ANNIHILATION THEOREM AND SEPARATION THEOREM FOR BASIC CLASSICAL LIE SUPERALGEBRAS

MARIA GORELIK

Abstract. In this article we prove that for a basic classical Lie superalgebra the annihilator of a strongly typical Verma module is a centrally generated ideal. For a basic classical Lie superalgebra of type I we prove that the localization of the enveloping algebra by a certain central element is free over its centre.

1. Introduction

1.1. Let 𝔤 be a complex semisimple Lie algebra, U be its universal enveloping algebra and Z(𝔤) be the centre of U. Consider U as a 𝔤-module with respect to the adjoint action. Separation Theorem of Kostant (see [Ko]) states the existence of a submodule H of U such that the multiplication map provides the bijection H ⊗ Z(𝔤) ⟶^{∼} U. Moreover the multiplicity of each simple finite dimensional module V in H is equal to the dimension of its zero weight space. Such an ad 𝔤-invariant subspace H is called a harmonic space. An easy proof of Separation Theorem was found by Bernstein and Lunts—see [BL]. This theorem is an important ingredient in the proof of the annihilation theorem of Duflo (see [D], 8.4.3) asserting that the annihilator of a Verma module is generated by its intersection with Z(𝔤). The annihilation theorem is reproven by Joseph and G. Letzter. They also generalize it to the quantum case— see [JL],[J3].

In this paper we obtain analogous theorems in the case of basic classical Lie superalgebras. This was done earlier for the completely reducible case— see [M1],[GL1].

Let 𝔤 = 𝔤_{0} ⊕ 𝔤_{1} be a basic classical Lie superalgebra, Ũ be its universal enveloping superalgebra and Z(𝔤) be the centre of Ũ. Let T be a special ghost element constructed in [G]. Call a highest weight module strongly typical if it is not annihilated by T.

We prove the following version of Annihilation Theorem.

1.1.1. Theorem. The annihilator of a strongly typical Verma 𝔤-module M̃ is a centrally generated ideal.

The author was partially supported by TMR Grant No. FMRX-CT97-0100. Research at MSRI is supported in part by NSF grant DMS-9701755.

Moreover, for a strongly typical Verma module M̃, we describe the quotient Ũ/ Ann M̃ as an ad 𝔤-module and show that the natural map from Ũ/ Ann M̃ to the locally finite part of End_{ℂ}(M̃) is bijective.

1.1.2. The proof of Theorem 1.1.1 goes as follows. As in [JL], we use the Parthasarathy– Ranga-Rao– Varadarajan (PRV) determinants. We generalize the notion of PRV determinants (see [PRV]) to basic classical Lie superalgebras. Our construction is based on the fact that the two-sided ideal ŨT = T Ũ, considered as ad 𝔤-module, is injective in a category of locally finite modules.

In the completely reducible case one assigns to each simple finite dimensional module Ṽ a PRV determinant which is a polynomial in S(𝔥).

On the contrary, for the non-completely reducible case the lack of harmonic space forces us to substitute the PRV determinant by a set of PRV determinants corresponding to the same Ṽ. However, we do not have to calculate these determinants, but only verify that they are non-zero. We use these determinants to show that if the locally finite part F (M̃, M̃) of the endomorphisms End_{ℂ}(M̃) of a strongly typical Verma module M̃ has “a right size” as an ad 𝔤-module and the natural map Ũ → F (M̃, M̃) is surjective, then Ann M̃ is centrally generated.

For type I we directly verify both conditions. The crucial point in the study of type II case is the construction in Section 8 of a perfect mate χ ∈ Max Z(𝔤_{0}) for each strongly typical χ̃ ∈ Max Z(𝔤). We call a maximal ideal χ ∈ Max Z(𝔤_{0}) a perfect mate for χ̃ ∈ Max Z(𝔤) if the following conditions are satisfied.

(i) For any Verma 𝔤-module annihilated by χ̃, its 𝔤_{0}-submodule annihilated by a power of χ is a Verma 𝔤_{0}-module.

(ii) Any 𝔤-module annihilated by χ̃ has a non-trivial 𝔤_{0}-submodule annihilated by χ.

The condition (ii) seems to be difficult to check. However, it turns out that it is enough to verify (ii) only for simple highest weight 𝔤-modules. This is deduced from [M2].

For 𝔤 = 𝔬𝔰𝔭(1, 2l) the annihilator of a Verma 𝔤-module M̃ is a centrally generated ideal iff M̃ is strongly typical—see [GL1]. In this paper we prove the similar equivalence for the basic classical Lie superalgebras of type I.

As it is shown in [PS1], if 𝔤 has type I then for any strongly typical χ̃ ∈ Z(𝔤) the algebra Ũ/(Ũχ̃) is the matrix algebra over U(𝔤_{0})/(U(𝔤_{0})χ) for a suitable χ ∈ Max Z(𝔤_{0}). As it was pointed out by V. Serganova this result implies Theorem 1.1.1 for type I case. The opposite implication is also easy (see 7.4).

1.2. Let 𝔤 be a basic classical Lie superalgebra which is not completely reducible. Then Ũ is not a domain (see [AL]) and Z(𝔤) is not Noetherian ([M1], 2.8). However all nonzero central elements are non-zero divisors and Z(𝔤) contains an element z such that the localized algebra Z(𝔤)[z^{−1}] is isomorphic to a localization of a polynomial algebra. One can take z := T^{2} where T is the element mentioned above. The element T is even; it commutes with the even elements of Ũ and anticommutes with the odd ones. Moreover, the image of T in the symmetric algebra S(𝔤) belongs to the top exterior power of 𝔤_{1}. These properties determine T up to a scalar.

It is easy to show (see [G], 4.5) that Ũ, considered as an ad 𝔤-module, does not admit a factorization of the form H ⊗ Z(𝔤). For type I Lie superalgebras, we prove the following version of Separation Theorem.

1.2.1. Theorem. For 𝔤 of the type I, there exists ad 𝔤-invariant subspace H of Ũ such that the multiplication map provides the bijection H ⊗ Z(𝔤)[T^{−2}] → Ũ[T^{−2}].

Clearly, Z(𝔤)[T^{−2}] coincides with the centre of Ũ[T^{−2}]. As an ad 𝔤-module, H is injective in an appropriate category of locally finite modules and for any simple finite dimensional module Ṽ one has dim Hom_{𝔤}(Ṽ , H) = dim Ṽ |_{0} where Ṽ |_{0} is the zero weight space of Ṽ. For a basic classical Lie superalgebra of type II, we obtain a weaker result, namely that the similar assertions hold if we substitute z by a certain subset S of Z(𝔤). This set S can be described in terms of the PRV determinants.

A natural conjecture is that one can always choose S equal to {T^{2}}. A possible way to prove this conjecture is to show that an irreducible factor of a PRV determinant is either a factor of Shapovalov form or is of the form (β^{∨} +β^{∨}(ρ)) for some odd coroot β. However it is not clear how to calculate these determinants if 𝔤 is not completely reducible.

1.3. Content of the paper. In Section 2 we recall some facts about the basic classical Lie superalgebras.

In Section 3 we define a category Fin of locally finite 𝔤-modules and provide some properties of Fin. We also recall the construction and properties of the element T.

In Section 4 we investigate the 𝔤-module structure of Ũ given by the adjoint action. We start with studying Hom_{𝔤}(Ṽ , Ũ) for a simple finite dimensional module Ṽ. For each Ṽ we construct a central element z such that the localized module Hom_{𝔤}(Ṽ , Ũ[z^{−1}]) = Hom_{𝔤}(Ṽ , Ũ)[z^{−1}] is a free Z(𝔤)[z^{−1}]-module whose rank is equal to dim Ṽ |_{0}. In 4.6 we generalize a notion of PRV determinants to the case of non-completely reducible Lie superalgebras. We also establish properties of these determinants which are similar to the properties of the original PRV determinants. In 4.7 we show that for a suitable subset S of Z(𝔤) the localized algebra Ũ[S^{−1}] is free over its centre Z(𝔤)[S^{−1}]. We describe the ad 𝔤-module structure of the corresponding “generic harmonic space” H. We show that one may choose H to be the ad 𝔤-module generated by H′T where H′ is a certain harmonic space of U(𝔤_{0}).

In Section 5 we establish a connection between PRV determinants and the annihilators of simple modules. We show that if all PRV determinants do not vanish at a point λ ∈ 𝔥^{∗} and if a simple module Ṽ (λ) is strongly typical then its annihilator is a centrally generated ideal. Moreover, for a simple strongly typical Verma module Ṽ (λ) all PRV determinants do not vanish at a point λ ∈ 𝔥^{∗} iff the natural map Ũ → F (M̃, M̃) is surjective and F (M̃, M̃) has a certain nice structure as ad 𝔤-module (it should be isomorphic to the “generic harmonic space” H).

Sections 6,7 are devoted to type I case. Preliminary facts are concentrated in Section 6. In Section 7 we prove that for a suitable ad 𝔤-stable H the multiplication map provides an isomorphism H ⊗ Z(𝔤)[T^{−2}] → Ũ[T^{−2}]. We also prove that the annihilator of a Verma module is centrally generated iff this module is strongly typical.

Sections 8,9 are devoted to type II case.

In Section 8 we describe, for each strongly typical χ̃ ∈ Max Z(𝔤) its perfect mate χ ∈ Max Z(𝔤_{0}). Note that in type I case for any strongly typical χ̃ ∈ Max Z(𝔤) and for any λ such that χ̃ annihilates M̃(λ), the ideal Ann_{Z(𝔤_{0})} M(λ) is a perfect mate for χ̃. This does not hold for type II. For certain “generic” χ̃, the ideal Ann_{Z(𝔤_{0})} M(λ) is a perfect mate if one chooses λ satisfying, apart from χ̃M̃(λ) = 0, also a kind of “dominance” condition. For B(m, n) and G(3) all strongly typical central characters are generic. For the remaining superalgebras D(m, n), D(2, 1, α) and F (4) we select perfect mates for nongeneric strongly typical central characters case by case.

In Section 9 we prove Theorem 1.1.1. The existence of a perfect mate for each strongly typical χ̃ ∈ Max Z(𝔤) enables us to show that for a Verma module M̃ with the central character χ̃ the natural map Ũ → F (M̃, M̃) is surjective and F (M̃, M̃) ≅ H. According Section 5, these two conditions imply that the annihilator of M̃ is centrally generated ideal.

In Section 10 we study the 𝔤_{0}-structure of Verma 𝔤-modules.

Appendix 11 contains some lemmas used in the main text. It also contains alternative proofs of some results (non-vanishing of the PRV determinant,Corollary 4.5.2 and Theorem 4.7.4). These proofs do not use Kostant’s Separation Theorem. Probably, these proofs may be useful in the case when separation theorem does not hold.

Acknowledgments. It is a pleasant duty to express the gratitude to my teacher A. Joseph. Many ideas of this paper owe to his book and to his courses given at Weizmann Institute. The author would like to thank M. Duflo, E. Lanzmann, I. Musson and I. Penkov for helpful discussions and V. Serganova for her extreme patience and useful comments. This work was done when the author was a postdoctoral fellow at MSRI. The author would like to thank MSRI and the organizers of the Noncommutative Algebra Program for their support and hospitality.

2. Preliminaries

In this paper the ground field is ℂ. Everywhere in the paper apart from Sections 6,7, 𝔤 = 𝔤_{0} ⊕𝔤_{1} denotes one (unless otherwise specified, an arbitrary one) of the basic classical complex Lie superalgebras 𝔤𝔩(m, n), 𝔰𝔩(m, n), 𝔬𝔰𝔭(m, n), 𝔭𝔰𝔩(n, n). Each of these Lie superalgebras possesses the following properties: it admits a 𝔤-invariant bilinear form which is non-degenerate on [𝔤, 𝔤] and the even part 𝔤_{0} is a reductive Lie algebra.

In Sections 6,7, 𝔤 = 𝔤\overline{0} ⊕𝔤\overline{1} denotes one of the basic classical complex Lie superalgebras of type I: 𝔤𝔩(m, n), 𝔰𝔩(m, n), 𝔬𝔰𝔭(2, n), 𝔭𝔰𝔩(n, n). We shall slightly change our notations since these superalgebras admit ℤ-grading 𝔤 = 𝔤_{−1} ⊕ 𝔤_{0} ⊕ 𝔤_{1}. The even part 𝔤\overline{0} coincides with 𝔤_{0} and the odd part 𝔤\overline{1} is the sum of two dual 𝔤_{0}-modules 𝔤_{−1} and 𝔤_{1}.

In this section we present the main preliminary facts about the structure of the basic classical complex Lie superalgebras and their representations, which we shall use in this paper.

2.1. Conventions. We denote by ℕ^{+} the set of positive integers and by #I the number of elements of the given set I. If A is an algebra, N is an A-module and X, Y are subsets of A and N respectively, we denote by XY the set of the products xy where x ∈ X, y ∈ Y.

For a ℤ_{2}-homogeneous element u of a superalgebra denote by d(u) its ℤ_{2}-degree. In all formulas where this notation is used, u is assumed to be ℤ_{2}-homogeneous.

For a superalgebra 𝔪 denote by U(𝔪) its universal enveloping superalgebra. Set Ũ := U(𝔤) and U := U(𝔤_{0}).

In this text all modules are assumed to be left modules. For given module N we denote by N^{⊕r} the direct sum of n-copies of N. We say that A-module N is locally finite if dim Av < ∞ for all v ∈ N.

The symbol Ṽ (resp., V) is always used for a simple 𝔤 (resp., 𝔤_{0}) module and the symbol M̃ (resp., M) for a Verma 𝔤 (resp., 𝔤_{0}) module.

2.2. Triangular decompositions. Triangular decompositions of the superalgebras are defined in [PS2] as follows. A Lie subsuperalgebra 𝔥 ⊂ 𝔤 is called a Cartan subsuperalgebra if 𝔥 is nilpotent and coincides with its centralizer in 𝔤. For the basic classical Lie superalgebras the set of Cartan subalgebras coincides with the set of Cartan subalgebras of 𝔤_{0}. Fix a Cartan subalgebra 𝔥; it acts semisimply on 𝔤:

𝔤 := ⊕_{μ∈𝔥^{∗}}𝔤|_{μ}, 𝔤|_{μ} := {a ∈ 𝔤| ∀h ∈ 𝔥, [h, a] = μ(a)}.

Denote by Δ the set of non-zero roots that is the set {α ∈ 𝔥^{∗}| 𝔤|_{α} ≠ 0} \ {0}. An element h ∈ 𝔥 is called regular if Re α(h) ≠ 0 for all α ∈ Δ. Any regular element determines the decomposition Δ = Δ^{+} ∐ Δ^{−} where

Δ^{+} := {α ∈ Δ| Re α(h) > 0}, Δ^{−} := {α ∈ Δ| Re α(h) < 0}.

Moreover it determines a decomposition 𝔤 = 𝔫^{−} ⊕ 𝔥 ⊕ 𝔫^{+} where

𝔫^{+} := ⊕_{α∈Δ^{+}}𝔤|_{α}, 𝔫^{−} := ⊕_{α∈Δ^{−}}𝔤|_{α}.

Such decompositions of 𝔤 are called triangular decompositions. It is clear that 𝔫^{±} are nilpotent Lie subsuperalgebras of 𝔤.

A Lie subsuperalgebra 𝔟 ⊂ 𝔤 is called a Borel subsuperalgebra if 𝔟 = 𝔥 ⊕ 𝔫^{+} for some triangular decomposition 𝔤 = 𝔫^{−} ⊕ 𝔥 ⊕ 𝔫^{+}. A Borel subsuperalgebra determines the triangular decomposition; we will add a lower index to designate the corresponding Borel subsuperalgebra in the case where the choice of triangular decomposition is not clear from the context. We denote by Δ(𝔟) the set of non-zero roots of 𝔟. We say that a vector of v of 𝔤-module is 𝔟-primitive if [𝔟, 𝔟]v = 0 and 𝔥v ∈ ℂv.

A triangular decomposition 𝔤 = 𝔫^{−} ⊕ 𝔥 ⊕ 𝔫^{+} induces the triangular decomposition of the even part 𝔤_{0} = 𝔫_{0}^{−} ⊕ 𝔥 ⊕ 𝔫_{0}^{+}. The group of inner automorphisms of 𝔤_{0} acts transitively on all triangular decompositions of 𝔤_{0} and the action of this group can be extended to 𝔤. Hence the theory does not depend from the choice of a triangular decomposition of 𝔤_{0}. In the sequel we fix a triangular decomposition 𝔤_{0} = 𝔫_{0}^{−} ⊕ 𝔥 ⊕ 𝔫_{0}^{+} and consider triangular decompositions of 𝔤 which induce this fixed triangular decomposition of 𝔤_{0}.

2.2.1. Denote by Δ_{0} the set of non-zero even roots of 𝔤 and by Δ_{1} the set of odd roots of 𝔤. Set Δ_{0}^{±} := Δ_{0} ∩ Δ^{±} and Δ_{1}^{±} := Δ_{1} ∩ Δ^{±}.

Denote by (−, −) a 𝔤-invariant bilinear form on 𝔤 which is non-degenerate on [𝔤, 𝔤] and the induced W -invariant bilinear form on 𝔥^{∗}. A root α ∈ Δ is called isotropic if (α, α) = 0. For a root α denote by α^{∨} the element of 𝔥 satisfying α^{∨}(μ) = (α, μ) for each μ ∈ 𝔥^{∗}.

Set

\overline{Δ}_{0}^{+}:= \{α ∈ Δ_{0}^{+}| α/2 ∉ Δ_{1}^{+}\}, \overline{Δ}_{1}^{+}:= \{β ∈ Δ_{1}^{+}| 2β ∉ Δ_{0}^{+}\}.

The set of isotropic roots coincides with \overline{Δ}_{1}^{+}.

Remark that ∑_{α∈Δ^{+}} n_{α}α = 0 for some n_{α} ∈ ℕ implies n_{α} = 0 for all α ∈ Δ^{+}. This allows us to define the standard partial order relation on 𝔥^{∗} by λ ≤ μ ⟺ μ − λ ∈ ∑_{α∈Δ^{+}} ℕα. One can easily sees that the minimal (with respect to this partial order) elements of Δ^{+} form a basis of simple roots.

Denote by π_{0} the basis of simple roots of 𝔤_{0} and by W the Weyl group of Δ_{0}. Denote by |W | the number of elements in W. For w ∈ W set sn(w) := (−1)^{l(w)} where l(w) is the length of w. For a non-isotropic root α define s_{α} ∈ Aut 𝔥^{∗} by setting

s_{α}(λ) := λ − 2\frac{(α, λ)}{(α, α)}α.

Evidently s_{kα} = s_{α} and so the subgroup of Aut 𝔥^{∗} generated by the s_{α} coincides with W.

Set

ρ_{0} := \frac{1}{2} ∑_{α∈Δ_{0}^{+}} α, ρ_{1} := \frac{1}{2} ∑_{α∈Δ_{1}^{+}} α, ρ:= ρ_{0} − ρ_{1}.

For a simple root α one has 2(α, ρ) = (α, α).

Define the translated action of W on 𝔥^{∗} and on the symmetric algebra S(𝔥) by the formulas:

w.λ := w(λ + ρ) − ρ, w.f(λ) := f(w^{−1}.λ), ∀λ ∈ 𝔥^{∗}, w ∈ W.

2.2.2. For a 𝔤_{0}-module N and an element μ ∈ 𝔥^{∗} set

N|_{μ} := {m ∈ M| hm = μ(h)m, ∀h ∈ 𝔥}.

We shall consider mainly 𝔥-diagonalizable modules that is satisfying N = ∑_{μ∈𝔥^{∗}} N|_{μ}. Set

Ω(N) := {μ ∈ 𝔥^{∗}| N|_{μ} ≠ 0}.

If dim N|_{μ} < ∞ for each μ ∈ 𝔥^{∗} , set ch N := ∑_{λ∈𝔥^{∗}}(dim N|_{μ})e^{μ}.

When we use the notation Ũ|_{μ}, the action of 𝔤 on Ũ is assumed to be the adjoint action.

2.3. An important property of the basic classical Lie superalgebras is the existence of a Cartan superantiautomorphism σ coming from the supertransposition of matrices. Recall that an even linear endomorphism ι of a Lie (resp., associative) superalgebra is called a superantiautomorphism if ι([x, y]) = (−1)^{d(x)d(y)}[ι(y), ι(x)] (resp., ι(xy) = (−1)^{d(x)d(y)}ι(y)ι(x)) for all homogeneous elements x, y. The Cartan superantiautomorphism σ has the following properties:

a) σ^{2}(g) = (−1)^{d(g)}g, ∀g ∈ 𝔤,
b) σ(𝔫^{+}) = 𝔫^{−},
c) σ(h) = h,  ∀h ∈ 𝔥.

The restriction of σ to 𝔤_{0} is a Cartan antiinvolution:

a) (σ^{2})|_{𝔤_{0}} = id,
b) σ(𝔫_{0}^{+}) = 𝔫_{0}^{−},
c) σ(h) = h, ∀h ∈ 𝔥.

2.3.1. Symmetric algebra. Denote by F the canonical filtration of Ũ given by F^{k} := 𝔤^{k}. This filtration is ad 𝔤-invariant and the associated graded superalgebra S(𝔤) inherits an ad 𝔤-module structure. The superalgebra S(𝔤) is supercommutative: it is the product of the symmetric (even) algebra S(𝔤_{0}) and the external superalgebra Λ𝔤_{1}.

For u ∈ Ũ denote by gr u its image in S(𝔤); identify U(𝔥) and its image S(𝔥).

2.3.2. Centre. By definition, the (super)centre Z(𝔤) := Ũ^{𝔤}. One has gr Z(𝔤) = S(𝔤)^{𝔤}.

Denote by gr P the projection S(𝔤) → S(𝔥) along S(𝔤) gr(𝔫^{−} + 𝔫^{+}). The restriction of gr P to S(𝔤)^{𝔤} provides a monomorphism ι : S(𝔤)^{𝔤} → S(𝔥)^{W}. As a consequence, all non-zero elements of S(𝔤)^{𝔤} (resp., Z(𝔤)) are non-zero divisors in S(𝔤) (resp., Ũ). The image of ι is described in [K2], [S1], [BZV]; ι is bijective iff 𝔤 = 𝔬𝔰𝔭(1, 2l).

Denote by P_{∅} the projection S(𝔤) → S(𝔤_{0}) with the kernel K := ∑_{i≥1} S(𝔤_{0})Λ^{i}𝔤_{1}. It is easy to see that K is ad 𝔤_{0}-invariant and so P_{∅} is an ad 𝔤_{0}-map. Moreover P_{∅} provides a monomorphism S(𝔤)^{𝔤} → S(𝔤_{0})^{𝔤_{0}} since gr P = gr P ∘ P_{∅} and so the injectivity of P_{∅} on S(𝔤)^{𝔤} follows from the injectivity of gr P.

2.3.3. Harish-Chandra projection. Denote by P the Harish-Chandra projection Ũ → S(𝔥) with respect to the decomposition Ũ = (Ũ𝔫^{+} + 𝔫^{−}Ũ) ⊕ U(𝔥). The restriction of P to Ũ|_{0} = Ũ^{𝔥} is an algebra homomorphism. An element a ∈ Ũ^{𝔥} acts on a primitive vector of weight μ by the multiplication by the scalar P(a)(μ).

The Harish-Chandra projection provides a monomorphism Z(𝔤) → S(𝔥)^{W.}. If Ñ is a 𝔤-module generated by a primitive vector of weight λ, then a central element z acts on Ñ by the multiplication by the scalar P(z)(λ).

Call χ̃ ∈ Max Z(𝔤) a central character of a 𝔤-module N if χ̃^{r}N = 0 for r >> 0.

2.3.4. Let z be an element of Z(𝔤). Since z has weight zero, z = P(z) + ∑_{i} u_{i}^{−}u_{i}^{+} where u_{i}^{−} ∈ U(𝔥^{−})𝔫^{−}, u_{i}^{+} ∈ U(𝔥^{+})𝔫^{+} for all i. One has σ(u_{i}^{−}u_{i}^{+}) = ±σ(u_{i}^{+})σ(u_{i}^{−}) ∈ U(𝔥^{−})𝔫^{−}U(𝔥^{+})𝔫^{+}. Therefore

σ(z) = σ(P(z)) + ∑_{i} ±σ(u_{i}^{+})σ(u_{i}^{−}) ∈ P(z) + Ũ𝔫^{+}

that is P(σ(z)) = P(z). Thus the superantiautomorphism σ stabilizes the central elements. This implies that Z(𝔤_{0}), Z(𝔤) ⊂ Ũ^{σ}.

2.3.5. For a ℤ_{2}-graded 𝔤_{0}-module L denote by Ind_{𝔤_{0}}^{𝔤} L a vector space Ũ ⊗_{U} L (here Ũ is considered as a right U-module and a left Ũ-module through the multiplication) equipped by the natural structure of a left Ũ-module. Denote by Coind_{𝔤_{0}}^{𝔤} L a vector space Hom_{U}(Ũ, L) (here Ũ is considered as a left U-module) equipped by the following structure of a left Ũ-module: (uf)(u′) := f(u′u) for any f ∈ Hom_{U}(Ũ, L), u, u′ ∈ Ũ. For a 𝔤-module Ñ and a 𝔤_{0}-module L one has the canonical bijections

Hom_{𝔤_{0}}(Ñ, L) ⟶^{∼} Hom_{𝔤}(Ñ, Coind_{𝔤_{0}}^{𝔤} L),

Hom_{𝔤_{0}}(L, Ñ) ⟶^{∼} Hom_{𝔤}(Ind_{𝔤_{0}}^{𝔤} L, Ñ).

By [BF], Ind_{𝔤_{0}}^{𝔤} L ≅ Coind_{𝔤_{0}}^{𝔤} L.

2.4. Hopf algebra structure. The enveloping algebra Ũ is a supercommutative Hopf superalgebra. This means, in particular, that the antipode S is a superantiautomorphism of Ũ and that the comultiplication Δ′ : Ũ → Ũ ⊗ Ũ is a homomorphism of superalgebras satisfying the relation s ∘ Δ′ = Δ′ where s is a linear map s : Ũ ⊗ Ũ → Ũ ⊗ Ũ given by s(u_{1} ⊗ u_{2}) := (−1)^{d(u_{1})d(u_{2})}(u_{2} ⊗ u_{1}). The Hopf algebra structure on Ũ is given by

\begin{aligned}Δ′(g) & = g ⊗ 1 + 1 ⊗ g, \\ ε(g) & = 0, \\ S(g) & = −g \end{aligned}

for any g ∈ 𝔤.

The Hopf superalgebra structure on Ũ gives a 𝔤-module structure on the tensor product N_{1} ⊗N_{2} := N_{1} ⊗_{ℂ}N_{2} of two 𝔤-modules N_{1}, N_{2}. The map n_{1} ⊗n_{2} ↦ (−1)^{d(n_{1})d(n_{2})} provides the canonical 𝔤-isomorphism N_{1} ⊗ N_{2} → N_{2} ⊗ N_{1}.

2.4.1. Throughout the paper we shall write “ad 𝔤-module” instead “𝔤-module with respect to the adjoint action”.

View Ũ as 𝔤-module through the adjoint action given by

(ad g)u := ug − (−1)^{d(g)d(u)}ug, ∀g ∈ 𝔤, u ∈ Ũ.

As ad 𝔤-module, Ũ is locally finite.

For any 𝔤-modules N_{1}, N_{2} view Hom(N_{1}, N_{2}) := Hom_{ℂ}(N_{1}, N_{2}) as a 𝔤-module through the adjoint action:

(ad g)ψ(v) = gψ(v) − (−1)^{d(g)d(ψ)}ψ(gv) ∀g ∈ 𝔤, ψ ∈ Hom(N_{1}, N_{2}).

