Twisted Localization of Weight Modules

Abstract

We discuss how the twisted localization functor leads to a classification of the simple objects and a description of the injectives in various categories of weight modules. The article is a survey on existing results for finite-dimensional simple Lie algebras and superalgebras, affine Lie algebras, and algebras of differential operators.

Partially supported by NSA grant H98230-13-1-0245.

1 Introduction

The study of weight modules of Lie algebras and superalgebras have attracted considerable attenion in the last 30 years. Examples of weight modules include parabolically induced modules (in particular, modules in the category $\mathcal{O}$), $\mathcal{D}$-modules, generalized Harish-Chandra modules, among others. The first major steps in the systematic study of weight modules were made in the 1980s and 1990s by G. Benkart, D. Britten, S. Fernando, V. Futorny, A. Joseph, F. Lemire, and others, [3, 6, 19, 20, 30]. Two remarkable results for simple finite-dimensional Lie algebras include the Fernando–Futorny parabolic induction theorem, [19, 20], reducing the classification of all simple weight modules with finite weight multiplicities to the classification of all simple cuspidal modules (defined in Sect. 3.5); and the result of Fernando [19] that cuspidal modules exist only for Lie algebras of type A and C. In 2000 O. Mathieu [32], classified the simple cuspidal modules, and in this way completed the classification of all simple weight modules with finite weight multiplicities of all finite-dimensional reductive Lie algebras.

An essential part of Mathieu’s classification result is the application of the twisted localization functor: the result states that every cuspidal module is a twisted localization of a highest weight module. The definition of a twisted localization functor for Lie algebras in [32] generalizes other fundamental constructions: Deodhar’s localization functors, and in particular, Enright’s completions, see [11] and [18]. The twisted localization functor is especially convenient on categories of bounded weight modules, i.e., those weight modules whose sets of weight multiplicities are uniformly bounded. The functor admits several important properties: it preserves central characters; it commutes with the parabolic induction and the translation functors; it generally maps injectives and projectives to injectives and projectives, respectively. After its inception in [32], it was obvious that the application of the functor is not limited only to finite-dimensional Lie algebras. For the last 10 years the twisting localization functor was used in numerous other important classification results and in particular for:

• Description of injective modules in categories of bounded and cuspidal weight modules [25–27];

• Classification of simple weight modules with finite weight multiplicities of affine Lie algebras and Schrödinger algebras [13, 17];

• Classification of simple and injective weight modules of algebras of twisted differential operators on the projective space [27].

A detailed account on the twisted localization functor can be found in Chapter 3 of [33].

The main goal of this paper is to present a survey on the applications of the twisted localization functor in the classification of the simple and injective objects of various categories of weight modules. Due to the technical nature of the classification results, the content of the paper is limited to the presentation of simple and indecomposable injective objects in terms of twisted localization, but it does not address the uniqueness of this presentation. One should note that such uniqueness is present for all results collected in the paper and the reader is referred to the corresponding references for details.

The content of the paper is as follows. Section 3 is devoted to the background material on weight modules of associative algebras. The notations and definitions of all categories of weight modules is introduced in this section. The cases of simple finite-dimensional Lie algebras and superalgebras, Weyl algebras of differential operators, and affine Lie algebras are considered as separate subsections. In Sect. 4 we first introduce the twisted localization functor in its most general setting and then following the subsection order of Sect. 3, we present the classification results on the simple and injective weight modules on a case-by-case basis.

2 Index of Notation

Below we list some notation that are frequently used in the paper under the section number they are first introduced.

3.1 supp, M ^λ, M ^(λ), $(\mathcal{U},\mathcal{H})$-mod, $^{\mathrm{f}}(\mathcal{U},\mathcal{H})$-mod, $^{\mathrm{w}}(\mathcal{U},\mathcal{H})$-mod, $^{\mathrm{wf}}(\mathcal{U},\mathcal{H})$- mod.

3.2 $\mathcal{D}(n)$, $\mathcal{D}$, $(\mathcal{D},\mathcal{H})$-mod, $^{\mathrm{b}}(\mathcal{D},\mathcal{H})$-mod, $(\mathcal{D},\mathcal{H})_{\nu }$-mod, $^{\mathrm{b}}(\mathcal{D},\mathcal{H})_{\nu }$-mod.

3.3 | ν | , E, $(\mathcal{D}^{E},\mathcal{H})^{a}$-mod, $(\mathcal{D}^{E},\mathcal{H})_{\nu }^{a}$-mod, $_{\mathrm{s}}(\mathcal{D},\mathcal{H})$-mod, $_{\mathrm{s}}(\mathcal{D}^{E},\mathcal{H})^{a}$-mod, $_{\mathrm{s}}^{\mathrm{b}}(\mathcal{D}^{E},\mathcal{H})^{a}$-mod, $_{\mathrm{s}}(\mathcal{D}^{E},\mathcal{H})_{\nu }^{a}$-mod, $_{\mathrm{s}}^{\mathrm{b}}(\mathcal{D}^{E},\mathcal{H})_{\nu }^{a}$-mod, $S_{\mathcal{H}'}$.

3.4 $\mathcal{D}(\infty )$

3.5 $\mathcal{B}$, $\mathcal{G}\mathcal{B}$, $\overline{\mathcal{B}}$, $\overline{\mathcal{G}\mathcal{B}}$, $\mathcal{B}_{\nu }$, $\mathcal{G}\mathcal{B}_{\nu }$, $\mathcal{B}^{\lambda }$, $\mathcal{G}\mathcal{B}^{\lambda }$, $\mathcal{B}_{\nu }^{\lambda }$, $\mathcal{G}\mathcal{B}_{\nu }^{\lambda }$, $\overline{\mathcal{B}}_{\nu }^{\lambda }$, $\overline{\mathcal{G}\mathcal{B}}_{\nu }^{\lambda }$.

3.6 $\mathfrak{h}_{\bar{0}}^{ss}$.

3.7 $\mathfrak{G}$, $\mathfrak{H}$, $\mathcal{L}(\mathfrak{g})$, $\mathcal{L}(\mathfrak{g},\sigma )$, $\mathcal{A}(\mathfrak{g})$, $\mathcal{A}(\mathfrak{g},\sigma )$.

4.1 D _F , $\varTheta _{F}^{\mathbf{x}}$, $\varTheta _{F}^{\mathbf{x}}$, $D_{F}^{\mathbf{x}}$.

4.2 $D_{i}^{+},D_{i}^{-}$, $D_{i}^{x,+},D_{i}^{x,-}$, $D_{i,j}$, $D_{i,j}^{x}$.

4.4 D _Γ , D _Γ ^x, γ, $\mathcal{F}_{\nu },\mathcal{F}_{\nu }^{\log }$, σ _J , $\mathcal{F}_{\nu }^{\log }(J)$, Int(ν), Γ _a , Φ, $\psi _{A},\varPsi _{A},\psi _{C},\varPsi _{C}$, $S_{\mathfrak{h}}$.

4.6 $\mathcal{L}_{a_{1},\ldots,a_{k}}(Y _{1} \otimes \ldots \otimes Y _{k})$, $\mathcal{L}_{a_{1},\ldots,a_{k}}^{\sigma }(Y _{ 1} \otimes \ldots \otimes Y _{k})$, $V _{a_{1},\ldots,a_{k}}(Y _{1} \otimes \ldots \otimes Y _{k})$.

3 Background

In this paper the ground field is $\mathbb{C}$ and $\mathbb{N}$ stands for the set of positive integers. All vector spaces, algebras, and tensor products are assumed to be over $\mathbb{C}$ unless otherwise stated.

3.1 Categories of Weight Modules of Associative Algebras

Let $\mathcal{U}$ be an associative unital algebra and let $\mathcal{H}\subset \mathcal{U}$ be a commutative subalgebra. We assume in addition that $\mathcal{H}$ is a polynomial algebra identified with the symmetric algebra of a vector space $\mathfrak{h}$, and that we have a decomposition

$$\displaystyle{\mathcal{U} =\bigoplus _{\mu \in \mathfrak{h}^{{\ast}}}\mathcal{U}^{\mu },}$$

where

$$\displaystyle{\mathcal{U}^{\mu } =\{ x \in \mathcal{U}\vert [h,x] =\mu (h)x,\forall h \in \mathfrak{h}\}.}$$

Let $Q_{\mathcal{U}} = \mathbb{Z}\varDelta _{\mathcal{U}}$ be the $\mathbb{Z}$-lattice in $\mathfrak{h}^{{\ast}}$ generated by $\varDelta _{\mathcal{U}} =\{\mu \in \mathfrak{h}^{{\ast}}\;\vert \;\mathcal{U}^{\mu }\neq 0\}$. We obviously have $\mathcal{U}^{\mu }\mathcal{U}^{\nu }\subset \mathcal{U}^{\mu +\nu }$.

We call a $\mathcal{U}$-module M a generalized weight $(\mathcal{U},\mathcal{H})$ -module if $M =\bigoplus _{\lambda \in \mathfrak{h}^{{\ast}}}M^{(\lambda )}$, where

$$\displaystyle{M^{(\lambda )} =\{ m \in M\vert (h -\lambda (h)\mbox{ Id})^{N}m = 0\,\text{for some}\,N > 0\,\text{and all}\,h \in \mathfrak{h}\}.}$$

Equivalently, generalized weight modules are those on which $\mathfrak{h}$ acts locally finitely. We call M ^(λ) the generalized weight space of M and $\mathrm{dim}M^{(\lambda )}$ the weight multiplicity of the weight λ. Note that

$$\displaystyle{ \mathcal{U}^{\mu }M^{(\lambda )} \subset M^{(\mu +\lambda )}. }$$

(1)

A generalized weight module M is called a weight module if $M^{(\lambda )} = M^{\lambda },$ where

$$\displaystyle{M^{\lambda } =\{ m \in M\vert (h -\lambda (h)\mbox{ Id})m = 0\,\text{ for all }h \in \mathfrak{h}\}.}$$

Equivalently, weight modules are those on which $\mathfrak{h}$ acts semisimply. Weight and generalized weight modules in similar situations and general setting are also studied in [15].

By $(\mathcal{U},\mathcal{H})$-mod and $^{\mathrm{w}}(\mathcal{U},\mathcal{H})$-mod we denote the category of generalized weight modules and weight modules, respectively. Furthermore, by $^{\mathrm{f}}(\mathcal{U},\mathcal{H})$-mod and $^{\mathrm{b}}(\mathcal{U},\mathcal{H})$-mod we denote the subcategories of $(\mathcal{U},\mathcal{H})$-mod consisting of modules with finite weight multiplicities and bounded set of weight multiplicities, respectively. By $^{\mathrm{wf}}(\mathcal{U},\mathcal{H})$-mod and $^{\mathrm{wb}}(\mathcal{U},\mathcal{H})$-mod we denote the subcategories of $^{\mathrm{w}}(\mathcal{U},\mathcal{H})$-mod that are in $^{\mathrm{f}}(\mathcal{U},\mathcal{H})$-mod and $^{\mathrm{b}}(\mathcal{U},\mathcal{H})$-mod, respectively.

