Simple Modules Over Lie Algebras

UPPSALA DISSERTATIONS IN MATHEMATICS 94

Simple Modules over Lie Algebras

Jonathan Nilsson

Department of Mathematics Uppsala University UPPSALA 2016 Dissertation presented at Uppsala University to be publicly examined in Häggsalen, Ångströmlaboratoriet, Lägerhyddsvägen 1, Uppsala, Wednesday, 1 June 2016 at 13:15 for the degree of Doctor of Philosophy. The examination will be conducted in English. Faculty examiner: Professor Erhard Neher (University of Ottawa).

Abstract Nilsson, J. 2016. Simple Modules over Lie Algebras. Uppsala Dissertations in Mathematics 94. 50 pp. Uppsala: Department of Mathematics. ISBN 978-91-506-2544-8.

Simple modules are the elemental components in representation theory for Lie algebras, and numerous mathematicians have worked on their construction and classification over the last century. This thesis consists of an introduction together with four research articles on the subject of simple Lie algebra modules. In the introduction we give a light treatment of the basic structure theory for simple finite dimensional complex Lie algebras and their representations. In particular we give a brief overview of the most well-known classes of Lie algebra modules: highest weight modules, cuspidal modules, Gelfand-Zetlin modules, Whittaker modules, and parabolically induced modules. The four papers contribute to the subject by construction and classification of new classes of Lie algebra modules. The first two papers focus on U(h)-free modules of rank 1 i.e. modules which are free of rank 1 when restricted to the enveloping algebra of the Cartan subalgebra. In Paper I we classify all such modules for the special linear Lie algebras sln+1(C), and we determine which of these modules are simple. For sl2 we also obtain some additional results on tensor product decomposition. Paper II uses the theory of coherent families to obtain a similar classification for U(h)-free modules over the symplectic Lie algebras sp2n(C). We also give a proof that U(h)-free modules do not exist for any other simple finite-dimensional algebras which completes the classification. In Paper III we construct a new large family of simple generalized Whittaker modules over the general linear Lie algebra gl2n(C). This family of modules is parametrized by non-singular nxn-matrices which makes it the second largest known family of gl2n-modules after the Gelfand-Zetlin modules. In Paper IV we obtain a new class of sln+2(C)- modules by applying the techniques of parabolic induction to the U(h)-free sln+1-modules we constructed in Paper I. We determine necessary and sufficient conditions for these parabolically induced modules to be simple.

Keywords: Lie algebra, Representation, Simple module, Non-weight module, Classification, Construction

Jonathan Nilsson, Department of Mathematics, Algebra and Geometry, Box 480, Uppsala University, SE-751 06 Uppsala, Sweden.

ISSN 1401-2049 ISBN 978-91-506-2544-8 urn:nbn:se:uu:diva-283061 (http://urn.kb.se/resolve?urn=urn:nbn:se:uu:diva-283061) List of papers

This thesis is based on the following papers, which are referred to in the text by their Roman numerals.

I Jonathan Nilsson; Simple sln+1-module structures on U (h). Journal of Algebra 424 (2015) 294–329.

II Jonathan Nilsson; U (h)-free modules and coherent families. Journal of Pure and Applied Algebra 220 (2016) 1475–1488.

III Jonathan Nilsson; A new family of simple gl2n(C)-modules. Paciﬁc Journal of Mathematics 283, No. 1, (2016) 1–19.

IV Yan-an Cai, Genqiang Liu, Jonathan Nilsson, Kaiming Zhao; Generalized Verma modules over sln+2 induced from U (hn)-free sln+1-modules. Manuscript.

Reprints were made with permission from the publishers.

Also published by the Author but not included in this thesis:

Jonathan Nilsson; Enumeration of Basic Ideals in Type B Lie Algebras. Journal of Integer Sequences, Volume 15 2012, Article 12.9.5.