We denote by F (N_{1}, N_{2}) the locally finite part of ad 𝔤-module Hom(N_{1}, N_{2}). Notice that F (N_{1}, N_{2}) coincides with the ad 𝔤_{0}-locally finite part of Hom(N_{1}, N_{2}), since Ũ is a finite extension of U.

Throughout the paper an action of 𝔤 on Ũ and on F (N_{1}, N_{2}) is assumed, by default, to be the adjoint action.

For a 𝔤-module N, the natural map Ũ → End(N), coming from the action of Ũ on N, is an ad 𝔤-homomorphism and its image lies in F (N, N).

2.4.2. Let L be a finite dimensional 𝔤-module. Equip the dual supervector space L^{∗} by 𝔤-module structure through the antipode S:

g.f(v) := (−1)^{d(g)d(f)}f(S(g)v) = (−1)^{d(g)d(f)}f(−gv), ∀v ∈ L, g ∈ 𝔤.

We shall use the following form of the Frobenius reciprocity

Hom_{𝔤}(L, Hom(N_{1}, N_{2})) ≅ Hom_{𝔤}(L ⊗ N_{1}, N_{2}) ≅ Hom_{𝔤}(N_{1}, N_{2} ⊗ L^{∗})

for any 𝔤-modules N_{1}, N_{2} and a finite dimensional 𝔤-module L.

2.5. The category Õ and Verma modules. Denote by O the full subcategory of finitely generated 𝔥-diagonalizable 𝔤_{0}-modules which are locally 𝔫_{0}^{+}-finite. Denote by Õ the similarly defined category of 𝔤-modules.

Since U(𝔫^{+}) is finite over U(𝔫_{0}^{+}), a 𝔤-module N belongs Õ iff as 𝔤_{0}-module N belongs to O. In particular, any module of category Õ has a finite length.

2.5.1. For λ ∈ 𝔥^{∗} denote by ℂ_{λ} a simple 𝔟-module such that 𝔫^{+}v = 0 and hv = λ(h)v for any h ∈ 𝔥, v ∈ ℂ_{λ}. Define a Verma module M̃(λ) by setting

M̃(λ) := Ũ ⊗_{U(𝔟)} ℂ_{λ}.

The module M̃(λ) has a unique simple quotient, which we denote by Ṽ (λ). Similarly, denote by M(λ) and V (λ) respectively, Verma and simple 𝔤_{0}-modules of the highest weight λ.

2.5.2. Definition. A weight λ ∈ 𝔥^{∗} is called typical if (λ + ρ, β) ≠ 0 for any isotropic β ∈ Δ_{1}.

2.5.3. If λ is typical then Ann_{Z(𝔤)} M̃(λ) = Ann_{Z(𝔤)} M̃(λ′) implies λ′ ∈ W.λ— see [K3], Theorem 2. In particular, if a typical weight λ is a minimal element in its orbit W.λ then M̃(λ) is simple.

On the other hand, if a typical weight λ is a maximal element in W.λ then M̃(λ) is projective in Õ. Indeed, take a short exact sequence 0 → Ñ′ → Ñ → M̃(λ) → 0 in Õ. One may assume that χ̃^{k}Ñ = 0 where χ̃ := Ann_{Z(𝔤)} M̃(λ) and k is a positive integer. Then the weight of a primitive vector of any simple subquotient of Ñ belongs to W.λ. Since λ is a maximal element of W.λ one has Ω(Ñ) ∩ {λ + ℤΔ^{+}} ⊆ {λ − ℕΔ^{+}}. Thus a preimage of a highest weight vector of M̃(λ) is primitive and the above exact sequence splits.

By a similar argument, a short exact sequence 0 → M̃(λ′) → Ñ → M̃(λ) → 0 in Õ splits if λ′ ≯ λ.

2.5.4. Definition. A weight λ ∈ 𝔥^{∗} is called strongly typical if (λ + ρ, β) ≠ 0 for any β ∈ Δ_{1}.

2.5.5. Call the highest weight modules Ṽ (λ), M̃(λ) typical (resp., strongly typical) if λ is typical (resp., strongly typical).

Apart from the cases B(m, n) (that is 𝔬𝔰𝔭(2m + 1, 2n)) and G(3), all odd roots of 𝔤 are isotropic and thus the notions of “typical” and “strongly typical” coincide.

2.5.6. Take N ∈ Õ. Equip the graded dual vector space N^{#} := ⊕_{μ∈𝔥^{∗}}(N|_{μ})^{∗} by the following 𝔤-module structure

u.f(v) := (−1)^{d(f)d(u)}f(σ(u)v)), ∀u ∈ Ũ, f ∈ N^{#}, v ∈ N

where σ : Ũ → Ũ is the superantiautomorphism defined in 2.3. One can easily sees that N ↦ N^{#} defines a duality functor on Õ.

Since the restriction of σ on 𝔥 is equal to identity, ch N^{#} = ch N. In particular, Ṽ (λ)^{#} ≅ Ṽ (λ) and it is isomorphic to the socle of M̃(λ)^{#}.

2.5.7. According to [LM], for any Zariski dense S ⊆ 𝔥^{∗}

∩_{λ∈S} Ann M̃(λ) = 0.

2.5.8. A Verma module is not irreducible iff its highest weight is a root of a Shapovalov form. Shapovalov forms are polynomials in S(𝔥) indexed by the weights of U(𝔫^{−}). Each polynomial admits a linear factorization which was established by Kac. The linear factors of these polynomials are

\begin{matrix}α^{∨} + (ρ, α) − n(α, α)/2 & n ∈ ℕ^{+}, α ∈ \overline{Δ}_{0}^{+}, \\ α^{∨} + (ρ, α) − n(α, α)/2 & n ∈ 2ℕ + 1, α ∈ (Δ_{1}^{+} \ \overline{Δ}_{1}^{+}), \\ α^{∨} + (ρ, α), & α ∈ \overline{Δ}_{1}^{+}. \end{matrix}

Hence M̃(λ) is not irreducible iff (λ+ρ, α) = n(α, α)/2 for a positive root α and a positive integer n which should be odd for odd α.

2.6. Finite dimensional modules. Necessary and sufficient conditions for Ṽ (λ) to be finite dimensional are given in [K1], Theorem 8. One can immediately sees from these conditions that any typical finite dimensional Ṽ (λ) is strongly typical. If Ṽ (λ), Ṽ are finite dimensional satisfying Ann_{Z(𝔤)} Ṽ (λ) = Ann_{Z(𝔤)} Ṽ and Ṽ is typical then Ṽ ≅ Ṽ (λ)— see [K2], Prop. 2.7.

The following character formula of a typical finite dimensional module is established by Kac (see [K2],[K3]):

ch Ṽ (λ) = D ∑_{w∈W} sn(w)e^{w.λ}, where D = ∏_{α∈Δ_{0}^{+}}(1 − e^{−α})^{−1} ∏_{β∈Δ_{1}^{+}}(1 + e^{−β}).  (1)

2.7. Odd reflections. The odd reflections were introduced by I. Penkov and V. Serganova— see [PS2], 3.1. If 𝔤 is a basic classical Lie superalgebra, the definition takes the following form. Two Borel subsuperalgebras 𝔟, 𝔟′ ⊂ 𝔤 are connected by an odd reflection along β iff β is a simple odd isotropic root of 𝔟′ and

Δ(𝔟) = {−β} ∪ Δ(𝔟′) \ {β}.

Note that Δ_{0}(𝔟) = Δ_{0}(𝔟′) and so 𝔟_{0} = 𝔟′_{0}. As it is shown in [PS2], 3.1, any Borel subsuperalgebras 𝔟, 𝔟′ ⊂ 𝔤 satisfying 𝔟_{0} = 𝔟′_{0} are connected by a chain of odd reflections.

Let 𝔟 and 𝔟′ be Borel subsuperalgebras connected by the odd reflection along an odd isotropic root β. Then 𝔟′ = 𝔟 ∩ 𝔟′ + ℂx where x is a non-zero element of 𝔤_{β}. If v is a 𝔟primitive vector such that xv = 0 then v is also a 𝔟′-primitive. This implies Ṽ_{𝔟}(λ) = Ṽ_{𝔟′}(λ) for λ satisfying (λ, β) = 0. If xv ≠ 0 then xv is a 𝔟′-primitive since [x, x] = 0. This implies Ṽ_{𝔟}(λ) = Ṽ_{𝔟′}(λ + β) and M̃_{𝔟}(λ) = M̃_{𝔟′}(λ + β) for (λ, β) ≠ 0. Remark that (ρ_{𝔟}, β) = 0, since β is a simple isotropic root, and so (λ, β) = 0 iff (λ + ρ_{𝔟}, β) = 0. Taking into account that ρ_{𝔟} = ρ_{𝔟′} + β, one concludes that for Borel subsuperalgebras 𝔟, 𝔟′ satisfying 𝔟_{0} = 𝔟′_{0} and a 𝔟-typical weight λ one has Ṽ_{𝔟}(λ) = Ṽ_{𝔟′}(λ′) and M̃_{𝔟}(λ) = M̃_{𝔟′}(λ′) where λ + ρ_{𝔟} = λ′ + ρ_{𝔟′}.

2.8. Completely reducible Lie superalgebras. Recall that a Lie superalgebra is called completely reducible if all its finite dimensional modules are completely reducible. According to the theorem of Djoković and Hochschild (see [Sch], p. 239), any finite dimensional completely reducible Lie superalgebra is a direct sum of semisimple Lie algebras and algebras 𝔬𝔰𝔭(1, 2l) (l ≥ 1). The superalgebra 𝔤 == 𝔬𝔰𝔭(1, 2l) has many features of the semisimple Lie algebra; in particular, Ũ is a domain and Z(𝔤) is a polynomial algebra.

2.8.1. Separation Theorem. In [Ko] Kostant establishes the following theorem which is called sometimes “Separation Theorem”.

Theorem. Let 𝔤 be a semisimple complex Lie algebra. There exists an ad 𝔤-invariant subspace H in U(𝔤) such that the multiplication map induces an isomorphism Z(𝔤) ⊗ H ⟶^{∼} U(𝔤). Moreover, for every simple finite dimensional module V, [H : V] = dim V_{0}.

Such an ad 𝔤-invariant subspace H is called a harmonic space.

In [M1] Musson proves the analogous theorem for 𝔤 = 𝔬𝔰𝔭(1, 2l).

2.8.2. Annihilation Theorem. In [D] Duflo proves the following theorem.

Theorem. Let 𝔤 be a semisimple complex Lie algebra. Then for any λ ∈ 𝔥^{∗}

Ann M(λ) = U(𝔤) Ann_{Z(𝔤)} M(λ).

Let 𝔤 be a semisimple complex Lie algebra and M be a Verma module. The multiplicity of each simple finite dimensional module in F (M, M) is equal to dim V |_{0} if M is a 𝔤_{0}- simple Verma module. Combining the above theorem and Theorem 2.8.1, one concludes that for such M the natural map Ũ/ Ann M(λ) → F (M, M) is an isomorphism. In [J1], 6.4 Joseph generalizes this result to any 𝔤_{0}-Verma module.

2.8.3. For the case 𝔤 = 𝔬𝔰𝔭(1, 2l) the following results are obtained in [GL1],[GL2].

Theorem. The annihilator of a Verma module M̃ coincides with Ũ Ann_{Z(𝔤)} M̃ iff M̃ is strongly typical.

Using Separation Theorem, one concludes that for a strongly typical M̃ one has Ũ/(Ann M̃) ≅ H.

To describe the non-centrally generated annihilators of Verma modules, it is convenient to substitute the centre Z(𝔤) by the algebra Z̃(𝔤) defined in 3.3.

Theorem. For any λ ∈ 𝔥^{∗}

Ann M̃(λ) = Ũ Ann_{_{Z}(𝔤)} M̃(λ).

If λ is not strongly typical then the ideal Ann_{_{Z}(𝔤)} M̃(λ) is a maximal ideal of the algebra Z̃(𝔤).

The algebra Ũ is free over Z̃(𝔤); more precisely, Ũ contains ad 𝔤_{0}-submodule K such that the multiplication map induces the ad 𝔤_{0}-isomorphism K ⊗ Z̃(𝔤) → Ũ. Moreover, as ad 𝔤_{0}-modules H ≅ K ⊕ K and K ≅ Ũ/(Ann M̃) if M̃ is not strongly typical .

3. Category Fin and a twisted adjoint action

3.1. Category Fin. Denote by Fin_{0} the full category of 𝔤_{0}-modules whose objects are sums of simple finite dimensional modules. Denote by Fin the full category of 𝔤-modules whose objects, considered as 𝔤_{0}-modules, belong to Fin_{0}. Since Ũ is a finite extension of U(𝔤_{0}), any module N ∈ Fin is locally finite. In other words, the objects of Fin are the locally finite 𝔥-diagonalizable modules (𝔤_{0} is reductive and so all locally finite 𝔥-diagonalizable are completely reducible).

Denote by Irr the set of isomorphism classes of simple finite dimensional 𝔤-modules and by Irr_{0} the set of isomorphism classes of simple finite dimensional 𝔤_{0}-modules. Note that Ũ considered as ad 𝔤-module belongs to Fin.

Throughout this section all modules are objects of Fin. Everywhere in the paper, injectivity and projectivity mean, by default, injectivity and projectivity in the category Fin.

3.1.1. Recall that the socle Soc N of the module N is the sum of its simple submodules. Since any module in Fin is locally finite, it has a finite dimensional submodule and so a non-trivial socle.

Recall that a module N is called an essential extension of its submodule N′ if for any non-zero submodule N′′ of N one has N′ ∩ N′′ ≠ 0. Any module in Fin is an essential extension of its socle.

3.1.2. For a homomorphism ψ : N → N′ denote by Soc ψ its restriction to Soc N. If Soc ψ is a monomorphism then ψ is also a monomorphism, since ker ψ ∩ Soc N = 0 and N is an essential extension of Soc N.

3.2. Injective and projective objects in Fin. Recall that an injective envelope E(N) of the module N is an injective module which is an essential extension of N. An injective envelope of a given module is unique up to isomorphism.

A typical simple finite dimensional module is injective and projective in Fin (see 2.5.2).

A standard reasoning of [McL], 3.11 shows that any submodule N of an injective module E has an injective envelope which is a direct summand of E. In Lemma 3.2.2 we show that any module in Fin has an injective envelope.

3.2.1. A finite dimensional 𝔤_{0}-module L is projective and injective in Fin_{0} and so Coind_{𝔤_{0}}^{𝔤} L ≅ Ind_{𝔤_{0}}^{𝔤} L is projective and injective in Fin— see 2.3.5.

Let Ṽ be a simple finite dimensional 𝔤-module and V be its simple 𝔤_{0}-submodule. Then Ṽ is a submodule of the injective module Coind_{𝔤_{0}}^{𝔤} V. Therefore Ṽ has an injective envelope E(Ṽ) which is a direct summand of Coind_{𝔤_{0}}^{𝔤} V. The last is projective and so E(Ṽ) is projective as well.

3.2.2. Lemma. Any module N ∈ Fin has an injective envelope E(N). Moreover E(N) ≅ E(Soc N).

Proof. By 3.1.1, N is an essential extension of Soc N. Therefore, by [McL], 3.11.1, there exists a monomorphism ι : N → E(Soc N) whose socle is the natural embedding Soc N → E(Soc N). Since E(Soc N) is an essential extension of Soc N, it is also essential extension of N. Hence E(Soc N) is an injective envelope of N.

3.2.3. Lemma. Any direct sum of injective modules in Fin is injective.

Proof. Let E_{i}, i ∈ I be a collection of injective modules. One has to check that for any monomorphism ι : A → B and any homomorphism ϕ : A → ⊕_{i∈I}E_{i} there exists a homomorphism ψ : B → ⊕_{i∈I}E_{i} such that ψι = ϕ. By a standard reasoning based on Zorn’s lemma, it is enough to verify the above assertion assuming B being cyclic. Any cyclic module in Fin is finite dimensional. If B is finite dimensional, then A is also finite dimensional and so ϕ(A) lies in a finite subsum ⊕_{i∈J}E_{i} (J is a finite subset of I). Since each E_{i} is injective, ⊕_{i∈J}E_{i} is injective and so there exists ψ : B → ⊕_{i∈J}E_{i} such that ψι(a) = ϕ(a) for all a ∈ A. The assertion follows.

3.2.4. Corollary.

E(⊕_{i∈I}N_{i}) = ⊕_{i∈I} E(N_{i})

3.2.5. Proposition. Any injective module in Fin is projective.

Proof. Take an injective module N and write Soc N = ∑_{i∈I} L_{i} where the L_{i} are simple. Combining Lemma 3.2.2 and Corollary 3.2.4, one obtains N ≅ ⊕_{i∈I} E(L_{i}). By 3.2.1, all E(L_{i}) are projective so N is also projective.

3.3. Twisted adjoint action. In this subsection we present some results from [G] which are used in the sequel.

Define a twisted adjoin action of 𝔤 on Ũ by setting

(ad′ g)u = gu − (−1)^{d(g)(d(u)+1)}ug, ∀g ∈ 𝔤, u ∈ Ũ.

Note that ad g = ad′ g for g ∈ 𝔤_{0}.

The anticentre A(𝔤) is the set of invariants of Ũ under the twisted adjoint action ad′ 𝔤. The product of two anticentral elements is central. For 𝔤 being a basic classical Lie superalgebra, A(𝔤) is even and so any anticentral element commutes with the even elements of Ũ and anticommutes with the odd ones. Therefore “the ghost centre”

Z̃(𝔤) := Z(𝔤) + A(𝔤)

is a commutative subalgebra of Ũ.

For 𝔤 = 𝔬𝔰𝔭(1, 2l), Z̃(𝔤) is a polynomial algebra and, moreover, Z̃(𝔤) = Z(𝔤) ⊕ T Z(𝔤) where T is the element defined in 3.3.2.

3.3.1. Let L be an ad 𝔤_{0}-submodule of U. Then the ad′ 𝔤-submodule generated by L in Ũ is isomorphic to the induced module Ind_{𝔤_{0}}^{𝔤} L. As ad′ 𝔤-module Ũ is generated by U and so Ũ ≅ Ind_{𝔤_{0}}^{𝔤} U. Using the isomorphism Ind_{𝔤_{0}}^{𝔤} U ≅ Coind_{𝔤_{0}}^{𝔤} U, one obtains a linear isomorphism Z(𝔤_{0}) ⟶^{∼} A(𝔤) given by z ↦ (ad′ u)z where u ∈ Ũ is such an element that uV (0) is the trivial 𝔤-submodule of Ind_{𝔤_{0}}^{𝔤} V (0).

An anticentral element z acts on a module generated by a primitive vector v in the following way. It acts as P(z)(λ) id on Ũ_{0}v and as (−P(z)(λ) id) on Ũ_{1}v (Ũ_{0}, Ũ_{1} are homogeneous components of Ũ). From 2.5.7 it follows that the Harish-Chandra projection provides a monomorphism A(𝔤) → S(𝔥)^{W.}. The image of this monomorphism is equal to tS(𝔥)^{W.} where

t := ∏_{β∈Δ_{1}^{+}}(β^{∨} + (ρ, β)).

Any non-zero element of A(𝔤) is a non-zero divisor in Ũ.

3.3.2. Denote by T the element of A(𝔤) such that P(T) = t. Remark that a Verma module is strongly typical iff its annihilator does not contain T.

Up to a non-zero scalar, T is equal to the image of 1 under the above linear isomorphism Z(𝔤_{0}) ⟶^{∼} A(𝔤). The image of T in the symmetric algebra is a non-zero element of the one-dimensional vector space Λ^{top}𝔤_{1}.

Remark that T A(𝔤) ⊂ Z(𝔤) and the restriction of the Harish-Chandra projection provides an isomorphism T A(𝔤) ⟶^{∼} t^{2}S(𝔥)^{W.}. For 𝔤 = 𝔬𝔰𝔭(1, 2l), this implies that A(𝔤) is a free module over Z(𝔤) generated by T.

It is easy to check that (ad g)(uT) = ((ad′ g)u)T. Since T is a non-zero divisor, the ideal ŨT considered as ad 𝔤-module is isomorphic to Ũ considered as ad′ 𝔤-module. Hence as ad 𝔤-module ŨT ≅ Ind_{𝔤_{0}}^{𝔤} U and, in particular, it is injective in Fin. Note that ŨT is a two-sided ideal. Remark that, apart from the case when 𝔤 is completely reducible, Ũ itself as ad 𝔤-module is neither injective nor projective in Fin: it contains, as a direct summand, a trivial representation generated by 1.

3.3.3. For any N ∈ Fin_{0} one has

dim Hom_{𝔤}(Ṽ (ν), Coind_{𝔤_{0}}^{𝔤} N) = dim Hom_{𝔤_{0}}(Ṽ (ν), N).

Therefore

Ind_{𝔤_{0}}^{𝔤} N ≅ Coind_{𝔤_{0}}^{𝔤} N = ⊕_{_{V}∈Irr} E(Ṽ)^{r(_{V})}, r(Ṽ) := dim(Hom_{𝔤_{0}}(Ṽ , N)).  (2)

This has the following useful consequence. Let H be a harmonic space of U(𝔤_{0}) that is an ad 𝔤_{0}-submodule of U(𝔤_{0}) such that the multiplication map provides an isomorphism H ⊗ Z(𝔤_{0}) → U(𝔤_{0})—see 2.8.1. Then H ≅ ⊕_{V ∈Irr_{0}}V^{⊕ dim V |_{0}}. Taking into account 3.3.1 and (2), one obtains

(ad′ Ũ)H ≅ Ind_{𝔤_{0}}^{𝔤} H = ⊕_{V ∈Irr_{0}} Coind_{𝔤_{0}}^{𝔤} V^{⊕ dim V |_{0}} = ⊕_{_{V}∈Irr} E(Ṽ)^{⊕ dim _{V}|_{0}}  (3)

since ∑_{V ∈Irr_{0}} dim V |_{0} · dim(Hom_{𝔤_{0}}(Ṽ , V)) = dim Ṽ |_{0} for any Ṽ ∈ Irr.

4. The structure Ũ as ad 𝔤-module.

In this section we study the ad 𝔤-module structure of Ũ. We start from the studying Hom_{𝔤}(Ṽ , Ũ) for a simple finite dimensional module Ṽ. For each Ṽ we construct a central element z such that the localized module Hom_{𝔤}(Ṽ , Ũ[z^{−1}]) = Hom_{𝔤}(Ṽ , Ũ)[z^{−1}] is a free Z(𝔤)[z^{−1}]-module whose rank is equal to dim Ṽ |_{0}. In 4.6 we define and study PRV determinants for non-completely reducible basic classical Lie superalgebras.

In 4.7 we show that for a suitable S ⊂ Z(𝔤) the localized algebra Ũ[S^{−1}] is free over its centre Z(𝔤)[S^{−1}] and that the corresponding harmonic space H (“generic harmonic space”) is injective (in Fin) ad 𝔤-module. Moreover, the multiplicity of a simple finite dimensional module Ṽ in Soc H is equal to dim Ṽ |_{0}.

4.1. The following lemma of A. Joseph and G. Letzter provides a connection between Hom_{𝔤}(Ṽ , Ũ) and Hom(Ṽ |_{0}, S(𝔥)).

4.1.1. Lemma. An ad 𝔤-submodule N of Ũ lies in the annihilator of Ṽ (λ) iff P(N|_{0})(λ) = 0.

Proof. The proof is the same as in [J3], 7.2. Remark that P(N) = P(N|_{0}). Let v_{λ} be a highest weight vector of Ṽ (λ) and Ṽ (λ)_{−}:= U(𝔫^{−})𝔫^{−}v. One has

P(N)(λ) = 0 ⟺ Nv_{λ} ⊆ Ṽ (λ)_{−}.

In particular, NṼ (λ) = 0 forces P(N)(λ) = 0.

For the inverse implication, assume that P(N|_{0})(λ) = 0 that is Nv_{λ} ⊆ Ṽ (λ)_{−}. The ad 𝔤-invariance of N implies U(𝔫^{−})N = NU(𝔫^{−}) and thus

NṼ (λ) = NU(𝔫^{−})v_{λ} = U(𝔫^{−})Nv_{λ} ⊆ U(𝔫^{−})Ṽ (λ)_{−} ⊆ Ṽ (λ)_{−} ⊂_{≠} Ṽ (λ).

The ad 𝔤-invariance of N implies also U(𝔤)N = NU(𝔤) and so NṼ (λ) is a submodule of Ṽ (λ). Hence NṼ (λ) = 0 as required.

4.1.2. Corollary. Take Ṽ ∈ Irr, a basis v_{1}, … v_{r} be of Ṽ |_{0} and λ ∈ 𝔥^{∗}. For any θ_{1}, …, θ_{k} ∈ Hom_{𝔤}(Ṽ , Ũ), the image of the space ∑_{i=1}^{k} θ_{i}(Ṽ) under the natural map Ũ → End(Ṽ (λ)) is isomorphic to Ṽ^{⊕m} where m is the rank of the matrix (P(θ_{j}(v_{i}))(λ))_{i=1,r}^{j=1,k}.

4.1.3. Combining 2.5.7 and Lemma 4.1.1 one concludes that for an ad 𝔤-submodule N of Ũ the equality P(N|_{0}) = 0 implies N = 0. Moreover N = 0 provided that P(N|_{0})(R) = 0 for a Zariski dense subset R of 𝔥^{∗}.

4.2. Notation. Fix Ṽ ∈ Irr and consider Hom_{𝔤}(Ṽ , Ũ) as a Z(𝔤)-module with respect to the action induced by the multiplication.

The Harish-Chandra projection induces the map Ψ : Hom_{𝔤}(Ṽ , Ũ) → Hom_{𝔤}(Ṽ |_{0}, S(𝔥)) given by

Ψ(ϕ(v) = P(ϕ(v)), ∀ϕ ∈ Hom_{𝔤}(Ṽ , Ũ), v ∈ Ṽ |_{0}.

This map is a monomorphism by 4.1.3. Denote by I(Ṽ) the image of Ψ.

Define the action of Z(𝔤) on S(𝔥) by setting zp := P(z)p. This action induces the structure of a Z(𝔤)-module on Hom(Ṽ |_{0}, S(𝔥)). Obviously, Ψ is a Z(𝔤)-map. Thus the study of Z(𝔤)-module structure of Hom_{𝔤}(Ṽ , Ũ) reduces to the study of Z(𝔤)-module structure of I(Ṽ).

The vector space Hom(Ṽ |_{0}, S(𝔥)) has the natural structure of S(𝔥)-module. We denote by I(Ṽ)S(𝔥) the S(𝔥)-span of I(Ṽ) inside Hom(Ṽ |_{0}, S(𝔥)).

4.3. Examples.

4.3.1. Example. Let 𝔤 be a completely reducible simple Lie superalgebra. In this case P(Z(𝔤)) = S(𝔥)^{W.}. Separation Theorem (see 2.8.1) state the existence of an ad 𝔤submodule H of Ũ such that the multiplication map induces isomorphism H ⊗ Z(𝔤) ⟶^{∼} Ũ. Thus for any Ṽ ∈ Irr, a basis of the vector space Hom_{𝔤}(Ṽ , H) is a free Z(𝔤)- basis of Hom_{𝔤}(Ṽ , Ũ). Consequently, Hom_{𝔤}(Ṽ , Ũ) is a free Z(𝔤)-module of the rank dim Hom_{𝔤}(Ṽ , H) = dim Ṽ |_{0} (see 2.8.1).

4.3.2. Example. Consider the case 𝔤 = 𝔰l(2, 1). Then 𝔥 = span{z, h} where z is a central element of the reductive algebra 𝔤_{0} = 𝔰𝔩(2) × ℂ and h is an element of the Cartan subalgebra of 𝔰l(2). Then Δ_{0} = {±α}, Δ_{1} = {±β; ±(α + β)}. Choose such a triangular decomposition that Δ^{+} = {α, β, α + β}. The subalgebras 𝔫_{1}^{±} = 𝔤_{1} ∩ 𝔫^{±} are supercommutative and are ad 𝔤_{0}-submodules of 𝔤. Let {x_{1}, x_{2}} (resp., {y_{1}, y_{2}}) be a basis of 𝔫_{1}^{+} (resp., 𝔫_{1}^{−}).