For any module M in $(\mathcal{U},\mathcal{H})$-mod we set

$$\displaystyle{\mbox{ supp}\,M:=\{\lambda \in \mathfrak{h}^{{\ast}}\;\vert \;M^{(\lambda )}\neq 0\}}$$

to be the support of M. It is clear from (1) that $\mathrm{Ext}_{\mathcal{A}}^{1}(M,N) = 0$ if $(\mathrm{supp}M + Q_{U}) \cap \mathrm{supp}N = \varnothing$, where $\mathcal{A}$ is any of the categories of generalized weight modules or weight modules defined above. Then we have

$$\displaystyle{ (\mathcal{U},\mathcal{H})\text{-}mod =\bigoplus _{\overline{\mu }\in \mathfrak{h}^{{\ast}}/Q}(\mathcal{U},\mathcal{H})_{\overline{\mu }}\mathrm{\text{-}mod}, }$$

(2)

where $(\mathcal{U},\mathcal{H})_{\overline{\mu }}$-mod denotes the subcategory of $(\mathcal{D},\mathcal{H})$-mod consisting of modules M with $\mathrm{supp}M \subset \overline{\mu } =\mu +Q$. We similarly define $^{\mathrm{w}}(\mathcal{U},\mathcal{H})_{\overline{\mu }}$-mod, $^{\mathrm{f}}(\mathcal{U},\mathcal{H})_{\overline{\mu }}$-mod, $^{\mathrm{b}}(\mathcal{U},\mathcal{H})_{\overline{\mu }}$-mod, $^{\mathrm{wf}}(\mathcal{U},\mathcal{H})_{\overline{\mu }}$-mod, and $^{\mathrm{wb}}(\mathcal{U},\mathcal{H})_{\overline{\mu }}$-mod, and obtain the corresponding support composition where the direct summands are parametrized by elements of $\mathfrak{h}^{{\ast}}/Q$. With a slight abuse of notation, for $\mu \in \mathfrak{h}^{{\ast}}$ we set $(\mathcal{U},\mathcal{H})_{\mu }\mathrm{\text{-}mod} = (\mathcal{U},\mathcal{H})_{\overline{\mu }}$-mod, etc.

3.2 Weight $\mathcal{D}(n)$-Modules

Let $\mathcal{D}(n)$ be the Weyl algebra, i.e., the algebra of polynomial differential operators of the ring $\mathbb{C}[t_{1},\ldots,t_{n}]$ and consider $\mathcal{U} = \mathcal{D}(n)$. When n ≥ 1 is fixed, we use the notation $\mathcal{D}$ for $\mathcal{D}(n)$. Let $\mathfrak{h} = \mbox{ Span}\{t_{1}\partial _{1},\ldots,t_{n}\partial _{n}\}$ and hence $\mathcal{H} = \mathbb{C}[t_{1}\partial _{1},\ldots,t_{n}\partial _{n}]$ is a maximal commutative subalgebra of $\mathcal{D}$. Note that the adjoint action of the abelian Lie subalgebra $\mathfrak{h}$ on $\mathcal{D}$ is semisimple. We identify $\mathbb{C}^{n}$ with the dual space of $\mathfrak{h} = \mbox{ Span}\{t_{1}\partial _{1},\ldots,t_{n}\partial _{n}\}$, and fix $\{\varepsilon _{1},\ldots,\varepsilon _{n}\}$ to be the standard basis of this space, i.e., $\varepsilon _{i}(t_{j}\partial _{j}) =\delta _{ij}$. Then $Q =\bigoplus _{ i=1}^{n}\mathbb{Z}\varepsilon _{i}$ is identified with $\mathbb{Z}^{n}$, and

$$\displaystyle{ \mathcal{D} =\bigoplus _{\mu \in \mathbb{Z}^{n}}\mathcal{D}^{\mu }. }$$

Here $\mathcal{D}^{0} = \mathcal{H}$ and each $\mathcal{D}^{\mu }$ is a free left $\mathcal{H}$-module of rank 1 with generator $\prod _{\mu _{i}\geq 1}t_{i}^{\mu _{i}}\prod _{ \mu _{j}<0}\partial _{j}^{-\mu _{j}}$.

From this we see that the simple objects of $^{\mathrm{f}}(\mathcal{D},\mathcal{H})$-mod (equivalently, of $^{\mathrm{wf}}(\mathcal{D},\mathcal{H})$-mod) are in $^{\mathrm{b}}(\mathcal{D},\mathcal{H})$-mod, i.e., have bounded sets of weight multiplicities. Using the description of these simple objects (see for example Theorem 4.5), we obtain that $^{\mathrm{b}}(\mathcal{D},\mathcal{H})\text{-}mod = ^{\mathrm{f}}(\mathcal{D},\mathcal{H})$-mod and $^{\mathrm{wb}}(\mathcal{D},\mathcal{H})\text{-}mod = ^{\mathrm{wf}}(\mathcal{D},\mathcal{H})$-mod. The latter category was studied in [2] and [25] and the former in [1].

The support of every $(\mathcal{D},\mathcal{H})$-module will be considered as a subset of $\mathbb{C}^{n}$ and we have a natural decomposition

$$\displaystyle{(\mathcal{D},\mathcal{H})\text{-}mod =\bigoplus _{\overline{\nu }\in \mathbb{C}^{n}/\mathbb{Z}^{n}}(\mathcal{D},\mathcal{H})_{\overline{\nu }}\text{-}mod,}$$

As before, for $\nu \in \mathbb{C}^{n}$ we write $(\mathcal{D},\mathcal{H})_{\nu }\text{-}mod = (\mathcal{D},\mathcal{H})_{\overline{\nu }}\text{-}mod$. The same applies for the subcategories $^{\mathrm{w}}(\mathcal{D},\mathcal{H})_{\overline{\nu }}\text{-}mod$, $^{\mathrm{b}}(\mathcal{D},\mathcal{H})_{\overline{\nu }}\text{-}mod = ^{\mathrm{f}}(\mathcal{D},\mathcal{H})_{\overline{\nu }}\text{-}mod$, and $^{\mathrm{wb}}(\mathcal{D},\mathcal{H})_{\overline{\nu }}\text{-}mod = ^{\mathrm{wf}}(\mathcal{D},\mathcal{H})_{\overline{\nu }}\text{-}mod$.

3.3 Weight $\mathcal{D}^{E}$-Modules

In this subsection $\mathcal{D} = \mathcal{D}(n + 1)$ and we assume n ≥ 1. Let $E =\sum _{ i=1}^{n+1}t_{i}\partial _{i}$ be the Euler vector field. Denote by $\mathcal{D}^{E}$ the centralizer of E in $\mathcal{D}$. Note that $\mathcal{D}$ has a $\mathbb{Z}$-grading $\mathcal{D} =\bigoplus _{m\in \mathbb{Z}}\mathcal{D}^{m}$, where $\mathcal{D}^{m} =\{ d \in \mathcal{D}\vert [E,d] = md\}$. It is not hard to see that the center of $\mathcal{D}^{E}$ is generated by E. The quotient algebra $\mathcal{D}^{E}/(E - a)$ is the algebra of global sections of twisted differential operators on $\mathbb{P}^{n}$.

Let $a \in \mathbb{C}$, let $(\mathcal{D}^{E},\mathcal{H})^{a}$-mod be the category of generalized weight $\mathcal{D}^{E}$-modules with locally nilpotent action of E − a and $^{\mathrm{b}}(\mathcal{D}^{E},\mathcal{H})^{a}$-mod be the subcategory of $^{\mathrm{b}}(\mathcal{D}^{E},\mathcal{H})^{a}$-mod consisting of modules with finite weight multiplicities. We have again a decomposition

$$\displaystyle{(\mathcal{D}^{E},\mathcal{H})^{a}\text{-}mod =\bigoplus _{ \vert \nu \vert =a}(\mathcal{D}^{E},\mathcal{H})_{\nu }^{a}\text{-}mod,}$$

where $(\mathcal{D}^{E},\mathcal{H})_{\nu }^{a}\text{-}mod$ is the subcategory of modules with support in $\nu +\sum _{i=1}^{n}\mathbb{Z}(\varepsilon _{i} -\varepsilon _{i+1})$ and $\vert \nu \vert:=\sum _{ i=1}^{n+1}\nu _{i}$.

Let $\mathcal{H}'$ be the subalgebra of $\mathcal{D}$ generated by $t_{i}\partial _{i} - t_{j}\partial _{j}$. We denote by $_{\mathrm{s}}(\mathcal{D},\mathcal{H})$-mod (respectively, $_{\mathrm{s}}(\mathcal{D}^{E},\mathcal{H})$-mod, $_{\mathrm{s}}^{\mathrm{b}}(\mathcal{D}^{E},\mathcal{H})$-mod) the subcategory of $(\mathcal{D},\mathcal{H})$-mod (resp., the subcategory of $(\mathcal{D}^{E},\mathcal{H})$-mod, $^{\mathrm{b}}(\mathcal{D}^{E},\mathcal{H})$-mod) consisting of all modules semisimple over $\mathcal{H}'$. Similarly we define the categories $_{\mathrm{s}}(\mathcal{D}^{E},\mathcal{H})^{a}$-mod, $_{\mathrm{s}}^{\mathrm{b}}(\mathcal{D}^{E},\mathcal{H})^{a}$-mod, $_{\mathrm{s}}(\mathcal{D}^{E},\mathcal{H})_{\nu }^{a}$-mod, and $_{\mathrm{s}}^{\mathrm{b}}(\mathcal{D}^{E},\mathcal{H})_{\nu }^{a}$-mod. These categories are studied in [27] in detail. In this paper we will limit our attention to the role that the simples and injectives of these categories play in the classification of the simples and injectives in categories of bounded and generalized bounded $\mathfrak{s}\mathfrak{l}(n + 1)$-modules (see Proposition 4.10).

Define the left exact functor $S_{\mathcal{H}'}: (\mathcal{D},\mathcal{H})$-mod $\rightarrow _{\mathrm{s}}(\mathcal{D},\mathcal{H})$-mod to be the one that maps M to its submodule consisting of all $\mathcal{H}'$-eigenvectors. One can easily verify that $S_{\mathcal{H}'}$ maps injectives to injectives and blocks to blocks. For the injectives, note that if α: X → I is a homomorphism of $\mathcal{D}$-modules, X is in $_{\mathrm{s}}(\mathcal{D},\mathcal{H})$-mod, and I is in $(\mathcal{D},\mathcal{H})$-mod, then α(X) is in $_{\mathrm{s}}(\mathcal{D},\mathcal{H})$-mod, hence a submodule of $S_{\mathcal{H}'}(I)$.

Finally, we define two functors between the categories of weight $\mathcal{D}$-modules and weight $\mathcal{D}^{E}$-modules. For any $M \in (\mathcal{D},\mathcal{H})$-mod, let

$$\displaystyle{\varGamma _{a}(M) =\bigcup _{l>0}\mathrm{Ker}(E - a)^{l}.}$$

Then Γ _a is an exact functor from the category $(\mathcal{D},\mathcal{H})$-mod to the category $(\mathcal{D}^{E},\mathcal{H})^{a}$-mod. The induction functor

$$\displaystyle{\varPhi (X) = \mathcal{D}\otimes _{\mathcal{D}^{E}}X}$$

is its left adjoint.

3.4 Weight $\mathcal{D}(\infty )$-Modules

Denote by $\mathbb{C}[t_{i}]_{i\in \mathbb{N}}$ the polynomial algebra in t _i , $i \in \mathbb{Z}$. In this section $\mathcal{U} = \mathcal{D}(\infty )$ is the infinite Weyl algebra, where $\mathcal{D}(\infty )$ is defined as the subalgebra of $\mbox{ End}(\mathbb{C}[t_{i}]_{i\in \mathbb{N}})$ generated by the operators t _i (multiplication by t _i ) and ∂ _i (derivative with respect to t _i ). We also let $\mathfrak{h}$ to be the space spanned by t _i ∂ _i , $i \in \mathbb{N}$, and hence $\mathcal{H} = S(\mathfrak{h}) = \mathbb{C}[t_{i}\partial _{i}]_{i\in \mathbb{N}}$. All other definitions from Sect. 3.2 transfer trivially to the infinite case.