Foreword

This is a thesis on representation theory for Lie algebras. Thanks to con- nections throughout mathematics and physics, this subject has been actively researched for the last hundred years but many questions still remain unre- solved. In representation theory we study abstract mathematical objects called modules. To investigate these modules it is useful to break them down into smaller, more simple components. To understand the principle, consider the following analogy: In basic chemistry we learn that there is a finite list of elements (the periodic table), and we know that the corresponding atoms can come together to form molecules. Moreover, some fundamental properties of a molecule, such as its weight, can be deduced from knowing its atom constituents. However there also exist isomers – different molecules with the same atom components – which shows that molecules are not completely determined by their components. The “atoms” of representation theory are called simple modules. Any module can analyzed in terms of its simple components – the simple modules which it is made up of. Just as in chemistry however, truly different modules can con- sist of the same simple components, so we only get a rough picture by knowing the simple constituents. While the regular periodic table contains 120 or so elements, the periodic table for the representation theory of a Lie algebra is unfortunately both incomplete and enormous: there are infinitely many simple modules. This thesis is concerned with finding and classifying new classes of simple modules. The thesis has two parts. In the first part we provide an overview of Lie theory and how it has developed over the years. In particular we discuss the most well-known classes of modules: highest weight modules, cuspidal modules, Whittaker modules, Gelfand-Zetlin modules, and parabolically induced modules. The second part of the thesis is about my own work on constructing and classifying new families of modules. Here you will find summaries of the four papers included at the end of the thesis, as well as a summary in Swedish. Throughout this text all Lie algebras, modules, and vector spaces are as- sumed to be over the complex numbers C unless otherwise stated. We shall write ei, j for the matrix having a single 1 in position (i, j) and zeroes every- + where else. N denotes the non-negative integers while N denotes the positive integers.

Contents

Foreword ...... 5

Part I: Introduction ...... 9

1 Lie algebras ...... 11 1.1 Background ...... 11 1.2 Deﬁnition ...... 12 1.3 Classiﬁcation simple Lie algebras ...... 12 1.4 Some linear Lie algebras ...... 16 1.5 The universal enveloping algebra ...... 17

2 Representation theory ...... 19 2.1 Background ...... 19 2.2 Deﬁnition ...... 19 2.3 Simple modules and composition series ...... 20 2.4 Categorical techniques ...... 20 2.4.1 Characters and block decomposition ...... 20 2.4.2 Tensor functor ...... 21 2.4.3 Duality functors ...... 21 2.4.4 Functors related to subalgebras ...... 22 2.4.5 Twisting functors ...... 22 2.4.6 Translation functors ...... 22

3 Known classes of modules ...... 24 3.1 Weight modules ...... 24 3.1.1 Highest weight modules ...... 24 3.1.2 Finite dimensional modules ...... 25 3.1.3 Category O ...... 26 3.1.4 Cuspidal modules and coherent families ...... 26 3.2 Whittaker modules ...... 26 3.2.1 Kostant’s modules ...... 26 3.2.2 Generalized Whittaker modules ...... 27 3.3 Gelfand-Zetlin modules ...... 27 3.4 Parabolically induced modules ...... 29 Part II: Results of the Thesis ...... 31

4 U (h)-free modules ...... 33

5 On Paper I ...... 35 5.1 Construction and classiﬁcation of U (h)-free modules ...... 35 5.2 Submodules and properties ...... 36 5.3 Clebsch-Gordan problem for sl2 ...... 37

6 On Paper II ...... 38 6.1 Weighting functor and coherent families ...... 38 6.2 Classiﬁcation in type C ...... 39 6.3 Applying translation functors ...... 39

7 On Paper III ...... 40 7.1 Decomposition of gl2n ...... 40 7.2 Module construction ...... 40

8 On Paper IV ...... 42 8.1 Generalized Verma modules ...... 42 8.2 Our construction ...... 42 8.3 Simplicity ...... 43

9 Summary in Swedish - Sammanfattning på svenska ...... 44 9.1 Bakgrund ...... 44 9.2 Kända klasser av enkla moduler ...... 44 9.2.1 Viktmoduler ...... 44 9.2.2 Ändligtdimensionella moduler ...... 45 9.2.3 Whittakermoduler ...... 45 9.2.4 Gelfand-Zetlinmoduler ...... 45 9.3 Avhandlingens resultat ...... 46 9.3.1 U (h)-fria moduler ...... 46 9.3.2 Generaliserade Whittakermoduler för gl2n ...... 47 9.3.3 Generaliserade Vermamoduler från U (h)-fria moduler ...... 47