The algebra S(𝔥)^{W.} is a polynomial algebra generated by z and t = (z − h)(z + h + 2). The image P(Z(𝔤)) in S(𝔥)^{W.} is spanned by 1 and the elements {t^{n}z^{k}, n > 0, k ≥ 0}.

Denote by v a highest weight vector of Ṽ := Ṽ (α + 2β); one has Ṽ |_{0} = ℂy_{1}y_{2}v. It is easy to see that a highest weight vector of any copy of Ṽ inside Ũ is of the form ux_{1}x_{2} where u ∈ Z(𝔤_{0}). Up to a scalar, P((ad y_{1}y_{2})(ux_{1}x_{2})) = (z − h)(z + h + 2)P(u) = tP(u). One has P(Z(𝔤_{0})) = S(𝔥)^{W.} and so I(Ṽ) = Hom(ℂ, S(𝔥)^{W.}t). Since S(𝔥)^{W.} is not free over P(Z(𝔤)), the Z(𝔤)-module I(Ṽ) ≅ Hom_{𝔤}(Ṽ , Ũ) is not free.

One might expect from the above example that I(Ṽ) is stable with respect to the multiplication on S(𝔥)^{W.}. However I(Ṽ (0)) = Hom(ℂ, P(Z(𝔤))) is not stable with respect to the multiplication on S(𝔥)^{W.} apart from the case when P(Z(𝔤)) = S(𝔥)^{W.}.

4.4. In this subsection we show that the Z(𝔤)-rank of the module Hom_{𝔤}(Ṽ , Ũ) is not greater than dim Ṽ |_{0}. The crucial point is Lemma 4.4.2 asserting that any elements of I(Ṽ) which are “linearly dependent” over S(𝔥) are “linearly dependent” over Z(𝔤). Throughout this subsection Ṽ ∈ Irr is fixed.

4.4.1. Basic definitions. Let A be a commutative domain. For an A-module N define an A-rank of N to be the dimension over the field of fractions Fract A of the localized module N ⊗_{A} Fract A. Call elements θ_{1}, …, θ_{k} ∈ N A-linearly independent if their images in the localized module N ⊗_{A} Fract A are linearly independent. Call the elements A-linearly dependent if they are not A-linearly independent. Call an A-basic system of N a collection θ_{1}, …, θ_{k} ∈ N such that the image of this collection in the localized module N ⊗_{A} Fract A forms its Fract A-basis.

Let L be a vector space and v_{1}, …, v_{r} be a basis of L. Let N be an A-submodule of Hom(L, A) and k be the A-rank of N. The collection θ_{1}, …, θ_{k} ∈ N is a A-basic system of N iff the matrix (θ_{j}(v_{i}))_{i=1,r}^{j=1,k} ∈ Mat_{(r×k)}(A) has a (k × k) non-zero minor. Call such a non-zero minor (which is an element of A) a minor of a basic system θ_{1}, … θ_{k}.

If p ∈ A is a minor of a basic system θ_{1}, … θ_{k} then for any θ ∈ N one has pθ = ∑_{j=1}^{k} p_{j}θ_{j} for certain (unique) collection p_{1}, …, p_{k} ∈ A. In particular, the localized module N ⊗_{A} A[p^{−1}] is a free A[p^{−1}]-module and the images of θ_{1}, … θ_{k} form a basis of this free module.

Recall that t^{2}S(𝔥)^{W.} ⊆ P(Z(𝔤)) (see 3.3.2).

4.4.2. Lemma.

(i) Elements of I(Ṽ) are Z(𝔤)-linearly independent iff they are S(𝔥)-linearly independent.

(ii) If θ_{1} …, θ_{s} ∈ I(Ṽ) are S(𝔥)-linearly independent and

∑_{j=0}^{s} p_{j}θ_{j} = 0

for some θ_{0} ∈ I(Ṽ) and p_{0}, …, p_{s} ∈ S(𝔥), then there exists z_{0}, …, z_{s} ∈ Z(𝔤) such that ∑_{j=0}^{s} z_{j}θ_{j} = 0 and P(z_{0}) = t^{2}q where q is a maximal W.-invariant divisor of p_{0}^{|W |}.

Proof. Obviously, elements of I(Ṽ) are Z(𝔤)-linearly independent provided that they are S(𝔥)-linearly independent. The inverse implication follows from (ii) because we always can choose a minimal (with respect to the inclusion) subset of S(𝔥)-linearly dependent elements in a set of S(𝔥)-linearly dependent elements.

Let us prove (ii). The equality ∑_{j=0}^{s} p_{j}θ_{j} = 0 means that for any μ ∈ 𝔥^{∗} one has

∀v ∈ Ṽ |_{0} 0 = ∑_{j=0}^{s} p_{j}(μ)θ_{j}(v)(μ) = ∑_{j=0}^{s} p_{j}(μ)P((Ψ^{−1}θ_{j})(v))(μ).

In the light of Lemma 4.1.1, this gives

∀μ ∈ 𝔥^{∗} (∑_{j=0}^{s} p_{j}(μ)Ψ^{−1}(θ_{j}))(Ṽ) ⊂ Ann Ṽ (μ);

here the sum belongs to Hom_{𝔤}(Ṽ , Ũ) because Ψ^{−1}(θ_{j}) ∈ Hom_{𝔤}(Ṽ , Ũ) and p_{j}(μ) ∈ ℂ.

Fix a root α ∈ π_{0}; let s_{α} ∈ W be the corresponding reflection. Assume that μ ∈ 𝔥^{∗} is such that Ṽ (μ) = M̃(μ)/M̃(s_{α}.μ). Then, for “sufficiently large” μ, any copy of Ṽ inside Ũ which annihilates Ṽ (μ), annihilates also M̃(μ) and so Ṽ (s_{α}.μ). To be more precise, choose ω ∈ ℚΔ such that (ω, α′) ≥ 0 for all α′ ∈ Δ^{+} and (ω, α) = 1. Set

R := {μ ∈ 𝔥| (a) n(μ) ∈ ℕ^{+}, (b) n(μ) > (−ν, ω), (c)∀β ∈ Δ^{+} \ {ℚα} (μ + ρ, β) ∉ ℚ}.

where ν stands for the lowest weight of Ṽ and n(μ) := 2(μ + ρ, α)/(α, α). It is easy to see that R is Zariski dense in 𝔥^{∗} and that any element in R is typical. Take μ ∈ R.

By 2.5.8, M̃(μ) is not simple because n(μ) ∈ ℕ^{+}. If M̃(μ′) is a subquotient of M̃(μ), then (μ′ − μ) ∈ ℕΔ^{−} and, by 2.5.3, μ′ = w.μ for some w ∈ W. Using the fact that any w ∈ W can be written as the product s_{β_{1}} ⋯ s_{β_{m}} where β_{1}, …, β_{m} ∈ Δ_{0}^{+} are linearly independent, it is easy to deduce from the condition (c) that (w.μ−μ) ∉ ℕΔ^{−} for any w ≠ id, s_{α}. Thus Ṽ (μ) = M̃(μ)/M̃(s_{α}.μ) and M̃(s_{α}.μ) is simple. Observe that s_{α}.μ = μ − n(μ)α and so for any ξ ∈ Ω(M̃(s_{α}.μ)) one has (ξ, ω) ≤ (μ, ω) − n(μ). Thus (μ + ν) ∉ Ω(M̃(s_{α}.μ)) due to the condition (a).

Let N be a copy of Ṽ inside Ann Ṽ (μ). Let u_{ν} be a lowest weight vector of N and v_{μ} be a highest weight vector of M̃(μ). Since u_{ν}Ṽ (μ) = 0 the vector u_{ν}v_{μ} belongs to M̃(s_{α}.μ). However the latter does not have non-zero elements of weight (μ + ν). Thus u_{ν}v_{μ} = 0. One has u_{ν}M̃(μ) = u_{ν}U(𝔫^{−})v_{μ} = U(𝔫^{−})u_{ν}v_{μ} because u_{ν} is a lowest weight vector that is (ad 𝔫^{−})u_{ν} = 0. Therefore u_{ν}M̃(μ) = 0. Since Ann M̃(μ) is ad 𝔤-stable, N ⊂ Ann M̃(μ) and, in particular, N ⊂ Ann Ṽ (s_{α}.μ).

Hence ∑_{j=0}^{s} p_{j}(μ)(Ψ^{−1}θ_{j})(Ṽ) annihilates Ṽ (s_{α}.μ) for any μ ∈ R. Then, by Lemma 4.1.1

(∑_{j=0}^{s}(s_{α}.p_{j})θ_{j}(v)) (μ′) = 0

for any v ∈ Ṽ |_{0} and any μ′ such that s_{α}.μ′ ∈ R. The terms ∑_{j=0}^{s}(s_{α}.p_{j})θ_{j}(v) are polynomials in S(𝔥). Since R is a Zariski dense subset of 𝔥^{∗}, one concludes that these polynomials are equal to zero. Consequently

∑_{j=0}^{s}(s_{α}.p_{j})θ_{j} = 0.

Taking into account that ∑_{j=0}^{s} p_{j}θ_{j} = 0 and that θ_{1}, … θ_{s} are S(𝔥)-linearly independent, one concludes that p_{j}/p_{0} = (s_{α}.p_{j})/(s_{α}.p_{0}) for all j = 1, …, s. Thus p_{j}/p_{0} is W.-invariant for all j = 1, …, s. Then, by Lemma 11.1.6, for each j = 0, …, s there exists q_{j} ∈ S(𝔥)^{W.} such that q_{j}/q = p_{j}/p_{0}. By 3.3.2, t^{2}S(𝔥)^{W.} ⊂ P(Z(𝔤)) and thus t^{2}q_{j} ∈ P(Z(𝔤)) for all j = 0, …, s. Since q_{j}/q = p_{j}/p_{0} one has

(t^{2}q)θ_{0} + ∑_{j=1}^{s}(t^{2}q_{j})θ_{j} = 0.

This completes the proof.

4.4.3. Proposition. Let θ_{1}, …, θ_{s} ∈ I(Ṽ) be a S(𝔥)-basic system of I(Ṽ)S(𝔥), p be a minor of this system and z ∈ Z(𝔤) be such that P(z) = t^{2}q where q is a maximal W.- invariant divisor of p^{|W |}. Then the localized module Hom_{𝔤}(Ṽ , Ũ[z^{−1}]) is freely generated over Z(𝔤)[z^{−1}] by {Ψ^{−1}θ_{1}, …, Ψ^{−1}θ_{s}}.

Proof. Recall that Ψ provides a Z(𝔤)-isomorphism from Hom_{𝔤}(Ṽ , Ũ) onto I(Ṽ). Therefore the required assertion is equivalent to the following statement: the localized module I(Ṽ) ⊗_{Z(𝔤)} Z(𝔤)[z^{−1}] is freely generated over Z(𝔤)[z^{−1}] by {θ_{1}, …, θ_{s}}. In other words, one has to show that for any θ_{0} ∈ I(Ṽ) there exists a unique collection z_{1}, …, z_{s} such that zθ_{0} + ∑_{1}^{s} z_{j}θ_{j} = 0 or, equivalently, that there exists a unique collection a_{1}, …, a_{s} ∈ P(Z(𝔤)) ⊂ S(𝔥) such that

(t^{2}q)θ_{0} + ∑_{1}^{s} a_{j}θ_{j} = 0.

The uniqueness immediately follows from the fact that the collection θ_{1}, …, θ_{s} is a S(𝔥)-basic system of I(Ṽ)S(𝔥). To prove the existence, recall that I(Ṽ)S(𝔥) is a S(𝔥)- submodule of Hom(Ṽ |_{0}, S(𝔥)). By 4.4.1, for any θ_{0} ∈ I(Ṽ) there exists a unique collection p_{1}, …, p_{s} ∈ S(𝔥) such that pθ_{0} + ∑_{j=1}^{s} p_{j}θ_{j} = 0. Now the required existence follows from Lemma 4.4.2.

4.4.4. Corollary. The Z(𝔤)-rank of Hom_{𝔤}(Ṽ , Ũ) is not greater than dim Ṽ |_{0}.

4.4.5. Remark. Set r = dim Ṽ |_{0}. Assume that Hom_{𝔤}(Ṽ , Ũ) contains the collection θ_{1}, …, θ_{r} possessing the following property: for a certain λ ∈ 𝔥^{∗} the images of the modules θ_{1}(Ṽ), …, θ_{r}(Ṽ) under the map Ũ → Ũ/ Ann Ṽ (λ) form a direct sum. We claim that θ_{1}, …, θ_{r} is a Z(𝔤)-basic system of Hom_{𝔤}(Ṽ , Ũ).

In fact, choose a basis v_{1} … v_{r} of Ṽ |_{0}. By Corollary 4.1.2, the rank of the matrix (P(θ_{j}(v_{i}))(λ))_{i=1,r}^{j=1,r} is equal to r. Hence the rank of the matrix

(Ψ(θ_{j}(v_{i})))_{i=1,r}^{j=1,r} = (P(θ_{j}(v_{i})))_{i=1,r}^{j=1,r} ∈ Mat_{(r×r)}S(𝔥)

is also equal to r and so Ψθ_{1}, …, Ψθ_{r} are S(𝔥)-linearly independent in I(Ṽ)S(𝔥). Since I(Ṽ)S(𝔥) is a S(𝔥)-submodule of Hom(Ṽ |_{0}, Ũ), its rank is not greater than r. Hence Ψθ_{1}, …, Ψθ_{r} is a S(𝔥)-basic system of I(Ṽ)S(𝔥). Therefore θ_{1}, …, θ_{r} is a Z(𝔤)-basic system of Hom_{𝔤}(Ṽ , Ũ) by Proposition 4.4.3.

4.5. In this subsection we show that the Z(𝔤)-rank of Hom_{𝔤}(Ṽ , Ũ) is equal to dim Ṽ |_{0}.

4.5.1. Separation Theorem 2.8.1 claims the existence an ad 𝔤_{0}-submodule H of U such that the multiplication map provides an isomorphism H ⊗ Z(𝔤_{0}) ⟶^{∼} U. In both proofs ([Ko], [BL]) one constructs, actually, an ad 𝔤_{0}-submodule H′ of the symmetric algebra S(𝔤_{0}) such that the multiplication map provides an isomorphism H′ ⊗ S(𝔤_{0})^{𝔤_{0}} ⟶^{∼} S(𝔤_{0}).

Lemma. Let H be an ad 𝔤_{0}-submodule of U such that the multiplication map provides an isomorphism gr H ⊗ S(𝔤_{0})^{𝔤_{0}} → S(𝔤_{0}). Then the ad 𝔤-module L := (ad Ũ)(HT) is isomorphic to

⊕_{_{V}∈Irr} E(Ṽ)^{⊕ dim _{V}|_{0}}

and the multiplication induces the monomorphism L ⊗ Z(𝔤) → Ũ.

Proof. Fix H satisfying the above condition. According to 3.3, L := (ad Ũ)(HT) = (ad′ Ũ)(H)T and as ad 𝔤-module L is isomorphic to the ad′ 𝔤-module (ad′ Ũ)(H) ≅ Ind_{𝔤_{0}}^{𝔤} H. Now the required isomorphism follows from (3).

Since L = (ad′ Ũ)(H)T and T is a non-zero divisor, the injectivity of the map L ⊗ Z(𝔤) → Ũ is equivalent to the injectivity of the map (ad′ Ũ)(H) ⊗ Z(𝔤) → Ũ (both maps are restrictions of the multiplication map). To check the injectivity of the map (ad′ Ũ)(H) ⊗ Z(𝔤) → Ũ it is enough to check the injectivity of its “graded version”:

gr((ad′ Ũ)(H)) ⊗ gr Z(𝔤) → S(𝔤).

Recall that gr Z(𝔤) = S(𝔤)^{𝔤}. Let {x_{i}}_{i∈I} be an ordered basis of 𝔤_{1}. For any J ⊆ I set x_{J} := ∏_{j∈J} x_{j} where the product is taken with respect to the given order. Assume that

∑_{J⊆I;k=1,... ,s} z_{k,J} gr((ad′ x_{J})h_{k}) = 0

for some non-zero h_{1}, …, h_{s} ∈ H and some z_{k,J} ∈ S(𝔤)^{𝔤}. To check the injectivity one has to show that all elements z_{k,J} are equal to zero. Suppose not. For x ∈ 𝔤_{1}, u ∈ Ũ one has (ad′ x)u = 2xu − (ad x)u. This implies gr((ad′ x_{J})u) = 2^{|J|} gr x_{J} gr u for any J ⊆ I and u ∈ U. For each J ⊆ I denote by P_{J} the projection S(𝔤) → S(𝔤_{0}) gr x_{J} with the kernel N_{J} := ∑_{J′⊆I,J′≠J} S(𝔤_{0}) gr x_{J′}. By 2.3.2, the restriction of P_{∅} to S(𝔤)^{𝔤} provides a monomorphism S(𝔤)^{𝔤} → S(𝔤_{0})^{𝔤_{0}}. Choose J ⊆ I such that z_{k,J} ≠ 0 for some k and J is a minimal (under inclusion) subset of I possessing this property. Then

0 = P_{J}( ∑_{J′⊆I,k=1,... ,s} z_{k,J′} gr((ad′ x_{J′})h_{k}))

= P_{J}( ∑_{J′⊆I,k=1,... ,s} 2^{|J′|}z_{k,J′} gr x_{J′} gr h_{k}))

= 2^{|J|} gr x_{J} ∑_{k=1,... ,s} P_{∅}(z_{k,J}) gr h_{k}.

Since gr H ⊗ S(𝔤_{0})^{𝔤_{0}} ⟶^{∼} S(𝔤_{0}), all elements z_{k,J} are equal to zero. The lemma is proven.

4.5.2. Corollary. The Z(𝔤)-rank of Hom_{𝔤}(Ṽ , Ũ) is equal to dim Ṽ |_{0}.

Proof. Fix Ṽ ∈ Irr and set r := dim Ṽ |_{0}. Choose H satisfying the assumption of Lemma 4.5.1. Then the vector space Hom_{𝔤}(Ṽ , L) has dimension r; let θ_{1}, …, θ_{r} be a basis of Hom_{𝔤}(Ṽ , L). The injectivity of the map L⊗Z(𝔤) → Ũ, induced by the multiplication, implies the Z(𝔤)- linear independence of θ_{1}, …, θ_{r} in Hom_{𝔤}(Ṽ , Ũ). Thus the Z(𝔤)-rank of Hom_{𝔤}(Ṽ , Ũ) is greater than or equal to r. Comparing with Corollary 4.4.4, one concludes the required assertion.

4.6. PRV determinants.

4.6.1. Definitions. Fix Ṽ ∈ Irr and set r := dim Ṽ |_{0}. Lemma 4.4.2 (i) implies that θ_{1}, …, θ_{r} is a Z(𝔤)-basic system of Hom_{𝔤}(Ṽ , Ũ) iff Ψθ_{1}, …, Ψθ_{r} is a S(𝔥)-basic system of I(Ṽ)S(𝔥). By 4.4.1, the last is equivalent to the condition that det C ≠ 0 where

C := (P(θ_{j}(v_{i}))) ∈ Mat_{(r×r)}(S(𝔥))

and {v_{1}, …, v_{r}} is a basis of Ṽ |_{0}.

Call such a matrix a PRV matrix (corresponding to the basic system θ_{1}, …, θ_{r}) and the determinant det C a PRV-determinant. Note that for different choices of a basis {v_{1}, …, v_{r}} of Ṽ |_{0} the PRV-matrix corresponding to θ_{1}, …, θ_{r} differ by the multiplication on an invertible scalar matrix.

Denote by J (Ṽ) the set of all PRV-matrices and by det J (Ṽ) the set of PRV determinants (for fixed Ṽ ∈ Irr). One has

det J (Ṽ) = {det(P(θ_{j}(v_{i})))| θ_{1}, …, θ_{r} ∈ Hom_{𝔤}(Ṽ , Ũ)} \ {0}.

For each ν ∈ 𝔥^{∗} such that dim Ṽ (ν) < ∞ set det J (ν) := det J (Ṽ (ν)). For λ ∈ 𝔥^{∗} we write det J (Ṽ)(λ) = 0 if (det C)(λ) = 0 for all C ∈ J (Ṽ).

If θ_{1}, …, θ_{r} is a Z(𝔤)-basic system of Hom_{𝔤}(Ṽ , Ũ), then for any non-zero z ∈ Z(𝔤) the collection zθ_{1}, θ_{2} …, θ_{r} is also a Z(𝔤)-basic system. Consequently, det J (Ṽ) is closed under the multiplication on the non-zero elements of P(Z(𝔤)).

4.6.2. Let N be an ad 𝔤-submodule of Ũ such that dim Hom_{𝔤}(Ṽ , N) = dim Ṽ |_{0}. By slightly abuse of notation, we shall denote by det J (Ṽ; N) the determinant of the matrix (P(θ_{j}(v_{i}))) where θ_{1}, …, θ_{r} (resp, v_{1}, …, v_{r}) is a basis of Hom_{𝔤}(Ṽ , N) (resp., Ṽ |_{0}). Note that det J (Ṽ; N) can be zero.

For different choices of a basis θ_{1}, …, θ_{r}, the values of det J (Ṽ; N) differ by a multiplicative constant. Hence det J (Ṽ; N) is a polynomial in S(𝔥) defined up to a non-zero scalar.

4.6.3. Remark. The original definition of PRV determinants given in [PRV] for the semisimple Lie algebras differ from our definition. Namely, the PRV determinant p(V) is equal to the polynomial det J (V; H) where H is a harmonic space (see 2.8.1). This definition works for completely reducible Lie superalgebras as well. In our notation det J (V) = S(𝔥)^{W.}p(V) \ {0}— this follows from Separation Theorem 2.8.1.

4.6.4. Take Ṽ ∈ Irr and p ∈ det J (Ṽ). Let θ_{1}, …, θ_{r} be a Z(𝔤)-basic system of Hom_{𝔤}(Ṽ , Ũ) such that the determinant of the corresponding PRV matrix is equal to p. Denote by q the maximal W.-invariant divisor of p. Combining Remark 4.4.5 and Proposition 4.4.3, one concludes that z ∈ Z(𝔤) satisfying P(z) = t^{2}q^{|W |} possesses the following property: for any θ ∈ Hom_{𝔤}(Ṽ , Ũ) one has zθ = ∑_{i=1}^{r} z_{i}θ_{i} for certain z_{i} ∈ Z(𝔤).

In particular, for a := P(z)^{r} = t^{2r}q^{r|W |} one has

a · det J (Ṽ) ⊂ P(Z(𝔤))p ⊆ S(𝔥)^{W.}p.

On the other hand, P(Z(𝔤))p ⊆ det J (Ṽ) by 4.6.1.

4.6.5. Corollary. Take Ṽ ∈ Irr. For any p ∈ det J (Ṽ) there exists a ∈ S(𝔥)^{W.} such that

P(Z(𝔤) \ {0})p ⊆ det J (Ṽ) ⊂ S(𝔥)^{W.}[a^{−1}]p.

4.6.6. Fix Ṽ ∈ Irr. Take a basic system θ_{1}, …, θ_{r} and denote by C the corresponding PRV matrix. Recall that det C ≠ 0. For each λ ∈ 𝔥^{∗} denote by C(λ) the complex matrix which is obtained from C by the evaluation of all entries at λ. Clearly det C(λ) = (det C)(λ). By Corollary 4.1.2,

corank C(λ) = dim Hom_{𝔤}(Ṽ , Ann Ṽ (λ) ∩ ∑_{j=1}^{r} θ_{j}(Ṽ)).

In particular, Ann Ṽ (λ) ∩ ∑_{j=1}^{r} θ_{j}(Ṽ) = 0 iff det C(λ) ≠ 0. Then, for f_{λ} being the natural map Ũ → End(Ṽ (λ)), one has

dim Hom_{𝔤}(Ṽ , f_{λ}(Soc Ũ)) = max{rank C(λ)|C ∈ J (Ṽ)}.  (4)

Combining 11.1.5 and (4) one obtains

4.6.7. Corollary. Assume that M̃(λ) is simple. Then the image of the socle of Ũ under the natural map Ũ → F (M̃(λ), M̃(λ)) coincides with the socle of F (M̃(λ), M̃(λ)) iff det J (Ṽ)(λ) ≠ 0 for any Ṽ ∈ Irr.

4.6.8. Change of Borel. In the definition of PRV matrices we use the Harish-Chandra projection. Therefore this definition depends on the choice of triangular decomposition. In the sequel we will add a lower index to designate the corresponding Borel subsuperalgebra in the cases when the choice of triangular decomposition is not clear from the context. For instance, J_{𝔟}(Ṽ) is the set of PRV matrices of the form (P_{𝔟}(θ_{j}(v_{i}))) where θ_{1}, …, θ_{r} is a Z(𝔤)-basic system of Hom_{𝔤}(Ṽ , Ũ) and v_{1}, …, v_{r} is a basis of Ṽ |_{0}.

Let 𝔟 and 𝔟′ be connected by the odd reflection along an odd isotropic root β. By 2.7, Ṽ_{𝔟}(λ) = Ṽ_{𝔟′}(λ′) where λ = λ′ if λ is such that (λ, β) = 0 and λ = λ′ + β otherwise. Then for any Ṽ ∈ Irr the equality (4) implies

max{rank C(λ)| C ∈ J_{𝔟_{1}}(Ṽ)} = max{rank C(λ′)| C ∈ J_{𝔟_{2}}(Ṽ)},
det J_{𝔟_{1}}(Ṽ)(λ) = 0 ⟺ det J_{𝔟_{2}}(Ṽ)(λ′) = 0.

4.6.9. Remark. The equality 4) has the following consequence: for any PRV determinant p ∈ det J (Ṽ) and any λ ∈ 𝔥^{∗}, the order of zero of p at the point λ is greater than or equal to the value (dim Ṽ |_{0} − dim Hom_{𝔤}(Ṽ , End(Ṽ (λ))).

In the case when 𝔤 is completely reducible, this property is used for the calculation of PRV determinants—see [JL],[J3],[GL1]. Recall that, in this case, the PRV determinant is a polynomial defined up to a non-zero scalar (see Remark 4.6.3).

4.7. Separation type theorem.

4.7.1. Definition. Call an ad 𝔤-submodule L of Ũ a generic harmonic space if the multiplication map provides an isomorphism L ⊗ Z(𝔤)[S^{−1}] ⟶^{∼} Ũ[S^{−1}] for certain S ⊂ (Z(𝔤) \ {0}).

In Theorem 4.7.4 we describe all generic harmonic spaces in Ũ. Of course, if L is a generic harmonic space then L ⊗ Z(𝔤)[S^{−1}] ⟶^{∼} Ũ[S^{−1}] for S := Z(𝔤) \ {0}. However it is always possible to find a smaller S; in Corollary 4.7.7 we describe such S in terms of PRV determinants. As it is shown in Section 7, for 𝔤 = 𝔰𝔩(m, n) or 𝔬𝔰𝔭(2, 2n) there is a generic harmonic space in Ũ such that one can take S = {T^{2}}.

4.7.2. Lemma. If an ad 𝔤-submodule L of Ũ is such that

(a) L ≅ ⊕_{_{V}∈Irr} E(Ṽ)^{⊕ dim _{V}|_{0}},
(b) ∀Ṽ ∈ Irr det J (Ṽ; L) ≠ 0

then it is a generic harmonic space.