3.5 Weight Modules of Lie Algebras

Let $\mathfrak{g}$ be a simple finite-dimensional Lie algebra and let $U = U(\mathfrak{g})$ be its universal enveloping algebra. We fix a Cartan subalgebra $\mathfrak{h}$ of $\mathfrak{g}$ and denote by ( , ) the Killing form on $\mathfrak{g}$. We apply the setting of Sect. 3.1 with $\mathcal{U} = U$ and $\mathcal{H} = S(\mathfrak{h})$. We will use the following notation: $\mathcal{G}\mathcal{B} = ^{\mathrm{b}}(\mathcal{U},\mathcal{H})\text{-}mod$, $\mathcal{B} = ^{\mathrm{wb}}(\mathcal{U},\mathcal{H})\text{-}mod$, $\mathcal{G}\mathcal{B}_{\mu } = \mathcal{G}\mathcal{B}_{\overline{\mu }} = ^{\mathrm{b}}(\mathcal{U},\mathcal{H})_{\overline{\mu }}\text{-}mod$, and $\mathcal{B}_{\mu } = \mathcal{B}_{\overline{\mu }} = ^{\mathrm{wb}}(\mathcal{U},\mathcal{H})_{\overline{\mu }}\text{-}mod$. By a result of Fernando [19] and Benkart, Britten and Lemire [3], infinite-dimensional simple objects of $\mathcal{B}$ and $\mathcal{G}\mathcal{B}$ exist only for Lie algebras of type A and C. The modules in $\mathcal{B}$ and $\mathcal{G}\mathcal{B}$ are called bounded and generalized bounded modules, respectively.

A generalized weight module M with finite weight multiplicities will be called a generalized cuspidal module if the nonzero elements of the root space $\mathfrak{g}^{\alpha }$ act injectively (and hence bijectively) on M for all roots α of $\mathfrak{g}$. If M is a weight cuspidal module we will call it simply cuspidal module. By $\mathcal{G}\mathcal{C}$ and $\mathcal{C}$ we will denote the categories of generalized cuspidal and cuspidal modules, respectively, and the corresponding subcategories defined by the supports will be denoted by $\mathcal{G}\mathcal{C}_{\overline{\mu }}$ and $\mathcal{C}_{\overline{\mu }}$. One should note that the simple objects of $\mathcal{B}$ and $\mathcal{G}\mathcal{B}$ (as well as those of $\mathcal{C}$ and $\mathcal{G}\mathcal{C}$) coincide. The category $\mathcal{C}$ was described in [26] and [34] for $\mathfrak{g} = \mathfrak{s}\mathfrak{l}(n + 1)$, and in [7] for $\mathfrak{g} = \mathfrak{s}\mathfrak{p}(2n)$. The category $\mathcal{G}\mathcal{C}$ was described in [34] for $\mathfrak{g} = \mathfrak{s}\mathfrak{l}(n + 1)$, and in [35] for $\mathfrak{g} = \mathfrak{s}\mathfrak{p}(2n)$.

The induced form on $\mathfrak{h}^{{\ast}}$ will be denoted by ( , ) as well. In this case $Q \subset \mathfrak{h}^{{\ast}}$ is the root lattice. By W we denote the Weyl group of $\mathfrak{g}$. Denote by Z: = Z(U) the center of U and let $Z':= \mbox{ Hom}(Z, \mathbb{C})$ be the set of all central characters (here Hom stands for homomorphisms of unital $\mathbb{C}$-algebras). By χ _λ ∈ Z′ we denote the central character of the irreducible highest weight module with highest weight λ. Recall that χ _λ = χ _μ iff λ +ρ = w(μ +ρ) for some element w of the Weyl group W, where, as usual, ρ denotes the half-sum of positive roots. Finally, recall that λ is dominant integral if $(\lambda,\alpha ) \in \mathbb{Z}_{\geq 0}$ for all positive roots α.

One should note that every generalized bounded module has finite Jordan–Hölder series (see Lemma 3.3 in [32]). Since the center Z of U preserves weight spaces, it acts locally finitely on the generalized bounded modules. For every central character χ ∈ Z′ let $\mathcal{G}\mathcal{B}^{\chi }$ (respectively, $\mathcal{B}^{\chi },\mathcal{G}\mathcal{C}^{\chi },\mathcal{C}^{\chi }$) denote the category of all generalized bounded modules (respectively, bounded, generalized cuspidal, cuspidal) modules M with generalized central character χ, i.e., such that for some $n\left (M\right ),\left (z -\chi \left (z\right )\right )^{n\left (M\right )} = 0$ on M for all $z \in Z$. It is clear that every generalized bounded module M is a direct sum of finitely many $M_{i} \in \mathcal{G}\mathcal{B}^{\chi _{i}}$. Thus, one can write

$$\displaystyle{ \mathcal{G}\mathcal{B} =\bigoplus _{{ \chi \in Z' \atop \bar{\mu }\in \mathfrak{h}^{{\ast}}/Q} }\mathcal{G}\mathcal{B}_{\bar{\mu }}^{\chi },\;\mathcal{G}\mathcal{C} =\bigoplus _{{ \chi \in Z' \atop \bar{\mu }\in \mathfrak{h}^{{\ast}}/Q} }\mathcal{G}\mathcal{C}_{\bar{\mu }}^{\chi },\;\mathcal{B} =\bigoplus _{{ \chi \in Z' \atop \bar{\mu }\in \mathfrak{h}^{{\ast}}/Q} }\mathcal{B}_{\bar{\mu }}^{\chi },\;\mathcal{C} =\bigoplus _{{ \chi \in Z' \atop \bar{\mu }\in \mathfrak{h}^{{\ast}}/Q} }\mathcal{C}_{\bar{\mu }}^{\chi },}$$

where $\mathcal{G}\mathcal{B}_{\bar{\mu }}^{\chi } = \mathcal{G}\mathcal{B}^{\chi }\cap \mathcal{G}\mathcal{B}_{\bar{\mu }}$, etc. Note that many of the direct summands above are trivial.

By $\chi _{\lambda }$ we denote the central character of the simple highest weight $\mathfrak{g}$-module with highest weight λ. For simplicity we put $\mathcal{G}\mathcal{B}^{\lambda }:= \mathcal{G}\mathcal{B}^{\chi _{\lambda }}$, $\mathcal{G}\mathcal{B}_{\bar{\mu }}^{\lambda }:= \mathcal{G}\mathcal{B}_{\bar{\mu }}^{\chi _{\lambda }}$, etc.

Let $\overline{\mathcal{B}}$ (respectively, $\overline{\mathcal{G}\mathcal{B}}$) be the full subcategory of all weight modules (respectively, generalized weight modules) consisting of $\mathfrak{g}$-modules M with countable dimensional weight spaces whose finitely generated submodules belong to $\mathcal{B}$ (respectively, $\mathcal{G}B$). With the aid of (2), it is not hard to see that every such M is a direct limit $\lim\limits_{\longrightarrow }M_{i}$ for some directed system {M _i   |  i ∈ I} such that each $M_{i} \in \mathcal{G}\mathcal{B}$ (respectively, $M_{i} \in \mathcal{B}$). It implies that the action of the center Z of the universal enveloping algebra U on M is locally finite and we have decompositions

$$\displaystyle{ \overline{\mathcal{B}} =\bigoplus _{{ \chi \in Z', \atop \bar{\mu }\in \mathfrak{h}^{{\ast}}/Q} }\overline{\mathcal{B}}_{\bar{\mu }}^{\chi },\;\overline{\mathcal{G}\mathcal{B}} =\bigoplus _{{ \chi \in Z' \atop \bar{\mu }\in \mathfrak{h}^{{\ast}}/Q} }\overline{\mathcal{G}\mathcal{B}}_{\overline{\mu }}^{\chi }.}$$

In a similar way we define $\overline{\mathcal{C}}$ and $\overline{\mathcal{G}\mathcal{C}}$ and obtain their block decompositions. Finally, we set $\overline{\mathcal{G}\mathcal{B}}_{\bar{\mu }}^{\lambda }:= \overline{\mathcal{G}\mathcal{B}}_{\bar{\mu }}^{\chi _{\lambda }}$, $\overline{\mathcal{B}}_{\bar{\mu }}^{\lambda }:= \overline{\mathcal{B}}_{\bar{\mu }}^{\chi _{\lambda }}$, and so on.

3.6 Weight Modules of Lie Superalgebras

In this case $\mathfrak{g}$ is a simple finite-dimensional Lie superalgebra, $\mathcal{U} = U(\mathfrak{g})$. Let $\mathfrak{h} = \mathfrak{h}_{\bar{0}} \oplus \mathfrak{h}_{\bar{1}}$ be a Cartan subalgebra of $\mathfrak{g}$, i.e., a self-normalizing nilpotent Lie subsuperalgebra of $\mathfrak{g}$. In particular (see [36, 37]), $\mathfrak{h}_{\bar{0}}$ is a Cartan subalgebra of $\mathfrak{g}_{\bar{0}}$ and $\mathfrak{h}_{\bar{1}}$ is the generalized weight space of weight 0 of the $\mathfrak{h}_{\bar{0}}$-module $\mathfrak{g}_{\bar{1}}$. We fix a Levi subalgebra $\mathfrak{h}_{\bar{0}}^{ss}$ of $\mathfrak{h}_{\bar{0}}$ and set $\mathcal{H} = S(\mathfrak{h}_{\bar{0}}^{ss})$. Note that in this case the vector space $\mathfrak{h}$ from Sect. 3.1 is $\mathfrak{h}_{\bar{0}}^{ss}$. If $\mathfrak{g}$ is a classical Lie superalgebra, then $\mathfrak{h}_{\bar{0}}^{ss} = \mathfrak{h}_{\bar{0}}$. For the four Cartan type series $\mathfrak{g}$, a list of the corresponding $\mathfrak{h}_{\bar{0}}^{ss}$ can be found for example in the appendix of [24]. Note that while the definitions of bounded and generalized bounded modules can be easily transferred from the Lie algebra case to the Lie superalgebra case (by replacing $\mathfrak{h}$ with $\mathfrak{h}_{\bar{0}}^{ss}$), the definitions of cuspidal and generalized cuspidal modules for Lie superalgbras require some additional conditions (see §1.5 in [14]).

3.7 Weight Modules of Affine Lie Algebras

In this section $\mathcal{U} = U(\mathfrak{G})$ and $\mathcal{H} = S(\mathfrak{H})$ where $\mathfrak{G}$ is an affine Lie algebra and $\mathfrak{H}$ is a Cartan subalgebra of $\mathfrak{G}$. For the reader’s convenience, we recall the construction of affine Lie algebras and fix notation. For more detail, see [31].