Acknowledgements ...... 48

References ...... 49 Part I: Introduction

1. Lie algebras

1.1 Background Sophus Lie was born in Norway in 1842. When he moved to Berlin for a scholarship he became close friends with the German mathematician Felix Klein and by reading the earlier works of Jacobi, they developed an interest for transformations of differential equations together. Their work and exchange of ideas resulted in an important paper by Lie about what he called continuous transformation groups. Here a transformation group meant a set of transformations, closed under composition, from some subset of n-dimensional complex space to itself. For Lie, the word continuous (as well as group) meant something different than it does today. Lie writes

"A group is called continuous when all of its transformations are generated by repeating inﬁnitesimal transformations inﬁnitely often.“

Lie’s brilliant idea was to – instead of analyzing the transformation group directly – study the space of such infinitesimal transformations. The modern generalization of these transformation groups are called Lie groups. They are important in mathematics and physics since they describe the sets of symmetries of objects (think of the symmetries of a sphere for example). Although Lie viewed infinitesimal transformations as special elements in the group itself, modern mathematics calls the space they generate the Lie algebra of the Lie group. The Lie algebra is a vector space and as such is easier to work with. Also, as Lie realized, the structure of the group can typically be reconstructed from the Lie algebra via an exponential map. One of Lie’s great projects was to try and classify all transformation groups. The German mathematician Wilhelm Killing suggested that it would be necessary to first classify all possible Lie algebras, and he devoted much time to this project himself during the late 19th century. Although Killing’s work was sometimes confusing, in the end he managed to show that aside from a few exceptional cases, there can only exist Lie algebras of the special, symplectic and orthogonal kind. The young French mathmatician Élie Cartan removed any doubt about Killings result with his complete classification of simple complex Lie algebras in his doctoral thesis which was published in 1894. Cartan continued on to also classify Lie algebras over the real numbers by realizing them as real forms of the complex algebras. A review of Killing and Cartan’s approach to Lie algebra classification is presented in Section 1.3. For more information about the early developments of Lie theory, see [Ha].

11 1.2 Deﬁnition Nowadays Lie algebras are often introduced abstractly without relying on a Lie group. A Lie algebra g is a vector space in which you can multiply vectors using a so called Lie bracket written [x,y]. Like regular multiplication of numbers, the bracket is required to be linear in each entry i.e. [x+y,z] = [x,z]+[y,z] etcetera. Unlike multiplication however, the bracket also has to satisfy [x,x] = 0 for all x which is equivalent to skew-symmetry:

[x,y] = −[y,x] for all x,y in g. (1.1) The bracket is not associative either, instead it satisﬁes the Jacobi-identity:

[x,[y,z]] + [z,[x,y]] + [y,[z,x]] = 0 for all x,y,z in g. (1.2)

Any bilinear operation satisfying (1.1) and (1.2) gives rise to a Lie algebra so there are many examples: • Let V be any vector space and define [x,y] = 0 for all x,y ∈ V. This operation is bilinear and satisfies, so this is a (trivial) Lie algebra structure. • Three-dimensional space R3 becomes a Lie algebra (over R) if we define the bracket operation by [x,y] := x × y (cross product of vectors). This Lie algebra structure is usually encountered by engineers and mathematicians in the first course in Linear algebra. • Let W be the infinite dimensional vector space with basis {ln|n ∈ Z} equipped with the bracket [lm,ln] = (m − n)lm+n. This Lie algebra is called the Witt algebra. Together with the related Virasoro-algebra it plays a part in theoretical physics. • Let gln be the set of n × n-matrices and define [A,B] := AB − BA, where the multiplication on the right side is usual matrix multiplication. This is called a general linear Lie algebra. • More generally, let g be any associative algebra with multiplication (x,y) 7→ x · y. By defining [x,y] := x · y − y · x we equip g with a Lie algebra structure.

1.3 Classification simple Lie algebras The elemental components of Lie algebras are called simple Lie algebras. The theory leading to the classification of simple finite dimensional Lie algebras was developed by Killing-Cartan, and the results are very beautiful. Their work is summarized in what follows. For a simple Lie algebra g, let h be a nilpotent self-normalizing subalgebra ∗ of g called a Cartan subalgebra. Now let h = HomC(h,C) be the space of ∗ linear maps h → C. For each nonzero α ∈ h we define the corresponding root

12 space gα in g to be

gα := {x ∈ g | [h,x] = α(h)x for all h ∈ h}.