Proof. Set 𝔥_{f}^{∗} := {ν| dim Ṽ (ν) < ∞}. For each ν ∈ 𝔥_{f}^{∗} choose a basis θ_{1}^{ν}, …, θ_{r(ν)}^{ν} of Hom_{𝔤}(Ṽ (ν), L). Recall that r(ν) = dim Ṽ (ν)|_{0}. By 4.6.1, the collection θ_{1}^{ν}, …, θ_{r(ν)}^{ν} forms a Z(𝔤)-basic system of Hom_{𝔤}(Ṽ (ν), Ũ). By 4.6.4, for a suitable z(ν) ∈ Z(𝔤) the collection θ_{1}^{ν}, …, θ_{r(ν)}^{ν} forms a free Z(𝔤)[z(ν)^{−1}]-basis of the localized module Hom_{𝔤}(Ṽ (ν), Ũ[z(ν)^{−1}]).

Set S := {z(ν), ν ∈ 𝔥_{f}^{∗}}. Denote by ψ the map L ⊗ Z(𝔤)[S^{−1}] → Ũ[S^{−1}] induced by the multiplication. For any ν ∈ 𝔥_{f}^{∗} the collection θ_{1}^{ν}, …, θ_{r(ν)}^{ν} forms a free Z(𝔤)[S^{−1}] basis of the localized module Hom_{𝔤}(Ṽ (ν), Ũ[S^{−1}]). This means that the restriction of ϕ on the space ∑_{i=1}^{s(ν)} θ_{i}^{ν}(Ṽ) ⊗ Z(𝔤)[S^{−1}] is a monomorphism and its image coincides with the isotypical component of Ṽ (ν) in the socle of Ũ[S^{−1}]. Then the restriction of ϕ on Soc L ⊗ Z(𝔤)[S^{−1}] is a monomorphism and Soc Ũ[S^{−1}] = ϕ(Soc L ⊗ Z(𝔤)[S^{−1}]). Using the equality Soc L ⊗ Z(𝔤)[S^{−1}] = Soc(L ⊗ Z(𝔤)[S^{−1}]) and 3.1.2, one concludes that ϕ is a monomorphism. From 3.2.3, it follows that L ⊗ Z(𝔤)[S^{−1}] is an injective module in Fin. Since the image of ψ contains Soc Ũ[S^{−1}], ψ is a bijection. The lemma is proven.

4.7.3. Corollary. If H satisfies the assumption of Lemma 4.5.1 then (ad Ũ)(HT) is a generic harmonic space.

Proof. By Lemma 4.5.1, L := (ad Ũ)(HT) fulfills the condition (a) of Lemma 4.7.2.

Fix Ṽ ∈ Irr and choose a basis θ_{1}^{,} …, θ_{r} of Hom_{𝔤}(Ṽ , L). Observe that r = dim Ṽ |_{0}. The injectivity of the map L ⊗ Z(𝔤) → Ũ, induced by the multiplication, implies the Z(𝔤)-linear independence of θ_{1}, …, θ_{r} in Hom_{𝔤}(Ṽ , Ũ). Since the Z(𝔤)-rank of the latter is equal to r, the collection θ_{1}, …, θ_{r} forms a Z(𝔤)-basic system of Hom_{𝔤}(Ṽ , Ũ). Then, by 4.6.1, det J (Ṽ; L) ≠ 0. By Lemma 4.7.2, (ad Ũ)(HT) is a generic harmonic space.

4.7.4. Theorem. An ad 𝔤-submodule L of Ũ is a generic harmonic space iff

(a) L ≅ ⊕_{_{V}∈Irr} E(Ṽ)^{⊕ dim _{V}|_{0}}
and one of the following conditions holds
(b) ∀Ṽ ∈ Irr det J (Ṽ; L) ≠ 0
(c) the multiplication map provides an embedding L ⊗ Z(𝔤) → Ũ.

Proof. By Lemma 4.7.2, L satisfying (a) and (b) is a generic harmonic space. Arguing as in Corollary 4.7.3 one concludes that L satisfying (a) and (b) is also a generic harmonic space.

It remains to show that any generic harmonic space L fulfills the conditions (a)-(c). The condition (c) obviously holds. Moreover, for any Ṽ ∈ Irr a basis θ_{1}, …, θ_{r} of Hom_{𝔤}(Ṽ , L) is a Z(𝔤)-basic system of Hom_{𝔤}(Ṽ , Ũ). Therefore det J (Ṽ; L) ≠ 0 and so (b) holds as well.

To verify the condition (a), let us show that all generic harmonic spaces are pairwise isomorphic as ad 𝔤-modules. Indeed, let L and L′ be generic harmonic spaces. Since Ũ is countably dimensional, one can choose the corresponding sets S, S′ ⊂ Z(𝔤) \ {0} having countable number of elements. Take a maximal ideal m of Z(𝔤) such that m∩(S∪S′) = ∅. Then as ad 𝔤-modules L ≅ Ũ/(mŨ) ≅ L′. In Corollary 4.7.3 we construct a generic harmonic space satisfying (a). Hence all generic harmonic spaces satisfy the condition (a). The theorem is proven.

4.7.5. Now we can formulate the following “receipt”. For each Ṽ ∈ Irr fix a Z(𝔤)-basic system θ_{V} ; 1, …, θ_{V} ; r of Hom_{𝔤}(Ṽ , Ũ). The module ∑_{_{V}∈Irr;j} T^{2}θ_{V} ; j(Ṽ) has an injective envelope inside an injective ad 𝔤-module ŨT. This injective envelope is a generic harmonic space by Theorem 4.7.4.

4.7.6. Combining 4.6.4 and the proof of Lemma 4.7.2, one obtains the 4.7.7. Corollary. Let L be a generic harmonic space and S ⊂ Z(𝔤) \ {0} satisfies the following property: for any Ṽ ∈ Irr there exists s ∈ S such that P(s) = t^{2}q where q is a

maximal W.-invariant divisor of (det J (Ṽ; L))^{|W |}. Then the multiplication map provides an isomorphism L ⊗ Z(𝔤)[S^{−1}] ⟶^{∼} Ũ[S^{−1}].

5. Application to the description of Verma module annihilators

In this section we prove Theorem 5.2 which provides a connection between PRV determinants and the annihilators of simple modules.

5.1. Proposition. For a simple strongly typical Verma moduleM̃(λ) the following conditions are equivalent:

(i) For any Ṽ ∈ Irr one has det J (Ṽ)(λ) ≠ 0.

(ii)The natural map Ũ → F (M̃(λ), M̃(λ)) is surjective and

F (M̃(λ), M̃(λ)) ≅ ⊕_{_{V}∈Irr} E(Ṽ)^{⊕ dim _{V}|_{0}}.

Proof. Fix a strongly typical λ such that M̃(λ) is simple. Denote by f the natural map Ũ → F (M̃(λ), M̃(λ)).

Let us show that (ii) implies (i). Indeed, if both conditions of (ii) hold then F (M̃(λ), M̃(λ)) is projective in Fin and so f has a left inverse f^{−1}. Denote by N the image of f^{−1}. Take Ṽ ∈ Irr and choose a basis θ_{1}, …, θ_{r} of Hom_{𝔤}(Ṽ , N). By Remark 4.4.5, θ_{1}, …, θ_{r} is a Z(𝔤)-basic system of Hom_{𝔤}(Ṽ , Ũ). Denote by C the corresponding PRV matrix. Since the restriction of f to N is a monomorphism corank C(λ) = 0, by 4.6.6. Hence det C(λ) ≠ 0 and so det J (Ṽ)(λ) ≠ 0 as required.

Let us show that (i) implies (ii). Suppose that (i) holds. By Corollary 4.6.7, f(Soc Ũ) coincides with the socle of F (M̃(λ), M̃(λ)). The socle of Ũ is a completely reducible 𝔤-module and so it contains a submodule L such that the restriction of f gives an isomorphism L ⟶^{∼} Soc F (M̃(λ), M̃(λ)). The central element T^{2} acts on M̃(λ) by a non-zero scalar t(λ)^{2} (because λ is strongly typical) and thus one can choose L lying in T^{2}Ũ. By 3.3.2, T Ũ is an injective ad 𝔤-submodule of Ũ and so it contains an injective envelope E(L) of L. The restriction of f to E(L) is a monomorphism because the restriction of f to L is a monomorphism. Therefore f(E(L)) is an injective module containing the socle of F (M̃(λ), M̃(λ)). Using 3.1.1, one concludes that the restriction of f provides an isomorphism E(L) ⟶^{∼} F (M̃(λ), M̃(λ)). In particular, f is surjective and the ad 𝔤-module F (M̃(λ), M̃(λ)) is injective in Fin. Combining these facts and 11.1.5, one obtains (ii).

5.2. Theorem. Suppose that λ ∈ 𝔥^{∗} is strongly typical and det J (Ṽ)(λ) ≠ 0 for all Ṽ ∈ Irr. Then Ann Ṽ (λ) = Ũ Ann_{Z(𝔤)} Ṽ (λ).

Proof. For each Ṽ (ν) ∈ Irr choose p_{ν} ∈ det J (ν) such that p_{ν}(λ) ≠ 0. Let θ_{1}^{ν}, …, θ_{r(ν)}^{ν} ∈ Hom_{𝔤}(Ṽ (ν), Ũ) be a Z(𝔤)-basic system such that the determinant of the PRV matrix corresponding to this system is equal to p_{ν}. Let q_{ν} be the maximal W.-invariant divisor of p_{ν}^{|W |} and z_{ν} ∈ Z(𝔤) be such that P(z_{ν}) = t^{2}q_{ν}. Note that z_{ν} ∉ Ann_{Z(𝔤)} Ṽ (λ) since (t^{2}p_{ν})(λ) ≠ 0.

Let S be the multiplicative closure of the set {T^{2}; z_{ν}| Ṽ (ν) ∈ Irr} . Denote by A the localization of Z(𝔤) by S and by Ũ_{A} the localization of Ũ by S. The action of Ũ on Ṽ (λ) can be canonically extended to the action of the localized algebra Ũ_{A} since S ∩ Ann_{Z(𝔤)} Ṽ (λ) = ∅ . The action of Ũ on Ṽ (λ) can be canonically extended to the action of the localized algebra Ũ_{A} acts on Ṽ (λ). Clearly, the ideal Ann_{A} Ṽ (λ) is maximal in A.

Let us show that Ann_{U_{A}} Ṽ (λ) = Ũ_{A} Ann_{A} Ṽ (λ). Combining 4.7.5 and Corollary 4.7.7, one concludes the existence of ad 𝔤-submodule H of Ũ such that

Soc H = ∑_{ν:_{V}(ν)∈Irr} ∑_{i=1}^{r(ν)} T^{2} Im θ_{ν}^{i}  (5)

and that the multiplication map induces an isomorphism H ⊗ A ⟶^{∼} Ũ_{A}. To verify Ann_{U_{A}} Ṽ (λ) = Ũ_{A} Ann_{A} Ṽ (λ) it is enough to check that H ∩ Ann Ṽ (λ) = 0. Observe that det J (Ṽ (ν); H) = p_{ν}t^{2r(ν)} and so det J (Ṽ (ν), H)(λ) ≠ 0 for all Ṽ (ν) ∈ Irr. Then, by 4.6.6,

(∑_{i=1}^{r(ν)} Im θ_{ν}^{i}) ∩ Ann Ṽ (λ) = 0.

that is Soc H ∩ Ann Ṽ (λ) = 0. Therefore H ∩ Ann Ṽ (λ) = 0 as required. Hence

Ann_{U_{A}} M̃(λ) = Ũ_{A} Ann_{A} M̃(λ).

Take u ∈ Ann_{U} Ṽ (λ). Write u = ∑_{i=1}^{m} u_{i}z_{i} where u_{i} ∈ H, z_{i} ∈ Ann_{A} Ṽ (λ). Choose z ∈ S such that zu_{i} ∈ Ũ and zz_{i} ∈ Z(𝔤) for all i = 1, …, m. Observe that zz_{i} ∈ Ann_{Z(𝔤)} Ṽ (λ) for all i = 1, …, m. Therefore

z^{2}u = ∑_{i=1}^{m}(zu_{i})(zz_{i}) ∈ Ũ Ann_{Z(𝔤)} Ṽ (λ).

Recall that S ∩ Ann_{Z(𝔤)} Ṽ (λ) = ∅ and so (z^{2} − c) ∈ Ann_{Z(𝔤)} Ṽ (λ) for a certain non-zero scalar c. Hence u ∈ Ũ Ann_{Z(𝔤)} Ṽ (λ). This completes the proof.

5.3. PRV determinants and Shapovalov form factorization. Assume that for some Ṽ ∈ Irr the set det J (Ṽ) contains a non-zero polynomial p whose all irreducible factors are either a divisor of a Shapovalov form (see 2.5.8) or a divisor of t. An irreducible factor of a Shapovalov form takes either a form (α^{∨} − (ρ, α) − n) (a first type factor) for some non-isotropic positive root α and some n ∈ ℕ^{+} or a form (α^{∨} − (ρ, α)) (a second type factor) for some isotropic positive root α. Note that the factors of the second type are divisors of t. Observe that for n ≠ 0 the element

s_{α}.(α^{∨} − (ρ, α) − n) = −α^{∨} + (ρ, α) − n

is neither a divisor of a Shapovalov form nor a divisor of t. This forces the maximal W.-invariant factor of p^{|W |} to be a divisor of t^{m} for some m ∈ ℕ.

Then 4.7.4– 4.7.5 imply

5.3.1. Theorem. Assume that for any Ṽ ∈ Irr there exists a non-zero element in det J (Ṽ) whose any irreducible factor is either a divisor of a Shapovalov form or a divisor of t. Then there exists an ad 𝔤-submodule H of Ũ such that

H ≅ ⊕_{_{V}∈Irr} E(Ṽ)^{⊕ dim _{V}|_{0}}

and the multiplication map induces an isomorphism H ⊗ Z(𝔤)[T^{−2}] ⟶^{∼} Ũ[T^{−2}].

Recall that M̃(λ) is simple iff all Shapovalov forms are non-zero at the point λ. Theorem 5.2 yields

5.3.2. Theorem. Assume that for any Ṽ ∈ Irr there exists a non-zero element in det J (Ṽ) whose any irreducible factor is either a divisor of a Shapovalov form or a divisor of t. Then the annihilator of M̃(λ) is centrally generated provided M̃(λ) being simple strongly typical.

As it is shown in [GL1] the condition of the last two theorems holds for 𝔤 = 𝔬𝔰𝔭(1, 2n). We will prove that this condition also holds for the basic classical Lie superalgebras of type I. Contrary to the case 𝔬𝔰𝔭(1, 2n), the proof is not based on the calculations of PRV determinants.

6. The basic classical Lie superalgebras of type I.

Throughout this section 𝔤 is a basic classical Lie superalgebras of type I that is 𝔤 = 𝔤𝔩(m, n), 𝔰𝔩(m, n), 𝔭𝔰𝔩(n, n) or 𝔤 = 𝔬𝔰𝔭(2, n). We describe some common properties of these superalgebras which are used in the next section.

6.1. Notation. The Lie superalgebras of type I are ℤ-graded. We denote by 𝔤_{r} (r ∈ ℤ) the corresponding homogeneous component of 𝔤 and by 𝔤\overline{0} (resp., 𝔤\overline{1}) the even (resp., the odd) part of 𝔤.

One has 𝔤_{0} = 𝔤\overline{0} and 𝔤\overline{1} = 𝔤_{1} + 𝔤_{−1}. In particular, both superalgebras 𝔤_{±1} are supercommutative and the exterior algebras Λ𝔤_{±1} are naturally embedded into Ũ. One has Ũ = Λ(𝔤_{−1})UΛ(𝔤_{1}). As ad 𝔤_{0}-modules 𝔤_{±1} are irreducible and dual one to another.

All odd roots are isotropic and so the notion of “typical” coincide with the notion of “strongly typical”.

Extend the above ℤ-grading on U(𝔤); evidently U(𝔤)_{r} = 0 if |r| > dim 𝔤_{1}. Moreover for r = dim 𝔤_{1} one has U(𝔤)_{±r} = UΛ^{r}(𝔤_{±1}). By default, “the degree” of an element of Ũ means its degree with respect to this ℤ-grading.

6.1.1. Distinguished triangular decomposition. Set 𝔫_{d}^{±} := 𝔫_{0}^{±} + 𝔤_{±1}. The decomposition 𝔤 = 𝔫_{d}^{−} ⊕ 𝔥 ⊕ 𝔫_{d}^{+} is called a distinguished triangular decomposition. By default, all highest weight modules, positive/negative roots, Harish-Chandra projection P and PRV matrices/determinants are constructed with respect to the distinguished triangular decomposition.

Note that (ρ, α) = (ρ_{0}, α) for any α ∈ Δ_{0} since Δ_{1}^{+} = Ω(𝔤_{1}) is W -invariant. In particular, w.μ = w(μ + ρ_{0}) − ρ_{0} for any w ∈ W and μ ∈ 𝔥^{∗}.

The Verma modules constructed using a distinguished triangular decomposition have the following nice structure: M̃(μ) = Ind_{𝔤_{0}⊕𝔫_{1}^{+}}^{𝔤} M(μ) where the action of 𝔫_{1}^{+} on M(μ) is assumed to be trivial.

6.1.2. For α ∈ Δ_{0}^{+} denote by e_{α} (resp., f_{α}) an element of the weight α (resp., −α) of 𝔤. For β ∈ Δ_{1}^{+} (resp., β ∈ Δ_{1}^{−}) denote by x_{β} (resp., y_{β}) an element of the weight β of 𝔤.

Denote by I the set of the positive odd roots with a fixed total order. Then {x_{i}}_{i∈I} (resp., {y_{i}}_{i∈I}) is a basis of 𝔤_{1} (resp., 𝔤_{−1}). For any J ⊆ I set x_{J} := ∏_{i∈J} x_{i}, y_{J} := ∏_{i∈J} y_{i} where the products are taken with respect to the total order on I. If we change the order of factors in the product y_{J} the result is equal either to y_{J} or to (−y_{J}), since 𝔤_{±1} are supercommutative. Note that yy_{I} = xx_{I} = 0 for any y ∈ 𝔤_{−1}, x ∈ 𝔤_{1}.

Evidently x_{I} ∈ Λ^{top}𝔤_{1}, y_{I} ∈ Λ^{top}𝔤_{−1} and so x_{I}, y_{I} are invariant with respect to the adjoint action of [𝔤_{0}, 𝔤_{0}]. Moreover x_{I}y_{I} is ad 𝔤_{0}-invariant due to the duality 𝔤_{1}^{∗} ≅ 𝔤_{−1}.

6.1.3. Sometimes we will deal with a non-distinguished triangular decomposition 𝔤 = 𝔫^{−} ⊕ 𝔥 ⊕ 𝔫^{+}. We shall use the following notation: 𝔟 := 𝔥 + 𝔫^{+}; P_{𝔟} will be the Harish-Chandra projection with respect to this triangular decomposition, Δ(𝔟) will be the set of non-zero roots of 𝔟 and also

\begin{aligned}I_{+} : & = I ∩ Δ(𝔟), \\ I_{−} : & = I \ I_{+}, \\ t_{𝔟} : & = P_{𝔟}(T) \end{aligned}

Note that

t_{𝔟} : = ∏_{β∈Δ_{1}∩Δ_{𝔟}}(β^{∨} + (β, ρ)).

As before, the triangular decomposition of 𝔤_{0} is assumed to be fixed and all triangular decompositions of 𝔤 which we consider are compatible with this triangular decomposition of 𝔤_{0}.

6.1.4. Case 𝔭𝔰𝔩(n, n). This case is rather special. A Cartan algebra of 𝔤 = 𝔭𝔰𝔩(n + nε) is “too small” and a distinguished triangular decomposition does not fit the definition given in 2.2. Moreover the restriction of the Harish-Chandra projection determined by the distinguished triangular decomposition to the zero weight space is not an algebra homomorphism: for instance, both y_{I}, x_{I} have zero weight, P(y_{I}) = P(x_{I}) = 0 however P(x_{I}y_{I}) = t—see Corollary 6.3.3.

A possible treatment to this problem is the following “enlargement of a Cartan subsuperalgebra”. The Lie superalgebra 𝔤 is an ideal in the Lie superalgebra 𝔤̂ := 𝔤𝔩(n, n)/(ℂX) where X stands for the identity matrix. One has 𝔤\overline{1} = 𝔤̂_{1} and 𝔤̂_{0} = 𝔤_{0} × ℂz where z is a central element of the Lie algebra 𝔤̂_{0}.

Let 𝔥̂ be a Cartan subsuperalgebra of 𝔤̂_{0} spanned by 𝔥 and z. One can easily sees that 𝔥̂ acts semisimply on 𝔤 and a distinguished triangular decomposition of 𝔤 is determined, in a sense of 2.2, by a certain regular element h ∈ 𝔥̂. For μ ∈ 𝔥̂^{∗} set Ũ|_{μ} := {u ∈ Ũ| [h, u] = μ(h)u, ∀h ∈ 𝔥̂}. Then the restriction of the Harish-Chandra projection determined by the distinguished triangular decomposition on Ũ|_{0} is an algebra homomorphism because Ũ|_{0} ∩ U(𝔫^{−} + 𝔥)𝔫^{−} = 0.

It is easy to see that for any weight 𝔤-module M one can extend (not uniquely) its 𝔤module structure to a 𝔤̂-module structure. This implies that Lemma 4.1.1 remains true for the distinguished triangular decomposition if we define Ũ|_{0} as above. For ad 𝔤-submodules of Ũ consider the natural 𝔤̂-module structure coming from the embedding Ũ into U(𝔤̂). In the sequel, we substitute the categories Irr and Õ for 𝔤 by the same categories for 𝔤̂ (these 𝔤̂-categories have the same sets of objects as their 𝔤-analogues but less morphisms). Under these conventions all propositions of Sections 4, 5 remain true for 𝔤 = 𝔭𝔰𝔩(n, n).

6.2. Useful assertions. The following lemma is an immediate consequence of the supercommutativity of 𝔤_{±1}.

6.2.1. Lemma. Let Ñ be a 𝔤-module and N be its 𝔤_{0}-submodule such that Ñ = ŨN. Then the canonical map Ind_{𝔤_{0}}^{𝔤} N → Ñ is an isomorphism iff for each collection {v_{J}}_{J⊆I} of elements of N one has

x_{I} ∑_{J⊆I} y_{J}v_{J} = 0 ⟹ v_{J} = 0, ∀J ⊆ I.

Recall that M̃(λ) is typical iff t(λ) ≠ 0 or, equivalently, T M̃(λ) ≠ 0. .

6.2.2. Lemma. A typical Verma module M̃(λ) contains a simple Verma submodule.

Proof. Recall that (ρ_{0}, α) = (ρ, α) for any α ∈ Δ_{0}. Comparing the factorizations of Shapovalov forms (see 2.5.8) for the Lie superalgebras 𝔤 and 𝔤_{0}, one concludes that a typical 𝔤-module M̃(λ) is simple iff the 𝔤_{0}-module M(λ) is simple. Let M̃(λ) be a typical Verma module and v be its highest weight vector. Then U(𝔫_{0}^{−})v ≅ M(λ) contains a simple submodule M(λ′). Therefore there exists u ∈ U(𝔫_{0}^{−}) such that 𝔫_{0}^{+}uv = 0. The ad 𝔤_{0}-invariance of 𝔤_{1} yields 𝔤_{1}(uv) = u𝔤_{1}v = 0. Thus the vector uv is primitive and so Ũ(uv) is a quotient of M̃(λ′). The non-zero elements of U(𝔤_{0}) are non-zero divisors in Ũ since the non-zero elements of S(𝔤_{0}) are non-zero divisors in S(𝔤) = S(𝔤_{0})Λ(𝔤\overline{1}). Thus Ũ(uv) ≅ M̃(λ′). Since M(λ′) is simple, M̃(λ′) is also simple. The lemma is proven.

6.3. Element T. For the classical Lie superalgebras of type I the element T takes a very simple form given by the following

6.3.1. Lemma. Up to a non-zero scalar T = (ad′ y_{I}x_{I})1.

Proof. Recall 3.3.2. It is enough to show that if v is a generator of a trivial 𝔤_{0}-module V (0) then y_{I}x_{I}v spans a trivial 𝔤-submodule of Ind_{𝔤_{0}}^{𝔤} V (0). In other words, one has to verify that 𝔤(y_{I}x_{I}) ⊂ Ũ𝔤_{0}. The ad 𝔤_{0}-invariance of y_{I}x_{I} forces 𝔤_{0}(y_{I}x_{I}) = (y_{I}x_{I})𝔤_{0}. Moreover 𝔤_{−1}(y_{I}x_{I}) = 0. Thus it remains to check that x_{β}y_{I}x_{I} ∈ Ũ𝔤_{0} for any β ∈ Δ_{1}^{+}. Setting J := I \ {β} one has

x_{β}y_{I}x_{I} = ±x_{β}y_{−β}y_{J}x_{I} = ±([x_{β}, y_{−β}]−y_{−β}x_{β})y_{J}x_{I} = ±([x_{β}, y_{−β}]y_{J}x_{I} −y_{−β}((ad x_{β}), y_{J})x_{I})

since x_{β}x_{I} = 0.

The term y_{I}x_{I} is of the zero weight and so the weight of y_{J}x_{I} is equal to β. The term [x_{β}, y_{−β}] lies in 𝔥; one has [[x_{β}, y_{−β}], x_{β}] = 0 since x_{β}^{2} = 0. Therefore β([x_{β}, y_{−β}]) = 0 and so [x_{β}, y_{−β}]y_{J}x_{I} = y_{J}x_{I}[x_{β}, y_{−β}] ∈ Ũ𝔤_{0}.

Let us show that y_{−β}((ad x_{β})y_{J})x_{I}) ∈ Ũ𝔤_{0}. For any β′ ∈ J one has (ad x_{β})y_{−β′} ∈ (𝔫_{0}^{−} + 𝔫_{0}^{+}). Since Λ^{r}(𝔤_{−1}) is ad 𝔤_{0}-invariant for all r, this implies that

(ad x_{β})y_{J} ∈ Λ^{l−2}(𝔤_{−1})(ℂ + 𝔫_{0}^{−} + 𝔫_{0}^{+})  (6)

where l := dim 𝔤_{1}. If y_{J′} ∈ Λ^{l−2}(𝔤_{−1}) has the same weight as the element (ad x_{β})y_{J}, then the set I \ J′ contains two elements whose sum is equal to 2β. Since the multiset I contains exactly one element equal to β, this implies β ∈ J′ that is y_{−β}y_{J′} = 0. Then (6) gives

y_{−β}((ad x_{β})y_{J}) ∈ y_{−β}Λ^{l−2}(𝔤_{−1})(𝔫_{0}^{−} + 𝔫_{0}^{+})

and so

y_{−β}((ad x_{β})y_{J})x_{I} ∈ Λ^{l−1}(𝔤_{−1})(𝔫_{0}^{−} + 𝔫_{0}^{+})x_{I} = Λ^{l−1}(𝔤_{−1})x_{I}(𝔫_{0}^{−} + 𝔫_{0}^{+}) ∈ Ũ𝔤_{0}

since x_{I} is ad([𝔤_{0}, 𝔤_{0}]-invariant. The assertion follows.

Take any triangular decomposition 𝔤 = 𝔫^{−} ⊕ 𝔫 ⊕ 𝔫^{+} and retain notation of 6.1.3.

6.3.2. Lemma. Up to a non-zero scalar one has

P_{𝔟}(x_{I_{+}}y_{I_{+}})P_{𝔟}(y_{I_{−}}x_{I_{−}}) = ∏_{β∈Δ_{1}(𝔟)}(β^{∨} + β^{∨}(ρ_{𝔟})).

Proof. The right-hand side is equal to t_{𝔟} = P_{𝔟}(T). By Lemma 6.3.1, up to a non-zero scalar T = (ad′ x_{I}y_{I})1. Thus one should verify that

P_{𝔟}((ad′ y_{I}x_{I})1) = P_{𝔟}(x_{I_{+}}y_{I_{+}})P_{𝔟}(y_{I_{−}}x_{I_{−}})

up to a non-zero scalar.

