Sapovalov elements and the Jantzen filtration for contragredient Lie superalgebras: A Survey

This is a survey of some recent results on Sapovalov elements and the Jantzen filtration for contragredient Lie superalgebras. The topics covered include the existence and uniqueness of the Sapovalov elements, bounds on the degrees of their coefficients and the behavior of Sapovalov elements when the Borel subalgebra is changed. There is always a unique term whose coefficient has larger degree than any other term. This allows us to define some new highest weight modules. If X is a set of orthogonal isotropic roots and $\lambda \in h^*$ is such that $\lambda +\rho$ is orthogonal to all roots in X, we construct highest weight modules with character $\epsilon^\lambda p_X$. Here $p_X$ is a partition function that counts partitions not involving roots in X. When |X|=1, these modules are used to give a Jantzen sum formula for Verma modules in which all terms are characters of modules in the category O with positive coefficients.


Introduction.
Throughout this paper we work over an algebraically closed field k of characteristic zero. If g is a semisimple Lie algebra, necessary and sufficient conditions for the existence of a non-zero homomorphism between Verma modules can be obtained by combining work of Verma [Ver] with work of Bernstein, Gelfand and Gelfand [BGG1], [BGG2]. Such maps can be described explicitly in terms of certain elements introduced byŠapovalov in [Šap]. Necessary and sufficient conditions for a simple highest weight module to be a composition factor of a Verma module were also obtained in [BGG1], [BGG2]. A more elementary proof of the latter result can be given using the Jantzen filtration and sum formula [Jan]. 1 However neitherŠapovalov elements nor the Jantzen filtration have received much attention for classical simple Lie superalgebras. In this paper we review some recent results onŠapovalov elements, the Jantzen filtration and sum formula in the super case. New phenomena arise due to the presence of isotropic roots. Proofs not found here are given in [Mus1] Chapter 9 and 10, [Mus2] or [Mus3].
Let g = g(A, τ ) be a finite dimensional contragredient Lie superalgebra with Cartan subalgebra h, and set of simple roots Π, see [Mus1] Chapter 5. Let ∆ + be the set of positive roots containing Π, and g = n − ⊕ h ⊕ n + 1 the corresponding triangular decomposition of g. We use the Borel subalgebras b = h ⊕ n + and b − = n − ⊕ h. The Verma module M (λ) with highest weight λ ∈ h * , and highest weight vector v λ is induced from b. Let ρ 0 (resp. ρ 1 ) be the half-sum of the positive even (resp. odd) positive roots and ρ = ρ 0 − ρ 1 .
Fix a non-degenerate invariant symmetric bilinear form ( , ) on h * as in [Mus1] Theorem 5.4.1, and for all α ∈ h * , let h α ∈ h be the unique element such that (α, β) = β(h α ) for all β ∈ h * . Then for all α ∈ ∆ + , choose elements e ±α ∈ g ±α such that We give bounds on the degrees of the coefficients of H π of theŠapovalov elements in Equation (1) below. The exact form of the coefficients depends on the way the positive roots are ordered. However there is always a unique coefficient of highest degree, and we determine the leading term of this coefficient up to a scalar multiple. These results appear to be new even for simple Lie algebras. The existence of a unique coefficient of highest degree is used to construct some new highest weight modules M γ (λ), where γ is an isotropic root and (λ + ρ, γ) = 0, [Mus3]. There is a version of the Jantzen sum formula in [Mus1] Theorem 10.3.1, see also [Gor], which involves infinite sums of Verma modules with alternating sign coefficients. The module M γ (λ) has character ǫ λ p γ see (8) for notation. The infinite sums can be replaced by a finite sum of characters of modules of this form leading to a formula where both sides are sums of characters in the category O. In type A there are explicit expressions forŠapovalov elements, see [Mus2] Section 7.

Preliminaries.
To simplify the exposition, we assume in this survey that any non-isotropic root of g is even. This should give the reader the flavor of the results in general, but the proofs and some of the statements should be modified if this is not the case.
Let P(η) be the set of partitions of η. The degree of a partition π is defined to be |π| = α∈∆ + π(α). Partitions are useful because they can be used to index a basis for U (n − ). Fix an ordering on the set ∆ + , and for π a partition, set the product being taken with respect to this order. Then the elements e −π , with π ∈ P(η) form a basis of U (n − ) −η .
For a non-isotropic root α, we set α ∨ = 2α/(α, α), and write s α for the reflection corresponding to α. As usual the Weyl group W is the subgroup of GL(h * ) generated by all such reflections. For u ∈ W set Suppose Π even is the set of even simple roots, and let W even be the subgroup of W generated by the reflections s α , where α ∈ Π even . Note that W even can be a proper subgroup of W . For example this happens when g = osp(2m, 2n). If Π = {α i |i = 1, . . . , t} is the set of simple roots, and γ is a positive root such that γ = t i=1 a i α i , then the height htγ of γ is htγ = t i=1 a i . Theorem 2.1. Let g be semisimple Lie algebra or a contragredient Lie superalgebra, and γ a positive root. If γ is isotropic assume that m = 1. Suppose γ = wβ for a simple root β and w ∈ W even , and for α ∈ N (w −1 ), set q(w, α) = (wβ, α ∨ ). Then and H mπ γ has leading term Let d mγ be the degree of H mπ γ .