Then x ∈ gα just means that x is an eigenvector for the operator [h,−] for all h ∈ h simultaneously, where the corresponding eigenvalue is given by the scalar α(h). Most spaces gα will be zero, and the nonzero gα will have dimension 1. It also turns out that every element of g can be written uniquely as a linear combination of root vectors and a Cartan element. In other words we have M g = h ⊕ gα , (1.3) α∈Φ ∗ where Φ is the set of roots i.e. nonzero α ∈ h such that gα 6= 0. It is now easy to check that

[h,gα ] ⊂ gα and that [gα ,gβ ] ⊂ gα+β , so the decomposition above tells us much about the internal structure of g. We now turn our attention to the geometry of the set Φ in h∗. The vector space g comes equipped with a symmetric bilinear form called the Killing form defined by κ(x,y) = tr([x,[y,−]]). It turns out that the restriction of the Killing form to h × h is non-degenerate ∗ which allows us to for each α ∈ h define tα ∈ h as the unique element satisfying κ(tα ,h) = α(h) for all h ∈ h. This lets us define a symmetric bilinear ∗ ∗ form on h by (α,β) := κ(tα ,tβ ). Using this form we define reflections in h in the natural way: for each nonzero α ∈ h∗ we define

(α,β) σ : h∗ → h∗ by σ (β) = β − 2 α. α α (α,α)

Geometrically, σα reﬂect vectors in the hyperplane orthogonal to α. In fact, the set Φ appearing in our Lie algebra decomposition (1.3) always satisﬁes the following symmetry properties.

• Φ is ﬁnite, does not contain 0, and spans h∗. • For α ∈ Φ we also have −α ∈ Φ, but no other multiples of α occur. • We have σα (Φ) = Φ for all α ∈ Φ. • 2(β,α)/(α,α) ∈ Z for all α,β ∈ Φ.

In other words, Φ forms a so called root system in h∗. The rank of the root system is deﬁned as the dimension of h∗. The subgroup W of GL(h∗) generated by {σα }α∈Φ is called the Weyl group of g. Here follows a picture of all possible root systems of rank 2.

13 O X1 F 11 11 11 11 o / o 1 1 / 11 1 A 11 2 A1×A1 1 11 Ö 1

_? O ? O ?? ?? ?? ?? fMMM X11 F qq8 ?? MM 1 qq ? MM 11 qq ?? MMM1 qqq o ?? / o qq qM1 1MM / ?? qq 11 MM ? B2 qq 1 MM ?? qqq 1 MMM ?? xq Ö & ?? ?? G2 ?

Instead of keeping track of all roots it is useful to introduce a base of the root system. A base is deﬁned as a subset ∆ ⊂ Φ such that each root β can + be uniquely expressed as β = ±∑α∈∆ cα α where all cα ∈ N . Here, the set of roots for which the plus sign is chosen are denoted Φ+. Its elements are called positive roots. The set Φ− of negative roots is deﬁned correspondingly. The elements of the base ∆ are called simple roots. For example, consider the following root system:

β fMM X1 F q8 MM 11 qq MMM 1 qqq MM11 qq o MqM1 q / α qq 1MMM qqq 11 MM qqq 1 MMM xqq Ö 1 M&

Here ∆ = {α,β} is a base for the root system, which gives positive roots Φ+ = {α, β, β + α, β + 2α, β + 3α, 2β + 3α}, while Φ− = −Φ+. Letting

M M n+ := gα and n− := gα , α∈Φ+ α∈Φ−

14 we have obtained a triangular decomposition of our Lie algebra:

g = n− ⊕ h ⊕ n+, (1.4) separating it into a negative part, a Cartan subalgebra, and a positive part. Many aspects of the representation theory for Lie algebras relies on this decomposition. In fact, a root system is almost completely determined by the angles between its simple roots as illustrated in the ﬁgure of rank 2 root systems. More- over, the only possible such angles are π/2, 2π/3, 3π/4, and 5π/6. This lets us to encode the combinatorial data of a root system in so called Coxeter graph i.e. a graph with a vertex for each simple root, and 0,1,2, or 3 edges between two vertices depending on whether the angle between the two corresponding roots are π/2, 2π/3, 3π/4, or 5π/6. Thus the four root systems of rank 2 listed above has the following Coxeter graphs:

A1 × A1 : ◦◦ ◦◦ A2 : ◦ ◦ B2 : ◦ ◦ G2 : ◦ ◦ In order to completely determine the root system however, it turns out that we need one more piece of information: for each double- and triple edge in the Coxeter graph we need to specify which of the two involved roots is the longest. We do this by for each such edge drawing an arrow pointing towards the longer root. Our new objects are called Dynkin diagrams, and they determine the root system completely. Not every Dynkin diagram we imagine can be derived from a root system though. The admissible Dynkin diagrams fall into four inﬁnite families, together with ﬁve special cases.

Theorem 1. Consider the following list of Dynkin diagrams where the sub- scripts indicate the number of vertices.

An (n ≥ 1) ◦ ◦ ◦ ◦ ◦

Bn (n ≥ 2) ◦ ◦ ◦ ◦ / ◦

Cn (n ≥ 3) ◦ ◦ ◦ ◦ o ◦

Dn (n ≥ 4) ◦ ◦ ◦ ◦ ◦

◦

15 E6 ◦ ◦ ◦ ◦ ◦

◦

E7 ◦ ◦ ◦ ◦ ◦ ◦

◦

E8 ◦ ◦ ◦ ◦ ◦ ◦ ◦

◦

F4 ◦ ◦ / ◦ ◦

G2 ◦o ◦ Every simple ﬁnite dimensional complex Lie algebra has one of these listed Dynkin diagrams, and conversely, each listed diagram determines the corresponding Lie algebra up to isomorphism.

One strength of having such a theorem is that many results can be proved by case-by-case investigations. This is illustrated in Paper I and Paper II of this thesis.

1.4 Some linear Lie algebras If V is a vector space, the space of linear maps V →V is a Lie algebra under the Lie bracket [ f ,g] = f ◦ g − g ◦ f . This is called the general linear Lie algebra and it is denoted gl(V). It is a basic object of study in Lie theory. When V is finite dimensional we may choose a basis consisting of n elements and identify gl(V) with a space of n × n matrices and we write gln for this algebra. The importance of this Lie algebra is displayed by Ado’s theorem which states that every finite dimensional Lie algebra can be embedded as a subalgebra of gln for some n. The Lie algebra gln falls outside the classification in Theorem 1; it is not simple as the identity-matrix spans an ideal. The four infinite-families of Lie algebras from Theorem 1 can be realized as subalgebras of gln as follows. Let sln+1 := {A ∈ gln+1 | tr(A) = 0}. This is the special linear Lie algebra. It corresponds to the root system of type An. Of particular interest is the first Lie algebra in this series, sl2. It has a basis 0 1 1 0 0 0 x = h = y = , (1.5) 0 0 0 −1 1 0

16 with respect to which the bracket is given by

[h,x] = 2x, [h,y] = −2y, [x,y] = h.

Although this is an easy algebraic object, many deep Lie theoretic results can be derived from its study, see for example [Maz2]. Next, the symplectic Lie algebra of rank n is deﬁned by

n AB o sp = ∈ gl BT = B, CT = C . 2n C −AT 2n

This is a simple Lie algebra of type Cn. The orthogonal Lie algebras fall into two classes with different properties depending on whether the rank of the Lie algebra is even or odd – we deﬁne

n AB o o = ∈ gl BT = −B, CT = −C , 2n C −AT 2n which is a simple Lie algebra of type Dn, and we deﬁne ( ! ) 0 a b T T T o2n+1 = −b AB ∈ gl2n+1 B = −B, C = −C , −aT C −AT which has type Bn.