Since 𝔤_{1} is supercommutative, (ad′ x_{I})1 = 2^{dim 𝔤_{1}}x_{I} and

P_{𝔟}((ad′ y_{I}x_{I})1) = 2^{dim 𝔤_{1}}P_{𝔟}((ad′ y_{I})x_{I}) = ±2^{dim 𝔤_{1}}P_{𝔟}((ad′ y_{I_{+}} ad′ y_{I_{−}})x_{I}).

Recall that for α ∈ I_{+} one has y_{−α} ∈ 𝔫^{−} and so P((ad′y_{−α})u) = ±P(uy_{−α}) for all u ∈ Ũ. Therefore, up to a non-zero scalar,

P_{𝔟}((ad′ y_{I}x_{I})1) = P_{𝔟}(((ad′ y_{I_{−}})x_{I})y_{I_{+}}).

Similarly for any α ∈ I_{−} one has y_{−α} ∈ 𝔫^{+} and so P_{𝔟}(uy_{−α}y_{I_{+}}) = P_{𝔟}(±uy_{I_{+}}y_{−α}) = 0. Therefore

P_{𝔟}(((ad′ y_{I_{−}})x_{I})y_{I_{+}}) = P_{𝔟}(y_{I_{−}}x_{I}y_{I_{+}}) = ±P_{𝔟}(y_{I_{−}}x_{I_{−}}x_{I_{+}}y_{I_{+}}) = ±P_{𝔟}(y_{I_{−}}x_{I_{−}})P_{𝔟}(x_{I_{+}}y_{I_{+}})

where the last equality is a consequence of the fact that the restriction of the Harish-Chandra projection to the zero weight space is an algebra homomorphism. Hence

P_{𝔟}((ad′ y_{I}x_{I})1) = ±2^{dim 𝔤_{1}}P_{𝔟}(x_{I_{+}}y_{I_{+}})P_{𝔟}(y_{I_{−}}x_{I_{−}}).

The lemma is proven.

6.3.3. Corollary. Up to a non-zero scalar one has P(x_{I}y_{I}) = t = ∏_{β∈Δ_{1}^{+}}(β^{∨}+β^{∨}(ρ)).

6.3.4. Lemma. If λ ∈ 𝔥^{∗} is such that P_{𝔟}(x_{I_{+}}y_{I_{+}})(λ) = 0 then x_{I} M̃_{𝔟}(λ) = 0. If P_{𝔟}(y_{I_{−}}x_{I_{−}})(λ) = 0 then y_{I} M̃_{𝔟}(λ) = 0.

Proof. Both assertions are similar. To verify the first one, fix λ such that P_{𝔟}(x_{I_{+}}y_{I_{+}})(λ) = 0 and set M̃ := M̃_{𝔟}(λ). Denote by v a highest weight vector of M̃. One has

x_{I} M̃ = x_{I}U(𝔫_{0}^{−}) ∑_{J_{1}⊆I_{+},J_{2}⊆I_{−}} x_{J_{2}}y_{J_{1}}v = U(𝔫_{0}^{−}) ∑_{J_{1}⊆I_{+}} x_{I}y_{J_{1}}v

since x_{I} is ad 𝔫_{0}^{−}-invariant and x_{I}x_{J} = 0 for J ≠ ∅. Note that Ũ_{k}v = 0 if k > #I_{−}. Therefore x_{I}y_{J_{1}}v = 0 if (#I − #J_{1}) > #I_{−} that is if #J_{1} < #I_{+}. Consequently

x_{I} M̃ = U(𝔫_{0}^{−})x_{I}y_{I_{+}}v = U(𝔫_{0}^{−})x_{I_{−}}x_{I_{+}}y_{I_{+}}v = U(𝔫_{0}^{−})x_{I_{−}}P_{𝔟}(x_{I_{+}}y_{I_{+}})(λ)v = 0

as required.

7. Separation and Annihilation theorems for type I case.

Throughout this section 𝔤 is a basic classical Lie superalgebras of type I that is 𝔤 = 𝔤𝔩(m, n), 𝔰𝔩(m, n)𝔭𝔰𝔩(n, n) or 𝔤 = 𝔬𝔰𝔭(2, n). In this section we establish separation and annihilation type theorems for these algebras.

Retain notation of Section 6. Until 7.3.2 we will deal only with a distinguished triangular decomposition.

7.1. Proposition. For a typical Verma module M̃ one has

(i) F (M̃, M̃) ≅ ⊕_{V ∈Irr_{0}}(Ind_{𝔤_{0}}^{𝔤} V)^{⊕ dim V |_{0}}.
(ii) For H being a harmonic space of U(𝔤_{0}) the restriction of the natural map
Ũ → F (M̃, M̃) to (ad Ũ)(Hx_{I}y_{I}) is a bijection.

Proof. Let H be a harmonic space of U(𝔤_{0}). Recall that dim_{𝔤_{0}}(V, H) = dim V |_{0} for any V ∈ Irr_{0}.

Define the map ψ : Ind_{𝔤_{0}}^{𝔤}(H) → (ad Ũ)(Hx_{I}y_{I}) by g ⊗ h ↦ (ad g)(hx_{I}y_{I}). Since x_{I}y_{I} is ad 𝔤_{0}-invariant, Hx_{I}y_{I} ≅ H as ad 𝔤_{0}-modules and so ψ is a 𝔤-epimorphism. Denote by f the natural map f : Ũ → F (M̃, M̃). To prove the theorem it is enough to show that the map f ∘ ψ : Ind_{𝔤_{0}}^{𝔤}(Hx_{I}y_{I}) → F (M̃, M̃) is a bijection.

Let μ be the highest weight of M̃ and v be a highest weight vector of M̃. Then M := U(𝔤_{0})v ≅ M(μ). By 2.8.2, the restriction of the natural map U(𝔤_{0}) → F (M, M) to H is bijective.

To prove that f ∘ ψ is a monomorphism recall Lemma 6.2.1. Let {a_{J}}_{J⊆I} ⊂ H be such that

bM̃ = 0 where b := (∑_{J⊆I}(ad x_{I}y_{J})(a_{J}x_{I}y_{I})).

We need to show that a_{J} = 0 for all J ⊆ I. One has (ad y_{J})(a_{J}x_{I}y_{I}) = y_{J}a_{J}x_{I}y_{I} since 𝔤_{−1}y_{I} = 0. Therefore

b = (ad x_{I})(∑_{J⊆I} y_{J}a_{J}x_{I}y_{I}).

Take u ∈ U(𝔫_{0}^{−}). The equality 𝔤_{−1}y_{I} = 0 implies

0 = b(y_{I}uv) = ((ad x_{I})(∑_{J⊆I} y_{J}a_{J}x_{I}y_{I}))y_{I}uv = ∑_{J⊆I} ±y_{J}a_{J}x_{I}y_{I}x_{I}y_{I}uv

since y_{I}x_{J′}y_{I} = 0 for J′ ≠ I (the degree of y_{I}x_{J′}y_{I} is equal to r := #J′ − #2I and Ũ_{r} = 0 for r < −#I). By Corollary 6.3.3, P(x_{I}y_{I}) = t and so x_{I}y_{I}uv = ux_{I}y_{I}v = t(μ)uv. Therefore b(y_{I}uv) = t(μ)^{2} ∑_{J⊆I} ±y_{J}a_{J}uv. Hence b(y_{I}M) = t(μ)^{2} ∑_{J⊆I} ±y_{J}a_{J}M. Since t(μ) ≠ 0 and a_{J}M ⊂ M, the equality b(y_{I}M) = 0 implies a_{J}M = 0 for all J ⊆ I. However the a_{J} belong to H and H ∩ Ann M = 0 by Theorem 2.8.2. Therefore all a_{J} are equal to zero. Hence f ∘ ψ is a monomorphism.

Let us show that f ∘ ψ is surjective. Recall that as 𝔤_{0}-module M̃ ≅ M ⊗ Λ𝔤_{−1}. Take a simple finite dimensional 𝔤_{0}-module V. Using Frobenius reciprocity and 2.8.2, one obtains

dim Hom_{𝔤_{0}}(V, F (M̃, M̃)) = dim Hom_{𝔤_{0}}(M̃ ⊗ V, M̃)
= dim Hom_{𝔤_{0}}(V ⊗ Λ𝔤_{−1} ⊗ M, Λ𝔤_{−1} ⊗ M)
= dim Hom_{𝔤_{0}}(V ⊗ Λ𝔤\overline{1}, F (M, M))
= dim Hom_{𝔤_{0}}(V ⊗ Λ𝔤\overline{1}, H) = dim Hom_{𝔤_{0}}(V, Λ𝔤\overline{1} ⊗ H)
= dim Hom_{𝔤_{0}}(V, Ind_{𝔤_{0}}^{𝔤} H) = [Im(f ∘ ψ) : V].

Since F (M̃, M̃) is a completely reducible ad 𝔤_{0}-module and the multiplicity [Im(f ∘ψ) : V] is finite, this gives F (M̃, M̃) = Im(f ∘ ψ) and completes the proof.

Retain notation of 4.6.2.

7.1.1. Corollary. Let H be a harmonic space of U(𝔤_{0}) and L := (ad Ũ)(Hx_{I}y_{I}). Then

L ≅ ⊕_{_{V}∈Irr} E(Ṽ)^{⊕ dim _{V}|_{0}}

and for any Ṽ ∈ Irr the determinant det J (Ṽ; L) admits a linear factorization. Moreover each linear factor is a factor of a Shapovalov form.

Proof. Combining Proposition 7.1 and (3) one obtains

L ≅ ⊕_{_{V}∈Irr_{f}^{0}} E(Ṽ)^{⊕ dim _{V}|_{0}}.

Fix Ṽ ∈ Irr and set p := P RV (Ṽ; L). Recall that M̃(μ) is simple iff μ is not a zero of a Shapovalov form. In particular, an atypical Verma module is not simple. Taking into account Corollary 4.1.2, one concludes from Proposition 7.1 that p(μ) ≠ 0 provided that M̃(μ) is simple. Thus any zero of p is a zero of a Shapovalov form. Since each Shapovalov form admits a factorization into linear factors, this implies that the set of zeroes of p lies in a union of countably many hyperplanes which correspond to the linear factors of the Shapovalov forms. Therefore p is a product of linear factors which are factors of Shapovalov forms.

7.2. Separation theorem. Combining Corollary 7.1.1, Theorem 4.7.4, Corollary 4.7.7 and 5.3 one obtains the following version of Separation Theorem

7.2.1. Theorem. Let H be a harmonic space of U(𝔤_{0}). Then the multiplication map provides an isomorphism ((ad Ũ)Hx_{I}y_{I}) ⊗ Z(𝔤)[T^{−2}] ⟶^{∼} Ũ[T^{−2}].

7.3. Annihilation theorem. In this subsection we prove that Ann M̃_{𝔟}(λ) is a centrally generated ideal iff M̃_{𝔟}(λ) is typical.

7.3.1. Combining Corollary 7.1.1 and Theorem 5.3.2, one concludes that Ann M̃(λ) is centrally generated if M̃(λ) is simple. By Lemma 6.2.2, any typical Verma module M̃(λ) contains a simple Verma submodule. This implies that Ann M̃(λ) is centrally generated if M̃(λ) is typical. Using 2.7, it is easy to generalize this statement to any Borel subsuperalgebra 𝔟. Indeed, a typical Verma module M̃_{𝔟}(λ) is isomorphic to a Verma module M̃(λ′). Recall that a Verma module M̃_{𝔟}(λ) is typical iff its annihilator does not contain T — see 3.3.2. Thus M̃(λ′) is typical and so Ann M̃_{𝔟}(λ) = Ann M̃(λ′) is a centrally generated ideal. Hence Ann M̃_{𝔟}(λ) is centrally generated provided that M̃_{𝔟}(λ) is typical.

In the rest of this section we will show that Ann M̃_{𝔟}(λ) ≠ Ũ Ann_{Z(𝔤)} M̃_{𝔟}(λ) if M̃_{𝔟}(λ) is atypical that is t_{𝔟}(λ) = 0.

7.3.2. The proof goes as follows. Take λ ∈ 𝔥^{∗} such that t_{𝔟}(λ) = 0. Set χ̃ := Ann_{Z(𝔤)} M̃_{𝔟}(λ). The idea is to find Ṽ ∈ Irr satisfying the following conditions

a) Hom_{𝔤}(Ṽ , Ũ) ≠ 0,
b) Ṽ is typical ,
c) ∀θ ∈ Hom_{𝔤}(Ṽ , Ũ) θ(Ṽ) ⊂ Ann M̃_{𝔟}(λ).

Assume that Ann M̃_{𝔟}(λ) is a centrally generated ideal. Then the above conditions imply the equality Hom_{𝔤}(Ṽ , Ũ) = χ̃ Hom_{𝔤}(Ṽ , Ũ) that yields

det J (𝔟, Ṽ) ⊆ P(χ̃)det J (𝔟, Ṽ).

Since det J (𝔟, Ṽ) is a non-zero subset of S(𝔥) and P(χ̃) is a subset of S(𝔥) which does not contain non-zero scalars, the last inclusion is impossible. Thus Ann M̃_{𝔟}(λ) is not a centrally generated ideal.

7.3.3. Retain notation of 6.1.2. There exists z ∈ 𝔥 such that ad z acts by zero on 𝔤_{0} and by identity (resp., minus identity) on 𝔤_{1} (resp., 𝔤_{−1}). Recall that ρ_{1} = \frac{1}{2} ∑_{β∈Δ_{1}^{+}} β and so 2ρ_{1}(z) = dim 𝔤_{1}.

7.3.4. Lemma. There exists ν ∈ 𝔥^{∗} such that

a) Hom_{𝔤}(Ṽ (ν), Ũ) ≠ 0,
b) Ṽ (ν) is typical,
c′) z(ν) = dim 𝔤_{1}.

Proof. Recall that the condition b) is equivalent to the inequality t(ν) ≠ 0 where t = ∏_{β∈Δ_{1}^{+}}(β^{∨} + (ρ, β)). Observe that z ≠ β^{∨} for any β ∈ Δ_{1}^{+}, since β^{∨}(β) = 0 ≠ z(β). Thus the restriction of the polynomial t on the hyperplane

S := {μ ∈ 𝔥^{∗}| z(μ) = dim 𝔤_{1}, (Z(𝔤) ∩ 𝔥)(μ) = 0}

is a non-zero polynomial (one has Z(𝔤) ∩ 𝔥 = 0 apart from the case 𝔤 = 𝔤𝔩(m, n)).

Consider the set

X := {μ ∈ 𝔥^{∗}| Hom_{𝔤_{0}}(V (μ), U(𝔤_{0})) ≠ 0}.

It is easy to see that X is a Zariski dense subset of the hyperplane

{μ ∈ 𝔥^{∗}| z(μ) = 0, (Z(𝔤) ∩ 𝔥)(μ) = 0} = S − 2ρ_{1}.

Thus t(ν′ + 2ρ_{1}) ≠ 0 for a certain ν′ ∈ X. Hence ν := ν′ + 2ρ_{1} fulfills the conditions b) and c’). Let show that ν satisfies a). Since ν′ ∈ X, there exists a copy V ≅ V (ν) inside U(𝔤_{0}). Let u be a highest weight vector of V. Then ux_{I} is a non-zero primitive (that is (𝔫_{0}^{+} + 𝔤_{1})-invariant) element of Ũ having weight ν. Therefore the ad 𝔤-submodule generated by ux_{I} is a finite dimensional quotient of M̃(ν). Since t(ν) ≠ 0, a unique finite dimensional quotient of M̃(ν) is isomorphic to Ṽ (ν). Hence Hom_{𝔤}(Ṽ (ν), Ũ) ≠ 0.

7.3.5. Remark. Similarly there exists a typical simple module Ṽ with the lowest weight ν such that Hom_{𝔤}(Ṽ , Ũ) ≠ 0 and z(ν) = − dim 𝔤_{1}.

7.3.6. Fix λ ∈ 𝔥^{∗} such that P_{𝔟}(x_{I_{+}}y_{I_{+}})(λ) = 0. Set χ̃ := Ann_{Z(𝔤)} M̃(λ) and assume that Ann M̃(λ) = Ũχ̃.

Fix ν satisfying the conditions a)—c’) of Lemma 7.3.4. We claim that ν fulfills the condition c) of 7.3.2.

Indeed, let v be a highest weight vector of Ṽ := Ṽ (ν). Take any θ ∈ Hom_{𝔤}(Ṽ , Ũ). By the assumption c’) zv = (dim 𝔤_{1})v that is [z, θ(v)] = (dim 𝔤_{1})θ(v). Therefore θ(v) has degree dim 𝔤_{1} that is θ(v) ∈ U(𝔤_{0})x_{I}. Lemma 6.3.4 asserts that x_{I} M̃_{𝔟}(λ) = 0 and so θ(v) ∈ M̃_{𝔟}(λ) = 0. Since Ann M̃_{𝔟}(λ) is ad 𝔤-stable, it contains θ(Ṽ).

By the assumption Ann M̃(λ) = Ũχ̃ and so the element θ(v) can be written in a form θ(v) = ∑ z_{i}u_{i} where all z_{i} belong to χ̃. Without loss of generality we can assume that each element u_{i} has the same weight and the same central character as θ(v) with respect to the adjoint action of Z(𝔤) ⊂ Ũ on Ũ. Since Ṽ is typical, Ṽ is a unique, up to isomorphism, cyclic module in Fin with this central character. Therefore ad 𝔤-span of each u_{i} is isomorphic to Ṽ that is u_{i} ∈ θ_{i}(Ṽ) for certain θ_{i} ∈ Hom_{𝔤}(Ṽ , Ũ). Since u_{i} has weight ν and v spans Ṽ |_{ν}, one can assume that u_{i} = θ_{i}(v). Then θ(v) = ∑ z_{i}θ_{i}(v) and so θ = ∑ z_{i}θ_{i}. Hence

Hom_{𝔤}(Ṽ , Ũ) = χ̃ Hom_{𝔤}(Ṽ , Ũ).

Using the fact that P_{𝔟}(zθ) = P_{𝔟}(z)P_{𝔟}(θ) for any z ∈ Z(𝔤), θ ∈ Hom_{𝔤}(Ṽ (ν), Ũ), one concludes det J_{𝔟}(Ṽ (ν)) ⊆ P(χ̃)det J_{𝔟}(Ṽ (ν)). Since det J_{𝔟}(Ṽ (ν)) is a non-zero subset of a polynomial algebra S(𝔥) and P(χ̃) does not contain non-zero scalars, the last inclusion is impossible. Hence Hence Ann M̃(λ) ≠ Ũχ̃.

In the case P_{𝔟}(y_{I_{−}}x_{I_{−}})(λ) = 0 the proof is similar: one may choose Ṽ as in Remark 7.3.5.

Hence we prove that for any triangular decomposition the following theorem holds.

7.3.7. Theorem. The annihilator of a Verma module is a centrally generated ideal iff this module is typical.

7.4. Denote by Max A the set of maximal ideals of an algebra A. In [PS1], the following theorem is proven.

7.4.1. Theorem. For any χ̃ ∈ Max Z(𝔤) not containing T^{2}, there exists χ ∈ Max Z(𝔤_{0}) such that the algebra Ũ/(Ũχ) is the matrix algebra over U(𝔤_{0})/(U(𝔤_{0})χ_{0}).

One can deduce from this theorem that the annihilator of a typical Verma module is centrally generated. On the other hand, one can deduce Theorem 7.4.1 from Proposition 7.1 and 7.3.1. In fact, take λ ∈ 𝔥^{∗} such that χ̃M̃(λ) = 0. Then t(λ) ≠ 0 and so

Ũ/(Ũχ̃) = F (M̃(λ), M̃(λ)) = F (M(λ) ⊗ Λ𝔤_{1}^{−}, M(λ) ⊗ Λ𝔤_{1}^{−}) =
F (M(λ), M(λ)) ⊗ End(Λ𝔤_{1}^{−}) = U(𝔤_{0})/(U(𝔤_{0})χ) ⊗ End(Λ𝔤_{1}^{−})

where χ := Ann_{Z}(𝔤_{0})M(λ). Hence Ũ/(Ũχ) is a matrix algebra over U(𝔤_{0})/(U(𝔤_{0})χ).

8. Perfect pairs

In this section we find for each maximal ideal of Z(𝔤) not containing T^{2} a perfect mate which is a maximal ideal in Z(𝔤_{0}) possessing certain properties.

8.1. Denote by Max A the set of maximal ideals of an algebra A and by A − Mod the full category of left A-modules. For an A-module N and χ ∈ Max A set

N_{χ} := {v ∈ N| χ^{r}v = 0, ∀r >> 0}.

Call a maximal ideal of Z(𝔤) a 𝔤-central character. For χ̃ ∈ Max Z(𝔤) (resp., χ ∈ Max Z(𝔤_{0})) set

Ũ_{χ} := Ũ/(Ũχ̃), U_{χ} := U/(Uχ).

We canonically identify the Ũ_{χ}-modules and the (non-graded) Ũ-modules annihilated by χ̃.

8.1.1. Definition. Call χ̃ ∈ Max Z(𝔤) strongly typical if T^{2} ∉ χ̃.

Throughout this section χ̃ denotes a strongly typical central character.

8.1.2. For a fixed triangular decomposition set

W (χ̃) := {λ ∈ 𝔥^{∗}| χ̃M̃(λ) = 0}.

Recall that P(Z(𝔤)) ⊃ P(T^{2})S(𝔥)^{W.} and so W (χ̃) forms a single W.-orbit.

Remark that χ̃ is strongly typical iff any λ ∈ W (χ̃) is strongly typical.

For any pair (χ̃, χ) ∈ Max Z(𝔤) × Max Z(𝔤_{0}) there is a functor Ψ_{_{χ},χ} : Ũ_{χ} − Mod → U_{χ} − Mod given by Ñ ↦ Ñ_{χ}. One of our goals is to prove that any strongly typical 𝔤-central character χ̃ has a “perfect mate” χ ∈ Max Z(𝔤_{0}) such that the above functor provides the equivalence of categories. For the basic classical Lie superalgebras of type I this is proven in cite [PS1]. In [G], we will prove it for the basic classical Lie superalgebras of type II.

For type I case, it turns out that for any λ ∈ W (χ̃) χ := Ann_{Z(𝔤_{0})} M(λ) is a perfect mate for χ̃ (provided that χ̃ is strongly typical)— see also 7.4. This does not hold for type II case.

It is easy to see that if the functor Ψ_{_{χ},χ} provides the equivalence of the categories, then χ has, at least, the following properties: for a Verma Ũ_{χ}-module M̃ the 𝔤_{0}-module M̃_{χ} is a Verma 𝔤_{0}-module and for any Ũ_{χ}-module Ñ the 𝔤_{0}-module Ñ_{χ} is non-zero. We call χ a mate of χ̃ if it possesses the first property and a perfect mate if it possesses both properties. As we shall show in [G], these two properties really ensure the equivalence of categories Ũ_{χ} − Mod and U_{χ} − Mod provided that χ̃ is strongly typical.

A pair (χ̃, χ) ∈ Max Z(𝔤) × Max Z(𝔤_{0}) is called a perfect pair if χ is a perfect mate for χ̃.

The goal of this section is to find a perfect mate for any strongly typical χ̃. This is done in the following way. Lemma 8.3.4 gives a combinatorial criterion on χ to be a perfect mate for χ̃. In 8.5 we fix, for each basic classical Lie superalgebra of type II a triangular decomposition, in terms of which we will describe a perfect mate χ for χ̃. In 8.6 we consider χ̃ satisfying a certain genericity condition. For such a “generic” χ̃ we show that for a suitable λ ∈ W (χ̃) the 𝔤_{0}-central character of M(λ) is a perfect mate for χ̃. The case when χ̃ is not generic is treated in 8.7. It turns out that for 𝔤 of the types B(m, n), G(3) any strongly typical χ̃ is “generic”. The remaining basic classical Lie superalgebras of type II are treated case by case.

8.2. Notation.

8.2.1. We say that a Ũ-module Ñ has a finite support supp_{Z(𝔤)} Ñ = {χ̃_{1}, …, χ̃_{k}} if for any v ∈ Ñ there exist r_{1}, …, r_{k} ∈ ℕ^{+} such that ∏_{i} χ̃_{i}^{r_{i}}v = 0. In this case,

Ñ = ⊕_{i}Ñ_{χ_{i}}

and each Ñ_{χ_{i}} is canonically isomorphic to the localization of the module Ñ by the maximal ideal χ̃_{i}. If Ñ has a finite support and 0 → Ñ′ → Ñ → Ñ′′ → 0 is an exact sequence then, for any χ̃′ ∈ Max Z(𝔤), the sequence 0 → Ñ′_{χ′} → Ñ_{χ′} → Ñ\begin{matrix}′ & ′ \\ & χ′ \end{matrix} → 0 is also exact.

For a 𝔤-module Ñ and a maximal ideal χ̃′ ∈ Max Z(𝔤) define χ̃′-depth of Ñ to be a minimal r ∈ ℕ ∪ ∞ such that (χ̃′)^{r}Ñ_{χ} = 0.

We adopt the similar notation for Z(𝔤_{0}) and U-modules.

8.2.2. Denote by Γ the set of subsets of Δ_{1}^{+}. For γ ⊆ Δ_{1}^{+} set

|γ| := ∑_{β∈γ} β.

Define the action of the Weyl group W on Γ by setting

w_{∗}γ := w(γ ∪ −(Δ_{1}^{+} \ γ)) ∩ Δ_{1}^{+}.

Then

|w_{∗}γ| = w(|γ| − ρ_{1}) + ρ_{1}.

8.2.3. As a 𝔤_{0}-module, a Verma module M̃(λ) has a filtration 0 = M_{0} ⊂ M_{1} ⊂ … ⊂ M_{r} = M̃(λ) such that the set of factors M_{i+1}/M_{i} coincides with the multiset {M(λ − γ) : γ ∈ Γ}— see [M1], 3.2.

It is easy to check that for any w ∈ W, γ ∈ Γ

w.λ − |w_{∗}γ| + ρ_{0} = w(λ − |γ| + ρ_{0}).  (7)

Therefore the 𝔤_{0}-central characters of M(w.λ − |w_{∗}γ|) and M(λ − |γ|) coincide. Thus for fixed χ̃ the multiset of 𝔤_{0}-central characters of {M(λ − γ) : γ ∈ Γ} does not depend on the choice of λ ∈ W (χ̃).

8.2.4. Definition. Call χ ∈ Max Z(𝔤_{0}) a mate for χ̃ ∈ Max Z(𝔤) if for some M̃(λ) satisfying χ̃M̃(λ) = 0, the 𝔤_{0}-module M̃(λ)_{χ} is isomorphic to a Verma module M(λ − |γ|) for some γ ∈ Γ.

One can easily deduce from 8.2.3 that if χ is mate for χ̃ then for any M̃(λ) satisfying χ̃M̃(λ) = 0, the 𝔤_{0}-module M̃(λ)_{χ} is isomorphic to a Verma module M(λ − |γ|) for some γ ∈ Γ. In particular, if χ is mate for χ̃ then for any M̃ satisfying χ̃M̃ = 0, the χ-depth of M̃ is equal to 1.

8.2.5. Definition. Let us call χ ∈ Max Z(𝔤_{0}) a perfect mate for χ̃ ∈ Max Z(𝔤) if it is a mate for χ̃ and for any non-zero Ũ_{χ}-module Ñ one has Ñ_{χ} ≠ 0.

Definition 8.2.4 is given in terms of the category Õ. However, 2.7 implies that if for some triangular decomposition χ is a mate for χ̃, then it is a mate also with respect to all other choices of triangular decomposition.

8.3. Throughout this subsection a strongly typical 𝔤-central character χ̃ is assumed to be fixed. Our first goal is to reformulate Definition 8.2.5 in terms of category Õ.