Let $\mathfrak{g}$ be a simple finite-dimensional Lie algebra with a nondegenerate invariant symmetric bilinear form ( , ). Denote by $\mathcal{L}(\mathfrak{g})$ the loop algebra $\mathfrak{g} \otimes \mathbb{C}[t,t^{-1}]$. The affine Lie algebra $\mathcal{A}(\mathfrak{g}) = \mathcal{L}(\mathfrak{g}) \oplus \mathbb{C}D \oplus \mathbb{C}K$ has commutation relations

$$\displaystyle{[x \otimes t^{m},y \otimes t^{n}] = [x,y] \otimes t^{m+n} +\delta _{ m,-n}m\,(x,y)K,\quad [D,x \otimes t^{m}] = mx \otimes t^{m},\quad [K,\mathcal{A}(\mathfrak{g})] = 0,}$$

where $x,y \in \mathfrak{g}$, $m,n \in \mathbb{Z}$, and δ _{i, j} is Kronecker’s delta. The form ( , ) extends to a nondegenerate invariant symmetric bilinear form on $\mathcal{A}(\mathfrak{g})$, still denoted by $(\,\,,\,): \mathcal{A}(\mathfrak{g}) \times \mathcal{A}(\mathfrak{g}) \rightarrow \mathbb{C}$, via

$$\displaystyle\begin{array}{rcl} & & (x \otimes t^{m},y \otimes t^{n}) =\delta _{ m,-n}(x,y),\quad (D,K) = 1, {}\\ & & (x \otimes t^{m},D) = (x \otimes t^{m},K) = (K,K) = (D,D) = 0. {}\\ \end{array}$$

If σ is a diagram automorphism of $\mathfrak{g}$ of order s, then σ extends to an automorphism of $\mathcal{A}(\mathfrak{g})$, still denoted by σ, via

$$\displaystyle{\sigma (x \otimes t^{m}) =\zeta ^{m}\sigma (x)t^{m},\quad \sigma (D) = D,\quad \sigma (K) = K,}$$

where ζ is a fixed primitive sth root of unity. The twisted affine Lie algebra $\mathcal{A}(\mathfrak{g},\sigma )$ is the Lie algebra $\mathcal{A}(\mathfrak{g})^{\sigma }$ of σ-fixed points of $\mathcal{A}(\mathfrak{g})$. Note that

$$\displaystyle{\mathcal{A}(\mathfrak{g},\sigma ) = \mathcal{L}(\mathfrak{g},\sigma ) \oplus \mathbb{C}D \oplus \mathbb{C}K,}$$

where $\mathcal{L}(\mathfrak{g},\sigma ) = \mathcal{L}(\mathfrak{g})^{\sigma }$ is the subalgebra of σ-fixed points of $\mathcal{L}(\mathfrak{g})$. One has that

$$\displaystyle{\mathcal{L}(\mathfrak{g},\sigma ) =\bigoplus _{j\in \mathbb{Z}}\mathfrak{g}_{\bar{j}} \otimes t^{j},}$$

where

$$\displaystyle{\mathfrak{g} =\bigoplus _{\bar{j}\in \mathbb{Z}/s\mathbb{Z}}\mathfrak{g}_{\bar{j}}}$$

is the decomposition of $\mathfrak{g}$ into σ-eigenspaces. The restriction of ( , ) to $\mathcal{A}(\mathfrak{g},\sigma )$ is a nondegenerate invariant symmetric bilinear form for which we will use the same notation.

If $\mathfrak{h}$ is a Cartan subalgebra of $\mathfrak{g}$, then $\mathfrak{h} \oplus \mathbb{C}D \oplus \mathbb{C}K$ is a Cartan subalgebra of $\mathcal{A}(\mathfrak{g})$. Furthermore, σ preserves $\mathfrak{h}$ and $\mathfrak{h}^{\sigma } \oplus \mathbb{C}D \oplus \mathbb{C}K$ is a Cartan subalgebra of $\mathcal{A}(\mathfrak{g},\sigma )$. For the rest of the paper $\mathfrak{h}$ denotes a fixed Cartan subalgebra of $\mathfrak{g}$, $\mathfrak{G}$ denotes $\mathcal{A}(\mathfrak{g})$ or $\mathcal{A}(\mathfrak{g},\sigma )$, and $\mathfrak{H}$ denotes the corresponding Cartan subalgebra of $\mathfrak{G}$.

The Lie algebra $\mathfrak{G}$ admits a root decomposition

$$\displaystyle{\mathfrak{G} = \mathfrak{H} \oplus \left (\bigoplus _{\alpha \in \varDelta }\mathfrak{G}^{\alpha }\right ).}$$

To describe the root system $\varDelta$ of $\mathfrak{G}$, let $\delta \in \mathfrak{H}^{{\ast}}$ denote the element with

$$\displaystyle{\delta (D) = 1,\quad \delta (h) =\delta (K) = 0\quad \text{for every}\quad h \in \mathfrak{h}.}$$

If $\mathfrak{G} = \mathcal{A}(\mathfrak{g})$, denote the root system of $\mathfrak{g}$ by $\mathring{\varDelta}$. If $\mathfrak{G} = \mathcal{A}(\mathfrak{g},\sigma )$, denote the nonzero weights of the $\mathfrak{g}_{\bar{0}}$-module $\mathfrak{g}_{\bar{j}}$ by $\mathring{\varDelta} _{\bar{j}}$ and set $\mathring{\varDelta} = \cup _{\bar{j}\in \mathbb{Z}/s\mathbb{Z}}\mathring{\varDelta} _{\bar{j}}$. Note that for $\mathfrak{G} = \mathcal{A}(\mathfrak{g},\sigma )\not\cong A_{2l}^{(2)}$, $\mathring{\varDelta} = \mathring{\varDelta} _{\bar{0}}$ is the root system of $\mathfrak{g}_{\bar{0}}$ and for $\mathfrak{G}\mathop{\cong}A_{2l}^{(2)}$, $\mathring{\varDelta}$ is the non–reduced root system BC _l .

The decomposition $\mathfrak{H} = \mathfrak{h} \oplus \mathbb{C}D \oplus \mathbb{C}K$ (respectively, $\mathfrak{H} = \mathfrak{h}^{\sigma } \oplus \mathbb{C}D \oplus \mathbb{C}K$) allows us to consider $\mathring{\varDelta}$ as a subset of $\mathfrak{H}^{{\ast}}$. The root system Δ decomposes as

$$\displaystyle{\varDelta =\varDelta ^{\mathrm{re}} \sqcup \varDelta ^{\mathrm{im}},}$$

where

$$\displaystyle{\varDelta ^{\mathrm{im}} =\{ n\delta \,\vert \,n \in \mathbb{Z}\setminus \{0\}\}}$$

are the imaginary roots of $\mathfrak{G}$ and the real roots $\varDelta ^{\mathrm{re}}$ are given as follows.

• (i) If $\mathfrak{G} = \mathcal{A}(\mathfrak{g})$, then

  $$\displaystyle{\varDelta ^{\mathrm{re}} =\{\alpha +n\delta \,\vert \,n \in \mathbb{Z},\,\alpha \in \mathring{\varDelta} \}.}$$

• (ii) If $\mathfrak{G} = \mathcal{A}(\mathfrak{g},\sigma )$, then

  $$\displaystyle{\varDelta ^{\mathrm{re}} =\{\alpha +n\delta \,\vert \,n \in \mathbb{Z},\,\alpha \in \mathring{\varDelta} _{\bar{ n}}\}.}$$

3.8 Parabolically Induced Module

In this subsection we have that $\mathcal{U} = U(\mathfrak{g})$ or $\mathcal{U} = U(\mathfrak{G})$, where $\mathfrak{g}$ is a simple finite dimensional Lie algebra or superalgebras, and $\mathfrak{G}$ is an affine Lie algebra, i.e., we have the setting of Sects. 3.5, 3.6, or 3.7. In this subsection, for simplicity, Δ will denote the root system of $\mathfrak{g}$ or $\mathfrak{G}$.

If Δ is symmetric (i.e., Δ = −Δ), then we call a proper subset P of Δ a parabolic set in Δ if

$$\displaystyle{\varDelta = P \cup (-P)\;\;\;\mbox{ and}\;\;\alpha,\beta \in P\;\text{ with }\;\alpha +\beta \in \varDelta \;\;\text{ implies }\;\;\alpha +\beta \in P.}$$

A detailed treatment of parabolic subsets of symmetric root systems can be found in [12]. If $\varDelta \neq -\varDelta$, then P ⊊ Δ will be called parabolic if $P =\tilde{ P}\cap \varDelta$ for some parabolic subset $\tilde{P}$ of $\varDelta \cup (-\varDelta )$.

For a symmetric root systems Δ and a parabolic subset of roots P of Δ, we call $L:= P \cap (-P)$ the Levi component of P, $N^{+}:= P\setminus (-P)$ the nilradical of P, and $P = L \sqcup N^{+}$ the Levi decomposition of P. If Δ ≠ −Δ, then we choose a parabolic subset $\tilde{P}$ of $\varDelta \cap (-\varDelta )$ such that $P =\tilde{ P}\cap \varDelta$, and define $\tilde{L} =\tilde{ P} \cap (-\tilde{P})\;\;\mbox{ and}\;\;\tilde{N}^{+} =\tilde{ P}\setminus (-\tilde{P}).$ We call $L:=\tilde{ L} \cap P$ a Levi component of P, $N^{+} =\tilde{ N}^{+} \cap P$ a nilradical of P, and $P = L \sqcup N^{+}$ a Levi decomposition of P. We note that in the nonsymmetric case the definition of a Levi component and nilradical of P essentially depends on the choice of a parabolic subset $\tilde{P}$ of $\varDelta \cap (-\varDelta )$. We refer the reader to Remarks 1.7 and 3.3 in [28] for examples.

We will call a subalgebra $\mathfrak{p}$ of $\mathfrak{g}$ (respectively, $\mathfrak{P}$ of $\mathfrak{G}$) parabolic if it is of the form $\mathfrak{p}_{P} = \mathfrak{h} \oplus \left (\bigoplus _{\mu \in P}\mathfrak{g}^{\mu }\right )$ (respectively, $\mathfrak{P}_{P} = \mathfrak{H} \oplus \left (\bigoplus _{\mu \in P}\mathfrak{G}^{\mu }\right )$) for some parabolic subset P of Δ. If L and N ⁺ are Levi component and a nilradical of a parabolic set P, respectively, then $\mathfrak{l} = \mathfrak{h} \oplus \left (\bigoplus _{\mu \in L}\mathfrak{g}^{\mu }\right )$ and $\mathfrak{n} = \mathfrak{h} \oplus \left (\bigoplus _{\mu \in N^{+}}\mathfrak{g}^{\mu }\right )$ are called a Levi subalgebra, and a nilradical of $\mathfrak{p}_{P}$, respectively. Similarly, we define a Levi subalgebra, and a nilradical of a parabolic subalgebra $\mathfrak{P}_{P}$ of $\mathfrak{G}$.

Let $\mathfrak{l}$ and $\mathfrak{n}$ be a Levi subalgebra and a nilradical of a parabolic subalgebra $\mathfrak{p}$. Every $\mathfrak{l}$-module S can be considered as a $\mathfrak{p}$-module with the trivial action of $\mathfrak{n}$. We define $M_{\mathfrak{p}}(S) = U(\mathfrak{g}) \otimes _{U(\mathfrak{p})}S$. Among the submodules of $M_{\mathfrak{p}}(S)$ which intersect trivially with $1 \otimes S$ there is a unique maximal one $Z_{\mathfrak{p}}(S)$. Set $V _{\mathfrak{p}}(S) = M_{\mathfrak{p}}(S)/Z_{\mathfrak{p}}(S)$. In this paper we will call a $\mathfrak{g}$-module parabolically induced if it isomorphic to $V _{\mathfrak{p}}(S)$ for some parabolic subalgebra $\mathfrak{p}$ of $\mathfrak{g}$ and some module S over a Levi component $\mathfrak{l}$ of $\mathfrak{p}$. We similarly define parabolically induced modules of $\mathfrak{G}$. For properties of the version of the parabolic induction functor used in the paper the reader is referred to [14].