1.5 The universal enveloping algebra Non-associative structures are in general harder to handle than associative ones. The universal enveloping algebra of a Lie algebra is a unital associative algebra which encodes the non-associative Lie bracket as the commutator operation. By passing to the universal enveloping algebra we obtain an al- gebraically easier object without loosing information about the representation theory. Recall that any algebra A becomes a Lie algebra by deﬁning [x,y] = xy − yx. We denote this Lie algebra by A0. The universal enveloping algebra of a Lie algebra g can be deﬁned abstractly as an associative algebra U together with a Lie algebra homomorphism i : g → U0 such that for any algebra A and for any Lie algebra homomorphism j : g → A0 there exists a unique algebra homomorphism ϕ : U → A such that ϕ ◦i = j. One can show that the universal enveloping algebra exists and is unique. It is denoted U (g). More concretely, U (g) can be constructed as follows. We start with the Tensor algebra T on g. This is the vector space spanned by elements of form x1 ⊗ x2 ⊗ ··· ⊗ xr with xi ∈ g and r ≥ 0. The algebra product on this basis is tensor product concatenation. Letting J be the two-sided ideal generated by all

17 elements of form x⊗y−y⊗x we deﬁne U (g) := T/J and we usually write its elements without tensor signs. A key observation here is that U (g) commutes up to terms of lower degree – this leads to the famous Poincaré-Birkhoff-Witt theorem which provides us with a basis for U (g).

Theorem 2. Let (x1,x2,...,xn) be any ordered basis of the vector space g. Then the set m1 m2 mn {x1 x2 ···xn |m1,m2,...,mn ∈ N} is a basis for U (g).

18 2. Representation theory

2.1 Background Parallell to the mathematical development of the Killing-Cartan theory in the beginning of the 20th century, a movement started among the many great physicists at the university of Göttingen. The goal was to develop a sound mathematical formalization of physics. At this time interest grew for the representations of Lie algebras i.e. how Lie algebras can act on vector spaces in a coherent fashion. At this time the mathematicians Hermann Weyl, Issai Schur, and Claude Chevalley made many contributions to representation theory for Lie algebras, such as the famous Schur’s lemma and Weyl’s character formula. With some help from Weyl’s work, Cartan managed to classify all simple ﬁnite dimensional representations in 1914. When Weyl moved from Göttingen to Zürich in 1913 he met a young Ein- stein who was working on his theory of general relativity. Weyl was very impressed with Einsteins work. He wrote

For myself I can say that the wish to understand what really is the mathematical substance behind the formal apparatus of relativity theory led me to the study of representations and invariants of groups.

Since the early 1900’s, countless developments have been made in Lie algebra representation theory. Many new classes of modules were discovered and I have tried to present most of them in Section 3.

2.2 Deﬁnition A representation of a Lie algebra g is a homomorphism g → gl(V) such that each element of g is represented as an Lie algebra endomorphism on V. It is often more useful to instead view a representation as a module – a vector space on which g acts. A module for a Lie algebra g is a vector space M equipped with a bilinear action g × M → M written (x,m) 7→ x · m such that x · (y · m) − y · (x · m) = [x,y] · m for all x,y ∈ g and m ∈ M. Alternatively we can consider M as a module over the universal enveloping algebra of g. The bilinear action of U (g) on M is then instead required to satisfy x · (y · m) = (xy) · m and 1 · m = m

19 for all x,y ∈ g and m ∈ M, which is the standard deﬁnition of a module over a ring. Since xy − yx = [x,y] for all x,y ∈ U (g) the two deﬁnitions are equivalent. This lets us identify g-modules with U (g)-modules.

2.3 Simple modules and composition series Modules can be broken down into their simple components in the following sense. A submodule N of the g-module M is a subspace N ⊂ M closed under the g-action. For each submodule we also have a corresponding quotient module – a canonical module structure on the quotient space M/N. A module can be studied in terms of its submodules and quotients. The trivial module {0} and M itself are always submodules of M, and if these are the only two submodules, M is called a simple module. A composition series for a module M is a chain of submodules

0 = M0 ⊂ M1 ⊂ M2 ⊂ ··· ⊂ Mr−1 ⊂ Mr = M, such that the quotient modules Li := Mi/Mi−1 are simple for all 1 ≤ i ≤ r. If M admits a composition series, the integer r (called the length of M) and the multi-set of simple components Li are the same in all composition series of M. To determine what kind of Lie algebra modules exist, it is therefore important to first determine what simple modules exist. For simple Lie algebras there is however no complete classification of simple modules (except arguably for sl2, see [Bl, Maz2]). Moreover, such a classification seems close to hopeless to ever obtain. There are however various known classes of modules, and the papers in this thesis contribute to Lie algebra representation theory by constructing and analyzing some new such classes of simple modules.