8.3.1. Lemma. There exist χ_{1}, …, χ_{k} ∈ Max Z(𝔤_{0}) and r_{1}, …, r_{k} ∈ ℕ^{+} such that for any Ũ_{χ}-module Ñ one has

Ñ = ⊕_{1}^{k}Ñ_{χ_{i}}

and χ^{r_{i}} Ñ_{χ_{i}} = 0.

Proof. Theorem 2.5 of [M1] implies that for any x ∈ Z(𝔤_{0}) there exist z_{0}, …, z_{l} ∈ Z(𝔤) (where l = #Γ) such that ∑_{0}^{l} z_{i}x^{i} = 0 and z_{l} = T^{2} (notice that 2.1 of [M1] contains a misprint; to correct it one has to substitute g by g^{2} in 2.1 and in Theorem 2.5). Therefore for any x ∈ Z(𝔤_{0}) there exists c_{0}, …, c_{l−1} ∈ ℂ such that x^{l} + ∑_{0}^{l−1} c_{i}x^{i} ∈ (Z(𝔤_{0}) ∩ Ũχ̃). Consequently, the ideal (Z(𝔤_{0}) ∩ Ũχ̃) has a finite codimension in Z(𝔤_{0}) and so there exist χ_{1}, …, χ_{k} ∈ Max Z(𝔤_{0}) and r_{1}, …, r_{k} ∈ ℕ^{+} such that

(Z(𝔤_{0}) ∩ Ũχ̃) ⊇ ∏_{i=1}^{k} χ_{i}^{r_{i}}.

Then, for any Ũ_{χ}-module Ñ

∏_{i=1}^{k} χ_{i}^{r_{i}} ⊆ Ann_{Z(𝔤_{0})} Ñ.

The assertion follows.

8.3.2. Corollary. If χ is a perfect mate for χ̃ then

Ñ = ŨÑ_{χ}

for any Ũ_{χ}-module Ñ.

Proof. Since supp_{Z(𝔤_{0})} Ñ is finite, one has (Ñ/(ŨÑ_{χ}))_{χ} = 0. Hence Ñ/(ŨÑ_{χ}) = 0 because χ is a perfect mate for χ̃.

8.3.3. Lemma. If χ ∈ Max Z(𝔤_{0}) is a mate for χ̃ ∈ Max Z(𝔤) and Ṽ (λ)_{χ} ≠ 0 for all λ ∈ W (χ̃), then χ is a perfect mate.

Proof. Suppose that Ṽ (λ)_{χ} ≠ 0 for all λ ∈ W (χ̃). One has to verify that Ñ_{χ} ≠ 0 for any non-zero Ũ_{χ}-module Ñ. Since Ũ is Noetherian, Ñ has a simple subquotient Ñ′. Lemma 8.3.1 implies that Ñ_{χ} ≠ 0 provided Ñ′_{χ} ≠ 0. Hence it is enough to check that Ñ_{χ} ≠ 0 for any simple Ũ_{χ}-module Ñ.

Take a simple Ũ_{χ}-module Ñ. The ideal Ann Ñ is primitive and so, according to [M2], Ann Ñ = Ann Ṽ where Ṽ is a simple highest weight module. Obviously, χ̃Ṽ = 0 and so, by our assumption, Ṽ_{χ} ≠ 0. As a 𝔤_{0}-module, Ṽ has a finite length. Therefore

Ann_{Z(𝔤_{0})} Ñ = Ann_{Z(𝔤_{0})} Ṽ = χ^{r_{1}}χ_{2}^{r_{2}} … χ_{k}^{r_{k}}

where {χ, χ_{2}, …, χ_{k}} = supp_{Z(𝔤_{0})} Ṽ , r_{1}, …, r_{k} ∈ ℕ^{+}. Then Ñ′ := χ_{2}^{r_{2}} … χ_{k}^{r_{k}} Ñ ≠ 0 and χ^{r_{1}} Ñ′ = 0. Hence Ñ_{χ} ≠ 0. The lemma is proven.

8.3.4. Lemma. Take λ ∈ W (χ̃), γ ∈ Γ and set χ := Ann_{Z(𝔤_{0})} M(λ − |γ|).

(i) ∀γ′ ∈ Γ \ {γ} (λ − |γ′| + ρ_{0}) ∉ W (λ − |γ| + ρ_{0}) ⟺ χ is a mate for χ̃
(ii) If χ is a mate for χ̃ and Stab_{W} (λ − |γ| + ρ_{0}) ⊆ Stab_{W} (λ + ρ) then χ is a perfect mate.

Proof. The equivalence (i) follows from 8.2.3.

For (ii) recall Lemma 8.3.3. Suppose that χ is a mate for χ̃ but it is not a perfect mate. Then M̃(λ)_{χ} = M(λ − |γ|) and Ṽ (w.λ)_{χ} = 0 for some w ∈ W. The equality (7), implies that for any y ∈ W

M̃(y.λ)_{χ} = M(y.λ − |y_{∗}γ|).

Therefore Ṽ (y.λ)_{χ} is a quotient of M(y.λ − |y_{∗}γ|).

The module Ṽ (w.λ) is a homomorphic image of M̃(w.λ). Denote the kernel of this homomorphism by Ñ. The module Ñ has a finite length and the factors of its Jordan-Gölder series have the form Ṽ (μ) for some μ ∈ W.λ satisfying μ < wλ. Since 0 = Ṽ (w.λ)_{χ} = (M̃/Ñ))_{χ} one concludes that the 𝔤_{0}-module Ñ_{χ} = M̃_{χ} = M(w.λ − |w_{∗}γ|) has a finite filtration such that each factor of this filtration is a quotient of M(y.λ − |y_{∗}γ|) for some y ∈ W satisfying y.λ < w.λ. Therefore

w.λ − |w_{∗}γ| = y.λ − |y_{∗}γ|

for some y ∈ W satisfying y.λ < w.λ. By (7), the above equality is equivalent to the condition y^{−1}w ∈ Stab_{W} (λ + ρ_{0} − |γ|). However, y.λ < w.λ implies y^{−1}w ∉ Stab_{W} (λ + ρ). Thus Stab_{W} (λ + ρ_{0} − |γ|) ⊈ Stab_{W} (λ + ρ) as required.

8.4. Throughout this subsection a strongly typical 𝔤-central character χ̃ is assumed to be fixed.

8.4.1. Right version. One may repeat the above reasonings for the right 𝔤-modules. Suppose now that (χ̃, χ) is a perfect pair in the “left sense”. We claim that it is a perfect pair in the “right sense”.

Indeed, the superantiautomorphism σ (see 2.3) provides the duality between the left and the right Ũ-modules given by N ↦ N^{σ}:

v^{σ}.a := (−1)^{d(a)d(v)}σ(a).v, a ∈ Ũ, v ∈ N.

By 2.3.4, σ stabilizes the elements of Z(𝔤); thus the restriction of the above duality gives the duality between the left and the right Ũ_{χ}-modules. Since σ also stabilizes the elements of Z(𝔤_{0}), in order to show that (χ̃, χ) is a perfect pair in the “right sense”, it is enough to check that χ is mate for χ̃ in the “right sense”.

Take a minimal element λ ∈ W (χ̃). Then M̃ := M̃(λ) is a simple Verma module and M̃^{σ} is a right simple module which is a Verma module with respect to a suitable triangular decomposition of 𝔤. Since χ is a mate for χ̃, M̃_{χ} is a Verma 𝔤_{0}-module, say M. Then

χM̃^{σ} := {v ∈ M̃^{σ}| v.χ^{r} = 0, r >> 0} = (M̃_{χ})^{σ} = M^{σ}

since σ stabilizes the elements of Z(𝔤_{0}). Therefore _{χ}M̃^{σ} is a (right) 𝔤_{0}-submodule of a (right) Verma 𝔤-module and, at the same time, it is dual to a Verma 𝔤_{0}-module. Since a Verma 𝔤-module is U(𝔫_{0}^{−})- torsion-free, _{χ}M̃^{σ} is a (right) Verma 𝔤_{0}-module. Hence χ is a mate for χ̃ in the “right sense”.

8.4.2. Let χ be a perfect mate for χ̃ and L be a non-zero Ũ-bimodule such that χ̃.L = L.χ̃ = 0. Let us show that

χL_{χ} := {v ∈ L| χ^{r}.v = v.χ^{r} = 0, r >> 0}

is non-zero.

Indeed, according to Lemma 8.3.1, there exist pair-wise distinct 𝔤_{0}-central characters χ_{1}, …, χ_{k} and positive integers r_{1}, …, r_{k} such that ∏_{1}^{k} χ_{i}^{r_{i}}.L = 0. Recall that L_{χ} ≠ 0 since (χ̃, χ) is a perfect pair. Thus χ = χ_{i} for a certain i; we can assume that χ = χ_{1}. One has χ_{1}^{r_{1}} + ∏_{2}^{k} χ_{i}^{r_{i}} = Z(𝔤_{0}) since χ_{1}, …, χ_{k} ∈ Max Z(𝔤_{0}) are pair-wise distinct. Therefore

0 ≠ L_{χ} = ∏_{2}^{k} χ_{i}^{r_{i}}.L

and it is a right submodule of L. Clearly

χL_{χ} = {v ∈ L_{χ}| v.χ^{r} = 0, r >> 0}.

Using 8.4.1, one obtains _{χ}L_{χ} ≠ 0 as required.

8.5. By 8.2.5, to prove the existence of a perfect mate for a strongly typical 𝔤-central character, one can choose any triangular decomposition. Below we describe, for each basic classical Lie superalgebra of type II, a triangular decomposition we choose for the proof. We use notation of Kac— see [K1], 2.5.4 for all cases except D(1, 2, α) where we use δ, ϵ_{1}, ϵ_{2} instead ϵ_{1}, ϵ_{2}, ϵ_{3}. The chosen triangular decomposition always corresponds to the first “simplest system of simple roots” in [K1].

B(m, n) : 𝔤_{0} = 𝔰𝔬(2m + 1) × 𝔰𝔭(2n)
π = {δ_{1} − δ_{2}, …, δ_{n} − ϵ_{1}, ϵ_{1} − ϵ_{2}, …, ϵ_{m−1} − ϵ_{m}, ϵ_{m}},
Δ_{0}^{+} = {δ_{i} ± δ_{j}; 2δ_{i}}_{1≤i<j≤n} ∪ {ϵ_{i} ± ϵ_{j},; ϵ_{i}}_{1≤i<j≤m},
Δ_{1}^{+} = {δ_{i}; δ_{i} ± ϵ_{j}}_{1≤i≤n,1≤j≤m},
ρ_{1} = (m + \frac{1}{2})(δ_{1} + δ_{2} + … + δ_{n})

where {δ_{1} − δ_{2}, …, 2δ_{n}} is a system of simple roots of 𝔰𝔭(2n) and {ϵ_{1} − ϵ_{2}, …, ϵ_{m−1} − ϵ_{m}, ϵ_{m}} is a system of simple roots of 𝔰𝔬(2m + 1).

D(m, n) : 𝔤_{0} = 𝔰𝔬(2m) × 𝔰𝔭(2n)
π = {δ_{1} − δ_{2}, …, δ_{n} − ϵ_{1}, ϵ_{1} − ϵ_{2}, …, ϵ_{m−1} − ϵ_{m}, ϵ_{m−1} + ϵ_{m}},
Δ_{0}^{+} = {δ_{i} ± δ_{j}, 2δ_{i}}_{1≤i<j≤n} ∪ {ϵ_{i} ± ϵ_{j}}_{1≤i<j≤m},
Δ_{1}^{+} = {δ_{i} ± ϵ_{j}}_{1≤i≤n,1≤j≤m},
ρ_{1} = m(δ_{1} + δ_{2} + … + δ_{n})

where {δ_{1} − δ_{2}, …, 2δ_{n}} is a system of simple roots of 𝔰𝔭(2n) and {ϵ_{1} − ϵ_{2}, …, ϵ_{m−1} − ϵ_{m}, ϵ_{m−1} + ϵ_{m}} is a system of simple roots of 𝔰𝔬(2m).

D(2, 1, α) : 𝔤_{0} = 𝔰𝔩(2) × 𝔰𝔩(2) × 𝔰𝔩(2),
π = {δ_{1} + ϵ_{1} + ϵ_{2}, −2ϵ_{1}, −2ϵ_{2}},
Δ_{0}^{+} = {2δ_{1}; −2ϵ_{1}, −2ϵ_{2}},
Δ_{1}^{+} = {δ_{1} ± ϵ_{1} ± ϵ_{2}},
ρ_{1} = 2δ_{1}.
F (4) :  𝔤_{0} = 𝔰𝔩(2) × 𝔰𝔬(7),
π = \{\frac{1}{2}(ϵ_{1} + ϵ_{2} + ϵ_{3} + δ_{1}), −ϵ_{1}, ϵ_{1} − ϵ_{2}, ϵ_{2} − ϵ_{3}\},
Δ_{0}^{+} = {δ_{1}} ∪ {−ϵ_{i}, 1 ≤ i ≤ 3; ±ϵ_{i} − ϵ_{j}, 1 ≤ i < j ≤ 3},
Δ_{1}^{+} = \{\frac{1}{2}(δ_{1} ± ϵ_{1} ± ϵ_{2} ± ϵ_{3})\},
ρ_{1} = 2δ_{1}

where δ_{1} is a simple root of 𝔰𝔩(2) and {−ϵ_{1}, ϵ_{1} − ϵ_{2}, ϵ_{2} − ϵ_{3}} is a system of simple roots of 𝔰𝔬(7).

G(3) : 𝔤_{0} = 𝔰𝔩(2) × G_{2},
π = {δ_{1} + ϵ_{1}, ϵ_{2}, ϵ_{3} − ϵ_{2}},
Δ_{0}^{+} = {2δ_{1}} ∪ {ϵ_{2}, ϵ_{3}, −ϵ_{1}, ϵ_{3} − ϵ_{2}, ϵ_{3} − ϵ_{1}, ϵ_{2} − ϵ_{1}},
Δ_{1}^{+} = {δ_{1}; δ_{1} ± ϵ_{i}, i = 1, 2, 3},
ρ_{1} = \frac{7}{2}δ_{1}

where ϵ_{1} + ϵ_{2} + ϵ_{3} = 0, {ϵ_{2}, ϵ_{3} − ϵ_{2}} is a system of simple roots of G(2) and 2δ_{1} is a simple root of 𝔰𝔩(2).

The restriction of the non-degenerate invariant bilinear form (−, −) on 𝔤_{0} is a nondegenerate invariant bilinear form. Thus

(δ_{i}, δ_{i′}) = (δ_{i}, ϵ_{j}) = 0 ∀i ≠ i′

and

(ϵ_{j}, ϵ_{j′}) = 0 ∀j ≠ j′

if 𝔤 is not of the type G(3).

Till the end of this section 𝔤 will denote one (unless otherwise specified, an arbitrary one) of the basic classical Lie superalgebras of type II with the fixed triangular decomposition described above.

8.5.1. More notation. For all above root systems denote by n the number of δ’s and by m the number of ϵ’s (that is n = 1 for D(1, 2, α), F (4), G(3) and m = 2 for D(1, 2, α), m = 3 for F (4), G(3)). Remark that for all above systems

ρ_{1} = p ∑_{i=1}^{n} δ_{i}

for a certain scalar p ∈ \frac{1}{2}ℕ^{+}.

Denote by W_{1} (resp., W_{2}) the subgroup of W which acts on {δ_{i}}_{i=1}^{n} (resp., on {ϵ_{j}}_{j=1}^{m}). Then W = W_{1} × W_{2}. Remark that W_{1} is always the group of signed permutations of {δ_{i}}_{i=1}^{n}: for B(m, n) and D(m, n) it is the Weyl group corresponding to 𝔰𝔭(2n) and for D(1, 2, α), F (4), G(3) it is the Weyl group corresponding to 𝔰𝔩(2).

For μ ∈ 𝔥^{∗} write

μ = ∑_{i=1}^{n} k_{i}δ_{i} + ∑_{j=1}^{m} l_{j}ϵ_{j}

and set μ_{δ_{i}} := k_{i}, μ_{ϵ_{j}} := l_{j}.

Say that χ̃ is generic if there exists λ ∈ W (χ̃) such that (λ + ρ)_{δ_{i}} ≠ 0 for i = 1, …, n. Remark that T^{2} ∉ χ̃ implies that χ̃ is generic if δ_{1} ∈ Δ_{1} that is in the cases B(m, n) and G(3).

Define a lexicographic order on ℂ by setting c_{1} > c_{2} if Re c_{1} > Re c_{2} or Re c_{1} = Re c_{2} and Im c_{1} > Im c_{2}.

One can easily sees that for any γ ∈ Γ and i = 1, …, n one has 0 ≤ |γ|_{σ_{i}} ≤ 2p and, moreover, γ = ∅ iff |γ|_{σ_{1}} = … = |γ|_{σ_{n}} = 0. We will use many times the following easy lemma.

8.5.2. Lemma. Let d; a_{1}, …, a_{s}; r_{1}, …, r_{s} be complex numbers such that d, a_{1}, …, a_{s} > 0 and 0 ≤ r_{1}, …, r_{s} ≤ 2d. Suppose there exists a signed permutation σ which maps (a_{1} + d, …, a_{s} + d) to (a_{1} + d − r_{1}, …, a_{s} + d − r_{s}). Then σ is a usual (non-signed) permutation and r_{1} = … = r_{s} = 0.

Proof. The permutation σ is non-signed because a_{i} + a_{j} > 0 ≥ −2d + r_{j} and so a_{i} + d ≠ −(a_{j} + d − r_{j}) for any indexes i, j. Since σ is a usual permutation,

∑_{i}^{s}(a_{i} + d) = ∑_{1}^{s}(a_{i} + d − r_{i})

that is ∑ r_{i} = 0. The inequalities r_{1}, …, r_{s} ≥ 0 imply the assertion.

8.6. Generic case. Fix a generic central character χ̃ . Since W_{1} acts by signed permutation on {δ_{i}}_{i=1}^{n}, there exists λ ∈ W (χ̃) such that (λ + ρ)_{δ_{i}} > 0 for all i = 1, …, n. Fix such a λ and let us show that the 𝔤_{0}-character of M(λ) is a perfect mate for χ̃.

To verify that the 𝔤_{0}-central character of M(λ) is a mate for χ̃, recall Lemma 8.3.4 (i). Suppose that

(λ + ρ_{0}) = w(λ + ρ_{0} − |γ|)

for some γ ∈ Γ, w ∈ W . Write

λ + ρ = ∑_{1}^{n} k_{i}δ_{i} + ∑_{1}^{m} l_{j}ϵ_{j}, |γ| = ∑_{1}^{n} s_{i}δ_{i} + ∑_{1}^{m} r_{j}ϵ_{j}.

Recall that k_{i} > 0 and 0 ≤ s_{i} ≤ 2p for i = 1, …, n. One has

λ + ρ_{0} = λ + ρ + ρ_{1} = ∑(k_{i} + p)δ_{i} + ∑ l_{j}ϵ_{j}

and

λ + ρ_{0} − |γ| = ∑(k_{i} + p − s_{i})δ_{i} + ∑(l_{j} − r_{j})ϵ_{j}.

Write w = w_{1}w_{2} where w_{1} ∈ W_{1}, w_{2} ∈ W_{2}. Then

∑(k_{i} + p)δ_{i} = w_{1}(∑(k_{i} + p − s_{i})δ_{i})

By Lemma 8.5.2, this implies s_{1} = … = s_{n} = 0 and thus γ = ∅. Hence the 𝔤_{0}-central character of M(λ) is a mate for χ̃.

One has

Stab_{W} (λ+ρ_{0}) = Stab_{W} (∑(k_{i}+p)δ_{i}+∑ l_{j}ϵ_{j}) = Stab_{W} (∑ k_{i}δ_{i}+∑ l_{j}ϵ_{j}) = Stab_{W} (λ+ρ)

since p; k_{1}, …, k_{n} > 0. Using Lemma 8.3.4 (ii), one concludes that the 𝔤_{0}-central character of M(λ) is a perfect mate for χ̃.

8.7. The case when χ̃ is not generic. Suppose that 𝔤 is of the type D(m, n), D(1, 2, α) or F (4) and χ̃ is not generic.

8.7.1. Case D(m, n). Since χ̃ is not generic, for any λ ∈ W (χ̃) there exists i ∈ {1, …, n} such that (λ + ρ)_{δ_{i}} = 0. On the other hand, T^{2} ∉ χ̃ implies (λ + ρ, β) ≠ 0 for any β ∈ Δ_{1}. In particular, (λ + ρ, δ_{i} + ϵ_{j}) ≠ 0 and so (λ + ρ)_{ϵ_{j}} ≠ 0 for j = 1, …, m. Taking into account that W_{1} acts on {δ_{i}}_{1}^{n} by signed permutations and W_{2} acts on {ϵ_{j}}_{1}^{m} by signed permutations changing even number of signs, one concludes the existence of λ ∈ W (χ̃) such that

λ + ρ = ∑_{i=1}^{n} k_{i}δ_{i} + ∑_{j=1}^{m} l_{j}ϵ_{j}

where

\begin{matrix}k_{1} ≥ k_{2} ≥ … ≥ k_{n−d} > k_{n−d+1} = k_{n−d+2} = … = k_{n} = 0, \\ l_{1} ≥ l_{2} ≥ … ≥ l_{m−1} > 0, l_{m−1} ≥ l_{m}, l_{m−1} ≥ −l_{m}, l_{m} ≠ 0. \end{matrix}

Fix λ as above and set

γ_{d} := {δ_{i} − ϵ_{j} : n − d < i ≤ n, 1 ≤ j ≤ m − 1} ∪ {δ_{i} − sn(l_{m})ϵ_{m} : n − d < i ≤ n}

where sn(l_{m}) := 1 if l_{m} > 0 and sn(l_{m}) := 1 if l_{m} < 0.

Let us show that the 𝔤_{0}-central character of M(λ − |γ_{d}|) is a perfect mate for χ̃. To verify that it is a mate suppose that (λ − γ_{d} + ρ_{0}) = w(λ − γ + ρ_{0}) for some γ ∈ Γ, w ∈ W. Write

|γ| = ∑_{1}^{n} s_{i}δ_{i} + ∑_{1}^{m} r_{j}ϵ_{j}.

Observe that s_{i} ∈ {0, 1, …, 2m} for i = 1, …, n. One has

\begin{aligned}λ − |γ|_{d} + ρ_{0} & = λ + ρ + (ρ_{1} − |γ_{d}|) \\ & = ∑_{1}^{n−d}(k_{i} + m)δ_{i} + ∑_{1}^{m−1}(l_{j} + d)ϵ_{j} + (l_{m} + sn(l_{m})d)ϵ_{m} \end{aligned}  (8)

and

λ + ρ_{0} − |γ| = ∑_{1}^{n−d}(k_{i} + m − s_{i})δ_{i} + ∑_{n−d+1}^{n}(m − s_{i})δ_{i} + ∑_{1}^{m}(l_{j} − r_{j})ϵ_{j}.

Write w = w_{1}w_{2} where w_{1} ∈ W_{1}, w_{2} ∈ W_{2}. Then

∑_{1}^{n−d}(k_{i} + m)δ_{i} = w_{1}(∑_{1}^{n−d}(k_{i} + m − s_{i})δ_{i} + ∑_{n−d+1}^{n}(m − s_{i})δ_{i}).

For any indexes i, i′ such that 1 ≤ i ≤ n − d < i′ ≤ n one has k_{i} + m > m ≥ ±(m − s_{i′}). Therefore w_{1} = w′_{1}w′′_{1} where w′_{1} (resp., w′′_{1}) is a signed permutation of {δ_{i}}_{1}^{n−d} (resp., of {δ_{i}}_{n−d+1}^{n}). Then ∑_{1}^{n−d}(k_{i} + m)δ_{i} = w′_{1}(∑_{1}^{n−d}(k_{i} + m − s_{i})δ_{i}) and so s_{1} = … = s_{n−d} = 0

by Lemma 8.5.2. On the other hand, 0 = w′′_{1}(∑_{n−d+1}^{n}(m − s_{i})δ_{i}) gives s_{n−d} = … = s_{n} = m.

It is easy to see that s_{1} = … = s_{n−d} = 0 implies r_{j} ∈ {0, ±1, …, ±d} for all j = 1, …, m.

Suppose that l_{m} > 0. Then

w_{2}(∑_{1}^{m}(l_{j} − r_{j})ϵ_{j}) = ∑_{1}^{m}(l_{j} + d)ϵ_{j}

and, by Lemma 8.5.2, r_{j} = −d for all j = 1, …, m. Thus |γ| = |γ_{d}|. It is easy to check that this implies the required equality γ = γ_{d}.

Suppose that l_{m} < 0. Then

w_{2}(∑_{1}^{m}(l_{j} − r_{j})ϵ_{j}) = ∑_{1}^{m−1}(l_{j} + d)ϵ_{j} + (l_{m} − d)ϵ_{m}

where w_{2} is a signed permutation of {ϵ_{j}}_{1}^{m}. Then

w′_{2}( ∑_{1}^{m−1}(l_{j} − r_{j})ϵ_{j} + (r_{m} − l_{m})ϵ_{m}) = ∑_{1}^{m−1}(l_{j} + d)ϵ_{j} + (−l_{m} + d)ϵ_{m}

for another signed permutation w′_{2} of {ϵ_{j}}_{1}^{m}. Since l_{1}, …, l_{m−1}, −l_{m} > 0, Lemma 8.5.2 gives r_{1} = … = r_{m−1} = d and r_{m} = −d. Thus again |γ| = |γ_{d}| and so γ = γ_{d}. Hence the 𝔤_{0}-central character of M(λ − |γ|_{d}) is a mate for χ̃.

One has

Stab_{W} (λ + ρ) = Stab_{W} ((∑_{1}^{n−d} k_{i}δ_{i} + ∑_{1}^{m} l_{j}ϵ_{j}) = Stab_{W} ((∑_{1}^{n−d}(k_{i} + m)δ_{i} + ∑_{1}^{m−1}(l_{j} + d)ϵ_{j} + (l_{m} + sn(l_{m})d)ϵ_{m}) = Stab_{W} (λ − |γ_{d}| + ρ_{0})

since m; d; k_{1}, …, k_{n}; l_{1}, …, l_{m−1} > sn(l_{m})l_{m} > 0. Using Lemma 8.3.4 (ii), one concludes that the 𝔤_{0}-central character of M(λ − |γ|_{d}) is a perfect mate for χ̃.

8.7.2. Case D(2, 1, α). Set δ := δ_{1}. Since χ̃ is not generic, (λ + ρ)_{δ} = 0 for any λ ∈ W (χ̃). On the other hand, T^{2} ∉ χ̃ implies (λ+ρ, β) ≠ 0 for any β ∈ Δ_{1}. Since Δ_{1}^{+} = {δ±ϵ_{1}±ϵ_{2}}, one obtains (λ + ρ)_{ϵ_{j}} ≠ 0 for j = 1 or j = 2. Recall that W ≅ ℤ_{2} × ℤ_{2} × ℤ_{2} where the copies of ℤ_{2} act by sign on δ; ϵ_{1}; ϵ_{2} respectively. Therefore there exists λ ∈ W (χ̃) such that

λ + ρ = l_{1}ϵ_{1} + l_{2}ϵ_{2}

where l_{1}, l_{2} ≥ 0 and at least one of l_{1}, l_{2} is non-zero. Fix such a λ. Both cases l_{1} > 0 and l_{2} > 0 are similar so we can assume l_{1} ≠ 0. Set

γ_{d} := {δ − ϵ_{1} − ϵ_{2}; δ − ϵ_{1} + ϵ_{2}}.

Let us show that the 𝔤_{0}-central character of M(λ − |γ_{d}|) is a perfect mate for χ̃.

One has ρ_{1} − |γ_{d}| = 2ϵ_{1} and

λ − |γ_{d}| + ρ_{0} = λ + ρ + ρ_{1} − |γ_{d}| = (l_{1} + 2)ϵ_{1} + l_{2}ϵ_{2}.