4 Twisted Localization

4.1 Twisted Localization in General Setting

Retain the notation of Sect. 3.1. Namely, $\mathcal{U}$ is an associative unital algebra, and $\mathcal{H} = S(\mathfrak{h})$ is a commutative subalgebra of $\mathcal{U} =\bigoplus _{\mu \in Q_{\mathcal{U}}}U^{\mu }$, in particular, $\mathrm{ad}(h)$ is semisimple on $\mathcal{U}$ for every $h \in \mathfrak{h}$. Now let $F =\{ f_{j}\;\vert \;j \in J\}$ be a subset of commuting elements of $\mathcal{U}$ such that ad (f), f ∈ F, are locally nilpotent endomorphisms of $\mathcal{U}$. In addition, we assume that for every $u \in \mathcal{U}$, uf = fu, for all but finitely many f ∈ F. Let 〈F〉 be the multiplicative subset of $\mathcal{U}$ generated by F, i.e., the 〈F〉 consists of the elements $f_{1}^{n_{1}}\ldots f_{k}^{n_{k}}$ for f _i ∈ F, $n_{i} \in \mathbb{N}$. By $D_{F}\mathcal{U}$ we denote the localization of $\mathcal{U}$ relative to 〈F〉. Note that 〈F〉 satisfies Ore’s localizability condition due to the fact that f _i are locally ad-nilpotent. The proof of that and more details on the 〈F〉-localization can be found in §4 of [32].

For a $\mathcal{U}$-module M, by $D_{F}M = D_{F}\mathcal{U}\otimes _{\mathcal{U}}M$ we denote the localization of M relative to 〈F〉. We will consider D _F M both as a $\mathcal{U}$-module and as a $D_{F}\mathcal{U}$-module. By $\theta _{F}: M \rightarrow D_{F}M$ we denote the localization map defined by $\theta _{F}(m) = 1 \otimes m$. Then

$$\displaystyle{\mbox{ ann}_{M}F:=\{ m \in M\;\vert \;sm = 0\mbox{ for some }s \in F\}}$$

is a submodule of M (often called, the torsion submodule with respect to F). Note that if $\mbox{ ann}_{M}F = 0$, then θ _F is an injection. In the latter case, we will say that M is F-torsion free, and M will be considered naturally as a submodule of D _F M. Note also that if $F = F_{1} \cup F_{2}$, then $D_{F_{1}}D_{F_{2}}M \simeq D_{F_{2}}D_{F_{1}}M \simeq D_{F}M$.

It is well known that D _F is a functor from the category of $\mathcal{U}$-modules to the category of $D_{F}\mathcal{U}$-modules. For any category $\mathcal{A}$ of $\mathcal{U}$-modules, by $\mathcal{A}_{F}$ we denote the category of $D_{F}\mathcal{U}$-modules considered as $\mathcal{U}$-modules are in $\mathcal{A}$. Some useful properties of the localization functor D _F are listed in the following lemma.

Lemma 4.1.

• (i) If $\varphi: M \rightarrow N$ is a homomorphism of $\mathcal{U}$ -modules, then $D_{F}(\varphi )\theta _{F} =\theta _{F}\varphi$.

• (ii) D _F is an exact functor.

• (iii) If N is a $D_{F}\mathcal{U}$ -module and $\varphi: M \rightarrow N$ is a homomorphism of $\mathcal{U}$ -modules, then there exists a unique homomorphism of $D_{F}\mathcal{U}$ -modules $\overline{\varphi }: D_{F}M \rightarrow N$ such that $\overline{\varphi }\theta _{F} =\varphi$ . If we identify N with D _F N, then $\overline{\varphi } = D_{F}(\varphi )$.

• (iv) Let $\mathcal{A}$ be any category of U-modules. If I is an injective module in $\mathcal{A}_{F}$, then I (considered as an $\mathcal{U}$ -module) is injective in $\mathcal{A}$ as well.

We now introduce the “generalized conjugation” in $D_{F}\mathcal{U}$ following §4 of [32]. For $\mathbf{x} \in \mathbb{C}^{J}$ define the automorphism $\varTheta _{F}^{\mathbf{x}}$ of $D_{F}\mathcal{U}$ in the following way. For u ∈ D _F U, f ∈ F, and $x \in \mathbb{C}$ set

$$\displaystyle\begin{array}{rcl} \varTheta _{\{f\}}^{x}(u):=\sum \limits _{ i\geq 0}\binom{x}{i}\,\mbox{ ad}(f)^{i}(u)\,f^{-i},& & {}\\ \end{array}$$

where $\binom{x}{i}:= x(x - 1)\ldots (x - i + 1)/i!$ for $x \in \mathbb{C}$ and $i \in \mathbb{Z}_{\geq 0}$. Note that the sum on the right-hand side is well defined since f is ad-nilpotent on $\mathcal{U}$. Now, for J ⊂ I and $\mathbf{x} = (x_{j})_{j\in J} \in \mathbb{C}^{J}$ define

$$\displaystyle\begin{array}{rcl} \varTheta _{F}^{\mathbf{x}}(u):=\prod _{ j\in J}\varTheta _{\{f_{j}\}}^{x_{j} }(u).& & {}\\ \end{array}$$

The product above is in fact finite since $\mbox{ ad}(f_{j})(u) = 0$ for all but finitely many f _j . Note that if $J =\{ 1,2,\ldots,k\}$ and $\mathbf{x} \in \mathbb{Z}^{k}$, we have $\varTheta _{F}^{\mathbf{x}}(u) = \mathbf{f}^{\mathbf{x}}u\mathbf{f}^{-\mathbf{x}}$, where $\mathbf{f}^{\mathbf{x}}:= f_{1}^{x_{1}}\ldots f_{k}^{x_{k}}$.

For a $D_{F}\mathcal{U}$-module N by $\varPhi _{F}^{\mathbf{x}}N$ we denote the $D_{F}\mathcal{U}$-module N twisted by $\varTheta _{F}^{\mathbf{x}}$. The action on $\varPhi _{F}^{\mathbf{x}}N$ is given by

$$\displaystyle{u \cdot v^{\mathbf{x}}:= (\varTheta _{ F}^{\mathbf{x}}(u) \cdot v)^{\mathbf{x}},}$$

where $u \in D_{F}\mathcal{U}$, v ∈ N, and w ^x stands for the element w considered as an element of $\varPhi _{F}^{\mathbf{x}}N$. In the case $J =\{ 1,2,\ldots,k\}$ and $\mathbf{x} \in \mathbb{Z}^{k}$, there is a natural isomorphism of $D_{F}\mathcal{U}$-modules $M \rightarrow \varPhi _{F}^{\mathbf{x}}M$ given by $m\mapsto (\mathbf{f}^{\mathbf{x}} \cdot m)^{\mathbf{x}}$ with inverse map defined by $n^{\mathbf{x}}\mapsto \mathbf{f}^{-\mathbf{x}} \cdot n$. In view of this isomorphism, for $\mathbf{x} \in \mathbb{Z}^{k}$, we will identify M with $\varPhi _{F}^{\mathbf{x}}M$, and for any $\mathbf{x} \in \mathbb{C}^{k}$ we will write $\mathbf{f}^{\mathbf{x}} \cdot m$ (or simply f ^x m) for m ^−x whenever m ∈ M. The action of Φ _F ^x on a homomorphism α: M → N of $D_{F}\mathcal{U}$-modules is defined by $\varPhi _{F}^{\mathbf{x}}(\alpha )(\mathbf{f}^{\mathbf{x}} \cdot m) = \mathbf{f}^{\mathbf{x}} \cdot (\alpha (m))$.

The basic properties of the twisting functor $\varPhi _{F}^{\mathbf{x}}$ on $D_{F}\mathcal{U}$-mod are summarized in the following lemma. The proofs are straightforward.

Lemma 4.2.

Let $F =\{ f_{1},\ldots,f_{k}\}$ be a set of locally ad -nilpotent commuting elements of $\mathcal{U}$, M and N be $D_{F}\mathcal{U}$ -modules, m ∈ M, $u \in \mathcal{U}$, and $\mathbf{x},\mathbf{y} \in \mathbb{C}^{k}$.

• (i) $\varTheta _{F}^{\mathbf{x}} \circ \varTheta _{F}^{\mathbf{y}} =\varTheta _{ F}^{\mathbf{x+y}}$, in particular $\mathbf{f}^{\mathbf{x}} \cdot (\mathbf{f}^{\mathbf{y}} \cdot m) = \mathbf{f}^{\mathbf{x+y}} \cdot m$ ;

• (ii) $\varPhi _{F}^{\mathbf{x}}\varPhi _{F}^{\mathbf{y}} =\varPhi _{ F}^{\mathbf{x+y}}$, in particular, $\varPhi _{F}^{\mathbf{x}}\varPhi _{F}^{-\mathbf{x}} = \mathrm{Id}$ on the category of $D_{F}\mathcal{U}$ -modules;

• (iii) $\mathbf{f}^{\mathbf{x}} \cdot (u \cdot (\mathbf{f}^{-\mathbf{x}} \cdot m)) =\varTheta _{ F}^{\mathbf{x}}(u) \cdot m$ ;

• (iv) Φ ^x_F is an exact functor;

• (v) M is simple (respectively, injective) if and only if $\varPhi _{F}^{\mathbf{x}}M$ is simple (respectively, injective);

• (vi) $\mathrm{Hom}_{\mathcal{U}}(M,N) = \mathrm{Hom}_{\mathcal{U}}(\varPhi _{F}^{\mathbf{x}}M,\varPhi _{F}^{\mathbf{x}}N)$.

For any $\mathcal{U}$-module M, and $\mathbf{x} \in \mathbb{C}^{J}$ we define the twisted localization $D_{F}^{\mathbf{x}}M$ of M relative to F and x by $D_{F}^{\mathbf{x}}M:=\varPhi _{ F}^{\mathbf{x}}D_{F}M$. The twisted localization is a exact functor from $\mathcal{U}$-mod to $D_{F}\mathcal{U}$-mod.

Some properties of the functor D _F ^x on the category of generalized weight $(\mathcal{U},\mathcal{H})$-modules are described in the following lemma.

Lemma 4.3.

Assume that $f_{i} \in \mathcal{U}^{\mathbf{a_{i}}}$ for $\mathbf{a_{i}} \in Q_{\mathcal{U}}$.

• (i) If M is a generalized weight $(\mathcal{U},\mathcal{H})$ -module, then D _F M is a generalized weight $(D_{F}\mathcal{U},\mathcal{H})$ -module.

• (ii) If N is a generalized weight $(D_{F}\mathcal{U},\mathcal{H})$ -module, then $\mathbf{f}^{\mathbf{x}}m \in N^{(\lambda +\mathbf{xa})}$ whenever $m \in N^{(\lambda )}$, where $\mathbf{xa} = x_{1}\mathbf{a_{1}} + \ldots + x_{k}\mathbf{a_{k}}$ . In particular, $\varPhi _{F}^{\mathbf{x}}N$ is a generalized weight $(D_{F}\mathcal{U},\mathcal{H})$ -module.

