finite generation of R to define the cohomological support varieties V_{(𝔢,𝔢¯0)}(M) and V_{(𝔤,𝔤¯0)}(M). In [BKN, Theorem 6.2.2] it was proved that V_{(𝔤,𝔤¯0)}(M) can be identified generically with V_{(𝔢,𝔢¯0)}(M)/W and conjectured that this should hold everywhere. The variety V_{(𝔢,𝔢¯0)}(M) can be identified via a “rank variety” description as a certain subvariety of 𝔢¯1 [BKN, Theorem 6.3.2]. The rank variety description enabled us to demonstrate that the representation theory for the superalgebra over ℂ has similar features to modular representations of finite groups over fields of characteristic two (cf. [BKN, Corollary 6.4.1]). When 𝔤 admits an appropriate bilinear form one can define the “atypicality” of a block and of a simple supermodule. Atypicality, due to Kac and Serganova, is a combinatorial invariant used to give a rough measure of the complications involved in the block structure. Evidence from examples led us to conjecture that if L(λ) is a finite dimensional simple 𝔤-supermodule then the atypicality of L(λ) (denoted by atyp(L(λ))) equals dim V_{(𝔢,𝔢¯0)}(L(λ)) (cf. [BKN, Conjecture 7.2.1]).

Date: August 23, 2007.

2000 Mathematics Subject Classification. Primary 17B56, 17B10; Secondary 13A50.

Research of the second author was partially supported by NSF grant DMS-0402916.

Research of the third author was partially supported by NSF grant DMS-0400548.

1.3. This paper is aimed at providing applications and concrete computations for the theory developed in [BKN]. In Section 3 we distinguish between Type I and Type II classical Lie superalgebras. Type I Lie superalgebras are the ones (such as 𝔤𝔩(m|n)) which admit a compatible ℤ-grading concentrated in degrees −1, 0, and 1. Otherwise 𝔤 is said to be of Type II. We prove for Type I Lie superalgebras that the category F of finite dimensional supermodules is a highest weight category as defined by Cline, Parshall and Scott [CPS1]. We also show that if K(λ) is a Kac supermodule (i.e., universal highest weight supermodule) for a Type I Lie superalgebra 𝔤, then V_{(𝔤,𝔤¯0)}(K(λ)) = V_{(𝔢,𝔢¯0)}(K(λ)) = {0}. On the other hand, by using work of Germoni [Ger1] on the Type II classical Lie superalgebras 𝔬𝔰𝔭(3|2), D(2, 1; α), and G(3), one sees that when 𝔤 is of Type II the category F need not be a highest weight category and there exist Kac supermodules with nontrivial support varieties. These results make clear the significant representation theoretic differences between Type I and Type II Lie superalgebras.

In Section 4 we apply results of [BKN], Duflo and Serganova [DS], and Serganova [Ser] to compute the support varieties V_{(𝔤,𝔤¯0)}(L(λ)) and V_{(𝔢,𝔢¯0)}(L(λ)) of the finite dimensional simple supermodules L(λ) when 𝔤 = 𝔤𝔩(m|n). For simple 𝔤𝔩(m|n)-supermodules our computations verify the conjectures mentioned in Section 1.2. Namely, we prove

V_{(𝔤,𝔤¯0)}(L(λ)) ≅ V_{(𝔢,𝔢¯0)}(L(λ))/W,

and that the atypicality of L(λ) equals dim V_{(𝔢,𝔢¯0)}(L(λ)). A remarkable outcome of our results is that one can now extend the definition of atypicality to all 𝔤𝔩(m|n)-supermodules in a functorial way by setting atyp(M) = dim V_{(𝔢,𝔢¯0)}(M).

In the final section we demonstrate that the codimension of the support variety V_{(𝔢,𝔢¯0)}(M) is directly related to the 2-divisibility of the dimension of M. In many ways our results can be thought of as block theoretic analogues of the celebrated Kac-Weisfeiler Conjecture for non-restricted representations of classical Lie algebras over fields of characteristic p > 0. The Kac-Weisfeiler Conjecture [KWe] connects the p-divisibility of modules with the size of an associated geometric object (i.e., the corresponding nilpotent orbit). Premet proved the conjecture in 1995 [Pre]. We also relate 2-divisibility to the defect and atypicality for the simple 𝔤𝔩(m|n)-supermodules.

The authors are grateful to Weiqiang Wang for bringing the paper [Ger1] to our attention. The second author would like to thank Calvin Burgoyne and Mitchell Rothstein for helpful conversations about the representation theory of Clifford algebras.

2. Notation

2.1. Throughout we work with the complex numbers ℂ as the ground field. Recall that a superspace is a ℤ_{2}-graded vector space and, given a superspace V and a homogeneous vector v ∈ V, we write \overline{v} ∈ ℤ_{2} for the parity (or degree) of v. Elements of V¯0 (resp. V¯1) are called even (resp. odd). The superdimension of a superspace V is the integer dim V¯0−dim V¯1. Note that if M and M′ are two superspaces, then the space Hom_{ℂ}(M, M′) is naturally ℤ_{2}-graded by declaring f ∈ Hom_{ℂ}(M, M′)_{r} (r ∈ ℤ_{2}) if f(M_{s}) ⊆ M′_{s+r} for all s ∈ ℤ_{2}.

A superalgebra is a ℤ_{2}-graded, unital, associative algebra A = A¯0 ⊕ A¯1 which satisfies A_{r}A_{s} ⊆ A_{r+s} for all r, s ∈ ℤ_{2}. A Lie superalgebra is a superspace 𝔤 = 𝔤¯0 ⊕ 𝔤¯1 with a bracket operation [ , ] : 𝔤⊗𝔤 → 𝔤 which preserves the ℤ_{2}-grading and satisfies graded versions of the usual Lie bracket axioms. In particular, we note that 𝔤¯0 is a Lie algebra under the bracket obtained by restricting the bracket of 𝔤. If 𝔤 is a Lie superalgebra, then one has a universal enveloping superalgebra U(𝔤) which is ℤ_{2}-graded and satisfies a PBW-type theorem. See, for example, [Kac1] for details and further background on Lie superalgebras.

We call a finite dimensional Lie superalgebra classical if there is a connected reductive algebraic group G¯0 such that Lie(G¯0) = 𝔤¯0, and an action of G¯0 on 𝔤¯1 which differentiates to the adjoint action of 𝔤¯0 on 𝔤¯1. In particular, if 𝔤 is classical, then 𝔤¯0 is a reductive Lie algebra and 𝔤¯1 is semisimple as a 𝔤¯0-module. Note that we do not assume that 𝔤 is simple. A basic classical Lie superalgebra is a classical Lie superalgebra with a nondegenerate invariant supersymmetric even bilinear form. The simple classical Lie superalgebras were classified by Kac [Kac1].

Given a Lie superalgebra 𝔤, let us describe the category of 𝔤-supermodules. The objects are all left U(𝔤)-modules which are ℤ_{2}-graded; that is, superspaces M satisfying U(𝔤)_{r}M_{s} ⊆ M_{r+s} for all r, s ∈ ℤ_{2}. If M is a 𝔤-supermodule, then by definition N ⊆ M is a subsupermodule if it is a supermodule which inherits its grading from M in the sense that M_{r} ∩ N = N_{r} for r ∈ ℤ_{2}. We say a supermodule is finitely semisimple if it decomposes into a direct sum of finite dimensional simple supermodules. Given 𝔤-supermodules M and N one can use the antipode and coproduct of U(𝔤) to define a 𝔤-supermodule structure on the contragradient dual M^{∗} and the tensor product M ⊗ N.

A morphism of U(𝔤)-supermodules is an element f ∈ Hom_{ℂ}(M, M′) satisfying f(xm) = (−1)^{\overline{f} \overline{x}}xf(m) for all m ∈ M and all x ∈ U(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, note that Hom_{U(𝔤)}(M, M′) inherits a ℤ_{2}-grading as a subspace of Hom_{ℂ}(M, M′).

The category of 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 interchanges the ℤ_{2}-grading of a supermodule, allows one to make use of the tools of homological algebra (cf. 3.5).

Given a category, C, of 𝔤-supermodules and objects M, N in C, we write

Ext_{C}^{d}(M, N)

for the degree d extensions between N and M in the category C.

As a special case of the above discussion, we always view a Lie algebra (e.g. the even part of a Lie superalgebra) as a Lie superalgebra concentrated in degree ¯0.

2.2. Relative Cohomology. Let us recall the definition of relative cohomology for Lie superalgebras. Let M be a 𝔤-supermodule. Let 𝔤 be a Lie superalgebra, 𝔱 ⊆ 𝔤 a Lie subsuperalgebra, and M a 𝔤-supermodule. Define

C^{p}(𝔤, 𝔱; M) = Hom_{𝔱}(∧_{s}^{p}(𝔤/𝔱), M).

Note that ∧_{s}^{p}(V) denotes the super (i.e. graded) wedge product of the superspace V. In particular, if V = V¯1 then Λ_{s}^{p}(V) = S^{p}(V), the ordinary symmetric product.

There is a differential d^{p} : C^{p}(𝔤, 𝔱; M) → C^{p+1}(𝔤, 𝔱; M) defined as for relative cohomology for Lie algebras (cf. [BKN, Section 2.2]) such that

H^{p}(𝔤, 𝔱; M) = Ker d^{p}/ Im d^{p−1}.

Let C(𝔤, 𝔱) denote the full subcategory of all 𝔤-supermodules which are finitely semisimple as 𝔱-supermodules. A key connection is that if M and N are objects of C(𝔤, 𝔱), then one has

Ext_{C(𝔤,𝔱)}^{•}(M, N) ≅ H^{•}(𝔤, 𝔱; M^{∗} ⊗ N).  (2.2.1)

2.3. Support Varieties. Let F = F(𝔤, 𝔤¯0) denote the full subcategory of all finite dimensional 𝔤-supermodules which are completely reducible as 𝔤¯0-supermodules. This is the category of interest in [BKN]. One then has the graded cohomology ring

R := Ext_{F}^{•}(ℂ, ℂ).

It was proven in [BKN, Theorem 2.5.1] that if M and N are objects in F, then

H^{•}(𝔤, 𝔤¯0; M^{∗} ⊗ N) ≅ Ext_{F}^{•}(M, N).  (2.3.1)

As a consequence, when 𝔤 is classical R is a finitely generated commutative ring. Furthermore, if M and N are objects of F, then Ext_{F}^{•}(M, N) is a finitely generated graded R-module [BKN, Theorem 2.5.3].

Set

I_{(𝔤,𝔤¯0)}(M, N) = Ann_{R}(Ext_{F}^{•}(M, N)),

the annihilator ideal of this module. We define the support variety of the pair (M, N) to be

V_{(𝔤,𝔤¯0)}(M, N) = MaxSpec(R/I_{(𝔤,𝔤¯0)}(M, N)).  (2.3.2)

In particular, when M = N we define the support variety of M to be

V_{(𝔤,𝔤¯0)}(M) = MaxSpec(R/I_{(𝔤,𝔤¯0)}(M, M)).  (2.3.3)

In the case when 𝔤 is a simple classical Lie superalgebra, the ring R turns out to always be a polynomial ring [BKN, Section 8.8] in, say, r variables. In this case one can view V_{(𝔤,𝔤¯0)}(M) as a closed, conical affine variety; namely, the subvariety of MaxSpec(R) ≅ 𝔸^{r} defined by the ideal I_{(𝔤,𝔤¯0)}(M, M).

2.4. Detecting Subalgebras. If 𝔤 is a classical Lie superalgebra which satisfies certain invariant theoretic restrictions (i.e. 𝔤 is stable and polar), then one can construct a classical subalgebra 𝔢 ⊆ 𝔤 such that the natural inclusion map 𝔢 ↪ 𝔤 induces the following isomorphism in cohomology,

R ≅ H^{•}(𝔢, 𝔢¯0; ℂ)^{W} ,

where W is a finite pseudoreflection group [BKN, Section 4.1].