Assume that (λ−|γ_{d}|+ρ_{0}) ∈ W (λ−|γ|+ρ_{0}) for some γ ∈ Γ. Then (λ−|γ|+ρ_{0})_{ϵ_{1}} = ±(l_{1}+2) that is |γ|_{ϵ_{1}} = l_{1}±(l_{1}+2). Taking into account that l_{1} > 0 and that |γ|_{ϵ_{1}} ∈ {0, ±1, ±2}, one concludes |γ|_{ϵ_{1}} = −2. This implies γ = γ_{d}. Hence the 𝔤_{0}-central character of M(λ − |γ_{d}|) is a mate for χ̃. Since l_{1} > 0

Stab_{W} (λ + ρ) = Stab_{W} (l_{1}ϵ_{1} + l_{2}ϵ_{2}) = Stab_{W} ((l_{1} + 2)ϵ_{1} + l_{2}ϵ_{2}) = Stab_{W} (λ − |γ_{d}| + ρ_{0}).

By Lemma 8.3.4, the 𝔤_{0}-central character of M(λ − |γ_{d}|) is a perfect mate for χ̃.

8.7.3. Case F (4). Set δ := δ_{1}. Since χ̃ is not generic, (λ + ρ)_{δ} = 0 for any λ ∈ W (χ̃). On the other hand, T^{2} ∉ χ̃ implies (λ + ρ, β) ≠ 0 for any β ∈ Δ_{1}. Since Δ_{1}^{+} = \{\frac{1}{2}(δ ± ϵ_{1} ± ϵ_{2} ± ϵ_{3}}, one obtains (λ + ρ)_{ϵ_{j}} ≠ 0 for some j ∈ {1, 2, 3}. Recall that W = W_{1} × W_{2} where W_{1} ≅ ℤ_{2} acts by sign on δ and W_{2} is the group of signed permutations of {ϵ_{j}}_{1}^{3}. Therefore there exists λ ∈ W (χ̃) such that

λ + ρ = l_{1}ϵ_{1} + l_{2}ϵ_{2} + l_{3}ϵ_{3}

where

l_{1} ≥ l_{2} ≥ l_{3} ≥ 0 & l_{1} ≠ 0.

Fix such a λ. Set

\begin{matrix}γ_{d} := \{\frac{1}{2}(δ − ϵ_{1} ± ϵ_{2} ± ϵ_{3})\} if l_{1} > l_{2}, \\ γ_{d} := \{\frac{1}{2}(δ − ϵ_{1} − ϵ_{2} − ϵ_{3}); \frac{1}{2}(δ + ϵ_{1} − ϵ_{2} − ϵ_{3}); \frac{1}{2}(δ − ϵ_{1} + ϵ_{2} − ϵ_{3}); \frac{1}{2}(δ − ϵ_{1} − ϵ_{2} + ϵ_{3})\} if l_{1} = l_{2}. \end{matrix}

We show below that the 𝔤_{0}-central character of M(λ − |γ_{d}|) is a perfect mate for χ̃.

Indeed, assume that (λ − |γ_{d}| + ρ_{0}) = w(λ − |γ| + ρ_{0}) for some γ ∈ Γ, w ∈ W. Write |γ| = sδ + ∑_{1}^{3} r_{j}ϵ_{j} and w = w_{1}w_{2} where w_{1} ∈ W_{1}, w_{2} ∈ W_{2}. One has

λ − |γ| + ρ_{0} = (2 − s)δ + ∑_{1}^{3}(l_{j} − r_{j})ϵ_{j}.

Suppose l_{1} > l_{2}. Then

|γ_{d}| = 2δ − 2ϵ_{1} that is λ − |γ_{d}| + ρ_{0} = (l_{1} + 2)ϵ_{1} + l_{2}ϵ_{2} + l_{3}ϵ_{3}.

The equality (λ − |γ_{d}| + ρ_{0}) = w(λ − |γ| + ρ_{0}) implies s = 2. Thus γ contains 4 elements and so r_{1}, r_{2}, r_{3} ∈ {0; ±1; ±2}. Then l_{1} + 2 > l_{j} − r_{j} for j = 1, 2 and l_{1} + 2 > −(l_{j} − r_{j}) for j = 1, 2, 3. Thus w_{2}(ϵ_{1}) = ϵ_{1} and so r_{1} = −2. Since γ contains 4 elements, this implies γ = γ_{d}. Hence the 𝔤_{0}-central character of M(λ − |γ_{d}|) is a mate for χ̃.

Suppose l_{1} = l_{2}. Then

|γ_{d}| = 2δ − ϵ_{1} − ϵ_{2} − ϵ_{3} that is λ − |γ_{d}| + ρ_{0} = ∑_{1}^{3}(l_{j} + 1)ϵ_{j}.

Again γ contains 4 elements and so r_{1}, r_{2}, r_{3} ∈ {0; ±1; ±2}. Assume that w_{2} is not a usual permutation that is w_{2}(ϵ_{j}) = −ϵ_{j′} for some j, j′. Then l_{j′} + 1 = −(l_{j} − r_{j}) that is r_{j} −1 = l_{j′} +l_{j} > 0 because l_{1} = l_{2} > 0 and l_{3} ≥ 0. Then r_{j} = 2 and, consequently, r_{j′′} = 0 for all j′′ ≠ j. In particular, w_{2} can change a sign of at most one of ϵ_{1}, ϵ_{2}, ϵ_{3}. Therefore l_{1} + 1 = l_{2} + 1 = l_{j′′} − r_{j′′} for some j′′ ≠ j. Then l_{1} + 1 = l_{j′} that contradicts to the inequality l_{1} ≥ l_{3}. Thus w_{2} is a usual permutation. This implies ∑_{1}^{3}(l_{j} + 1) = ∑_{1}^{3}(l_{j} − r_{j}) that is r_{1} + r_{2} + r_{3} = −3. Taking into account that r_{j} ∈ {0; ±1; ±2} and that the equality of the form r_{j} = ±2 implies r_{j′} = 0 for j′ ≠ j, one concludes r_{1} = r_{2} = r_{3} = −1 that is |γ| = |γ_{d}|. It is easy to check that this gives the required equality γ = γ_{d}. Hence the 𝔤_{0}-central character of M(λ − |γ_{d}|) is a mate for χ̃.

If l_{1} > l_{2} ≤ l_{3} ≤ 0 then

Stab_{W} (λ+ρ) = Stab_{W} (l_{1}ϵ_{1}+l_{2}ϵ_{2}+l_{3}ϵ_{3}) = Stab_{W} ((l_{1}+2)ϵ_{1}+l_{2}ϵ_{2}+l_{3}ϵ_{3}) = Stab_{W} (λ−|γ_{d}|+ρ_{0}).

If l_{1} = l_{2} > 0 then l_{3} ≠ 0 since (λ + ρ, δ + ϵ_{1} − ϵ_{2} + ϵ_{3}) ≠ 0. Therefore

Stab_{W} (λ + ρ) = Stab_{W} (l_{1}ϵ_{1} + l_{2}ϵ_{2} + l_{3}ϵ_{3})

= Stab_{W} ((l_{1} + 1)ϵ_{1} + (l_{2} + 1)ϵ_{2} + (l_{3} + 1)ϵ_{3}) = Stab_{W} (λ − |γ_{d}| + ρ_{0}).

Hence the 𝔤_{0}-central character of M(λ − |γ_{d}|) is a perfect mate for χ̃.

9. Annihilation Theorem

This section is devoted to the proof of Theorem 9.5.

9.1. Lemma. Let M̃ be a strongly typical Verma module and v ∈ M̃ be a primitive vector. Then Ũv is a Verma module. In particular, M̃ contains a simple Verma submodule.

Proof. Set χ̃ := Ann_{Z(𝔤)} M̃. Since M̃ is strongly typical, χ̃ is also strongly typical and so for χ̃ there exists a perfect mate χ ∈ Max Z(𝔤_{0})— see Section 8. Let Ñ be a submodule of M̃ generated by a primitive vector. Then Ñ is a quotient of a Verma module M̃′ and χ̃M̃′ = 0. By 8.2.4, M′ := M̃′_{χ} is a Verma 𝔤_{0}-module. Since χ is a perfect mate for χ̃, the 𝔤_{0}-module Ñ_{χ} is a non-zero quotient of M′. Taking into account that Ñ_{χ} ⊂ M̃ is torsion-free over U(𝔫_{0}^{−}), one concludes Ñ_{χ} = M′. Thus (M̃′/Ñ)_{χ} = 0 and so Ñ = M̃′ is a Verma module.

Recall that M̃ has a finite length and so it contains a simple submodule Ñ. A highest weight vector of Ñ is primitive. Hence Ñ is a simple Verma module.

9.2. Lemma. Let M̃ be a strongly typical Verma module and M̃′ be a simple submodule of M̃. Then the natural maps F (M̃, M̃) → F (M̃′, M̃) and F (M̃′, M̃′) ⟶^{∼} F (M̃′, M̃) are 𝔤-bimodule isomorphisms.

Proof. Denote by ι the natural map F (M̃, M̃) → F (M̃′, M̃) and by ι′ the natural map F (M̃′, M̃′) → F (M̃′, M̃) (both maps are induced by the embedding M̃′ to M̃). Both maps are 𝔤-bimodule homomorphisms.

To show that ι′ is a bijection and that ι is a monomorphism, we use the following reasoning which is essentially the same as in [J1].

By [J1], F (N_{1}, N_{2}) = 0 if the GK-dimension of any simple quotient of N_{1} is greater than GK-dimension of N_{2} or if the GK-dimension of N_{1} is less than GK-dimension of any simple submodule of N_{2}. Moreover, the GK-dimension of a Verma module is equal to the GK-dimension of the algebra U(𝔫^{−}).

Let v be a highest weight vector of M̃ and u ∈ U(𝔫^{−}) be such that uv is a highest weight vector of M̃′. Since M̃′ is a Verma module, u is a non-zero divisor in U(𝔫^{−}). This implies

GK dim M̃′ = GK dim M̃ > GK dim M̃/M̃′.

The only simple quotient of M̃′ is M̃′ itself; thus the inequality GK dim M̃′ > GK dim M̃/M̃′ implies F (M̃′, M̃/M̃′) = 0. Therefore ι′ is an epimorphism. Obviously, ι′ is a monomorphism. Hence ι′ is bijective. By Lemma 9.1, any simple submodule of M̃ is a Verma module. Thus the GK-dimension of M̃/M̃′ is less than the GK-dimension of any simple submodule of M̃ and so F (M̃/M̃′, M̃) = 0. Consequently, ι is a monomorphism.

It remains to verify the surjectivity of ι. Since ι′ is bijective, it is enough to check the surjectivity of the composed map ι′′ := (ι′)^{−1} ∘ ι : F (M̃, M̃) → F (M̃′, M̃′). Denote by N the cokernel of the map ι′′. This is a 𝔤-bimodule and its left and right annihilators contain χ̃ := Ann_{Z(𝔤)} M̃. Let χ ∈ Max Z(𝔤_{0}) be a perfect mate for χ̃. Using notation of 8.4.2, one has

χF (M̃, M̃)_{χ} ⟶^{∼} F (M̃_{χ}, M̃_{χ}), _{χ}F (M̃′, M̃′)_{χ} ⟶^{∼} F (M̃′_{χ}, M̃′_{χ}).

Since χ is a mate for χ̃, both M := M̃_{χ} and M′ := M̃′_{χ} are Verma 𝔤_{0}-modules. For any V ∈ Irr_{0} one has Hom_{𝔤_{0}}(V, F (M, M)) == dim V |_{0} = Hom_{𝔤_{0}}(V, F (M′, M′))— see [J1], 6.4. Taking into account the injectivity of ι′′, one concludes that ι′′(F (M, M)) = F (M′, M′) and so _{χ}N_{χ} = 0. Thus N = 0 by 8.4.2. Hence ι, ι′ are isomorphisms. The lemma is proven.

Using Lemma 9.1 one obtains the

9.2.1. Corollary. Let M̃, M̃′ be strongly typical Verma modules and M̃′ be a submodule of M̃. Then the 𝔤-bimodules F (M̃, M̃) and F (M̃′, M̃′) are isomorphic.

9.3. Proposition. If M̃ is a strongly typical projective (in Õ) Verma module then

F (M̃, M̃) ≅ ⊕_{_{V}∈Irr} E(Ṽ)^{⊕ dim _{V}|_{0}}.

Proof. Step 1. Let us show that for any 𝔤_{0}-modules N_{1}, N_{2} the following ad 𝔤-modules

L := F (Ind_{𝔤_{0}}^{𝔤} N_{1}, Ind_{𝔤_{0}}^{𝔤} N_{2}), L′ := Coind_{𝔤_{0}}^{𝔤}(F (N_{1}, N_{2}) ⊗ Λ𝔤_{1})

are isomorphic.

Indeed, using the Frobenius reciprocity, one obtains

Hom_{𝔤_{0}}(V, F (Ind_{𝔤_{0}}^{𝔤} N_{1}, Ind_{𝔤_{0}}^{𝔤} N_{2})) ≅ Hom_{𝔤_{0}}(V, F (N_{1}, N_{2}) ⊗ Λ𝔤_{1} ⊗ Λ𝔤_{1})
≅ Hom_{𝔤_{0}}(V, Coind_{𝔤_{0}}^{𝔤}(F (N_{1}, N_{2}) ⊗ Λ𝔤_{1}))

for any V ∈ Irr_{0}. Hence L ≅ L′ as ad 𝔤_{0}-modules. Note that dim Hom_{𝔤_{0}}(V, L) < ∞ for all V ∈ Irr_{0}.

On the other hand, for any Ṽ ∈ Irr one has

Hom_{𝔤}(Ṽ , F (Ind_{𝔤_{0}}^{𝔤} N_{1}, Ind_{𝔤_{0}}^{𝔤} N_{2})) ≅ Hom_{𝔤}(Ind_{𝔤_{0}}^{𝔤} N_{1}, Ind_{𝔤_{0}}^{𝔤} N_{2} ⊗ Ṽ^{∗})

≅ Hom_{𝔤_{0}}(N_{1}, Ind_{𝔤_{0}}^{𝔤} N_{2} ⊗ Ṽ^{∗}) ≅ Hom_{𝔤_{0}}(N_{1}, N_{2} ⊗ Λ𝔤_{1} ⊗ Ṽ^{∗})

≅ Hom_{𝔤_{0}}(Ṽ , F (N_{1}, N_{2}) ⊗ Λ𝔤_{1})
≅ Hom_{𝔤}(Ṽ , Coind_{𝔤_{0}}^{𝔤}(F (N_{1}, N_{2}) ⊗ Λ𝔤_{1})).

Hence Soc L ≅ Soc L′. Since L′ is injective in Fin, it contains a submodule isomorphic to L. Taking into account the ad 𝔤_{0}-isomorphism L ≅ L′, one concludes that L ≅ L′ as ad 𝔤-modules.

Step 2. Let χ be a perfect mate of Ann_{Z(𝔤)} M̃. Then M := M̃_{χ} is a Verma 𝔤_{0}-module and M̃ = ŨM by Corollary 8.3.2. Thus M̃ is a quotient of Ind_{𝔤_{0}}^{𝔤} M. Since M̃ is projective, it is a direct summand of Ind_{𝔤_{0}}^{𝔤} M Then the ad 𝔤-module F (M̃, M̃) is a direct summand of the ad 𝔤-module F (Ind_{𝔤_{0}}^{𝔤} M, Ind_{𝔤_{0}}^{𝔤} M). The last is isomorphic to Coind_{𝔤_{0}}^{𝔤}(F (M, M)⊗Λ𝔤_{1}) and so is injective in Fin. Hence the ad 𝔤-module F (M̃, M̃) is injective in Fin.

By Lemma 9.1, M̃ contains a simple Verma submodule M̃′. Combining Corollary 9.2.1 and 11.1.5 one obtains

Soc F (M̃, M̃) ≅ Soc F (M̃′, M̃′) ≅ ⊕_{_{V}∈Irr}Ṽ^{⊕ dim _{V}|_{0}}.

Now the injectivity of F (M̃, M̃) implies the required assertion.

9.4. Proposition. If M̃ is a strongly typical Verma module then the natural map Ũ → F (M̃, M̃) is surjective.

Proof. Denote by N the cokernel of the natural map f : Ũ → F (M̃, M̃). This is a 𝔤bimodule and its left and right annihilators contain χ̃ := Ann_{Z(𝔤)} M̃. Let χ ∈ Max Z(𝔤_{0}) be a perfect mate for χ̃. Using notation of 8.4.2 one has _{χ}F (M̃, M̃)_{χ} ⟶^{∼} F (M̃_{χ}, M̃_{χ}). Since χ is a mate for χ̃, M := M̃_{χ} is a Verma 𝔤_{0}-module. As a 𝔤_{0}-module, M̃ has a finite length and so

Ann_{Z(𝔤_{0})} M̃ = χ ∏_{i=1}^{k} χ_{i}^{r_{i}}

where {χ, χ_{1}, …, χ_{k}} = supp_{Z(𝔤_{0})} M̃ and r_{1}, …, r_{k} ∈ ℕ^{+}. Any element of ∏_{i=1}^{k} χ_{i}^{r_{i}} annihilates ∑_{i=1}^{k} M̃_{χ_{i}} and acts by scalar on M̃_{χ}. Taking into account that the natural map U → F (M, M) is surjective (see [J1], 6.4), one concludes that

f(U ∏_{i=1}^{k} χ_{i}^{r_{i}}) =_{χ}F (M̃, M̃)_{χ}.

and thus _{χ}N_{χ} = 0. By 8.4.2, N = 0 as required.

9.5. Theorem. For a strongly typical Verma 𝔤-module M̃

Ann M̃ = Ũ Ann_{Z(𝔤)} M̃.

Proof. Set χ̃ := Ann_{Z(𝔤)} M̃. Let M̃′′ be a projective (in Õ) Verma module such that χ̃M̃′′ = 0 and let M̃′ be a simple submodule of M̃′′ (by 9.1, M̃′ is a Verma module). Combining 9.2.1 and 9.3, one obtains

F (M̃′, M̃′) ≅ ⊕_{_{V}∈Irr} E(Ṽ)^{⊕ dim _{V}|_{0}}.

Summarizing 5.1,5.2 and 9.4, one concludes that Ann M̃′ = Ũχ̃ and that the natural map f′ : Ũ/(Ũχ̃) → F (M̃′, M̃′) is bijective.

Denote by f the natural map Ũ/(Ũχ̃) → F (M̃, M̃) and by p the composition map f ∘ (f′)^{−1} : F (M̃′, M̃′) → F (M̃, M̃). Obviously p is a Ũ-bimodule map. By 9.4, p is surjective. Let us show that p is bijective.

Let χ be a perfect mate for χ̃; set M := M̃_{χ}, M′ := M̃′_{χ}. Using notation of 8.4.2, one has

χF (M̃, M̃)_{χ} = F (M, M), _{χ}F (M̃′, M̃′)_{χ} = F (M′, M′)

and so _{χ}F (M̃, M̃)_{χ},_{χ} F (M̃′, M̃′)_{χ} are isomorphic ad 𝔤_{0}-modules. It is easy to see that

χ(Im p)_{χ} = p(_{χ}F (M̃′, M̃′)_{χ}).

Therefore p(_{χ}F (M̃′, M̃′)_{χ}) =_{χ} F (M̃, M̃)_{χ} since p is surjective. Taking into account that the multiplicity of each simple 𝔤_{0}-module V in F (M, M) is finite, one concludes that the restriction of p to _{χ}F (M̃′, M̃′)_{χ} is a monomorphism. Thus _{χ}(ker p)_{χ} = 0 and ker p = 0 by 8.4.2. This means that p is bijective and so f : Ũ/(Ũχ̃) → F (M̃, M̃) is bijective as well. The assertion follows.

9.6. Corollary. For a strongly typical central character χ̃

Ũ/(Ũχ̃) ≅ ⊕_{_{V}∈Irr} E(Ṽ)^{⊕ dim _{V}|_{0}}.

10. Remark about Verma modules.

In this section we study the 𝔤_{0}-structure of Verma 𝔤-modules.

10.1. Retain notation of Section 8. Take strongly typical χ̃ ∈ Max Z(𝔤) and λ ∈ W (χ̃). Recall that as an 𝔤_{0}-module M̃(λ) has a finite filtration with the factors {M(λ − |γ|), γ ∈ Γ}. Therefore

Ũχ̃ ∩ Z(𝔤_{0}) = Ann M̃(λ) ∩ Z(𝔤_{0}) = ∏_{i} χ_{i}^{r_{i}}  (9)

where the χ_{i} are the pairwise distinct elements of the multiset {Ann_{Z(𝔤_{0})} M(λ−|γ|), γ ∈ Γ} and the r_{i} are positive integers. Thus for any Ũ_{χ}-module Ñ

Ñ = ⊕_{i}Ñ_{χ_{i}}, χ_{i}^{r_{i}} Ñ_{χ_{i}} = 0.

Suppose that M̃(λ) = ⊕_{γ∈Γ}M(λ − γ) and so Z(𝔤_{0}) acts semisimply on M̃(λ). If, in addition, λ is strongly typical then (9) implies that Z(𝔤_{0}) acts semisimply on any Ũ_{χ}-module for χ̃ = Ann_{Z(𝔤)} M̃(λ).

10.2. Proposition. For M̃(λ) being a simple Verma module the following conditions are equivalent

(i) M̃ = ⊕_{γ∈Γ}M(λ − |γ|)
(ii) ∀γ, γ′ ∈ Γ (|γ| − |γ′|) ∈ ℤΔ_{0} \ {0} ⟹ (λ − |γ| + ρ_{0}) ∉ W (λ − |γ′| + ρ_{0})
(iii) M(λ − |γ|) is simple for any γ ∈ Γ.

Proof. The implication (ii) ⟹ (i) follows from the fact that any exact sequence 0 → M(μ) → N → M(μ) → 0 in O splits (see, for instance, 2.5.3).

Let us verify the implication (i) ⟹ (iii). Recall 2.5.6. Since M̃(λ) is simple it is isomorphic to M̃(λ)^{#} = ⊕_{γ∈Γ}M(λ − |γ|)^{#}. The module M̃(λ) is U(𝔫_{0}^{−})-torsion-free; thus all M(λ − |γ|)^{#} are also U(𝔫_{0}^{−})-torsion-free. This forces (iii).

The implication (iii) ⟹ (ii) follows from the fact that if M(μ), M(μ′) are simple Verma 𝔤_{0}-modules with the same 𝔤_{0}-central character and (μ − μ′) ∈ ℤΔ_{0} then μ = μ′. This fact can be deduced from [J2], A.1.14 and A.1.1 (vii).

Hence the conditions (i),(ii) and (iii) are equivalent provided that M̃(λ) is simple.

10.3. Corollary. If M̃(λ) contains a simple typical Verma submodule then the conditions (i) and (ii) are equivalent.

Proof. The implication (ii) ⟹ (i) follows from the same argument as in Proposition 10.2. To verify the implication (i) ⟹ (ii), assume that M̃(λ) = ⊕M_{i} is a direct sum of Verma 𝔤_{0}-modules and that

λ − |γ′| + ρ_{0} = w(λ − |γ′′| + ρ_{0})

for some w ∈ W and γ′, γ′′ ∈ Γ satisfying (|γ′′| − |γ′|) ∈ ℤΔ_{0} \ {0}. Take w′ ∈ W such that M̃(w′.λ) is a simple submodule of M̃(λ). In the notation of 8.2.2 one has

\begin{aligned}w′.λ − |w′_{∗}γ′| + ρ_{0} & = w′(λ − |γ′| + ρ_{0}) = w′w(λ − |γ′′| + ρ_{0}) \\ & = w′w(w′)^{−1}(w′.λ − |w′_{∗}γ′′| + ρ_{0}) \end{aligned}

by (7). Furthermore |w′_{∗}γ′| − |w′_{∗}γ′′| = w(|γ′′| − |γ′|) ∈ ℤΔ_{0} \ {0}. Thus the condition (ii) does not hold for a simple Verma module M̃(w′.λ). Therefore M̃(w′.λ) has a 𝔤_{0}- submodule N which is isomorphic to a non-splitting extension of M(μ) by M(μ′) for some μ, μ′ ∈ {λ − |γ| : γ ∈ Γ}. Since M̃(w′.λ) is a submodule of M̃(λ), N is a 𝔤_{0}- submodule of M̃(λ). Since N is indecomposable, it is isomorphic to a submodule of a Verma module M_{i}. However the Gelfand-Kirillov dimension of any proper quotient of a Verma 𝔤_{0}-module is strictly less than the Gelfand-Kirillov dimension of M(μ′)—see [J1]. This gives the required contradiction.

Remark. Recall that a strongly typical Verma module contains a simple strongly typical Verma submodule.

10.4. Denote by Γ_{0} (resp., Γ_{1}) the set of subsets of Δ_{1}^{+} containing an even (resp., an odd) number of elements. Take an arbitrary λ ∈ 𝔥^{∗} and fix a ℤ_{2}-grading on a Verma module M̃(λ) in such a way that a highest weight vector becomes even. As a 𝔤_{0}-module, M̃ = M̃_{0} ⊕ M̃_{1}. Let us show that each M̃(λ)_{j} has a finite filtration with factors {M(λ − γ)|γ ∈ Γ_{j}} (j = 0, 1).

Set

Q_{0}(π) := {∑_{α∈Δ} k_{α}α| k_{α} ∈ ℤ, ∑_{α∈Δ_{1}} k_{α} is even},
Q_{1}(π) := {∑_{α∈Δ} k_{α}α| k_{α} ∈ ℤ, ∑_{α∈Δ_{1}} k_{α} is odd}.

Note that both Q_{0}(π), Q_{1}(π) are W -stable. We claim that Q_{0}(π) ∩ Q_{1}(π) = ∅ (for 𝔤 = 𝔭𝔰𝔩(n, n) we substitute 𝔥 by 𝔥̂—see 6.1.4). Indeed, if 𝔤 is of the type I then 𝔥 ∩ Z(𝔤_{0}) contains an element z such that z(α) = 1 for any α ∈ Δ_{1}^{+}. Therefore

∑_{α∈Δ} k_{α}α(z) = ∑_{α∈Δ_{1}^{+}}(k_{α} − k_{−α})

and so μ(z) is an even (resp., an odd) integer for μ ∈ Q_{0}(π) (resp., μ ∈ Q_{1}(π)). Let 𝔤 be of the type II. Retain notation of 8.5. One can immediately sees that for 𝔤 ≠ F (4) the sum ∑_{1}^{n} μ_{δ_{i}} is an even (resp., an odd) integer for μ ∈ Q_{0}(π) (resp., μ ∈ Q_{1}(π)). For the remaining case 𝔤 = F (4), μ_{δ_{1}} is integer if μ ∈ Q_{0}(π) and belongs to the set ℤ + \frac{1}{2} if μ ∈ Q_{1}(π). This implies our claim.

The weights of M̃(λ)_{0} (resp., M̃(λ)_{1}) belong to the set (λ−Q_{0}(π)) (resp., (λ−Q_{1}(π))). The weights of M(λ − |γ|) belong to (λ − Q_{j}(π)) iff γ ∈ Γ_{j} (j = 0, 1). This proves that M̃(λ)_{j} has a finite filtration with the factors {M(λ − γ)|γ ∈ Γ_{j}} (j = 0, 1).

Using arguments as above, it is easy to show that for 𝔤 of the type II

Q_{0}(π) = ℤΔ_{0}

and for 𝔤 of the type I

Q_{0}(π) ∩ {μ ∈ 𝔥^{∗}| μ(z) = 0} = ℤΔ_{0}.

This implies that the condition (ii) of Proposition 10.2 is equivalent to the condition

∀γ′, γ′′ ∈ Γ_{j} (λ − |γ′| + ρ_{0}) ∈ W (λ − |γ′′| + ρ_{0}) ⟹ |γ′| = |γ′′|  (10)

for j = 0, 1.