3.Šapovalov Elements and their Coefficients.
A finite dimensional contragredient Lie superalgebra g has, in general several conjugacy classes of Borel subalgebras, and this both complicates and enriches the representation theory of g.
The complications are partially resolved by at first fixing a Borel subalgebra b (or equivalently a basis of simple roots for g) with special properties. The effect of changing the Borel subalgebra is studied in detail in [Mus3], see also the next Section.
In [Kac] Table VI, Kac gave a particular diagram in each case that we will call distinguished.
The corresponding set of simple roots and Borel subalgebra are also called distinguished.
The distinguished Borel subalgebra contains at most one simple isotropic root vector. Unless g = osp(1, 2n), or g = osp(2, 2n) there is exactly one other Borel subalgebra with this property up to conjugacy in Aut g. A representative of this class (and its set of simple roots) will be called anti-distinguished. We assume that b is either distinguished or anti-distinguished.
Theorem 2.1 is proved by looking at the proofs given in [Hum] or [Mus1] and keeping track of the coefficients. Given λ ∈ h * and Let (γ, m) be as in the statement of the Theorem and set H = H γ,m . The idea of the proof is to construct elements θ λ ∈ U (n − ) −mγ for all λ in a dense subset of H such that θ λ v λ is a highest weight vector in M (λ) λ−mγ , and that where a π,λ is a polynomial function of λ ∈ Λ satisfying suitable conditions. For π ∈ P(mγ), the assignment λ → a π,λ for λ ∈ Λ determines a polynomial map from H to U (n − ) −mγ , so there exists an element H π ∈ U (h) uniquely determined modulo I(H) such that H π (λ) = a π,λ for all λ ∈ Λ. We define the element θ ∈ U (b − ) by setting Note that θ(λ) = θ λ . TheŠapovalov element in Theorem 2.1 is constructed inductively using the next Lemma, see for example [Hum] Section 4.13 or [Mus1] Theorem 9.4.3.
Lemma 3.1. Let γ be a positive root, and m a positive integer which is equal to 1 if γ is isotropic. Suppose that α ∈ Π even , and set Using Lemma 3.1 it follows that the coefficients of theŠapovalov elements θ γ,m are obtained by taking k-linear combinations of products of coefficients of the θ γ ′ ,m , with binomial coefficients.

Changing the Borel Subalgebra.
Using adjacent Borel subalgebras (equivalently odd reflections), it is possible to give an alternative construction ofŠapovalov elements corresponding to an isotropic root γ, provided γ is a simple root for some Borel subalgebra. This condition always holds in type A, but for other types, it is quite restrictive: if g = osp(2m, 2n + 1), (an algebra that does not satisfy the restrictions imposed at the start of section 2) the assumption only holds for roots of the form ±(ǫ i − δ j ), while if g = osp(2m, 2n) it holds only for these roots and the root ǫ m + δ n .
Suppose that b is the distinguished Borel subalgebra, and let b ′ be another Borel subalgebra with the same even part as b. Consider a sequence of Borel subalgebras Assume there are isotropic roots α i such that g α i ⊂ b (i−1) , g −α i ⊂ b (i) for 1 ≤ i ≤ r, and α 1 , . . . , α r are distinct positive roots of b.
Theorem 4.1. Set F (γ) = {i|1 ≤ i ≤ r and (γ, α i ) = 0}. There is a nonzero c ∈ k such that for all λ ∈ H γ , 5. The Square of aŠapovalov Element. When γ is an isotropic root we write H γ and θ γ in place of H γ,1 and θ γ,1 respectively. Here we record an elementary but important property of theŠapovalov element θ γ corresponding to such a root.