In what follows we will treat each of the special cases of $\mathcal{U}$ and $\mathcal{H}$ considered in the previous section separately. We will show in particular that in all cases every simple object of $^{\mathrm{f}}(\mathcal{U},\mathcal{H})$-mod (equivalently, in $^{\mathrm{wf}}(\mathcal{U},\mathcal{H})$-mod) is either parabolically induced or it is isomorphic to a twisted localization of a well understood module (for example, highest weight module, loop module, etc.).

4.2 Twisted Localization for $\mathcal{D}(n)$

Consider now $\mathcal{U} = \mathcal{D}(n)$ and $\mathcal{H} = \mathbb{C}[t_{1}\partial _{1},\ldots,t_{n}\partial _{n}]$. Obvious choices for F are {t _i }, {∂ _i }, {t _i ∂ j}, i ≠ j. We set for convenience $D_{i}^{+} = D_{\{\partial _{i}\}}$, $D_{i}^{-} = D_{\{t_{i}\}}$, $D_{i}^{x,-}:= D_{\{t_{i}\}}^{x}$, $D_{i}^{x,+} = D_{\{\partial _{i}\}}^{x}$, and $D_{i,j}^{x} = D_{\{t_{i}\partial _{j}\}}^{x}$ for $x \in \mathbb{C}$ and i ≠ j.

Let us first focus on the case n = 1, i.e., $\mathcal{D} = \mathcal{D}(1)$. We fix t ₁ = t and ∂ ₁ = ∂. For $\nu \in \mathbb{C}$ we set

$$\displaystyle{\mathcal{F}_{\nu } = t^{\nu }\mathbb{C}[t,t^{-1}]}$$

and consider $\mathcal{F}_{\nu }$ as a $\mathcal{D}$-module with the natural action of $\mathcal{D}$. It is easy to check that $\mathcal{F}_{\nu }\in ^{\mathrm{b}}(\mathcal{D},\mathcal{H})_{\mu }\mathrm{-mod}$ and $\mathcal{F}_{\nu }$ is simple iff $\nu \notin \mathbb{Z}$. By definition, $\mathcal{F}_{\nu }\simeq \mathcal{F}_{\mu }$ if and only if $\mu -\nu \in \mathbb{Z}$. So, if $\nu \in \mathbb{Z}$ we may assume ν = 0. One easily checks that $\mathcal{F}_{0}$ has length 2 and one has the following nonsplit exact sequence

$$\displaystyle{0 \rightarrow \mathcal{F}_{0}^{+} \rightarrow \mathcal{F}_{ 0} \rightarrow \mathcal{F}_{0}^{-}\rightarrow 0,}$$

where $\mathcal{F}_{0}^{+} = \mathbb{C}[t]$ and $\mathcal{F}_{0}^{-}$ is a simple quotient. Moreover, if σ denotes the automorphism of $\mathcal{D}$ defined by $\sigma (t) = \partial,\sigma (\partial ) = -t$, then $\mathcal{F}_{0}^{-}\simeq (\mathcal{F}_{0}^{+})^{\sigma }$. As follows for instance from [25], any simple object in $(\mathcal{D},\mathcal{H})$-mod is isomorphic to $\mathcal{F}_{\nu }$ for some non-integer ν, $\mathcal{F}_{0}^{-}$ or $\mathcal{F}_{0}^{+}$. A classification of all simple $\mathcal{D}(1)$-modules (not necessarily generalized weight ones) can be found in [5]. Also, an alternative approach that leads to a classification of the simple weight modules of a more general class of Weyl type algebras can be found in [16].

Now let $\mathcal{D} = \mathcal{D}(n)$. To describe the simple and injective modules in the category $(\mathcal{D},\mathcal{H})$-mod we use the fact that $\mathcal{D}(n)$ is the tensor product of n copies of $\mathcal{D}(1)$. With this in mind, for $\nu \in \mathbb{C}^{n}$, we set $\mathcal{F}_{\nu } = \mathcal{F}_{\nu _{1}} \otimes \ldots \otimes \mathcal{F}_{\nu _{n}}$, $\mathcal{F}^{+} = \mathcal{F}_{0}^{+} \otimes \ldots \otimes \mathcal{F}_{0}^{+}$, $\mathcal{F}_{\nu }^{\log } = \mathcal{F}_{\nu _{1}}^{\log } \otimes \ldots \otimes \mathcal{F}_{\nu _{ n}}^{\log }$, and $\mathcal{F}_{0}^{+} = \mathbb{C}[t_{1},\ldots,t_{n}]$. If μ has the property that μ _i ≥ 0 whenever $\mu _{i} \in \mathbb{Z}$, by $\mathcal{F}_{\mu }^{+}$ we denote the submodule of $\mathcal{F}_{\mu }$ generated by t ^μ. One can check that $\mathcal{F}_{\mu }^{+}$ is the unique simple submodule of $\mathcal{F}_{\mu }$. For a nonempty subset J of {1, 2, …. , n} let σ _J be the automorphism of $\mathcal{D}(n)$ defined as σ on the j-th $\mathcal{D}(1)$-component of $\mathcal{D}(n)$ for j ∈ J, and as identity for all other $\mathcal{D}(1)$-components. We set $\mathcal{F}_{\nu }^{\log }(J) = (\mathcal{F}_{\nu }^{\log })^{\sigma _{J}}$ and $\mathcal{F}_{\nu }(J) = (\mathcal{F}_{\nu })^{\sigma _{J}}$.

The application of the twisted localization functors D _i ^{x, ±} on the modules $\mathcal{F}_{\nu }^{\log }(J)$ is described in the following lemma. Recall that $\varepsilon _{i} \in \mathbb{C}^{n}$ are defined by $(\varepsilon _{i})_{j} =\delta _{ij}$.

Lemma 4.4.

The following isomorphisms hold.

$$\displaystyle{D_{i}^{x,+}(\mathcal{F}_{\nu }^{\log }(J)) \simeq \mathcal{F}_{\nu -x\varepsilon _{i}}^{\log }(J \cup i),\;D_{i}^{x,-}(\mathcal{F}_{\nu }^{\log }(J)) \simeq \mathcal{F}_{\nu +x\varepsilon _{i}}^{\log }(J\setminus i).}$$

One can show that for $\nu = (\nu _{1},\ldots,\nu _{n})$ every simple module in $(\mathcal{D},\mathcal{H})_{\nu }$-mod is the tensor product $S_{1} \otimes \ldots \otimes S_{n}$ where each S _i is a simple module in $(\mathcal{D}(1),\mathcal{H}(1))_{\nu _{i}}$-mod (see for example [25]).

Let Int(ν) be the set of all i such that $\nu _{i} \in \mathbb{Z}$ and $\mathcal{P}(\nu )$ be the power set of Int(ν). For every $J \in \mathcal{P}(\nu )$ set

$$\displaystyle{\mathcal{S}_{\nu }(J):= S_{1} \otimes \ldots \otimes S_{n},}$$

where $S_{i} = \mathcal{F}_{\nu _{i}}$ if $i\notin \mathrm{Int}(\nu )$, $S_{i} = \mathcal{F}_{0}^{+}$ if $i \in \mathrm{Int}(\nu )\setminus J$ and $S_{i} = \mathcal{F}_{0}^{-}$ if i ∈ J. As discussed above, $\mathcal{S}_{\nu }(J)$ are exactly the simple objects of $(\mathcal{D},\mathcal{H})$-mod. On the other hand, as follows from Proposition 5.2 in [27], $\mathcal{F}_{\nu }^{\log }(J)$ is injective in $(\mathcal{D},\mathcal{H})$-mod. Combining these results with the lemma above, one has the following theorem.

Theorem 4.5

Every simple object M in $(\mathcal{D},\mathcal{H})$ - mod (or, equivalently, in $^{\mathrm{w}}(\mathcal{D},\mathcal{H})$ - mod , in $^{\mathrm{f}}(\mathcal{D},\mathcal{H})$ - mod , and in $^{\mathrm{wf}}(\mathcal{D},\mathcal{H})$ - mod ) is isomorphic to some $\mathcal{S}_{\nu }(J)$ and the injective envelope of $\mathcal{S}_{\nu }(J)$ in $(\mathcal{D},\mathcal{H})$ - mod is isomorphic to $\mathcal{F}_{\nu }^{\log }(J)$ . In particular, every such simple module M is isomorphic to the σ _J -twist of a twisted localized module $D_{F}^{\mathbf{x}}\mathcal{F}_{0}^{+}$, and the injective envelope of M in $(\mathcal{D},\mathcal{H})$ - mod is isomorphic to the σ _J -twist of $D_{F}^{\mathbf{x}}\mathcal{F}_{0}^{\log }$.

4.3 Twisted Localization for $\mathcal{D}(\infty )$

Recall that $\mathbb{N}$ stands for the set of positive integers. The $\mathcal{D}(\infty )$-analogs of the $\mathcal{D}(n)$-modules $\mathcal{F}_{\nu }$ can be defined with the aid of twisted localization. Indeed, for $\nu \in \mathbb{C}^{\mathbb{N}}$ and $T =\{ t_{i}\;\vert \;i \in \mathbb{N}\}$, set $\mathcal{F}_{\nu } = D_{T}^{\nu }\mathcal{F}_{0}^{+}$, where, as before, $\mathcal{F}_{0}^{+} = \mathbb{C}[t_{i}]_{i\in \mathbb{N}}$. In particular, one can show that $\mathcal{F}_{0} \simeq \mathbb{C}[t_{i}^{\pm 1}]_{i\in \mathbb{N}}$. One way to think of $\mathcal{F}_{\nu }$ is as the space with basis $\mathbf{t}^{\nu }\mathbf{t}^{\mathbf{z}},$ where $\mathbf{z} = (z_{1},z_{2},\ldots )$ runs through all sequences of integers, such that all but finitely many z _i are zero, and $\mathbf{t}^{\mathbf{z}} =\prod _{i>0}t_{i}^{z_{i}}$. As in the case of $\mathcal{D}(n)$, for any $J \subset \mathbb{N}$, we may define an automorphism σ _J of $\mathcal{D}(\infty )$. The following theorem is proved in [21].

Theorem 4.6

• (i) For every $\nu \in \mathbb{C}^{\mathbb{N}}$, the module $\mathcal{F}_{\nu }$ has a unique simple submodule $\mathcal{F}_{\nu }^{+}$ . If ν is such that ν _i ≥ 0 whenever $\nu _{i} \in \mathbb{Z}$, then the module $\mathcal{F}_{\nu }^{+}$ is the submodule of $\mathcal{F}_{\nu }$ generated by t ^ν.

• (ii) Every simple module in $^{\mathrm{w}}(\mathcal{D}(\infty ),\mathcal{H})$ -mod is isomorphic to the σ _J -twist $(\mathcal{F}_{\nu }^{+})^{\sigma _{J}}$ of some $\mathcal{F}_{\nu }^{+}$ . Here J is a subset of Int (ν) and ν can be chosen so that ν _i ≥ 0 whenever $\nu _{i} \in \mathbb{Z}$.

• (iii) The injective envelope of $(\mathcal{F}_{\nu }^{+})^{\sigma _{J}}$ in $^{\mathrm{w}}(\mathcal{D}(\infty ),\mathcal{H})$ -mod is $(\mathcal{F}_{\nu })^{\sigma _{J}}$.