10.5. Remark. The case 𝔤 = 𝔬𝔰𝔭(1, 2l) was treated in [M1], 3.7 and [GL2], 7.2.

11. Appendix

This section contains some lemmas used in the main text. We also give an alternative proof of the fact that the Z(𝔤)-rank of Hom_{𝔤}(Ṽ , Ũ) is equal to dim Ṽ |_{0} for any Ṽ ∈ Irr. This alternative proof does not use Separation Theorem 2.8.1.

11.1. In this subsection we prove some lemmas which were used in the main text.

11.1.1. Recall that a simple 𝔤_{0}-module V (λ) is finite dimensional iff λ + ρ_{0} > w(λ + ρ_{0}) for any w ∈ W, w ≠ id. In principle, the similar fact does not hold for simple finite dimensional 𝔤-modules. For instance, there are triangular decompositions such that the corresponding ρ is equal to 0 and so w.0 = 0 for all w ∈ W even though Ṽ (0) is onedimensional.

However, if Ṽ (λ) is finite dimensional and strongly typical then λ > w.λ for all w ∈ W, w ≠ id. This can be checked in the following way. Fix a strongly typical weight ν such that Ṽ (ν) is finite dimensional. Write the character formula (1) in the form

D′ ch Ṽ (ν) = ∑_{w∈W} sn(w)e^{w.ν}, where D′ := (ch M̃(0))^{−1}.  (11)

Combining the facts that Stab_{W} (ν +ρ) is generated by the reflections it contains and that ch Ṽ (ν) ≠ 0, one obtains w.ν = w′.ν iff w = w′. A strongly typical Verma module M̃(w.ν) has a finite filtration with the factors of the form Ṽ (w′.ν) where w′.ν ≤ w.ν. Therefore ch M̃(w.ν) = ∑_{w′} a_{w,w′} ch Ṽ (w′.ν) where (a_{w,w′}) form “an upper triangular matrix” that is a_{w,w′} = 0 if w′.ν ≰ w.ν and a_{w,w} = 1. Consequently,

D′ ch Ṽ (w.ν) = ∑_{w′} b_{w,w′}e^{w′.ν}

where b_{w,w′} = 0 if w′.ν ≰ w.ν and b_{w,w} = 1. Comparing the last equality with (11), one obtains w′.ν < ν for all w′ ∈ W, w′ ≠ id.

11.1.2. Lemma. Assume that N_{1}, N_{2} are finite dimensional 𝔥-diagonalizable 𝔤-modules and all simple subquotients of N_{1} ⊗N_{2} are typical. Then N_{1} ⊗N_{2} is a completely reducible module.

Proof. Since central characters separate non-isomorphic typical finite dimensional modules, it is enough to show that N_{1} ⊗ N_{2} does not contain non-trivial extensions Ṽ by Ṽ for any typical finite dimensional module Ṽ. Let N be such an extension. Then the highest weight subspace of N is two dimensional and admits a basis {v_{1}, v_{2}} such that v_{1} is primitive and v_{1} ∈ Ũv_{2}. Then v_{1} ∈ U(𝔥)v_{2} and so the action of 𝔥 on N is not semisimple. Hence N_{1} ⊗ N_{2} does not contain submodule isomorphic to N. The assertion follows.

11.1.3. Lemma. For any simple finite dimensional module Ṽ and any μ ∈ 𝔥^{∗}, the set of λ ∈ 𝔥^{∗} such that Ṽ (λ) is finite dimensional and

dim Hom_{𝔤}(Ṽ , Hom(Ṽ (λ − μ), Ṽ (λ)) = dim Ṽ |_{μ}

is Zariski dense in 𝔥^{∗}.

Proof. Fix a simple finite dimensional module Ṽ. Let R be the subset of 𝔥^{∗} consisting of the weights λ such that Ṽ (λ) is a typical finite dimensional module and the tensor product Ṽ^{∗} ⊗ Ṽ (λ) is the direct sum of typical simple modules. Let us show that R is Zariski dense in 𝔥^{∗}.

Indeed, take a finite dimensional 𝔤_{0}-module V (λ). The induced module Ind_{𝔤_{0}}^{𝔤} V (λ) has a simple submodule Ṽ (λ′) which is finite dimensional. The 𝔤_{0}-module V (λ′) is a 𝔤_{0}- submodule of Ṽ (λ′) and so is a 𝔤_{0}-submodule of Ind_{𝔤_{0}}^{𝔤} V (λ). As a 𝔤_{0}-module Ind_{𝔤_{0}}^{𝔤} V (λ) ≅ V (λ)⊗Λ𝔤_{1}. For finite dimensional 𝔤_{0}-modules L, V (μ), the inequality Hom_{𝔤_{0}}(V (ν), V (μ)⊗ L) ≠ 0 implies (ν − μ) ∈ Ω(L). Therefore (λ − λ′) ∈ Ω(Λ𝔤_{1}). Thus for any λ ∈ 𝔥^{∗} such that V (λ) is simple, there exists λ′ ∈ λ + Ω(Λ𝔤_{1}) such that Ṽ (λ′) is simple. Taking into account that the set of λ’s such that dim V (λ) < ∞ is Zariski dense in 𝔥^{∗}, one concludes that the set of λ’s such that dim Ṽ (λ) < ∞ is also Zariski dense in 𝔥^{∗}.

The condition on weight to be atypical is polynomial (see 2.5.2) and so the set of λ’s such that Ṽ (λ) is typical finite dimensional is also Zariski dense. A module Ṽ (λ) is a submodule of Coind_{𝔤_{0}}^{𝔤} V (λ) and so Ṽ^{∗} ⊗Ṽ (λ) is a submodule of Ṽ^{∗} ⊗Coind_{𝔤_{0}}^{𝔤} V (λ). Again Hom_{𝔤_{0}}(V (ν), Ṽ^{∗}⊗Coind_{𝔤_{0}}^{𝔤} V (λ)) ≠ 0 implies (ν−λ) ∈ Ω(Ṽ^{∗}⊗Λ𝔤_{1}). Therefore if Ṽ^{∗}⊗Ṽ (λ) has a subquotient isomorphic to Ṽ (ν), then (ν − λ) ∈ Ω(Ṽ^{∗} ⊗ Λ𝔤_{1}). Using Lemma 11.1.2 and the fact that the set Ω(Ṽ^{∗} ⊗ Λ𝔤_{1}) is finite, we conclude that R is Zariski dense in 𝔥^{∗}.

Frobenius reciprocity gives

Hom_{𝔤}(Ṽ , Hom(Ṽ (λ − μ), Ṽ (λ))) ≅ Hom_{𝔤}(Ṽ (λ − μ), Ṽ^{∗} ⊗ Ṽ (λ)).

Take λ ∈ R and denote by m_{ν} the multiplicity of Ṽ (ν) in the completely reducible module Ṽ^{∗} ⊗ Ṽ (λ). The character formula (1) gives

D ch(Ṽ^{∗} ⊗ Ṽ (λ)) = ∑_{w∈W} sn(w)e^{w.λ} ch Ṽ^{∗} = ∑_{ν} m_{ν} ∑_{w∈W} sn(w)e^{w.ν}.

For typical finite dimensional modules Ṽ (ν), Ṽ (ν′) the equality w.ν = w′.ν′ implies ν = ν′ and w = w′ (see 11.1.1). Therefore dim Hom_{𝔤}(Ṽ (λ − μ), Ṽ (λ) ⊗ Ṽ^{∗}) = m_{λ−μ} is equal to the coefficient of the term e^{λ−μ} in the expression ∑_{w∈W} sn(w)e^{w.λ} ch Ṽ^{∗}. For “sufficiently large” λ ∈ R this coefficient is equal to dim Ṽ^{∗}|_{−μ}. More precisely, take λ ∈ R, w ≠ id and α ∈ π_{0} such that s_{α}w < w. Then w^{−1}α ∈ Δ_{0}^{−} and so (w(λ + ρ), α) = (λ + ρ, w^{−1}α) < 0 by 11.1.1. Then

(λ − μ − w.λ, α) = (λ + ρ − μ, α) − (w(λ + ρ), α) > (λ + ρ − μ, α).

As a consequence, for any λ belonging to the set

R_{1} := {λ ∈ R| (λ, α) > (ξ + μ − ρ, α) ∀ξ ∈ Ω(Ṽ^{∗}), α ∈ π_{0}}

the inclusion (λ−μ) ∈ (w.λ+Ω(Ṽ^{∗})) implies w = id. Hence m_{λ−μ} = dim Ṽ^{∗}|_{−μ} = dim Ṽ |_{μ} for any λ ∈ R_{1}.

For any λ ∈ R and α ∈ π_{0}, the value (λ, α) belongs to ℕ^{+}, since 𝔤_{0}-module V (λ) is finite dimensional. Thus R_{1} is obtained from R by removing the points lying at finitely many hyperplanes. Taking into account that R is Zariski dense in 𝔥^{∗}, one concludes that R_{1} is also Zariski dense. This completes the proof.

Remark. In Proposition 11.3.1 we prove a stronger assertion for the particular case μ = 0.

11.1.4. Lemma. For all λ, ν ∈ 𝔥^{∗} and all simple finite dimensional Ṽ one has

dim Hom_{𝔤}(Ṽ , Hom(M̃(λ − μ), M̃(λ)^{#})) = dim Ṽ |_{μ}

Proof. Frobenius reciprocity gives

Hom_{𝔤}(Ṽ , Hom(M̃(λ − μ), M̃(λ)^{#})) ≅ Hom_{𝔤}(M̃(λ − μ), M̃(λ)^{#} ⊗ Ṽ^{∗}).

Using notation of 2.5.1 one has

Hom_{𝔤}(M̃(λ − μ), M̃(λ)^{#} ⊗ Ṽ^{∗}) ≅ Hom_{𝔟}(ℂ_{λ−μ}, M̃(λ)^{#} ⊗ Ṽ^{∗}).  (12)

All Verma modules are isomorphic as 𝔫^{−}-modules. Therefore their duals are isomorphic as 𝔫^{+}-modules. Furthermore for any λ′ ∈ 𝔥^{∗} the 𝔟-modules M̃(λ)^{#} and M̃(λ′)^{#} ⊗ ℂ_{λ−λ′} are isomorphic. Taking into account (12), one concludes that

k := dim Hom_{𝔤}(Ṽ , Hom(M̃(λ − μ), M̃(λ)^{#}))

does not depend on λ (for fixed Ṽ and μ).

Recall that Ṽ (λ − μ) is a quotient of M̃(λ − μ) and Ṽ (λ) is a submodule of M̃(λ)^{#}. Consequently k ≥ dim Hom_{𝔤}(Ṽ (λ − μ), Ṽ (λ) ⊗ Ṽ^{∗}) for all λ ∈ 𝔥^{∗}. Using Lemma 11.1.3, one obtains k ≥ dim Ṽ |_{μ}.

To verify that k ≤ dim Ṽ |_{μ} fix λ such that M̃(λ)^{#} is simple. Denote by m_{1} (resp., m_{2}) a highest weight vector of M̃(λ − μ) (resp., M̃(λ)^{#}). Consider a map

ι : Hom_{𝔤}(Ṽ , Hom(M̃(λ − μ), M̃(λ)^{#})) → Ṽ^{∗}|_{−μ} s.t
ι(ψ)(v)m_{2} = ψ(v)(m_{1}) ∀v ∈ Ṽ |_{μ}, ψ ∈ Hom_{𝔤}(Ṽ , Hom(M̃(λ − μ), M̃(λ)^{#}))

Let us show that ι is a monomorphism. Take a non-zero element ψ ∈ Hom_{𝔤}(Ṽ , Hom(M̃(λ−

μ), M̃(λ)^{#})). The vector space ψ(Ṽ)(M̃(λ − μ)) is a non-zero 𝔤-submodule of a simple module M̃(λ)^{#}; thus it coincides with M̃(λ)^{#}. One has

ψ(Ṽ)(M_{1}) = ψ(Ṽ)(U(𝔫^{−})m_{1}) = U(𝔫^{−})ψ(Ṽ)(m_{1})

since ψ(Ṽ) is ad 𝔤-stable. Therefore ψ(Ṽ)(m_{1}) contains the highest weight vector m_{2} that is ψ(v)(m_{1}) = m_{2} for a certain v ∈ Ṽ. Obviously one can choose v be a weight vector; then v ∈ Ṽ |_{μ} and ι(ψ)(v) = 1. Hence ι is a monomorphism and so k ≤ dim Ṽ^{∗}|_{−μ} = dim Ṽ |_{μ} as required.

11.1.5. Remark. If M̃(λ) is simple, the above lemma gives

dim Hom_{𝔤}(Ṽ , Hom(M̃(λ), M̃(λ)) = dim Ṽ |_{0}

for any simple finite dimensional Ṽ.

11.1.6. Lemma. Let A be a polynomial algebra and W be a finite group acting on A. Assume that p, p′ ∈ A are such that p′/p is W -invariant and let q be a maximal W -invariant divisor of p^{|W |}. Then there exist q′ ∈ A^{W} such that p′/p = q′/q.

Proof. Any non-zero polynomial has a unique factorization into irreducible ones. Let a/b be a reduced form of the fraction p′/p. For any s ∈ W one has s(a)/s(b) = a/b and so b/s(b) is a scalar. Since W is finite, 1 = (b/s(b))^{|W |} = b^{|W |}/s(b^{|W |}). Hence b^{|W |} ∈ A^{W}. Since p is divisible by b, p^{|W |} is divisible by b^{|W |}. Therefore q is divisible by b^{|W |}. Then there exist q′ ∈ A such that q′/q = a/b = p′/p. The W -invariance of both q and p′/p implies the W -invariance of q′.

11.2. In this subsection we present alternative proofs of Corollary 4.5.2 and Theorem 4.7.4.

Another proof of Corollary 4.5.2.

11.2.1. Fix a subset Y of Irr and set X := {ν ∈ 𝔥^{∗}| Ṽ (ν) ∈ Y }. For each ν ∈ X choose a Z(𝔤)-basic system θ_{1}^{ν}, …, θ_{s(ν)}^{ν} of Hom_{𝔤}(Ṽ (ν), Ũ). Let z_{ν} ∈ Z(𝔤) be such an element that Hom_{𝔤}(Ṽ (ν), Ũ[z_{ν}^{−1}]) is a free Z(𝔤)[z_{ν}^{−1}]-module generated by θ_{1}^{ν}, …, θ_{s(ν)}^{ν} (such an element exists by Proposition 4.4.3). Let S ⊂ Z(𝔤) \ {0} be a multiplicative closed set containing {T^{2}; z_{ν}, ν ∈ X}. Denote by A the localization of Z(𝔤) on S and by Ũ_{A} the localization of Ũ on S. Both actions ad 𝔤 and ad′ 𝔤 can be canonically extended to Ũ_{A}. Note that as ad 𝔤-module (resp., ad′ 𝔤-module) Ũ_{A} belongs to Fin. For any ν ∈ X the localized algebra Z(𝔤)[z_{ν}^{−1}] is a subalgebra of A and so Hom_{𝔤}(Ṽ (ν), Ũ_{A}) is a free A-module generated by θ_{1}^{ν}, …, θ_{s(ν)}^{ν}. Clearly Ũ_{A} inherits the structure of a superalgebra and its centre is equal to A. Now we are ready to formulate the

11.2.2. Lemma. There exist ad 𝔤-submodules H, N of Ũ_{A} such that

(i) the multiplication map induces a monomorphism H ⊗ A → Ũ_{A},
(ii) Ũ_{A} = HA ⊕ N,
(iii) H ≅ ⊕_{ν∈X} E(Ṽ (ν)^{s_{ν}}),
(iv) Soc H = ∑_{ν∈X} ∑_{i=1}^{s(ν)} θ_{i}^{ν}(Ṽ (ν)),
(v) Hom_{𝔤}(Ṽ (ν), N) = 0 ∀ν ∈ X.

Proof. Since θ_{1}^{ν}, …, θ_{s(ν)}^{ν} is a Z(𝔤)-basic system of Hom_{𝔤}(Ṽ (ν), Ũ), the sum

L := ∑_{ν∈X} ∑_{i=1}^{s(ν)} θ_{i}^{ν}(Ṽ (ν))

is direct. By 3.3.2, the ad 𝔤-module T Ũ is injective. Since T^{2} ∈ Z(𝔤), the ad 𝔤-module T^{−1}Ũ is an injective submodule of Ũ_{A}. Therefore T^{−1}Ũ contains an injective envelope H of L and H ≅ ⊕_{ν∈X} E(Ṽ (ν)^{s_{ν}}). The multiplication map induces a 𝔤-homomorphism ϕ : H ⊗ A → Ũ_{A}.

Recall that Hom_{𝔤}(Ṽ (ν), Ũ[z_{ν}^{−1}]) is a free Z(𝔤)[z_{ν}^{−1}]-module generated by θ_{1}^{ν}, …, θ_{s(ν)}^{ν} for any ν ∈ X. Therefore the module Hom_{𝔤}(Ṽ (ν), Ũ_{A}) is a free A-module generated by θ_{1}^{ν}, …, θ_{s(ν)}^{ν}. This means that for any ν ∈ X the restriction of ϕ on ∑_{i=1}^{s(ν)} θ_{i}^{ν}(Ṽ) ⊗ A is a monomorphism and its image coincides with the isotypical component of Ṽ (ν) in the socle of Ũ_{A}. Then the restriction of ϕ on L⊗A is a monomorphism and Soc Ũ_{A} = ϕ(L⊗A)⊕N′ where Hom_{𝔤}(Ṽ (ν), N′) = 0 for any ν ∈ X. Recall that Soc H = L and so Soc(H ⊗ A) = L ⊗ A. Therefore ϕ is a monomorphism by 3.1.2. From 3.2.3, it follows that H ⊗ A is an injective module in Fin. Therefore Ũ_{A} contains a submodule N such that Ũ_{A} = HA ⊕ N. Then Soc Ũ_{A} = HA ⊕ Soc N and thus Soc N ≅ N′. This completes the proof.

11.3. In this subsection we prove that the Z(𝔤)-rank of the module Hom_{𝔤}(Ṽ , Ũ) is equal to dim Ṽ |_{0} for any Ṽ ∈ Irr. Recall that, by Remark 4.4.5, in order to prove this assertion it is suffices to find λ ∈ 𝔥^{∗} such that the image of Soc Ũ in End_{ℂ}(Ṽ (λ)) contains an ad 𝔤-submodule L ≅ Ṽ^{⊕ dim _{V}|_{0}}.

To find this element λ, we use Density Theorem stating the surjectivity of the natural map Ũ → End(Ṽ) for any Ṽ ∈ Irr. As it is shown in Lemma 11.1.3, for any Ṽ ∈ Irr there exists λ such that End_{ℂ}(Ṽ (λ)) contains a submodule Ṽ^{⊕ dim _{V}|_{0}}. This would imply the assertion when Ũ = Soc Ũ that is for completely reducible Lie superalgebras (however for these algebras the assertion immediately follows from Separation theorems). The general case requires a certain extra work: one should choose λ in such a way that any copy of Ṽ inside the socle of End_{ℂ}(Ṽ (λ)) lies in the image of Soc Ũ. This can be done with the help of Proposition 11.3.1.

11.3.1. Proposition. For any Ṽ ∈ Irr the set of λ ∈ 𝔥^{∗} such that dim Ṽ (λ) < ∞ and there exists a monomorphism

E(Ṽ)^{⊕ dim _{V}|_{0}} → End_{ℂ}(Ṽ (λ)).

is a Zariski dense subset of 𝔥^{∗}.

Proof. Fix Ṽ ∈ Irr. For any λ ∈ 𝔥^{∗} denote by f_{λ} the natural homomorphism Ũ → End_{ℂ}(Ṽ (λ)) and by C_{V}(λ) the isotypical component of Ṽ in the socle of End_{ℂ}(Ṽ (λ)).

Let X be the set of ν ∈ 𝔥^{∗} such that dim Ṽ (ν) ∈ Irr and E(Ṽ (ν)) has a subquotient isomorphic to Ṽ. We claim that X is finite. Indeed, E(Ṽ (ν)) is a submodule of Coind V (ν). As 𝔤_{0}-module, Coind V (ν) ≅ V (ν) ⊗ Λ𝔤_{1} and so the weight of any 𝔤_{0}-primitive weight vector of Coind V (ν) belongs to the set ν + Ω(Λ𝔤_{1}). Thus the highest weight of Ṽ belongs to this set for any ν ∈ X. Therefore X is finite.

Retain notation of 4.2. For each ν ∈ X choose a Z(𝔤)-basic system θ_{1}^{ν}, …, θ_{s(ν)}^{ν} of Hom_{𝔤}(Ṽ , Ũ). The collection Ψθ_{1}^{ν}, …, Ψθ_{s(ν)}^{ν} is a Z(𝔤)-basic system of I(Ṽ) and so it is also a S(𝔥)-basic system of I(Ṽ)S(𝔥) (see Lemma 4.4.2). Denote p^{ν} a minor of this S(𝔥)-basic system. Recall that X is finite and take z ∈ Z(𝔤) such that P(z) = t^{2} ∏_{ν∈X} ∏_{w∈W} w.p^{ν}. By Proposition 4.4.3, for any ν ∈ X the localized module Hom_{𝔤}(Ṽ (ν), Ũ)[z^{−1}] is freely generated over Z(𝔤)[z^{−1}] by {θ_{1}^{ν}, …, θ_{s(ν)}^{ν}}. Therefore one can apply Lemma 11.2.2 to the set X and the algebra A := Z(𝔤)[z^{−1}]. This gives Ũ_{A} = (HA)⊕N where Hom_{𝔤}(Ṽ (ν), N) = 0 for any ν ∈ X. Thus Soc N = ⊕_{i∈I} Ṽ (μ_{i}) where μ_{i} ∉ X for each i ∈ I. By 3.2.2, N is isomorphic to a submodule of E(Soc N) ≅ ⊕_{i∈I} E(Ṽ (μ_{i})). Each module E(Ṽ (μ_{i})) does not have a subquotient isomorphic to Ṽ because μ_{i} ∉ X. Therefore N also does not have a subquotient isomorphic to Ṽ.

From Lemma 11.1.3 it follows that the set

R := {λ ∈ 𝔥^{∗}| dim Ṽ (λ) < ∞, C_{V}(λ) = Ṽ^{⊕_{V}|_{0}}, P(z)(λ) ≠ 0}

is Zariski dense in 𝔥^{∗}. Let us show that for any λ ∈ R there exists an injective map required in the proposition.

Take λ ∈ R. The module Ṽ (λ) is finite dimensional and, by Density Theorem, the map f_{λ} is surjective. Since z acts on Ṽ (λ) by a non-zero scalar, one can extend f_{λ} to the epimorphism Ũ_{A} → End_{ℂ}(Ṽ (λ)). Taking into account that N has no subquotient isomorphic to Ṽ, one concludes that C_{V}(λ) ⊆ f_{λ}(HA) = f_{λ}(H).

Let us show that the restriction of f_{λ} to H is a monomorphism. By 3.1.2, it is enough to verify that the restriction of f_{λ} to

Soc H = ∑_{ν∈X} ∑_{i=1}^{s(ν)} θ_{i}^{ν}(Ṽ (ν))

is a monomorphism. Since Ṽ (ν) are pairwise non-isomorphic for different ν, it suffices to check that the restriction of f_{λ} to ∑_{i=1}^{s(ν)} θ_{i}^{ν}(Ṽ (ν)) is a monomorphism for each ν ∈ X. This follows from Corollary 4.1.2. Indeed, fix ν ∈ X. Recall that p_{ν} is a non-zero s(ν) × s(ν)-minor of the matrix (P(θ_{j}(v_{i})))_{i=1,r}^{j=1,s(ν)} where v_{1}, …, v_{r} is basis of Ṽ (ν)|_{0}. Since P(z)(λ) ≠ 0 one has p^{ν}(λ) ≠ 0 and so, by Corollary 4.1.2, the restriction of f_{λ} on ∑_{i=1}^{s(ν)} θ_{i}^{ν}(Ṽ (ν)) is a monomorphism.

Combining the facts that H is injective and that C_{V}(λ) ⊆ f_{λ}(H) ≅ H, one completes the proof.

11.3.2. Corollary. For any simple finite dimensional module Ṽ the Z(𝔤)-rank of Hom_{𝔤}(Ṽ , Ũ) is equal to dim Ṽ |_{0}.

Proof. Set r := dim Ṽ |_{0}. By Proposition 11.3.1 one can choose λ ∈ 𝔥^{∗} such that Ṽ (λ) is finite dimensional and there exists a monomorphism

ϕ : L := E(Ṽ)^{⊕r} ⟶ End_{ℂ}(Ṽ (λ)).

By Density Theorem the natural map f_{λ} : Ũ → End_{ℂ}(Ṽ (λ)) is surjective. Since L is projective in Fin, Ũ contains a submodule L′ such that the restriction of f_{λ} to L′ provides an isomorphism L′ ⟶^{∼} L. In the light of Remark 4.4.5, a basis θ_{1}, …, θ_{r} of Hom_{𝔤}(Ṽ , L) is a Z(𝔤)-basic system of Hom_{𝔤}(Ṽ , Ũ).

Another proof of Theorem 4.7.4 11.3.3. Theorem. An ad 𝔤-submodule L of Ũ is a generic harmonic space iff

(a) L ≅ ⊕_{_{V}∈Irr} E(Ṽ)^{⊕ dim _{V}|_{0}}
and one of the following conditions holds
(b) ∀Ṽ ∈ Irr det J (Ṽ; L) ≠ 0
(c) the multiplication map provides an embedding L ⊗ Z(𝔤) → Ũ.

Proof. Applying Lemma 11.2.2 for Y := Irr and S := (Z(𝔤) \ {0}), we conclude the existence of a generic harmonic space satisfying the condition (a).

Let us show that all generic harmonic spaces are pairwise isomorphic as ad 𝔤-modules. Indeed, let L and L′ be generic harmonic spaces. Since Ũ is countably dimensional, one can choose the corresponding sets S, S′ ⊂ Z(𝔤) \ {0} having countable number of elements. Take a maximal ideal m of Z(𝔤) such that m ∩ (S ∪ S′) = ∅. Then as ad 𝔤-modules L ≅ Ũ/(mŨ) ≅ L′. Hence all generic harmonic spaces satisfy the condition (a).

Let L be a generic harmonic space. Then the condition (c) obviously holds. Moreover, for any Ṽ ∈ Irr a basis θ_{1}, …, θ_{r} of Hom_{𝔤}(Ṽ , L) is a Z(𝔤)-basic system of Hom_{𝔤}(Ṽ , Ũ). Therefore det J (Ṽ; L) ≠ 0 and so (b) holds as well.

Fix L satisfying (a) and (b). For Ṽ ∈ Irr denote by q(Ṽ) a maximal W.-invariant divisor

of (det J (Ṽ; L))^{|W |}. Take S ⊂ Z(𝔤) such that P(S) consists of the elements t^{2}q(Ṽ) where Ṽ ∈ Irr. It is easy to deduce from Proposition 4.4.3 that the multiplication map provides an isomorphism Soc L ⊗ Z(𝔤)[S^{−1}] ⟶^{∼} Soc(Ũ[S^{−1}]). Since L is injective, L ⊗ Z(𝔤)[S^{−1}] is also injective and so L ⊗ Z(𝔤)[S^{−1}] ⟶^{∼} Ũ[S^{−1}]. Hence L is a generic harmonic space.

Finally, fix L satisfying (a) and (c). For any Ṽ ∈ Irr a basis of Hom_{𝔤}(Ṽ , L) contains dim Ṽ |_{0} elements due to the condition (a) and these elements Z(𝔤)-linearly independent due to condition (c). Hence these elements form a Z(𝔤)-basic system of Hom_{𝔤}(Ṽ , Ũ) that is det J (Ṽ; L) ≠ 0. Thus L fulfills the condition (b) as well. This completes the proof.

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  email:  gorelik@mpim-bonn.mpg.de