As 𝔢 is classical, one can define support varieties just as in (2.3.3) for objects of F(𝔢, 𝔢¯0). The variety V_{(𝔢,𝔢¯0)}(M) shares a number of features of the classical theory of support varieties and admits the following rank variety description [BKN, Theorem 6.3.2]. As a matter of notation, given a homogeneous element x ∈ 𝔢, let ⟨x⟩ denote the Lie subsuperalgebra generated by x. Define the rank variety of M to be

V_{(𝔢,𝔢¯0)}^{rank} (M) = {x ∈ 𝔢¯1 | M is not projective as an ⟨x⟩-supermodule} ∪ {0}.

Then V_{(𝔢,𝔢¯0)}(M) ≅ V_{(𝔢,𝔢¯0)}^{rank}(M). We also remark that if M is an object of F(𝔤, 𝔤¯0), then one can view M as an object of F(𝔢, 𝔢¯0) by restriction. In particular, one then has the following induced map of varieties,

res^{∗} : V_{(𝔢,𝔢¯0)}(M) → V_{(𝔤,𝔤¯0)}(M).

Furthermore, one has (cf. [BKN, (6.1.3)]),

V_{(𝔢,𝔢¯0)}(M)/W ≅ res^{∗} (V_{(𝔢,𝔢¯0)}(M)) ⊆ V_{(𝔤,𝔤¯0)}(M).  (2.4.1)

3. Type I and Type II Lie Superalgebras

A Lie superalgebra is said to be of Type I if it admits a consistent ℤ-grading concentrated in degrees −1, 0, and 1. Otherwise it is of Type II (cf. Section 3.1). The distinction between these two types was first made by Kac [Kac1]. In this section we will see that this distinction is also significant from the point of view of representation theory and support varieties.

3.1. Let 𝔤 = 𝔤¯0 ⊕ 𝔤¯1 be a classical Lie superalgebra with a ℤ-grading

𝔤 = 𝔤_{−1} ⊕ 𝔤_{0} ⊕ 𝔤_{1}

which satisfies [𝔤_{i}, 𝔤_{j}] ⊆ 𝔤_{i+j} for all i, j ∈ ℤ, 𝔤¯0 = 𝔤_{0}, and 𝔤¯1 = 𝔤_{−1} ⊕ 𝔤_{1}. That is, in the terminology of [Kac1], the ℤ-grading is consistent since 𝔤¯0 = ∑_{k} 𝔤_{2k} and 𝔤¯1 = ∑_{k} 𝔤_{2k+1}. As 𝔤 is concentrated in degrees −1, 0, and 1, by definition 𝔤 is of Type I. Let

𝔭^{+} = 𝔤_{0} ⊕ 𝔤_{1}  and  𝔭^{−} = 𝔤_{0} ⊕ 𝔤_{−1}.

We observe that, because of the ℤ-grading, 𝔤_{1} is an ideal of 𝔭^{+}. Fix a Cartan subalgebra 𝔥 ⊆ 𝔤¯0 and Borel subalgebras 𝔟¯0 ⊆ 𝔤¯0 and 𝔟 ⊆ 𝔤 so that 𝔥 ⊆ 𝔟¯0 and

𝔟 = 𝔟¯0 ⊕ 𝔤_{1}.  (3.1.1)

Let ≤ denote the usual dominance order on 𝔥^{∗} given by the choice of 𝔟. Note that the Lie superalgebra 𝔤𝔩(m|n) and the simple Lie superalgebras of types A(m, n) and C(n) are all of Type I (cf. [Kac1, Section 2]).

Let X+ ¯0 ⊆ 𝔥^{∗} denote the parameterizing set of highest weights for the simple finite dimensional 𝔤¯0-supermodules with respect to the pair (𝔥, 𝔟¯0). Given λ ∈ X+ ¯0, let L¯0(λ) denote the simple 𝔤¯0-supermodule of highest weight λ (concentrated in degree ¯0). We view L¯0(λ) as a simple 𝔭^{+}-supermodule via inflation through the canonical quotient 𝔭^{+} ↠ 𝔤¯0. Let

K(λ) = U(𝔤) ⊗_{U(𝔭^{+})} L¯0(λ),

be the Kac supermodule of highest weight λ. A standard argument using (3.1.1) and the fact that 𝔤_{1} is an ideal of 𝔭^{+} shows that K(λ) is the universal highest weight supermodule in F of highest weight λ. By highest weight arguments, K(λ) has simple head L(λ) and the set

{L(λ) | λ ∈ X+ ¯0}

is a complete irredundant collection of simple objects in F. Note that since L(λ) is generated by a one dimensional highest weight space, it follows by Schur’s Lemma for Lie superalgebras that Hom_{𝔤}(L(λ), L(λ)) is one dimensional and consists of maps given by scalar multiplication.

3.2. The following theorem shows that the relative cohomology of a Kac supermodule can be nonzero in only finitely many degrees. We remind the reader of the relevant notation and results which were established in Section 2.2.

Theorem 3.2.1. Let λ ∈ X+ ¯0 and M ∈ F. Then Ext_{F}^{n}(K(λ), M) = 0 for n ≫ 0.

Proof. We first require some preliminary results. By (2.2.1), (2.3.1), and Frobenius reciprocity (cf. [Kum, Lemma 3.1.14]) we have

Ext_{F}^{j}(K(λ), M) ≅ Ext_{C(𝔤,𝔤¯0)}^{j}(K(λ), M) ≅ Ext_{C(𝔭^{+},𝔤¯0)}^{j} (L¯0(λ), M)  (3.2.1)

for j ≥ 0. Since 𝔤_{1} is an ideal of 𝔭^{+}, one can apply the Lyndon-Hochschild-Serre spectral sequence for the pair (𝔤_{1}, {0}) in (𝔭^{+}, 𝔤¯0) (cf. [BW, Theorem 6.5]):

E_{2}^{i,j} = Ext_{C(𝔤¯0,𝔤¯0)}^{i}(L¯0(λ), Ext_{C(𝔤_{1},{0})}^{j}(ℂ, M)) ⇒ Ext_{C(𝔭^{+},𝔤¯0)}^{i+j}(L¯0(λ), M).

Since Ext_{C(𝔤¯0,𝔤¯0)}^{i}(−, −) calculates extensions in the category of 𝔤¯0-supermodules which are completely reducible, the higher Ext’s vanish and the spectral sequence collapses. This yields the following isomorphism:

Hom_{𝔤¯0} (L¯0(λ), Ext_{C(𝔤_{1},{0})}^{j}(ℂ, M)) ≅ Ext_{C(𝔭^{+},𝔤¯0)}^{j} (L¯0(λ), M) ,  (3.2.2)

for all j ≥ 0. Combining (3.2.1) and (3.2.2), we obtain

Ext_{F}^{j}(K(λ), M) ≅ Hom_{𝔤¯0} (L¯0(λ), Ext_{C(𝔤_{1},{0})}^{j}(ℂ, M)) ,  (3.2.3)

for all j ≥ 0.

By the definition of relative cohomology for the pair (𝔤_{1}, {0}), Ext_{C(𝔤_{1},{0})}^{j}(ℂ, M) is a subquotient of S^{j}(𝔤_{1}^{∗})⊗M. We observe that since M is finite dimensional it has only finitely many nonzero weight spaces. Furthermore, by our assumptions in Section 3.1, 𝔤_{1} consists of positive root spaces so 𝔤_{1}^{∗} has a weight space decomposition consisting of negative roots. Therefore for j sufficently large the 𝔤¯0-supermodule S^{j}(𝔤_{1}^{∗}) ⊗ M cannot have a nontrivial λ weight space. That is, for j sufficently large L¯0(λ) does not appear as a composition factor of S^{j}(𝔤_{1}^{∗})⊗M as a 𝔤¯0-supermodule. Hence it also does not appear in Ext_{C(𝔤_{1},{0})}^{j}(ℂ, M) and, therefore, for j ≫ 0 the Hom space in (3.2.3) is zero. This implies the desired result. □

3.3. Support Varieties for Kac Supermodules. The previous theorem allows one to compute the relative support varieties and support varieties of Kac supermodules. We continue with the assumptions of the previous section.

Corollary 3.3.1. Let K(λ) be a Kac supermodule of highest weight λ ∈ X+ ¯0 and let M ∈ F. Then,

(a) V_{(𝔤,𝔤¯0)}(K(λ), M) = {0};

(b) V_{(𝔤,𝔤¯0)}(K(λ)) = {0};

(c) V_{(𝔢,𝔢¯0)}(K(λ)) = {0};

(d) K(λ) is projective upon restriction to U(𝔢).

Proof. We first prove (a). By Theorem 3.2.1 we can fix N ≥ 0 so that Ext_{F}^{j}(K(λ), M) = 0 for all j ≥ N. Then, given an element x ∈ R of positive degree, it follows that x^{N} annihilates Ext_{F}^{•}(K(λ), M) as it is a graded R-module. That is, all postive degree elements of R are contained in the radical of the ideal

I_{(𝔤,𝔤¯0)} (K(λ), M) = Ann_{R} (Ext_{F}^{•}(K(λ), M)) .

However, an ideal and its radical ideal define the same variety. Therefore, by (2.3.2), V_{(𝔤,𝔤¯0)}(K(λ), M) = {0}. Part (b) follows immediately from (a) by setting M = K(λ). To prove (c), one notes that the map in (2.4.1), res^{∗} : V_{(𝔢,𝔢¯0)}(K(λ)) → V_{(𝔤,𝔤¯0)}(K(λ)), is finiteto-one. Since V_{(𝔢,𝔢¯0)}(K(λ)) is a conical variety, it follows that it must be {0}. Finally, part (d) follows from [BKN, Theorem 6.4.2(b)]. □

In the case of modular representations of finite groups and restricted Lie algebras, one has that the support variety is trivial if and only if the module is projective. In contrast, despite Corollary 3.3.1(b), K(λ) need not be a projective 𝔤-supermodule. For example, this is already true in the principal block of 𝔤𝔩(1|1).

3.4. Filtrations by Kac Supermodules. We now show that for a Type I Lie superalgebra the category F is a highest weight category in the sense of [CPS1]. This was proven for 𝔤𝔩(m|n) by Brundan [Bru1] (see also [Ger2, Section 3.6]), and is implicit for 𝔬𝔰𝔭(2|2n) in work of Cheng, Wang, and Zhang [CWZ]. We provide a general proof which includes both of these as cases.

We continue to assume that 𝔤 is a Type I Lie superalgebra which satisfies the assumptions of Section 3.1. Given λ ∈ X+ ¯0, let P (λ) denote the projective cover in F for the simple 𝔤-supermodule L(λ), and let

Q(λ) = U(𝔤) ⊗_{U(𝔤¯0)} L¯0(λ)  and  K′(λ) = Hom_{U(𝔭^{−})} (U(𝔤), L¯0(λ)) ,

where L¯0(λ) is viewed as a U(𝔭^{−})-supermodule by inflation through the canonical quotient 𝔭^{−} ↠ 𝔤¯0. Note that since L¯0(λ) is projective in the category of finite dimensional 𝔤¯0supermodules (which are completely reducible over 𝔤¯0) and the functor U(𝔤) ⊗_{U(𝔤¯0)} − is exact, Q(λ) is a projective object in F. By Frobenius reciprocity one sees that P (λ) appears as a direct summand of Q(λ).

Given a supermodule M in F, say M admits a K-filtration if there is a filtration

M = M_{0} ⊋ M_{1} ⊋ M_{2} ⊋ ⋯

as 𝔤-supermodules so that for each k ≥ 0 such that M_{k} ≠ 0, one has M_{k}/M_{k+1} ≅ K(μ) for some μ ∈ X+ ¯0. If M admits a K-filtration, then write (M : K(μ)) for the number of times K(μ) appears. As we will see, this number is independent of the choice of filtration.