2.4 Categorical techniques For a ﬁxed Lie algebra g, the category of all modules over g is denoted g-Mod or U (g)-Mod. Its objects are g-modules, and its morphisms are g-module homomorphisms. It is often useful to study g-Mod through subcategories and functors.

2.4.1 Characters and block decomposition A central character is a Lie algebra homomorphism χ : Z(g) → C from the center of U (g) to the scalars. A module M is said to have central character χ provided that

z · v = χ(z)v for all z ∈ Z(g) and v ∈ M.

20 While not all modules have central character it is easy to prove that simple modules always do. Moreover, there can exist no nontrivial homomorphisms between modules with different central characters. We shall write χM for the central character of a module M. An important result is that every character can be realized as a character of a simple highest weight module L(λ) (see Section 3.1.1). In other words, if M has central ∗ character we have χM = χL(λ) for some λ ∈ h . One usually writes just χλ for χL(λ). More generally, a module is said to have generalized central character if each of its vectors is annihilated by a large enough power of (z − χ(z)) for every central z. The full subcategory of U (g)-Mod consisting of modules with generalized central character χ is denoted U (g)-Mod(χ), and we have projection functors

Pr(χ) : g-Mod → g-Mod(χ) M 7→ M(χ), where

M(χ) = {m ∈ M|∀z ∈ Z(g),(z − χ(z))N = 0 for large enough N}.

The category of all U (g) modules on which the action of the center is locally ﬁnite is denoted U (g)-ModZ f . This category decomposes completely into blocks depending on central characters – the so called block decomposition: M (χ) U (g)-ModZ f = U (g)-ModZ f . (2.1) χ∈Θ Here Θ is the set space of central characters.

2.4.2 Tensor functor For any two Lie algebra modules M and N there is a natural g-module structure on M ⊗C N given by x · (m ⊗ n) = (x · m) ⊗ n + m ⊗ (x · n).

The assignment M 7→ M ⊗ N is in fact functorial and the corresponding end- ofunctor on g-Mod is usually written − ⊗ N. The tensor functor interacts additively with central character in the sense that χM⊗N = χM + χN whenever M and N has central character.

2.4.3 Duality functors If M is a left g-module there is a natural right module structure on M∗ = HomC(M,C) given by ( f · x)(m) = f (x · m). It is however desirable to stay

21 in the category of left g-modules so we can instead pick a Lie algebra anti- homomorphism τ and deﬁne a left g-structure on M∗ by (x · f )(m) = f (τ(x) · m) for all x ∈ g and f ∈ M∗. Here there are two natural choices for τ: either the involution mapping x to −x for all x ∈ g, or the Chevalley involution, an involution exchanging root spaces gα → g−α .

2.4.4 Functors related to subalgebras Let a be a ﬁxed subalgebra of g. First we have the induction functor g Inda : a-Mod → g-Mod, which maps an a-module M to U (g) ⊗U (a) M (where the g-action is the natural U (g)-multiplication on the left). A functor going in the other direction is the restriction functor g Resa : g-Mod → a-Mod, which maps modules to themselves but with action restricted from g to a. Induction and restriction functors form an adjoint pair i.e. for each M ∈ a- Mod and N ∈ g-Mod there is a natural vector space isomorphism g g Homg(IndaM,N) ' Homa(M,ResaN).

2.4.5 Twisting functors For each Lie algebra automorphism ϕ : g → g we have a corresponding twisting functor ϕ Fϕ : g-Mod → g-Mod Fϕ : M 7→ M. Here ϕ M is identiﬁed with M as a vector space but with a new twisted g-action given by x•m = ϕ(x)·m. It is clear that Fϕ ◦Fϕ−1 is isomorphic to the identity functor so Fϕ is an equivalence on g-Mod.

2.4.6 Translation functors Translation functors are compositions of the tensor functors and projection functors discussed in 2.4.2 and 2.4.1 above. They provide categorical equivalences between blocks of different central character. More precisely, let λ and µ be weights such that λ − µ is dominant and integral (i.e. L(λ − µ) is ﬁnite dimensional, see Section 3.1.1). Then the functor T(µ,λ) : U (g)-Mod(χµ ) → U (g)-Mod(χλ ) M 7→ (M ⊗ L(λ − µ))(χλ )

22 is an equivalence of categories. Translation functors were independently introduced Zuckerman and Jantzen see [Zu, Ja2]. This was based on previous work by Bernstein, Gelfand, and Gelfand [BGG1]. See also [BG] which investigates projective functors – a generalization of translation functors.