Remark 4.7

The above theorem can be extended to the category $(\mathcal{D}(\infty ),\mathcal{H})$-mod. Namely, every simple object in $(\mathcal{D}(\infty ),\mathcal{H})$-mod is also of the form $(\mathcal{F}_{\nu }^{+})^{\sigma _{J}}$ and the injective envelope of $(\mathcal{F}_{\nu }^{+})^{\sigma _{J}}$ in $(\mathcal{D}(\infty ),\mathcal{H})$-mod is isomorphic to $(\mathcal{F}_{\nu }^{\log })^{\sigma _{J}}$. Here $\mathcal{F}_{\nu }^{\log } = D_{T}^{\nu }\mathcal{F}_{0}^{\log }$, and $\mathcal{F}_{0}^{\log } = \mathbb{C}[t_{i}^{\pm 1},\log t_{i}]_{i\in \mathbb{N}}$. To prove that $(\mathcal{F}_{\nu }^{\log })^{\sigma _{J}}$ is injective in $(\mathcal{D}(\infty ),\mathcal{H})$-mod, it is sufficient to prove it for $J = \varnothing$, and for the latter we follow the steps of the proof of the same statement in $(\mathcal{D}(n),\mathcal{H})$-mod (see Proposition 5.2 in [27]).

4.4 Twisted Localization for Finite-Dimensional Lie Algebras

In this case we consider a simple finite-dimensional Lie algebra $\mathfrak{g}$ with a fixed Cartan subalgebra $\mathfrak{h}$, $\mathcal{U} = U(\mathfrak{g})$ and $\mathcal{H} = S(\mathfrak{h})$. The multiplicative sets will be always of the form $F =\langle e_{\alpha }\:\vert \:\alpha \in \varGamma \rangle$, where Γ is a set of k commuting roots and e _α is in the α-root space of $\mathfrak{s}\mathfrak{l}(n + 1)$. For $\mathbf{x} \in \mathbb{C}^{k}$, we write D _Γ and D _Γ ^x for D _F and $D_{F}^{\mathbf{x}}$, respectively. The following theorem is proved in [32].

Proposition 4.8

Every simple module in $^{\mathrm{wf}}(\mathcal{U},\mathcal{H})$ (equivalently, in $^{\mathrm{f}}(\mathcal{U},\mathcal{H})$ ) is either parabolically induced, or it is cuspidal. Every cuspidal module is isomorphic to D _Γ ^x L for some simple highest weight bounded module L, a set Γ of n commuting roots, and $\mathbf{x} \in \mathbb{C}^{n}$.

For the classification of the simple highest weight bounded modules we refer the reader to Sects. 8 and 9 in [32]. Below we describe the injectives in the categories of bounded and generalized bounded modules using equivalence of categories in each of the two cases $\mathfrak{g} = \mathfrak{s}\mathfrak{l}(n + 1)$ and $\mathfrak{g} = \mathfrak{s}\mathfrak{p}(2n)$.

4.4.1 The Case $\mathfrak{g} = \mathfrak{s}\mathfrak{l}(n + 1)$

In this case $\mathfrak{h}^{{\ast}}$ is identified with the subspace of $\mathbb{C}^{n+1}$ spanned by the simple roots $\varepsilon _{1} -\varepsilon _{2},\ldots,\varepsilon _{n} -\varepsilon _{n+1}$. By γ we denote the projection $\mathbb{C}^{n+1} \rightarrow \mathfrak{h}^{{\ast}}$ with one-dimensional kernel $\mathbb{C}(\varepsilon _{1} +\ldots +\varepsilon _{n+1})$.

Consider the homomorphism $\psi _{A}: U(\mathfrak{s}\mathfrak{l}(n + 1)) \rightarrow \mathcal{D}(n + 1)$ defined by $\psi (E_{ij}) = t_{i}\partial _{j}$, i ≠ j, where E _ij is the elementary (ij)th matrix in $\mathfrak{s}\mathfrak{l}(n + 1)$. The image of ψ _A is contained in $\mathcal{D}^{E}$. Using lift by ψ _A any $\mathcal{D}^{E}$-module becomes $\mathfrak{s}\mathfrak{l}(n + 1)$-module. Since $\psi _{A}(U(\mathfrak{h})) \subset \mathcal{H}$, one has a functor $\varPsi _{A}: (\mathcal{D}^{E},\mathcal{H})\text{-}mod \rightarrow \overline{\mathcal{G}\mathcal{B}}$. One can check that Ψ _A is exact and that we have the following commutative diagram.

where $S_{\mathfrak{h}}$ stands for the functor $\mathcal{G}\mathcal{B}^{\gamma (a\varepsilon _{1})} \rightarrow \mathcal{B}^{\gamma (a\varepsilon _{1})}$ mapping a module to its submodule consisting of all $\mathfrak{h}$-eigenvectors.

Using translation functors (in terms of [4]), one can show that every block $\mathcal{G}\mathcal{B}^{\mu }$ and $\mathcal{B}^{\mu }$, of $\mathcal{G}\mathcal{B}$ and $\mathcal{B}$, respectively, is equivalent to $\mathcal{G}\mathcal{B}^{\gamma (a\varepsilon _{1})}$ and $\mathcal{B}^{\gamma (a\varepsilon _{1})}$, respectively, for some $a \in \mathbb{C}$. The case $a\notin \mathbb{Z}$ corresponds to a nonintegral central character, a = −1, …, −n to singular central character, and all remaining a to a regular integral central character.

The following theorem is proved in [27].

Theorem 4.9

Assume that $a\notin \mathbb{Z}$ or a = −1,…,−n. Then Ψ _A provides an equivalence between $^{\mathrm{b}}(\mathcal{D}^{E},\mathcal{H})^{a}\text{-}mod$ and $\mathcal{G}\mathcal{B}^{\gamma (a\varepsilon _{1})}$ and between $(\mathcal{D}^{E},\mathcal{H})^{a}\text{-}mod$ and $\overline{\mathcal{G}\mathcal{B}}^{\gamma (a\varepsilon _{1})}$.

Moreover, Ψ _A provides an equivalence between $_{\mathrm{s}}^{\mathrm{b}}(\mathcal{D}^{E},\mathcal{H})^{a}\text{-}mod$ and $\mathcal{B}^{\gamma (a\varepsilon _{1})}$, as well as, between $_{\mathrm{s}}(\mathcal{D}^{E},\mathcal{H})^{a}\text{-}mod$ and $\overline{\mathcal{B}}^{\gamma (a\varepsilon _{1})}$.

An important part of the proof of the above proposition is the description of the injectives and the application of the twisted localization functor. More precisely, we have the following.

Proposition 4.10

Assume that $a\notin \mathbb{Z}$ or a = −1,…,−n, and that $\nu \in \mathbb{C}^{n+1}$ . Every indecomposable injective in $\overline{\mathcal{G}\mathcal{B}}_{\gamma (\nu )}^{\gamma (a\varepsilon _{1})}$ (respectively, in $\overline{\mathcal{B}}_{\gamma (\nu )}^{\gamma (a\varepsilon _{1})}$ ) is isomorphic to $\varPsi _{A}(\varGamma _{a}(\mathcal{F}_{\nu }^{\log }(J)))$ (respectively, to $\varPsi _{A}(\varGamma _{a}(S_{\mathcal{H}'}(\mathcal{F}_{\nu }^{\log }(J))))$ ) for some subset J of {1,2,…,n + 1}. In particular, if ν = 0 and $J = \varnothing$, the injective envelope of $\varPsi _{A}(\varGamma _{a}(\mathcal{F}_{0}^{+}))$ in $\overline{\mathcal{G}\mathcal{B}}^{\gamma (a\varepsilon _{1})}$ is $\varPsi _{A}(\varGamma _{a}(\mathcal{F}_{0}^{\log }))$, while the injective envelope of $\varPsi _{A}(\varGamma _{a}(\mathcal{F}_{0}^{+}))$ in $\overline{\mathcal{B}}^{\gamma (a\varepsilon _{1})}$ is $\varPsi _{A}(\varGamma _{a}(S_{\mathcal{H}'}(\mathcal{F}_{0}^{\log })))$.

It is not hard to see that

$$\displaystyle{S_{\mathcal{H}'}(\mathcal{F}_{\nu }) = \mathcal{F}_{\nu },\;S_{\mathcal{H}'}(\mathcal{F}_{\nu }^{\log }) = \mathcal{F}_{\nu }\otimes \mathbb{C}[u],}$$

where $u =\log (t_{1}t_{2}\cdots t_{n+1})$.

One can easily show that the twisted localization functor commutes with the functors $\varGamma _{a},\varPhi,\varPsi _{A}$, more precisely, that the following diagram is commutative.

where $\mu =\nu +x(\varepsilon _{i} -\varepsilon _{j})$. With this and Lemma 4.2. (v) in mind, one can describe the injectives $\overline{\mathcal{B}}$ and $\overline{\mathcal{G}\mathcal{B}}$ (for singular and nonintegral central characters) as twisted localization of the injectives in $\overline{\mathcal{C}}$ and $\overline{\mathcal{G}\mathcal{C}}$. Note that $\mathcal{B}$ and $\mathcal{G}\mathcal{B}$ do not have injectives for $\mathfrak{g} = \mathfrak{s}\mathfrak{l}(n + 1)$, and this is the main reason one needs to introduce the categories $\overline{\mathcal{B}}$ and $\overline{\mathcal{G}\mathcal{B}}$.

4.4.2 The Case $\mathfrak{g} = \mathfrak{s}\mathfrak{p}(2n)$

In contrast with $\mathfrak{g} = \mathfrak{s}\mathfrak{l}(n + 1)$, for $\mathfrak{g} = \mathfrak{s}\mathfrak{p}(2n)$, the category $\mathcal{B}$ does have enough injectives. In fact one has (as proved in [25]) the following.

Theorem 4.11

The injective envelope of a simple object L in $\mathcal{B}$ is isomorphic to $D_{F}^{\mathbf{x}}L$ for some set of n commuting long roots F and $\mathbf{x} \in \mathbb{C}^{n}$.

To describe the injectives in $\overline{\mathcal{G}\mathcal{B}}$ (note that $\mathcal{G}\mathcal{B}$ does not have injectives) one needs to use an equivalence Ψ _C of categories analogous to the functor Ψ _A described above. To define Ψ _C , we use the presentation of every element $X \in \mathfrak{g}$ as a block matrix of the form

$$\displaystyle{ \left [\begin{array}{*{10}c} A& B\\ C &-A^{t} \end{array} \right ]}$$

where A is an arbitrary n × n-matrix, and B and C are symmetric n × n-matrices. Then the maps

$$\displaystyle{ B\mapsto \sum _{i\leq j}b_{ij}t_{i}t_{j}\text{, }C\mapsto \sum _{i\leq j}c_{ij}\partial _{i}\partial _{j}}$$

extend to a homomorphism of Lie algebras

$$\displaystyle{ \mathfrak{g} \rightarrow \mathcal{D}(n)}$$

which induces a homomorphism of associative algebras

$$\displaystyle{ \omega: U\left (\mathfrak{g}\right ) \rightarrow \mathcal{D}(n).}$$