Proposition 3.4.1. The following statements hold true in F.

(a) dim Hom_{𝔤}(K(λ), K′(μ)) = δ_{λ,μ} for all λ, μ ∈ X+ ¯0.

(b) Ext_{F}^{1}(K(λ), K′(μ)) = 0 for all λ, μ ∈ X+ ¯0.

(c) A ℤ-graded object of F has a K-filtration if and only if it is a ℤ-graded free U(𝔤_{−1})- supermodule.

(d) P (λ) admits a K-filtration.

(e) If M is an object of F with a K-filtration, then,

(M : K(μ)) = dim Hom_{𝔤}(M, K′(μ)).

(f) In particular, one has the reciprocity formula

(P (λ) : K(μ)) = [K′(μ) : L(λ)].

Furthermore, as the weights of K′(μ) are all dominated by μ, one has that (P (λ) : K(λ)) = 1 and (P (λ) : K(μ)) = 0 unless μ ≥ λ.

Proof. Parts (a) and (b) are proved in [Bru2, Lemma 3.6] and part (c) is proved in [Bru2, Lemma 4.2].

To prove (d), consider the ℤ-graded 𝔭^{+}-supermodule

N := U(𝔭^{+}) ⊗_{U(𝔤¯0)} L¯0(λ)

whose grading is obtained from the canonical ℤ-grading on U(𝔭^{+}) and by viewing L_{0}(λ) as concentrated in degree 0. For k ≥ 0, set N_{k} to be the U(𝔭^{+})-supermodule generated by all elements of degree greater than or equal to k. Then one has the filtration

N = N_{0} ⊃ N_{1} ⊃ ⋯

as 𝔭^{+}-supermodules, where N_{k}/N_{k+1} is a finite dimensional completely reducible 𝔭^{+}-super-module. Apply the functor U(𝔤) ⊗_{U(𝔭^{+})} − to this filtration. By the PBW theorem U(𝔤) is a free right U(𝔭^{+})-supermodule so the functor is exact. This along with the Tensor Identity implies that one obtains a filtration

Q(λ) = P_{0} ⊃ P_{1} ⊃ ⋯

where P_{k} ≅ U(𝔤) ⊗_{U(𝔭^{+})} N_{k}. Furthermore, one has that

P_{k}/P_{k+1} ≅ U(𝔤) ⊗_{U(𝔟^{+})} (N_{k}/N_{k+1}) ≅ ⨁ K(ν)

where the direct sum runs over all L¯0(ν) which appear as composition factors of the 𝔭^{+}- supermodule N_{k}/N_{k+1}. Thus Q(λ) is a ℤ-graded supermodule in F which admits a Kfiltration. Arguing as in [HN, Theorem 3.3], one sees that P (λ) is a ℤ-graded direct summand of Q(λ). Applying (c) it follows that P (λ) admits a K-filtration.

To prove (e), fix μ ∈ X+ ¯0 and say M = M_{0} ⊃ M_{1} ⊃ ⋯ is a fixed K-filtration. One then argues by induction on the length of the K-filtration of M, the long exact sequence induced by the functor Hom_{𝔤} (−, K′(μ)) , and parts (a) and (b). Finally, to prove (f), note that both sides of the equality are counted by dim Hom_{𝔤}(P (λ), K′(μ)). □

3.5. Highest Weight Categories. Let us now additionally assume there exists an automorphism τ : 𝔤 → 𝔤 such that τ(𝔤_{i}) = 𝔤_{−i} (i ∈ ℤ) and, when restricted to 𝔤_{0}, τ coincides with the Chevalley automorphism. For Type I simple classical Lie superalgebras such an automorphism exists by the proof of [Kac1, Proposition 2.5.3]. For 𝔤𝔩(m|n) one can take τ to be the supertranspose map [Bru1, Section 4-a].

Since morphisms are not required to preserve the ℤ_{2}-grading (cf. Section 2.1), F is not an abelian category. However one can consider the underlying even category F^{ev} which consists of the same objects as F but only the homomorphisms which preserve the ℤ_{2}-grading (i.e. the even morphisms of Section 2.1). Then F^{ev} is an abelian category. Furthermore, one has the parity change functor

Π : F → F

which is defined by ΠM = M as a vector space, the ℤ_{2}-grading is given by

(ΠM)_{r} = M_{r+1}

for r ∈ ℤ_{2}, and action given by

x.m = (−1)\overline{x}xm

for all homogeneous x ∈ 𝔤 and m ∈ M. One checks that for M, N in F,

Hom_{F}(M, N)¯1 ≅ Hom_{F}(M, ΠN)¯0 = Hom_{F^{ev}}(M, ΠN).

Thus one can reconstruct the category F from F^{ev}.

In particular, one can choose a projective resolution P_{•} → M where all maps are even and, hence, the differentials in the complex Hom_{F}(P_{•}, N) are degree preserving. The chain complex then decomposes as

Hom_{F}(P_{•}, N) = Hom_{F}(P_{•}, N)¯0 ⊕ Hom_{F}(P_{•}, N)¯1 ≅ Hom_{F^{ev}}(P_{•}, N) ⊕ Hom_{F^{ev}}(P_{•}, ΠN).

This in turn induces the following decomposition:

Ext_{F}^{•}(M, N) = Ext_{F}^{•}(M, N)¯0 ⊕ Ext_{F}^{•}(M, N)¯1 ≅ Ext_{F^{ev}}^{•}(M, N) ⊕ Ext_{F^{ev}}^{•}(M, ΠN). (3.5.1)

Since L(λ) is generated by a one dimensional highest weight space, Hom_{𝔤} (L(λ), L(λ)) is one dimensional and elements are given by scalar multiplication. It follows that L(λ) and ΠL(λ) are not isomorphic via a grading preserving map. Thus a complete irredundant set of simple supermodules in F^{ev} is given by

{L(λ), ΠL(λ) | λ ∈ X+ ¯0} .

That is, via the correspondence (λ, r) ↔ Π^{r}L(λ) =: L(λ, r), one can naturally parameterize the simple supermodules in F^{ev} by the set

X^{++} := X+ ¯0 × ℤ_{2}.

Let P (λ, r) = Π^{r}P (λ) be the projective cover of L(λ, r) in F^{ev} and let K(λ, r) = Π^{r}K(λ). Define a partial order ≤_{ev} on X^{++} using the lexicographic order obtained by taking the dominance order on X+ ¯0 given by the choice of Borel subalgebra 𝔟 and the trivial partial order on ℤ_{2}; that is, the one for which ¯0 and ¯1 are incomparable.

Given a supermodule M in the category F^{ev}, define the contravariant dual of M, M^{τ} , to be M^{∗} with the action of 𝔤 twisted by the automorphism τ chosen above. In particular, one sees that (M^{τ})^{τ} ≃ M and L(λ, r)^{τ} ≃ L(λ, r) for (λ, r) ∈ X^{++}. Thus, M ↦ M^{τ} defines a duality in the sense of [CPS2].

Theorem 3.5.1. The category F^{ev} with duality M ↦ M^{τ} , partial order ≤_{ev} and objects L(λ, r), K(λ, r), and P (λ, r) ((λ, r) ∈ X^{++}) is a highest weight category with duality in the sense of [CPS2].

Proof. The main thing to prove is that P (λ, r) admits a K-filtration with (P (λ, r) : K(μ, s)) ≠ 0 only if (μ, s) ≥_{ev} (λ, r). However, this follows by Proposition 3.4.1(f) and the definition of the partial order ≤_{ev}. Using this one can now verify that F^{ev} is a highest weight category with respect to the partial order ≤_{ev} . □

As mentioned earlier, Brundan proved a similar result for 𝔤𝔩(m|n). Namely, that one has the decomposition

F(𝔤𝔩(m|n), 𝔤𝔩(m|n)¯0) = F(𝔤𝔩(m|n), 𝔤𝔩(m|n)¯0)¯0 ⊕ F(𝔤𝔩(m|n), 𝔤𝔩(m|n)¯0)¯1,

where F(𝔤𝔩(m|n), 𝔤𝔩(m|n)¯0)¯0 and F(𝔤𝔩(m|n), 𝔤𝔩(m|n)¯0)¯1 are isomorphic highest weight categories [Bru1, Theorem 4.47]. One can show that Brundan’s result is equivalent to Theorem 3.5.1 for 𝔤 = 𝔤𝔩(m|n). In the case of 𝔤 = 𝔬𝔰𝔭(2|2n), Cheng, Wang, and Zhang [CWZ, Section 3.1] use but do not explicitly prove that F is a highest weight category. The above theorem includes both of these cases.

3.6. Type II Lie Superalgebras. The work of Germoni [Ger1] shows that Type I Lie superalgebras are rather special here. In particular, Germoni demonstrates that for the Type II Lie superalgebras 𝔬𝔰𝔭(3|2), D(2, 1; α), and G(3) the category F^{ev} is not a highest weight category with respect to the partial order ≤_{ev}. Furthermore, examples illustrate that for Type II Lie superalgebras the Kac supermodules may have nontrivial support variety. This can also be deduced from [Ger1] but for the sake of completeness we include the calculation of an example here.

Let 𝔤 = 𝔬𝔰𝔭(3|2). It has a consistent ℤ-grading,

𝔤 = 𝔤_{−2} ⊕ 𝔤_{−1} ⊕ 𝔤_{0} ⊕ 𝔤_{1} ⊕ 𝔤_{2},

but no consistent grading concentrated in degrees −1, 0, and 1. Thus it is of Type II. Since 𝔤¯0 is semisimple, the category F = F(𝔤, 𝔤¯0) is simply the category of all finite dimensional 𝔤-supermodules. Fix a Cartan subalgebra 𝔥 ⊆ 𝔤_{0} and Borel subalgebras 𝔟_{0} ⊆ 𝔤_{0}, 𝔟¯0 ⊆ 𝔤¯0, and 𝔟 ⊆ 𝔤 so that 𝔥 ⊆ 𝔟_{0}, 𝔟¯0 = 𝔟_{0} ⊕ 𝔤_{2}, and 𝔟 = 𝔟_{0} ⊕ 𝔤_{1} ⊕ 𝔤_{2}.

Let L_{0}(λ) be a simple finite dimensional 𝔤_{0}-supermodule concentrated in degree ¯0 and of highest weight λ ∈ 𝔥^{∗} with respect to the choice of the pair (𝔥, 𝔟_{0}). It can then be viewed as a simple supermodule for the Lie superalgebra 𝔭 := 𝔤_{0} ⊕ 𝔤_{1} ⊕ 𝔤_{2} via inflation through the canonical qoutient 𝔭 ↠ 𝔤_{0}. Define

K̃(λ) = U(𝔤) ⊗_{U(𝔭)} L_{0}(λ).

Using the argument used in the proof of [Bru2, Lemma 7.3] one sees that K̃(λ) admits a finite composition series. Thus, it makes sense to define

K(λ) = K̃(λ)/U  (3.6.1)

as the maximal finite dimensional quotient of K̃(λ). As with the Kac supermodules of Section 3.1, one can show that K(λ) is the universal highest weight supermodule in F of highest weight λ with respect to the choice of the pair (𝔥, 𝔟). That is, K(λ) is the Kac supermodule for 𝔤 of highest weight λ. Note that K(λ) is nonzero if and only if λ is the highest weight of a simple finite dimensional 𝔤¯0-supermodule with respect to the pair (𝔥, 𝔟¯0).