Modules with Prescribed Characters.
We introduce some new highest weight modules whose characters are given by generating functions for certain kinds of partitions. If X is a set of pairwise orthogonal isotropic positive roots, set P X (η) = {π ∈ P(η)|π(α) = 0 for all α ∈ X}.
and p X (η) = |P X (η)|. Set p X = p X (η)ǫ −η . We have If X is empty, set p = p X , and if X = {α} is a singleton write P α (η), p α (η), and p α instead of P X (η), p X (η), and p X . For a module M in the BGG category O, the character of M is defined by ch M = η∈h * dim k M η ǫ η . Recall that the Verma module M (λ) has character ǫ λ p.
Theorem 6.1. Suppose that X is an isotropic set of positive roots and λ ∈ H γ for all γ ∈ X.
Then there exists a factor module M X (λ) of M (λ) such that ch M X (λ) = ǫ λ p X .
If X = {γ} we write M γ (λ) in place of M X (λ). The construction of the modules M X (λ) involves a process of deformation and specialization. First we extend scalars to A = k[T ] and B = k(T ). If R is either of these algebras we set U (g) R = U (g) ⊗ R. Choose ξ ∈ h * such that (ξ, γ) = 0 for all γ ∈ X, and (ξ, α ∨ ) / ∈ Z for all even roots α. Next consider the U (g) B -module M ( λ) B with highest weight λ = λ + T ξ, and form the factor module of M ( λ) B obtained by setting θ γ v λ equal to zero for γ ∈ X. Then take a suitable U (g) A -submodule of this factor module and reduce mod T to obtain the module M X (λ). In more detail, we set Then Based on Corollary 2.2 we can show Lemma 6.2. Let M = U (g) B v be a module with highest weight λ and highest weight vector v. Suppose that θ γ v = 0 for all γ ∈ X. Then (a) for all η the weight space M λ−η is spanned over B by all e −π v where π ∈ P X (η).
Proof of Theorem 6.1 when X = {γ}. We set Then the module M γ (λ) defined by (9) is generated by the image v of v which is a highest weight vector of weight λ. Also by Theorem 5.1, θ γ u = 0, so we can apply Lemma 6.2 to both N B and M B /N B . This gives Since , and p γ (η−γ)+p γ (η) = p(η), it follows that equality holds in (10). Now we obtain the result from the following considerations applied to the weight spaces of the modules M γ ( λ) R for R = A, B. If K is an A-submodule of a finite dimensional B-module L such that K A ⊗ A B = L, then dim k K/T K = dim B L. Remark 6.3. If M ′ is the kernel of the natural map M (λ) −→ M γ (λ), then U (g)θ γ v λ ⊆ M ′ , but the inclusion can be strict. Indeed this happens when g = sl(2, 1) and λ = −ρ [Mus3], see also [Mus1] Exercise 10.5.4. 7. The Jantzen Sum Formula. The Jantzen sum formula for a semisimple Lie algebra expresses the sum of the characters of the terms in the Jantzen filtration as a sum of characters of Verma modules. There is a version of the formula for contragredient Lie superalgebras in [Mus1] Theorem 10.3.1, but it contains some terms that are not characters of Verma modules. Here we see that these extra terms are actually characters of the modules M γ (λ) introduced in Theorem 6.1.
. Now we state our improved version of the Jantzen sum formula. At the same time, rather than using characters as in [Mus1], it is useful to rewrite the result using the Grothendieck group K(O) of the category O. We define K(O) to be the free abelian group generated by the symbols where |M : L(λ)| is the multiplicity of the composition factor L(λ) in M .
The advantage of using this version of the formula is that K(O) has a natural partial order. For
Lemma 8.1. For λ ∈ H γ ∩ H γ ′ there are only finitely many c ∈ k such that Proof. Set λ = λ + T ξ. It follows from (4) and Corollary 2.2 that when θ γ ( λ − γ)θ γ ′ ( λ)v λ is written as a A-linear combination of terms e −π v λ , the coefficient of e −γ e γ ′ v λ is a polynomial in T of degree d γ + d γ ′ . Hence there are only finitely many c such that Equation (12) holds, and a similar argument applies to Equation (13).
It follows from Lemma 8.1 that the set For λ ∈ Λ, the Jantzen sum formula (11) reads Next we quote a result from [Mus3] without proof.
Corollary 8.5. There is a rational function p of λ ∈ H γ ∩ H γ ′ such that Proof. The highest weight vectors referred to in the last sentence of the proof are necessarily proportional. Hence (16) holds since the coefficients of e −γ ′ e −γ in these highest weight vectors (when written as linear combinations of the e −π with π ∈ P(γ + γ ′ )) are polynomials in λ.
Corollary 8.5 is applied in the proof of Theorem 6.1 in the case |X| ≥ 2. We sketch the main new ingredient when X = {γ, γ ′ } as in the Corollary. In this case M ( λ) B has a series of submodules where From the Corollary and Theorem 5.1 we deduce that θ γ θ γ ′ v λ ∈ W 2 , θ γ ′ θ γ v λ ∈ W 3 and θ γ ′ θ γ θ γ ′ v λ = 0. Thus from Theorem 5.1 and Lemma 6.2 we obtain bounds on the dimensions of the weight spaces of the factors in the series (17) which are analogous to (10). The statement about characters follows as before.
Remark 9.2. The most interesting case of Equation (18) arises when p = 0, since then we have an inclusion between submodules of a Verma module obtained by multiplying the highest weight vector v λ by θ γ and θ γ ′ . Similarly the most interesting case of Equation (19) is when p = 1.