The image of ω coincides with the $\mathcal{D}(n)^{\mathrm{ev}}:=\bigoplus _{\vert \nu \vert \in 2\mathbb{Z}}\mathcal{D}(n)^{\nu }$ (recall that $\vert \nu \vert =\nu _{1} + \ldots +\nu _{n}$). For convenience in this subsubsection we set $\mathcal{D} = \mathcal{D}(n)$ and $\mathcal{D}^{\mathrm{ev}} = \mathcal{D}(n)^{\mathrm{ev}}$. With the aid of the homomorphism ω we obtain equivalence of categories $^{\mathrm{wb}}(\mathcal{D}^{\mathrm{ev}},\mathcal{H})\text{-}mod \rightarrow \mathcal{B}^{\chi }$ and $(\mathcal{D}^{\mathrm{ev}},\mathcal{H})\text{-}mod \rightarrow \overline{\mathcal{G}B}^{\chi }$, where χ stands for any central character of a bounded $\mathfrak{g}$-module (translation functors provided equivalence between all blocks $\mathcal{B}^{\chi }$). By definition, the functor Ψ _C is the one providing the second equivalence, i.e., $(\mathcal{D}^{\mathrm{ev}},\mathcal{H})\text{-}mod \rightarrow \overline{\mathcal{G}B}^{\chi }$. The first equivalence is established in §5 of [25]. For the second equivalence one needs to prove the $\mathfrak{s}\mathfrak{p}(2n)$-analogs of Lemmas 8.8 and 8.9 in [27]. More precisely, one first proves that Ψ _C is an equivalence on the cuspidal blocks by using the description of the cuspidal blocks in the category $\overline{\mathcal{G}\mathcal{B}}$ provided by [35]. Then applying twisted localization functors to the cuspidal injectives we obtain all injectives in $\overline{\mathcal{G}\mathcal{B}}$. To describe the injectives in $(\mathcal{D}^{\mathrm{ev}},\mathcal{H})\text{-}mod$ we introduce the modules $\mathcal{F}_{\nu }^{\mathrm{ev}}$, $\mathcal{F}_{\nu }^{\mathrm{ev,+}}$, $\mathcal{F}_{\nu }^{\mathrm{ev},\log }$, and $\mathcal{F}_{\nu }^{\mathrm{ev},\log }(J)$ which are the even degree components of $\mathcal{F}_{\nu }$, $\mathcal{F}_{\nu }^{+}$, $\mathcal{F}_{\nu }^{\log }$, and $\mathcal{F}_{\nu }^{\log }(J)$, respectively. Let χ ⁺ be the central character of the module $\varPsi _{C}(\mathcal{F}_{0}^{\mathrm{ev,+}})$.

Theorem 4.12

Every indecomposable injective object in $\overline{\mathcal{G}B}^{\chi ^{+} }$ is isomorphic to $\varPsi _{C}(\mathcal{F}_{\nu }^{\mathrm{ev},\log }(J))$ for some ν and a subset J of {1,2,…,n}. In particular, the injective envelope of $\varPsi _{C}(\mathcal{F}_{0}^{\mathrm{ev,+}})$ in $\overline{\mathcal{G}B}^{\chi ^{+} }$ is $\varPsi _{C}(\mathcal{F}_{0}^{\mathrm{ev},\log })$.

4.5 Twisted Localization for Finite-Dimensional Lie Superalgebras

In this case we consider a simple finite-dimensional Lie superalgebra $\mathfrak{g}$, $\mathcal{U} = U(\mathfrak{g})$ and $\mathcal{H} = S(\mathfrak{h}_{\bar{0}}^{ss})$. We consider multiplicative subsets consisting of root elements e _α in $\mathfrak{g}^{\alpha }$ for even roots α.

The following theorem is proved in [23].

Proposition 4.13

Let $\mathfrak{g}$ be a classical Lie superalgebra. Every simple module in $^{\mathrm{f}}(\mathcal{U},\mathcal{H})$ (equivalently, in $^{\mathrm{wf}}(\mathcal{U},\mathcal{H})$ ) is isomorphic to $D_{\varGamma }^{\mathbf{x}}L$ for some simple highest weight module L, a set Γ of n even commuting roots, and $\mathbf{x} \in \mathbb{C}^{n}$.

The parabolic induction theorem for any $\mathfrak{g}$ (including the Cartan type series) is proved in [14], where a classification of all cuspidal modules of $\mathfrak{g}$ is provided for all $\mathfrak{g}$ except for $\mathfrak{g} = \mathfrak{o}\mathfrak{s}\mathfrak{p}(m\vert 2n)$, $m = 1,3,4,5,6$, $\mathfrak{g} = \mathfrak{p}\mathfrak{s}\mathfrak{q}(n)$, $\mathfrak{g} = D(\alpha )$, and the Cartan type series. The simple cuspidal modules of $\mathfrak{p}\mathfrak{s}\mathfrak{q}(n)$ are classified in [22], while those of $\mathfrak{g} = D(\alpha )$, are classified in [29]. The classification of the simple cuspidal modules for $\mathfrak{g} = \mathfrak{o}\mathfrak{s}\mathfrak{p}(m\vert 2n)$, m = 1, 3, 4, 5, 6, and for the Cartan type series remains an open question.

4.6 Twisted Localization for Affine Lie Algebras

In this case, we consider commuting subsets of real roots $\varGamma \subset \varDelta ^{\mathrm{re}}$ and the multiplicative subsets of $U(\mathfrak{G})$ are $F =\{ e_{\alpha }\;\vert \;\alpha \in \varGamma \}$, where $e_{\alpha } \in \mathfrak{g}^{\alpha }$. As in the Lie algebra case we will write D _Γ for D _F . An important part of the classification of the simple objects of $^{\mathrm{wf}}(\mathcal{U},\mathcal{H})$ achieved in [13] is the constructions of loop modules and their relations with twisted localization.

We recall the definition and some properties of loop modules. For more detail, see [8–10]. Let $\mathfrak{G} = \mathcal{A}(\mathfrak{g})$, let $Y _{1},\ldots,Y _{k}$ be weight $\mathfrak{g}$-modules, and let $a_{1},\ldots,a_{k}$ be nonzero scalars. Following [9], we define the loop module $\mathcal{L}_{a_{1},\ldots,a_{k}}(Y _{1} \otimes \ldots \otimes Y _{k})$ in the following way: the underlining vector space of $\mathcal{L}_{a_{1},\ldots,a_{k}}(Y _{1} \otimes \ldots \otimes Y _{k})$ is

$$\displaystyle{(Y _{1} \otimes \ldots \otimes Y _{k}) \otimes \mathbb{C}[t,t^{-1}],}$$

$X \otimes t^{n} \in \mathcal{A}(\mathfrak{g})$ acts as

$$\displaystyle{(X \otimes t^{n}) \cdot ((v_{ 1} \otimes \ldots \otimes v_{k}) \otimes t^{s}) =\sum _{ i=1}^{k}a_{ i}^{n}(v_{ 1} \otimes \ldots \otimes X \cdot v_{i} \otimes \ldots \otimes v_{k}) \otimes t^{n+s},}$$

D acts as $t \frac{d} {dt}$, and K acts trivially. If the scalars $a_{1},\ldots,a_{k}$ are distinct, then $\mathcal{L}_{a_{1},\ldots,a_{k}}(Y _{1} \otimes \ldots \otimes Y _{k})$ is completely reducible with finitely many simple components. Furthermore, the simple components of $\mathcal{L}_{a_{1},\ldots,a_{k}}(Y _{1} \otimes \ldots \otimes Y _{k})$ are isomorphic simple $\mathcal{L}(\mathfrak{g})$-modules. Denote by $V _{a_{1},\ldots,a_{k}}(Y _{1} \otimes \ldots \otimes Y _{k})$ the simple $\mathcal{L}(\mathfrak{g})$-module which is a component of $\mathcal{L}_{a_{1},\ldots,a_{k}}(Y _{1} \otimes \ldots \otimes Y _{k})$. Considered as $\mathcal{A}(\mathfrak{g})$-modules, the constituents of $\mathcal{L}_{a_{1},\ldots,a_{k}}(Y _{1} \otimes \ldots \otimes Y _{k})$ differ only by a shift of the action of D. By a slight abuse of notation we denote by $V _{a_{1},\ldots,a_{k}}(Y _{1} \otimes \ldots \otimes Y _{k})$ any shift of a simple $\mathcal{A}(\mathfrak{g})$–component of $\mathcal{L}_{a_{1},\ldots,a_{k}}(Y _{1} \otimes \ldots \otimes Y _{k})$. A relation between loop modules and twisted localization is provided in the following result in [13].

Proposition 4.14

Let $\mathfrak{G} = \mathcal{L}(\mathfrak{g})$ be an untwisted affine Lie algebra, let $\alpha \in \mathring{\varDelta}$, and let $\mathcal{L}_{a_{0},\ldots,a_{k}}(S \otimes F_{1} \otimes \ldots \otimes F_{k})$ be the loop $\mathfrak{G}$ -module for which $a_{i} \in \mathbb{C}$, F _i are simple finite-dimensional $\mathfrak{g}$ -modules, and S is a simple e _α -torsion free $\mathfrak{g}$ -module. Then $\mathcal{L}_{a_{0},\ldots,a_{k}}(S \otimes F_{1} \otimes \ldots \otimes F_{k})$ is e _α+rδ -torsion free and

$$\displaystyle{D_{\alpha +r\delta }\mathcal{L}_{a_{0},\ldots,a_{k}}(S \otimes F_{1} \otimes \ldots \otimes F_{k}) \simeq \mathcal{L}_{a_{0},\ldots,a_{k}}(D_{\alpha }S \otimes F_{1} \otimes \ldots \otimes F_{k})}$$

for every integer r.

If $\mathfrak{G} = \mathcal{A}(\mathfrak{g},\sigma )$, then $\mathcal{L}_{a_{1},\ldots,a_{k}}(Y _{1} \otimes \ldots \otimes Y _{k})$ admits an endomorphism compatible with σ if and only if the modules $Y _{1},\ldots,Y _{k}$ come in r-tuples of isomorphic modules and for each r-tuple the corresponding scalars $a_{1},\ldots,a_{r}$ are multiples (with the same scalar) of the rth roots of unity. We denote the corresponding endomorphism of $\mathcal{L}_{a_{1},\ldots,a_{k}}(Y _{1} \otimes \ldots \otimes Y _{k})$ by σ again and let $\mathcal{L}_{a_{1},\ldots,a_{k}}^{\sigma }(Y _{ 1} \otimes \ldots \otimes Y _{k})$ denote the fixed points of σ. Similarly, we can define $V _{a_{1},\ldots,a_{k}}^{\sigma }(Y _{ 1} \otimes \ldots \otimes Y _{k})$.

The following is the main theorem in [13].

Theorem 4.15

Every simple module M of $\mathfrak{G}$ in $^{\mathrm{f}}(\mathcal{U},\mathcal{H})$ (equivalently, in $^{\mathrm{wf}}(\mathcal{U},\mathcal{H})$ ) either is parabolically induced or is isomorphic (up to a shift) to

$$\displaystyle{V _{a_{0},a_{1},\ldots,a_{k}}(N \otimes F_{1} \otimes \ldots \otimes F_{k}),}$$

where $a_{0},\ldots,a_{k}$ are distinct nonzero scalars, N is a cuspidal $\mathfrak{g}$ -module, and $F_{1},\ldots,F_{k}$ are finite-dimensional $\mathfrak{g}$ -modules. In the latter case $\mathfrak{G}\mathop{\cong}A_{l}^{(1)}$ or $\mathfrak{G}\mathop{\cong}C_{l}^{(1)}$ . In particular, M is isomorphic to $D_{F}^{\mathbf{x}}V _{a_{0},a_{1},\ldots,a_{k}}(L \otimes F_{1} \otimes \ldots \otimes F_{k}),$ for some bounded highest weight $\mathfrak{g}$ -module L, commuting set of ℓ roots F, and $x \in \mathbb{C}^{\ell}$.

The study of the category of bounded modules of $\mathfrak{G}$, and in particular, describing the injectives in that category, is a largely open question.