Let v_{λ} denote a fixed nonzero element of the one dimensional λ weight space of K(λ). By the PBW theorem, note that K(λ) is spanned by elements of the form

x^{r}y^{s}v_{λ},  (3.6.2)

where x ∈ 𝔤_{−1}, y ∈ 𝔤_{−2}, r ∈ {0, 1}, and s ≥ 0.

We now consider the case when λ = 0. Since K(0) is completely reducible as a 𝔤¯0supermodule and v_{0} is a highest weight vector of weight 0 with respect to the pair (𝔥, 𝔟¯0), it follows that v_{0} spans a trivial 𝔤¯0-supermodule in K(0). That is, y^{s}v_{0} = 0 for all y ∈ 𝔤_{−2} and s ≥ 1. Furthermore, given x ∈ 𝔤_{−1} one can use [Kac2, Proposition 1.3] (or the matrix presentation of 𝔤 provided in [Kac1, Section 2]) to choose y ∈ 𝔤_{−2} and z ∈ 𝔤_{1} such that [z, y] = x. But yv_{0} = 0 by the previous remark and zv_{0} = 0 as K(0) is a quotient of K̃(0). Thus one has

xv_{0} = [z, y]v_{0} = zyv_{0} − yzv_{0} = 0  (3.6.3)

for any x ∈ 𝔤_{−1}. Taken together with (3.6.2) this implies that K(0) is one dimensional and spanned by v_{0}. That is, one has

K(0) = L(0) = ℂ.  (3.6.4)

Applying the calculations of [BKN, Section 8], one has

V_{(𝔤,𝔤¯0)}(K(0)) = V_{(𝔤,𝔤¯0)}(ℂ) ≅ 𝔸^{1},
V_{(𝔢,𝔢¯0)}(K(0)) = V_{(𝔢,𝔢¯0)}(ℂ) ≅ 𝔸^{1}.

Thus the support varieties of a Kac module for a Type II Lie superalgebra need not be trivial.

Another consequence of (3.6.4) is the fact that K(0) = L(0) despite the fact that λ = 0 is not a typical weight. However, in [Kac2, Theorem 1] it is asserted that for a basic classical Lie superalgebra such as 𝔬𝔰𝔭(3|2), K(λ) = L(λ) if and only if λ is typical. This counterexample may be known to experts but we could not find a suitable reference in the literature. We expect that the result in [Kac2] is correct for Type I Lie superalgebras.

4. Support Varieties for the Simple Supermodules of 𝔤𝔩(m|n)

4.1. In this section the results of [BKN], [DS], and [Ser] are combined to compute the support varieties of the simple 𝔤𝔩(m|n)-supermodules. Before doing so, let us fix various choices and develop the necessary notation and background.

Let 𝔤 = 𝔤𝔩(m|n) be the set of all (m + n) × (m + n) matrices over the complex numbers. If E_{i,j} (1 ≤ i, j ≤ m + n) denotes the matrix unit with a one in the (i, j) position, then the ℤ_{2}-grading is obtained by setting \overline{E}_{i,j} = ¯0 if 1 ≤ i, j ≤ m or m + 1 ≤ i, j ≤ m + n, and \overline{E}_{i,j} = ¯1, otherwise. The bracket is given by the super commutator,

[A, B] = AB − (−1)^{\overline{A} \overline{B}}BA,

for homogeneous A, B ∈ 𝔤𝔩(m|n). One can fix a consistent ℤ-grading,

𝔤 = 𝔤_{−1} ⊕ 𝔤_{0} ⊕ 𝔤_{1},

by setting 𝔤_{−1} to be the span of {E_{i,j} | m + 1 ≤ i ≤ m + n, 1 ≤ j ≤ m}, 𝔤_{0} to be the span of {E_{i,j} | 1 ≤ i, j ≤ m, or m + 1 ≤ i, j ≤ m + n}, and 𝔤_{1} to be the span of {E_{i,j} | 1 ≤ i ≤ m, m + 1 ≤ j ≤ m + n}. Note that 𝔤 is of Type I. Let 𝔥 ⊆ 𝔤¯0, 𝔟¯0 ⊆ 𝔤¯0, and 𝔟 ⊆ 𝔤 denote the subalgebras of all diagonal matrices, all upper triangular even matrices, and all upper triangular matrices, respectively.

Let G¯0 denote the connected reductive group with Lie(G¯0) = 𝔤¯0. Then G¯0 ≅ GL(m) × GL(n). We identify G¯0 as the subgroup of the supergroup GL(m|n) embedded diagonally as block matrices (cf. [Kuj, Section 2]). In particular, G¯0 acts on 𝔤 by conjugation. Let H ⊆ G¯0 denote the maximal torus such that Lie(H) = 𝔥. Similarly, let B¯0 ⊆ G¯0 denote the Borel subgroup such that Lie(B¯0) = 𝔟¯0.

With respect to the choice of the pair (𝔥, 𝔟) the root system of 𝔤 can be described as follows. Let ε_{i} ∈ 𝔥^{∗} be the linear functional which picks out the ith entry of a diagonal matrix. With respect to these choices we then have the roots, positive roots, even roots, and odd roots, respectively:

Φ = {ε_{i} − ε_{j} | 1 ≤ i, j ≤ m + n} ,
Φ^{+} = {ε_{i} − ε_{j} | 1 ≤ i < j ≤ m + n} ,
Φ¯0 = {ε_{i} − ε_{j} | 1 ≤ i, j ≤ m, or m + 1 ≤ i, j ≤ m + n} ,
Φ¯1 = {ε_{i} − ε_{j} | 1 ≤ i ≤ m, m + 1 ≤ j ≤ m + n, or m + 1 ≤ i ≤ m + n, 1 ≤ j ≤ n} .

One can define a bilinear form on 𝔥^{∗} by

(ε_{i}, ε_{j}) = \begin{cases}δ_{i,j}, & if 1 ≤ i ≤ m; \\ −δ_{i,j}, & if m + 1 ≤ i ≤ m + n. \end{cases}

Note that the odd roots are isotropic with respect to this bilinear form.

4.2. Defect and Atypicality. By definition the defect of 𝔤, which we denote by def(𝔤), is the maximal size of a set of pairwise orthogonal, isotropic roots of 𝔤. One can verify that def(𝔤) = min(m, n); we write r = def(𝔤) for short. Given λ ∈ X+ ¯0 (cf. Section 3.1), the atypicality of λ, which we denote atyp(λ), is defined to be the maximal size of a set of pairwise orthogonal, isotropic roots which are also orthogonal to λ + ρ, where

2ρ := ∑_{α∈Φ^{+}∩Φ¯0} α − ∑_{α∈Φ^{+}∩Φ¯1} α.

By definition one has

atyp(λ) ≤ def(𝔤).  (4.2.1)

Furthermore, note that atypicality is invariant under choice of Borel subalgebra so it makes sense to define the atypicality of L(λ) to be atyp(L(λ)) = atyp(λ). See [BKN, Section 7] for a more detailed discussion of defect and atypicality.

4.3. When the atypicality of a simple 𝔤𝔩(m|n)-supermodule L(λ) is zero, then necessarily L(λ) is projective in F [Kac2, Theorem 1]. In this case H^{t}(𝔤, 𝔤¯0; L(λ)^{∗} ⊗ L(λ)) = 0 for t ≥ 1 and by (2.3.2) one has V_{(𝔢,𝔢¯0)}(L(λ)) = V_{(𝔤,𝔤¯0)}(L(λ)) = {0}. Therefore we will primarily be interested in the case when the atypicality of L(λ) is strictly greater than zero. In this case one can assume λ is integral up to tensoring by some one dimensional representation. As it has no effect on support varieties by Proposition 4.6.2, we will freely and implicitly assume λ is integral. In particular, this is relevent to the use of the results of [Ser] in which this reduction is implicitly used.

4.4. Detecting Subalgebra for 𝔤𝔩(m|n). Let F = F(𝔤, 𝔤¯0) and, recalling the relative cohomology introduced in Section 2.1, let

R = H^{•}(𝔤, 𝔤¯0; ℂ),

be the cohomology ring of the category F. In [BKN, Section 8] we showed that 𝔤 admits a detecting subalgebra 𝔢 in the sense of Section 2.4 and one can use it to calculate R explicitly. Let us sketch how this is done as it will be needed in what follows.

As in [BKN, Section 8.10], one can identify 𝔢¯1 ⊆ 𝔤¯1 as the subspace spanned by the distinguished basis

x_{t} := E_{m+1−t,m+t} + E_{m+t,m+1−t}  for t = 1, …, r.  (4.4.1)

Let 𝔢¯0 = Stab_{𝔤¯0}(𝔢¯1). Then 𝔢 = 𝔢¯0 ⊕ 𝔢¯1 is a detecting subalgebra of 𝔤.

Let W = ℤ_{2}^{r} ⋊ Σ_{r}, where Σ_{r} is the symmetric group on r letters and Σ_{r} acts on ℤ_{2}^{r} by place permutation. Let X_{i} ∈ 𝔢∗ ; ¯1 be given by X_{i}(x_{j}) = δ_{i,j}. Then the symmetric algebra S(𝔢∗ ; ¯1) is isomorphic to the polynomial ring ℂ[X_{1}, …, X_{r}]. The group W naturally acts on ℂ[X_{1}, …, X_{r}] by letting the ith element of ℤ_{2}^{r} act by X_{i} ↦ −X_{i}, and by letting Σ_{r} act by permuting X_{1}, …, X_{r}. By [BKN, Theorem 4.4.1] the canonical restriction map defines the following isomorphism of graded rings

R ≅ H^{•}(𝔢, 𝔢¯0; ℂ)^{W} = S(𝔢∗ ; ¯1)^{W} = ℂ[X_{1}, …, X_{r}]^{W} = ℂ[Ẋ_{1}, …, Ẋ_{r}],  (4.4.2)

where Ẋ_{1}, …, Ẋ_{r} denote the elementary symmetric polynomials in the variables X_{1}^{2}, …, X_{r}^{2}. The ℤ-grading is given by Ẋ_{t} being of degree 2t, t = 1, …, r.

Note that R is a polynomial ring and, in particular, has no nonzero nilpotent elements. This point will be crucial in the proof of Proposition 4.7.3.

4.5. The Case of Full Atypicality. We first consider the special case of 𝔤 = 𝔤𝔩(k|k) and a simple supermodule L(λ) of atypicality k = def(𝔤). Our analysis will rely on the rank variety description of the 𝔢 support variety which was discussed in Section 2.4, and a result from [DS].

Proposition 4.5.1. Let L(λ) be a simple 𝔤𝔩(k|k)-supermodule of atypicality k. Then

V_{(𝔞,𝔞¯0)}(L(λ)) = V_{(𝔞,𝔞¯0)}(ℂ)

where 𝔞 stands for either 𝔤 or 𝔢.

Proof. We consider the case 𝔞 = 𝔢. The case 𝔞 = 𝔤 then follows immediately from (2.4.1) and the fact that res^{∗}(V_{(𝔢,𝔢¯0)}(ℂ)) = V_{(𝔤,𝔤¯0)}(ℂ) [BKN, Section 6]. Moreover, for the purposes of the proof one can take 𝔢¯1 to be spanned by the matrices E_{t,k+t}+E_{k+t,t} (t = 1, …, k) because this choice of 𝔢¯1 is conjugate under the Weyl group of 𝔤¯0 to the one given in Section 4.4.

Let D = diag(d_{1}, …, d_{k}) be an invertible diagonal k × k matrix and let

x_{D} = (\begin{matrix}0 & D \\ D & 0 \end{matrix}) ∈ 𝔢¯1.

We will prove that L(λ) contains a trivial direct summand as an ⟨x_{D}⟩-supermodule for all invertible diagonal matrices D. By the rank variety description (cf. Section 2.4), this will imply x_{D} ∈ V_{(𝔢,𝔢¯0)}(L(λ)) for all such D. This then implies the result as the set of all such x_{D} is dense in V_{(𝔢,𝔢¯0)}(ℂ) and V_{(𝔢,𝔢¯0)}(L(λ)) is a closed subset of V_{(𝔢,𝔢¯0)}(ℂ).

We first require some preliminaries. Let

x_{D}^{+} = (\begin{matrix}0 & D \\ 0 & 0 \end{matrix}) x_{D}^{−} = (\begin{matrix}0 & 0 \\ D & 0 \end{matrix}) ,

and let

K_{D}^{±} = \{(\begin{matrix}A & 0 \\ 0 & D^{∓1}AD^{±1} \end{matrix}) ∈ G¯0 | A is an invertible k × k matrix\} ≅ GL(k).

One checks that

Stab_{G¯0} (x_{D}^{±}) = K_{D}^{±}.  (4.5.1)

Let φ_{D} : G¯0 → G¯0 be the automorphism

(\begin{matrix}A & 0 \\ 0 & B \end{matrix}) ↦ (\begin{matrix}A & 0 \\ 0 & D^{−1}BD \end{matrix}) .

In particular, if K is the image in G¯0 of GL(k) under the diagional embedding, then

K_{D}^{+} = φ_{D}(K)  K_{D}^{−} = φ_{D^{−1}}(K).  (4.5.2)

If M is a G¯0-module, then let M^{φ_{D}} denote the twist of M by φ_{D}. One then observes the following.

(∗) If L is a simple G¯0-module, then L^{φ_{D}} ≅ L as G¯0-modules. Furthermore, the isomorphism is given by multiplication by some nonzero scalar.

The above claim follows immediately from the observation that twisting by φ_{D} fixes H pointwise and stabilizes B¯0. That is, twisting by φ_{D} sends a highest weight vector to a highest weight vector of the same highest weight. The result then follows by Schur’s Lemma for G¯0.

Applying (∗) one has the following observation. Since the simple 𝔤-supermodule L(λ) is completely reducible as a G¯0-module it follows that L(λ) and L(λ)^{φ_{D}} are isomorphic as G¯0-modules and, hence, as K-modules. That is, by (4.5.2), L(λ) has the same structure as K and K_{D}^{±}-module. In particular, L(λ) will have a trivial K_{D}^{±}-submodule if and only if it has a trivial K-submodule. Applying this observation and results on the K-module structure of L(λ) in [DS, Lemma 10.4], one obtains the following key fact.

(∗∗) The supermodule L(λ) is a ℤ-graded 𝔤-supermodule,

L(λ) = L(λ)_{0} ⊕ L(λ)_{−1} ⊕ L(λ)_{−2} ⊕ ⋯,

where the above direct sum decomposition is as G¯0-modules. In this decomposition L(λ)_{0} contains a trivial K_{D}^{±}-submodule whereas L(λ)_{−1} does not contain a trivial K_{D}^{±}-submodule. Furthermore, since the isomorphism in (∗) is given by scalar multiplication, one can choose the same fixed vector v ∈ L(λ)_{0} to provide the trivial submodule for K_{D}^{+} and K_{D}^{−}.

We are now prepared to prove the statement of the proposition. Let v ∈ L(λ)_{0} be as in (∗∗). We claim that x_{D}v = 0 and that there does not exist w ∈ L(λ) such that x_{D}w = v. By the structure of indecomposable ⟨x_{D}⟩-supermodules provided in [BKN, Proposition 5.2.1], this proves that L(λ) contains a trivial direct summand as a ⟨x_{D}⟩-supermodule.

Both assertions in the previous paragraph follow by using the ℤ-grading on L(λ) given in (∗∗), the fact that multiplication by x_{D}^{±} provides a K_{D}^{±}-module homomorphism between ℤ-graded pieces of L(λ) (by (4.5.1)), and by the structural information about L(λ) as a K_{D}^{±}-module provided by (∗∗).

Namely, note that x_{D}^{−} ∈ 𝔤_{−1} and x_{D}^{+} ∈ 𝔤_{1}, so by the ℤ-grading on L(λ) one has that x_{D}v = x_{D}^{−}v. If x_{D}^{−}v ≠ 0 then this would imply that there is a trivial K_{D}^{−}-module in L(λ)_{−1} spanned by x_{D}^{−}v, contradicting (∗∗). Therefore, x_{D}v = 0. On the other hand, if there is a w ∈ L(λ) with x_{D}w = v, then by the ℤ-grading one can assume w ∈ L(λ)_{−1} and x_{D}^{+}w = v. Thus, w would span a trivial K_{D}^{+}-submodule in L(λ)_{−1}, again contradicting (∗∗). □

4.6. Translation Functors. By, for example, [Ser, Section 3] the category F decomposes into blocks given by central characters. Thus one has

F = ⨁_{χ} F_{χ},  (4.6.1)

where the direct sum runs over the central characters of U(𝔤) and where F_{χ} denotes the block corresponding to the character χ. Furthermore, since each simple supermodule in a given fixed block has the same atypicality (cf. [Ser, Corollary 3.3]), it makes sense to define the atypicality of a block to be the atypicality of a simple supermodule in the block. The first main result of [Ser] is the following theorem.

Theorem 4.6.1. [Ser, Theorem 3.5] The blocks F_{χ} and F_{χ′} are equivalent as categories if and only if they have the same atypicality.

Let us mention how one constructs the functors which provide this equivalence of categories. Let pr_{χ} : F → F_{χ} denote the canonical projection functor given by the direct sum decomposition (4.6.1). Letting E denote a simple supermodule in F, one can define translation functors

T_{E}^{χ,χ′} : F_{χ} → F_{χ′}  (4.6.2)

by

M ↦ pr_{χ′}(M ⊗ E),

for M an object of F_{χ}. When χ and χ′ are “adjacent” in a sense made precise in [Ser, Lemma 5.5], one can make a suitable choice of simple supermodule E so that the translation functors T_{E}^{χ,χ′} and T_{E^{∗}}^{χ′,χ} provide the equivalence between neighboring blocks. The composition of such equivalences yields Theorem 4.6.1.

In the next proposition we will see that the support varieties of [BKN] behave well with respect to these translation functors. Before doing so, recall the following facts about support varieties. Let 𝔞 denote either 𝔤 or 𝔢 and let M, N be objects in F(𝔞, 𝔞¯0). Then by the argument from finite groups [Ben, Proposition 5.7.5] one has

V_{(𝔞,𝔞¯0)}(M ⊕ N) = V_{(𝔞,𝔞¯0)}(M) ∪ V_{(𝔞,𝔞¯0)}(N).  (4.6.3)

Using the fact that as functors one has

− ⊗ (M ⊗ N) ≅ (− ⊗ N) ∘ (− ⊗ M) ≅ (− ⊗ M) ∘ (− ⊗ N) ,

it follows that

V_{(𝔞,𝔞¯0)}(M ⊗ N) ⊆ V_{(𝔞,𝔞¯0)}(M) ∩ V_{(𝔞,𝔞¯0)}(N).  (4.6.4)

Proposition 4.6.2. Let χ, χ′, and E be such that T_{E}^{χ,χ′} and T_{E^{∗}}^{χ′,χ} provide an equivalence of categories between F_{χ} and F_{χ′}. If M (resp. N) is an object of F_{χ} (resp. F_{χ′}), then

V_{(𝔞,𝔞¯0)} (T_{E}^{χ,χ′}M) = V_{(𝔞,𝔞¯0)}(M)  and  V_{(𝔞,𝔞¯0)} (T_{E^{∗}}^{χ′,χ}N) = V_{(𝔞,𝔞¯0)}(N).

Here 𝔞 denotes either 𝔤 or 𝔢.

Proof. Taking (4.6.3) and (4.6.4) together, if M is an object of F_{χ}, then one has

V_{(𝔞,𝔞¯0)} (T_{E}^{χ,χ′}M) ⊆ V_{(𝔞,𝔞¯0)}(M ⊗ E) ⊆ V_{(𝔞,𝔞¯0)}(M) ∩ V_{(𝔞,𝔞¯0)}(E) ⊆ V_{(𝔞,𝔞¯0)}(M).  (4.6.5)

Similarly, one has

V_{(𝔞,𝔞¯0)} (T_{E^{∗}}^{χ′,χ}N) ⊆ V_{(𝔞,𝔞¯0)}(N ⊗ E^{∗}) ⊆ V_{(𝔞,𝔞¯0)}(N) ∩ V_{(𝔞,𝔞¯0)}(E^{∗}) ⊆ V_{(𝔞,𝔞¯0)}(N).

Since T_{E^{∗}}^{χ′,χ} (T_{E}^{χ,χ′}M) ≅ M one can apply (4.6.5), to obtain

V_{(𝔞,𝔞¯0)}(M) = V_{(𝔞,𝔞¯0)} (T_{E^{∗}}^{χ′,χ} (T_{E}^{χ,χ′}M)) ⊆ V_{(𝔞,𝔞¯0)} (T_{E}^{χ,χ′}M) ⊆ V_{(𝔞,𝔞¯0)}(M).

The first equality follows. The second equality follows by switching χ and χ′, E and E^{∗}, and M and N. □

4.7. Restriction Functors. For clarity and brevity, set F(m|n) = F(𝔤𝔩(m|n), 𝔤𝔩(m|n)¯0). Denote the block containing the trivial supermodule by F(m|n)_{χ_{ℂ}}. The second main result of Serganova [Ser, Theorem 3.6] is the following theorem.

Theorem 4.7.1. Let F(m|n)_{χ} be a block of atypicality k. Then F(m|n)_{χ} and F(k|k)_{χ_{ℂ}} are equivalent categories.

The equivalence of these two categories is provided by a functor

Φ : F(m|n)_{χ} → F(k|k)_{χ_{ℂ}}

which, very roughly speaking, is of the form

Res_{μ} ∘T_{t} ∘ ⋯ ∘ T_{1}  (4.7.1)

where the functors T_{i} are translation functors which give equivalances between blocks of F(m|n) as in Theorem 4.6.1, and where Res_{μ} is a functor which refines restriction. By Proposition 4.6.2 the 𝔤 and 𝔢 support varieties are invariant under the functors T_{i}. Thus the main task is to understand how they behave with respect to the functor Res_{μ} .

First, let us define Res_{μ} . Fix a block F(m|n)_{χ} of atypicality k and note that by (4.2.1) one has k ≤ m, n. Consequently we can naturally identify 𝔤𝔩(k|k) as the subalgebra of 𝔤𝔩(m|n) spanned by {E_{i,j} | m − k + 1 ≤ i, j ≤ m + k}. Under this identification the Cartan subalgebra 𝔥_{m|n} ⊆ 𝔤𝔩(m|n) naturally decomposes,

𝔥_{m|n} = 𝔥_{k|k} ⊕ 𝔥′,

where 𝔥′ is the linear span of E_{t,t} (t = 1, …, m − k, m + k + 1, …, m + n). Dualizing this decomposition one obtains,

𝔥_{m|n}^{∗} = 𝔥_{k|k}^{∗} ⊕ (𝔥′)^{∗}.

As in Section 4.4, we identify 𝔢¯1 ⊆ 𝔤¯1 as the subspace spanned by the distinguished basis

x_{t} := E_{m+1−t,m+t} + E_{m+t,m+1−t}  for t = 1, …, r.

Note that this is compatible with the subalgebra 𝔤𝔩(k|k) in the sense that if ẽ is the detecting subalgebra for 𝔤𝔩(k|k), then 𝔢̃¯1 ⊆ 𝔢¯1 and 𝔢̃¯1 is spanned by the elements x_{1}, …, x_{k}. Furthermore, one then has W̃ ⊆ W as the subgroup of elements which fix X_{k+1}, …, X_{r}. Here X_{j} (j = 1, …, r) is as in Section 4.4.

Fix a block F(m|n)_{χ} of atypicality k and μ ∈ (𝔥′)^{∗}. Let F(m|n)_{χ}^{μ} denote the full subcategory of F(m|n)_{χ} consisting of all supermodules whose simple composition factors are isomorphic to L(λ + μ) for some λ ∈ 𝔥_{k|k}^{∗}. Let F(k|k)_{χ_{ℂ}}^{μ} denote the full subcategory of F(k|k)_{χ_{ℂ}} of all supermodules whose simple composition factors are isomorphic to L(λ) for some λ ∈ 𝔥_{k|k}^{∗} such that L(λ + μ) is an object of F(m|n) (i.e., λ + μ ∈ X+ ¯0).

One can then define the refined restriction functor

Res_{μ} : F(m|n)_{χ}^{μ} → F(k|k)_{χ_{ℂ}}^{μ}

by

Res_{μ} M = {x ∈ M | hx = μ(h)x for all h ∈ 𝔥′} ,

viewed as a 𝔤𝔩(k|k)-supermodule by restriction. Note that 𝔥′ and 𝔤𝔩(k|k) are commuting subalgebras of 𝔤𝔩(m|n) so Res_{μ} M is in fact a 𝔤𝔩(k|k)-supermodule. A key lemma is the following result from [Ser, Lemma 6.3].

Proposition 4.7.2. Let F(m|n)_{χ} be a block of atypicality k and μ ∈ (𝔥′)^{∗}. Let F(m|n)_{χ}^{μ} and F(k|k)_{χ_{ℂ}}^{μ} be as above. Then the functor

Res_{μ} : F(m|n)_{χ}^{μ} → F(k|k)_{χ_{ℂ}}^{μ}

defines an equivalence of categories.

The functor Φ alluded to in (4.7.1) is then defined as follows. First one fixes a certain specific block F(m|n)_{χ_{0}} of atypicality k (recalling that they are all equivalent by Theorem 4.6.1). One chooses a suitable sequence of elements μ_{1}, μ_{2}, … ∈ (𝔥′)^{∗}; a sequence of central characters χ_{1}, χ_{2}, …; and a sequence of translation functors T_{1}, T_{2}, … so that T_{i} : F(m|n)_{χ_{i−1}} → F(m|n)_{χ_{i}} is an equivalence between blocks of atypicality k. The functor Φ is then defined by

Φ = lim_{i→∞} Res_{μ_{i}} ∘T_{i} ∘ ⋯ ∘ T_{1}.  (4.7.2)

Note that one has to verify that the limit makes sense by verifying that for M ∈ F_{χ_{0}} one has (T_{i} ∘ ⋯ ∘ T_{1}) M ∈ F(m|n)_{χ_{i}}^{μ_{i}} for i ≫ 0, and by verifying the appropriate compatibility condition. This is done in [Ser, Lemma 6.4] along with a more precise description of the functor Φ (e.g. the choice of F(m|n)_{χ_{0}} and the sequences μ_{1}, μ_{2}, …; χ_{1}, χ_{2}, …; and T_{1}, T_{2}, …).

We now turn to understanding the precise relationship between the functor Res_{μ} and support varieties. We continue to use the notation for relative cohomology introduced in Section 2.3. The inclusion 𝔤𝔩(k|k) ↪ 𝔤𝔩(m|n) induces a map

res : H^{•}(𝔤𝔩(m|n), 𝔤𝔩(m|n)¯0; M) → H^{•}(𝔤𝔩(k|k), 𝔤𝔩(k|k)¯0; M)

for any M in F(m|n). Note that this coincides with the map induced by the restriction functor, Res : F(m|n) → F(k|k). We then have the following commutative diagram.

I_{m|n}(M) ↪ H^{•}(𝔤𝔩(m|n), 𝔤𝔩(m|n)¯0; ℂ) ⟶^{m_{1}} H^{•}(𝔤𝔩(m|n), 𝔤𝔩(m|n)¯0; M ⊗ M^{∗})
res^{ℂ}↓  ↓^{res}
I_{k|k}(M) ↪ H^{•}(𝔤𝔩(k|k), 𝔤𝔩(k|k)¯0; ℂ) ⟶^{m_{2}} H^{•}(𝔤𝔩(k|k), 𝔤𝔩(k|k)¯0; M ⊗ M^{∗})

(4.7.3)

Here the horizontal maps are those induced by the functor − ⊗ M, and I_{m|n}(M) (resp. I_{k|k}(M)) is the kernel of this map; that is, as with finite groups [Ben], it is an ideal whose zero set is V_{(𝔤𝔩(m|n),𝔤𝔩(m|n)¯0)}(M) (resp. V_{(𝔤𝔩(k|k),𝔤𝔩(k|k)¯0)}(M)).

Also, for the purposes of uniformity in our notation we shall capitalize the names of functors and when they induce maps in cohomology call the induced maps by the same name but in lower case. For example, if F : C_{1} → C_{2} is an exact functor between categories of supermodules for Lie superalgebras, then we write

f : Ext_{C_{1}}^{•} (M, N) → Ext_{C_{2}}^{•} (F M, F N)

for the induced map of ℤ-graded superspaces. Thus, in (4.7.3) res_{ℂ} denotes the map induced by the exact functor Res (with coefficients in the trivial supermodule).

Let χ_{0} be the fixed central character of atypicality k, μ_{1}, μ_{2}, … the elements of (𝔥′)^{∗}, χ_{1}, χ_{2} … the central characters, and T_{1}, T_{2}, … the sequence of translation functors, all chosen as discussed above. The following proposition records certain properties of (4.7.3).

Proposition 4.7.3. Let J denote the kernel of the map res_{ℂ} and fix d ≥ 0 so that J is generated by elements of degree no more than d. Then the following statements about (4.7.3) hold true.

(a) The map res_{ℂ} is a surjective algebra homomorphism.

(b) Let M = L(λ) be a simple supermodule in a block F(m|n)_{χ} of atypicality k. Then the map m_{2} is injective.

(c) Assume M is an object of F(m|n)_{χ}^{μ} for some block F(m|n)_{χ} of atypicality k and some μ ∈ (𝔥′)^{∗}. Further assume that

Ext_{F(m|n)_{χ}}^{i}(M, M) = Ext_{F(m|n)_{χ}^{μ}}^{i}(M, M)

for i = 0, …, d. Then J ⊆ I_{m|n}(M).

Proof. One proves (a) as follows. Let us write 𝔢 ⊆ 𝔤𝔩(m|n) and 𝔢̃ ⊆ 𝔤𝔩(k|k) for the detecting subalgebras given in Section 4.4. By [BKN, Theorem 3.3.1(a)] (see also (4.4.2)), restriction induces isomorphisms H^{•}(𝔤𝔩(m|n), 𝔤𝔩(m|n)¯0; ℂ) → S(𝔢∗ ; ¯1)^{W} and H^{•}(𝔤𝔩(k|k), 𝔤𝔩(k|k)¯0; ℂ) →

S(𝔢̃∗ ; ¯1)W̃.

From the identification 𝔢̃¯1 ⊆ 𝔢¯1, one has the canonical homomorphism S(𝔢∗ ; ¯1) → S(𝔢̃∗ ; ¯1) given by restriction of functions. From the explicit description given in Section 4.4 one sees that this map corresponds to setting X_{k+1}, …, X_{r} to zero. Restriction of this map in turn induces a map

ρ : S(𝔢∗ ; ¯1)^{W} → S(𝔢̃∗ ; ¯1)W̃.

The explicit description given in (4.4.2) allows one to verify that this map is surjective. As all maps are induced by restrictions, one has the following commutative diagram.

H^{•}(𝔤𝔩(m|n), 𝔤𝔩(m|n)¯0; ℂ) ⟶^{≃} S(𝔢∗ ; ¯1)^{W}
res^{ℂ}↓  ↓^{ρ}
H^{•}(𝔤𝔩(k|k), 𝔤𝔩(k|k)¯0; ℂ) ⟶^{≃} S(𝔢̃∗ ; ¯1)W̃

Therefore, the map res_{ℂ} is surjective.

To prove (b) one argues as follows. Fix μ ∈ (𝔥′)^{∗} so that L(λ) lies in F(m|n)_{χ}^{μ}. For objects in F(m|n)_{χ}^{μ} one then has the decomposition

Res(M) = Res_{μ}(M) ⊕ G_{μ}(M)

where the functor G_{μ} : F(m|n)_{χ}^{μ} → F(k|k) is defined by

G_{μ}(N) = ∑_{\begin{matrix}ν∈(𝔥′)^{∗} \\ ν≠μ \end{matrix}} \{x ∈ N | hx = ν(h)x for all h ∈ 𝔥′\} .  (4.7.4)

By the additivity of the bifunctor H^{•}(𝔤𝔩(k|k), 𝔤𝔩(k|k)¯0; − ⊗ −) it follows that

H^{•}(𝔤𝔩(k|k), 𝔤𝔩(k|k)¯0;L(λ) ⊗ L(λ)^{∗})  (4.7.5)
≅ H^{•}(𝔤𝔩(k|k), 𝔤𝔩(k|k)¯0; Res_{μ}(L(λ)) ⊗ Res_{μ}(L(λ))^{∗}) ⊕ (∗∗),

where (∗∗) denotes the appropriate complementary superspace. By Proposition 4.7.2, Res_{μ}(L(λ)) is a simple supermodule of atypicality k in F(k|k). Hence by Proposition 4.5.1 one has that V_{𝔤𝔩(k|k)} (Res_{μ}(L(λ))) = V_{𝔤𝔩(k|k)} (ℂ) . That is, since H^{•}(𝔤𝔩(k|k), 𝔤𝔩(k|k)¯0; ℂ) has no nonzero nilpotent elements,

Ann_{H^{•}(𝔤𝔩(k|k),𝔤𝔩(k|k)¯0;ℂ)} (H^{•}(𝔤𝔩(k|k), 𝔤𝔩(k|k)¯0; Res_{μ}(L(λ)) ⊗ Res_{μ}(L(λ))^{∗})) = (0).

Consequently,

I_{k|k}(L(λ)) = Ann_{H^{•}(𝔤𝔩(k|k),𝔤𝔩(k|k)¯0;ℂ)} (H^{•}(𝔤𝔩(k|k), 𝔤𝔩(k|k)¯0; L(λ) ⊗ L(λ)^{∗})) = (0)

by (4.7.5). The result then follows.

To prove (c), we first prove the following claim.

(∗) The map

res : Ext_{F(m|n)_{χ}}^{i}(M, M) → Ext_{F(k|k)}^{i}(M, M)

is injective for i = 0, …, d.

By assumption there is a μ ∈ (𝔥′)^{∗} such that,

Ext_{F(m|n)_{χ}^{μ}}^{i}(M, M) = Ext_{F(m|n)_{χ}}^{i}(M, M)

for i = 0, …, d. Thus it suffices to show

res : Ext_{F(m|n)_{χ}^{μ}}^{i}(M, M) → Ext_{F(k|k)}^{i}(M, M)

is injective for i = 1, …, d.

To prove this, let C ⊆ F(k|k) denote the full subcategory consisting of all objects of F(k|k) which are 𝔤𝔩(k|k) ⊕ 𝔥′-supermodules and semisimple as 𝔥′-supermodules. One then has a restriction functor Res : F(m|n) → C which, when composed with the functor which forgets the action of 𝔥′, yields the functor Res : F(m|n) → F(k|k). Let P_{μ} : C → F(k|k) be the functor given by

P_{μ}(N) = {x ∈ N | hx = μ(h)x for all h ∈ 𝔥′} .

Since 𝔥′ and 𝔤𝔩(k|k) are commuting superalgebras, this is a supermodule for 𝔤𝔩(k|k). One then has the following factorization of Res_{μ} :

Res_{μ} = P_{μ} ∘ Res .  (4.7.6)

Now, since Res_{μ} : F(k|k)_{χ}^{μ} → F(k|k)_{χ_{ℂ}}^{μ} is an equivalence of categories by Proposition 4.7.2, it follows that the induced map

res_{μ} : Ext_{F(m|n)_{χ}^{μ}}^{i}(M, M) → Ext_{F(k|k)_{χ_{ℂ}}^{μ}}^{i}(M, M) ↪ Ext_{F(k|k)}^{i}(M, M)

is injective for all i ≥ 0. But (4.7.6) implies the induced maps satisfy res_{μ} = p_{μ} ∘ res . This along with the fact that the forgetful functor C → F(k|k) induces an injective map in cohomology implies the map

res : Ext_{F(m|n)_{χ}^{μ}}^{i}(M, M) → Ext_{F(k|k)}^{i}(M, M)

is injective. This proves (∗).

Now we can prove (c). Let a ∈ J be an element of degree less than or equal to d. By the commutativity of (4.7.3) one has

res(m_{1}(a)) = m_{2}(res_{ℂ}(a)) = 0.

Note, however, that the map m_{1} is grading preserving so m_{1}(a) is of degree no more than d. By (∗), res is injective in this range so m_{1}(a) = 0. That is, a ∈ I_{m|n}(M). Since J is generated by such elements, it follows that J ⊆ I_{m|n}(M). □

4.8. Support Varieties for Simple Supermodules. In this section the support varieties for the simple supermodules of 𝔤𝔩(m|n) will be computed. We continue with our fixed choice of a block of atypicality k, F(m|n)_{χ_{0}}, and sequences μ_{1}, μ_{2}, …; χ_{1}, χ_{2}, …; and T_{1}, T_{2}, ….

Before proceeding we first make an observation which will allow us to reduce to the situation of Proposition 4.7.3. The main concern is to ensure that we are working in F_{χ_{i}}^{μ_{i}} rather than F_{χ_{i}} so that Proposition 4.7.3(c) can be applied. As mentioned after (4.7.2), if M is an object of F(m|n)_{χ_{0}} then for some N > 0 (depending on M) one has that

(T_{i} ∘ ⋯ ∘ T_{1})M ∈ F(m|n)_{χ_{i}}^{μ_{i}}  (4.8.1)

for all i ≥ N.

Since the translation functors T_{i} are exact, one can lift (4.8.1) to Ext groups as follows. Fix t ≥ 0. Say

E := (0 → M → M_{1} → ⋯ → M_{t} → M → 0)

represents an element of Ext_{F(m|n)_{χ_{0}}}^{t} (M, M). Then, by (4.8.1), one can choose N > 0 (depending on M_{1}, …, M_{t}) so that

(T_{i} ∘ ⋯ ∘ T_{1})M_{r} ∈ F(m|n)_{χ_{i}}^{μ_{i}}

for all all i ≥ N and r = 1, …, t. That is, the induced linear map

t_{i} ∘ ⋯ ∘ t_{1} : Ext_{F(m|n)_{χ_{0}}}^{t} (M, M) → Ext_{F(m|n)_{χ_{i}}}^{t} ((T_{i} ∘ ⋯ ∘ T_{1})M, (T_{i} ∘ ⋯ ∘ T_{1})M)

satisfies

t_{i} ∘ ⋯ ∘ t_{1}(E) ∈ Ext_{F(m|n)_{χ_{i}}^{μ_{i}}}^{t} ((T_{i} ∘ ⋯ ∘ T_{1})M, (T_{i} ∘ ⋯ ∘ T_{1})M).  (4.8.2)

Since Ext_{F(m|n)_{χ_{0}}}^{t} (M, M) is finite dimensional we can fix a basis E_{1}, …, E_{q}. Applying (4.8.2), one can choose N > 0 (depending on E_{1}, …, E_{q}) so that the induced map

t_{i} ∘ ⋯ ∘ t_{1} : Ext_{F(m|n)_{χ_{0}}}^{t} (M, M) → Ext_{F(m|n)_{χ_{i}}}^{t} ((T_{i} ∘ ⋯ ∘ T_{1})M, (T_{i} ∘ ⋯ ∘ T_{1})M)

satisfies

t_{i} ∘ ⋯ ∘ t_{1} (Ext_{F(m|n)_{χ_{0}}}^{t} (M, M)) ⊆ Ext_{F(m|n)_{χ_{i}}^{μ_{i}}}^{t} ((T_{i} ∘ ⋯ ∘ T_{1})M, (T_{i} ∘ ⋯ ∘ T_{1})M) (4.8.3)
⊆ Ext_{F(m|n)_{χ_{i}}}^{t} ((T_{i} ∘ ⋯ ∘ T_{1})M, (T_{i} ∘ ⋯ ∘ T_{1})M)

for all i ≥ N. However, by assumption each of the translation functors T_{j} is an equivalence of categories and, hence, each of the induced maps t_{j} is an isomorphism. That is,

t_{i} ∘ ⋯ ∘ t_{1} : Ext_{F(m|n)_{χ_{0}}}^{t} (M, M) ⟶^{≃} Ext_{F(m|n)_{χ_{i}}}^{t} ((T_{i} ∘ ⋯ ∘ T_{1})M, (T_{i} ∘ ⋯ ∘ T_{1})M).

Therefore, (4.8.3) becomes

t_{i} ∘ ⋯ ∘ t_{1} : Ext_{F(m|n)_{χ_{0}}}^{t} (M, M) ⟶^{≃} Ext_{F(m|n)_{χ_{i}}^{μ_{i}}}^{t} ((T_{i} ∘ ⋯ ∘ T_{1})M, (T_{i} ∘ ⋯ ∘ T_{1})M) (4.8.4)
= Ext_{F(m|n)_{χ_{i}}}^{t} ((T_{i} ∘ ⋯ ∘ T_{1})M, (T_{i} ∘ ⋯ ∘ T_{1})M)

for all i ≥ N.

We can finally state the result we require. Given a fixed d ≥ 0, one can use (4.8.4) to choose N > 0 (depending on M and d) so that

t_{i} ∘ ⋯ ∘ t_{1} : Ext_{F(m|n)_{χ_{0}}}^{t} (M, M) ⟶^{≃} Ext_{F(m|n)_{χ_{i}}^{μ_{i}}}^{t} ((T_{i} ∘ ⋯ ∘ T_{1})M, (T_{i} ∘ ⋯ ∘ T_{1})M) (4.8.5)
= Ext_{F(m|n)_{χ_{i}}}^{t} ((T_{i} ∘ ⋯ ∘ T_{1})M, (T_{i} ∘ ⋯ ∘ T_{1})M)

for t = 0, …, d and all i ≥ N. We now prove one of the main results of the paper.

Theorem 4.8.1. Let 𝔤 = 𝔤𝔩(m|n), r = def(𝔤), and L(λ) be a simple 𝔤-supermodule of atypicality k. Let 𝔢̃¯1 ⊆ 𝔢¯1 be chosen as in Section 4.7. Then,

(a)

res^{∗}(𝔢̃¯1) = res^{∗} (V_{(𝔢,𝔢¯0)} (L(λ))) = V_{(𝔤,𝔤¯0)} (L(λ)) ≅ 𝔸^{k}.  (4.8.6)

(b)

V_{(𝔢,𝔢¯0)}(L(λ)) = W · 𝔢̃¯1.  (4.8.7)

In particular, V_{(𝔢,𝔢¯0)}(L(λ)) is the union of \binom{r}{k} k-dimensional subspaces. In terms of coordinates, if x_{1}, …, x_{r} is the distinguished basis of 𝔢¯1 given in (4.4.1), then

V_{(𝔢,𝔢¯0)}(L(λ)) = {∑_{t=1}^{r} a_{t}x_{t} | a_{t} ∈ ℂ and at least r − k of the a_{t} are zero} .  (4.8.8)

Proof. One proves (a) as follows. First note that by the results of Section 4.6 one may assume without a loss of generality that L(λ) lies in the block of atypicality k, F(m|n)_{χ_{0}}, fixed before Proposition 4.7.3. By using Proposition 4.6.2 and (4.8.1) we can replace L(λ) with (T_{i} ∘ ⋯ ∘ T_{1})L(λ) for sufficently large i and assume without loss that L(λ) is a simple supermodule of atypicality k which lies in F(m|n)_{χ_{i}}^{μ_{i}}. Furthermore, by (4.8.5) we can further assume (choosing an even larger i if necessary) that

Ext_{F(m|n)_{χ}}^{t}(L(λ), L(λ)) = Ext_{F(m|n)_{χ}^{μ_{i}}}^{t} (L(λ), L(λ))

for t = 0, …, d; here, as in Proposition 4.7.3, d is fixed so that Ker(res_{ℂ}) is generated by elements of degree no more than d.

Now by Proposition 4.7.3(c) we have that Ker(res_{ℂ}) ⊆ I_{m|n}(L(λ)). On the other hand, it follows by the commutativity of (4.7.3) and the injectivity of m_{2} (Proposition 4.7.3(b)) that I_{m|n}(L(λ)) ⊆ Ker(res_{ℂ}). Therefore, I_{m|n}(L(λ)) = Ker(res_{ℂ}). Using the surjectivity of res_{ℂ} and the description of H^{•}(𝔤𝔩(k|k), 𝔤𝔩(k|k)¯0; ℂ)) given in (4.4.2), one has

V_{(𝔤𝔩(m|n),𝔤𝔩(m|n)¯0)}(L(λ)) ≅ MaxSpec (H^{•}(𝔤𝔩(m|n), 𝔤𝔩(m|n)¯0; ℂ)/ Ker(res_{ℂ}))
≅ MaxSpec (H^{•}(𝔤𝔩(k|k), 𝔤𝔩(k|k)¯0; ℂ))
≅ MaxSpec (ℂ[Ẋ_{1}, …, Ẋ_{k}])
≅ 𝔸^{k}.

Now consider V_{(𝔢,𝔢¯0)}(L(λ)). Recall that 𝔢̃¯1 ⊆ 𝔢¯1. Since L(λ) ∈ F(m|n)_{χ_{i}}^{μ_{i}}, it follows from Proposition 4.7.2 that L(λ) contains a simple 𝔤𝔩(k|k)-supermodule of atypicality k as a direct summand (namely, Res_{μ_{i}} L(λ)). By Proposition 4.5.1 and the rank variety description of 𝔢̃ support varieties discussed in Section 2.4, it must be that for any x ∈ 𝔢̃¯1, L(λ) is not projective as an ⟨x⟩-supermodule. Here ⟨x⟩ denotes the Lie subsuperalgebra generated by x. This statement is equally true if we view x as an element of 𝔢¯1. Thus, we have 𝔢̃¯1 ⊆ V_{(𝔢,𝔢¯0)}(L(λ)). Therefore by (2.4.1) one has,

res^{∗}(𝔢̃¯1) ⊆ res^{∗}(V_{(𝔢,𝔢¯0)}(L(λ)) ⊆ V_{(𝔤,𝔤¯0)}(L(λ)) ≅ 𝔸^{k}.  (4.8.9)

However, by (2.4.1) the map res^{∗} is finite-to-one so res^{∗} \binom{}{𝔢¯1} is a k-dimensional closed subset of 𝔸^{k}. However 𝔸^{k} is a k-dimensional irreducible variety. Therefore res^{∗} \binom{}{𝔢¯1} = 𝔸^{k} and all the containments in (4.8.9) must be equalities. This proves (a).

To prove (b), one recalls from [BKN, (6.1.3)] that the fibers of the map res^{∗} are precisely the orbits of W. This along with (4.8.9) implies (4.8.7). To obtain (4.8.8), one uses (4.8.7) and the explicit description of the action of W on X_{1}, …, X_{r} provided in Section 4.4. □

4.9. Note that the above theorem confirms several conjectures for the simple supermodules of 𝔤𝔩(m|n). The first observation is that the second equality in (4.8.6) affirms a speculation of the authors in [BKN, Section 6.2].

Second, observe that

dim V_{(𝔢,𝔢¯0)}(L(λ)) = k = atyp(L(λ)).

This agrees with [BKN, Conjecture 7.2.1], where it was conjectured that the dimension of the 𝔢 support variety of a simple supermodule should equal its atypicality. The verification of this conjecture justifies the general (and functorial) definition of atypicality for finite dimensional supermodules of 𝔤𝔩(m|n) by setting atyp(M) := dim V_{(𝔢,𝔢¯0)}(M) for all M in F(m|n).

Finally recall that the superdimension of a supermodule M is the integer dim M¯0 − dim M¯1. In [KW, Conjecture 3.1] Kac and Wakimoto conjectured that for a simple basic classical Lie superalgebra, 𝔤, the superdimension of a simple supermodule L is nonzero if and only if atyp(L) = def(𝔤). As was discussed in [BKN, Section 7.3], the validity of [BKN, Conjecture 7.2.1] proved above for 𝔤𝔩(m|n) implies the “only if” direction of the Kac– Wakimoto conjecture in this case. It should be noted that this direction of the conjecture was also recently proved in [DS].

5. Clifford Superalgebras, Superdimension, and Divisibility

5.1. In this section we show that the codimension of the 𝔢 support variety of a supermodule M is closely related to the 2-divisibility of the dimension of M. Another consequence is that if V_{(𝔢,𝔢¯0)}(M) is a proper subset of V_{(𝔢,𝔢¯0)}(ℂ), then the superdimension of M is necessarily zero. First we establish some general results. Let 𝔠 = 𝔠¯0 ⊕ 𝔠¯1 be a Lie superalgebra which satisfies the following assumptions:

(1) 𝔠¯0 = [𝔠¯1, 𝔠¯1] and is an abelian Lie algebra;

(2) [𝔠¯0, 𝔠¯1] = 0.

Let F = F(𝔠, 𝔠¯0) be the category of all finite dimensional 𝔠-supermodules which are finitely semisimple as 𝔠¯0-supermodules. Note that if 𝔠¯1 is a subspace of 𝔢¯1 (where 𝔢 is the detecting subalgebra discussed in Section 2.4), then 𝔠 := [𝔠¯1, 𝔠¯1] ⊕ 𝔠¯1 satisfies the above assumptions. This is the context where the general theory will be applied.

Given χ ∈ 𝔠∗ ; ¯0, let ℂ_{χ} denote the unique simple 𝔠¯0-supermodule of dimension 1 (concentrated in degree ¯0) with action x.v = χ(x)v for all x ∈ 𝔠¯0 and v ∈ ℂ_{χ}. Given χ ∈ 𝔠∗ ; ¯0, let F_{χ} denote the full subcategory of F consisting of all supermodules M such that all composition factors are isomorphic to ℂ_{χ} when viewed as a 𝔠¯0-supermodule by restriction. By [BKN, Lemma 5.1.2] one has the following decomposition of the category F,

F = ⨁_{χ∈𝔠∗ ; ¯0} F_{χ}.

5.2. Clifford Superalgebras. Given a χ ∈ 𝔠∗ ; ¯0 one can define a Clifford superalgebra as follows. Define a bilinear form

( , ) : 𝔠¯1 ⊗ 𝔠¯1 → ℂ

by

(x, y) = χ([x, y]).

Since [x, y] = [y, x] for all x, y ∈ 𝔠¯1, this bilinear form is symmetric. Let A_{χ} denote the Clifford superalgebra defined by

A_{χ} = T (𝔠¯1)/I_{χ},  (5.2.1)

where T (𝔠¯1) denotes the tensor superalgebra on the superspace 𝔠¯1 and I_{χ} is the ideal of T (𝔠¯1) generated by the set

{x ⊗ y + y ⊗ x − (x, y) | x, y ∈ 𝔠¯1} .

The ℤ_{2}-grading on A_{χ} is obtained by setting \overline{x} = ¯1 for all x ∈ 𝔠¯1. Write A_{χ}-smod for the category of all finite dimensional A_{χ}-supermodules. The key result is the following proposition which was used by Penkov [Pen] in his study of the Lie superalgebra 𝔮(n) (see also [Fri, Section 2.2]).

Proposition 5.2.1. The category F_{χ} is isomorphic to the category A_{χ}-smod.

Proof. Let M be an object in F_{χ}. Since 𝔠¯1 generates A_{χ}, and for x, y ∈ 𝔠¯1 and m ∈ M one has

x.(y.m) + y.(x.m) = [x, y].m = χ([x, y])m = (x, y)m,

one has a well defined action of A_{χ} on M. The ℤ_{2}-grading on M is compatible with this action so in fact M is an A_{χ}-supermodule.

Conversely, let M be an A_{χ}-supermodule. We then have an action by 𝔠¯1 on M via the canonical inclusion 𝔠¯1 ↪ A_{χ}. This extends to an action of 𝔠¯0 by having [x, y] ∈ 𝔠¯0 act by [x, y].m = χ([x, y])m for all m ∈ M. It is straightforward to verify that this action along with its given ℤ_{2}-grading makes M into a 𝔠-supermodule.

One also can verify that a linear map which defines a supermodule homomorphism in one category defines a homomorphism in the other category. That is, the morphisms in the two categories concide. □

Recall that Schur’s Lemma in this setting states that dim Hom_{𝔠}(L, L) equals 1 or 2 for any simple 𝔠-supermodule L in F. We say L is type Q if the Hom-space is two dimensional, and type M otherwise.

Proposition 5.2.2. Let χ ∈ 𝔠∗ ; ¯0. Set z = dim { x ∈ 𝔠¯1 | (x, y) = 0 for all y ∈ 𝔠¯1 }, n = dim 𝔠¯1 − z, and ñ = ⌊(n + 1)/2⌋. Then the following statements hold for the category F_{χ}.

(a) There is a unique simple supermodule in F_{χ}, which we denote by S(χ), of type M if n is even, and of type Q otherwise. The supermodule S(χ) has dimension 2ñ. If n > 0, then S(χ) has superdimension 0.

(b) Write P (χ) for the projective cover of S(χ). Then P (χ) is also injective and of dimension

dim P (χ) = \begin{cases}2^{dim(𝔠¯1)−ñ}, & if S(χ) is of type M; \\ 2^{dim(𝔠¯1)−ñ+1}, & if S(χ) is of type Q. \end{cases}

Furthermore, P (χ) has superdimension zero.

Proof. By applying Proposition 5.2.1, the results in part (a) and the statement on injectivity in part (b) follow from [BK, Pen, Fri]. It only remains to calculate the dimension of P (χ).

Since ℂ_{χ} is projective in F(𝔠¯0, 𝔠¯0) and the induction functor is exact, U(𝔠) ⊗_{U(𝔠¯0)} ℂ_{χ} is projective in F_{χ}. Hence,

dim U(𝔠) ⊗_{U(𝔠¯0)} ℂ_{χ} = a dim P (χ)

for some positive integer a. By Frobenius reciprocity, one has

dim Hom_{𝔠}(U(𝔠) ⊗_{U(𝔠¯0)} ℂ_{χ}, S(χ)) = dim Hom_{𝔠¯0}(ℂ_{χ}, S(χ)) = dim S(χ) = 2ñ.

On the other hand,

dim Hom_{𝔠}(U(𝔠) ⊗_{U(𝔠¯0)} ℂ_{χ}, S(χ)) = a dim Hom_{𝔠}(P (χ), S(χ))
= \begin{cases}a, & if S(χ) is of type M; \\ 2a, & if S(χ) is of type Q. \end{cases}

By the PBW theorem dim U(𝔠) ⊗_{U(𝔠¯0)} ℂ_{χ} = 2^{dim(𝔠¯1)}. Combining this with the above calculations one obtains

dim P (χ) = \begin{cases}2^{dim(𝔠¯1)−ñ}, & if S(χ) is of type M; \\ 2^{dim(𝔠¯1)−ñ+1}, & if S(χ) is of type Q. \end{cases}  (5.2.2)

Lastly, if n > 0, then the simple supermodule in F_{χ} has superdimension zero and so all supermodules in F_{χ} also have superdimension 0. In particular, this holds for P (χ). If n = 0 then P (χ) = U(𝔠) ⊗_{U(𝔠¯0)} ℂ_{χ}, which has superdimension 0. □

5.3. 2-Divisibility. Let 𝔢 be a detecting Lie superalgebra for 𝔤 as discussed in Section 2.4. The following theorem relates the codimension of the support variety of a supermodule in F(𝔢, 𝔢¯0) to the 2-divisibility of its dimension and to its superdimension.

Theorem 5.3.1. Let 𝔤 be classical Lie superalgebra with detecting subalgebra 𝔢. Let M be an object of F(𝔢, 𝔢¯0) and let

d = dim V_{(𝔢,𝔢¯0)}(ℂ) − dim V_{(𝔢,𝔢¯0)}(M)

denote the codimension of the variety V_{(𝔢,𝔢¯0)}(M). Then,

2^{⌊d/2⌋} | dim M.

Furthermore, if d > 0, then M has superdimension 0.

Proof. The case d = 0 is vacuously true so we assume d > 0. As a consequence of the Noether Normalization Theorem (e.g. [Kun, Theorem 3.1]) one can choose a subspace H ⊆ V_{(𝔢,𝔢¯0)}(ℂ) = 𝔢¯1 of dimension d such that H ∩ V_{(𝔢,𝔢¯0)}(M) = {0}. Let 𝔠 denote the Lie subsuperalgebra of 𝔢 generated by H; that is, 𝔠¯1 = H and 𝔠¯0 = [H, H]. Observe that 𝔠 is a Lie superalgebra of the type considered in Sections 5.1-5.2. Using the rank variety description (cf. Section 2.4) and the fact that 𝔠¯1 ∩ V_{(𝔢,𝔢¯0)}(M) = {0}, one has V_{(𝔠,𝔠¯0)}(M) = 0. It then follows by [BKN, Theorem 6.4.2(b)] that M is projective as a 𝔠-supermodule. Thus M decomposes as a direct sum of P (χ) for various χ ∈ 𝔠∗ ; ¯0. The dimension of each P (χ) is a power of two and is at its smallest when n = d = dim(𝔠¯1) and S(χ) is of type M. Therefore, one has

dim P (χ) ≥ 2^{⌊d/2⌋}.

The first statement of the theorem follows from this inequality. Furthermore, M has superdimension 0 because each P (χ) has superdimension 0. □

Corollary 5.3.2. Let L(λ) be the finite dimensional simple 𝔤𝔩(m|n)-supermodule of highest weight λ, let r = min(m, n), the defect of 𝔤𝔩(m|n), and let a = atyp(L(λ)). Then

2^{⌊(r−a)/2⌋} | dim L(λ).

Furthermore, if the atypicality is strictly less than the defect, then the superdimension of L(λ) is zero.

Proof. By [BKN, Section 8.8] and Theorem 4.8.1, one has that r = dim V_{(𝔢,𝔢¯0)}(ℂ) and a = dim V_{(𝔢,𝔢¯0)}(L(λ)). The result then follows by the previous theorem. □

We remark that a closed formula for the dimension of L(λ) is given by Su and Zhang [SZ, Theorem 4.14]. However, their formula is quite complicated and it does not seem that the above divisibility statement can easily be recovered from their work.

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Department of Mathematics, University of Georgia, Athens, GA 30602

E-mail address: brian@math.uga.edu

Department of Mathematics, University of Oklahoma, Norman, OK 73019

E-mail address: kujawa@math.ou.edu

Department of Mathematics, University of Georgia, Athens, GA 30602

E-mail address: nakano@math.uga.edu