23 3. Known classes of modules

3.1 Weight modules

Let M be a module over a Lie algebra g with triangular decomposition n− ⊕ ∗ h ⊕ n+. For all λ ∈ h we deﬁne the corresponding weight space

Mλ := {m ∈ M|h · m = λ(h)m for all h ∈ h}. L M is called a weight module if M = λ∈h∗ Mλ , and the λ for which Mλ is nonzero are called the weights of M. The action of root vectors on weight spaces satisﬁes gα · Mλ ⊂ Mλ+α which shows that if a module is simple, all its weights lie in a single coset of h∗/Φ. Many results have been proved about weight modules, see for example [Di, Hu, DMO, Ve].

3.1.1 Highest weight modules Choosing a base ∆ for our root system also equips the space h∗ with a partial order relation ≺ generated by deﬁning α ≺ β whenever β −α ∈ ∆. M is called a highest weight module provided that M is generated by a weight vector vλ of weight λ such that n+vλ = 0. This means that M is a weight module where λ is maximal among its weights with respect to the order ≺. Verma modules are universal objects among the highest weight modules. They were originally studied by Verma [Ve] and can be constructed as follows: ∗ For each λ ∈ h let Cλ be the one dimensional h ⊕ n+ module, where h ∈ h acts by the scalar λ(h), and all of n+ acts by 0. Now let

M(λ) := U (g) ⊗U (h⊕n+) Cλ .

This is a U (g)-module under the natural left action. Note that M(λ) is isomorphic to U (n−) as a vector space (and as a n−-module). Verma modules have a unique maximal submodule and a corresponding unique simple quotient denoted L(λ). One can show that any highest weight module is a quotient of a Verma module, and so the assignment

λ 7→ L(λ) parametrizes the simple highest weight modules.

24 3.1.2 Finite dimensional modules In fact one can show that every ﬁnite dimensional module is a highest weight module with weights invariant under the Weyl group in the sense that

dimMλ = dimMw(λ), for all λ ∈ h∗, w ∈ W. Moreover, every simple ﬁnite dimensional module is completely determined by its highest weight. This situation becomes especially easy for g = sl2 where we can easily ∗ identify h with C via λ 7→ λ(h) with notation as in (1.5). In this case it turns out that the simple module L(λ) is ﬁnite dimensional if and only if λ ∈ N. Moreover, for n ∈ N, L(n) is (n + 1)-dimensional and its set of weights are

{n,n − 2,n − 4,...,−(n − 2),−n}.

Moreover, x and y acts on weight vectors by increasing the weight by ±2. For example, the module L(4) can be visualized as follows:

h h h h h x x x x + + + + -4 k -2 k 0 k 2 k 4 y y y y

Here the boxes indicate one-dimensional weight spaces of weight λ(h) indicated by the number. The following diagram illustrates an example of the occurring weights spaces of a simple ﬁnite dimensional weight module when the Lie algebra is of type ∗ A2. Here the positions of the circles indicate the occurring weights in h , while the numbers indicates the dimension of the corresponding weight space...... x. x. x. x...... x. y. y. y. x...... ×. . . . x. y. y. x...... x. y. x...... x. x......

25 3.1.3 Category O A natural generalization of the category of highest weight modules is category O. This is the full subcategory of U (g)-Mod consisting of ﬁnitely generated weight modules V on which the action of n+ is locally ﬁnite:

dimU (n+)v < ∞ for all v in V. Category O enjoys many nice properties which makes it attractive to work in. All its modules are Noetherian and have finite dimensional weight spaces, the category is abelian, and has enough projectives and injectives. Moreover every module in O is Z(g)-finite which makes category O subject to the block decomposition (2.1): M O = O(χ). χ∈Θ Category O was first defined in the paper [BGG2] where the authors extended much of the former work by Verma and Jantzen [Ve, Ja1]. See also [Hu] for a comprehensive summary.

3.1.4 Cuspidal modules and coherent families

A weight module M is called cuspidal provided that all root vectors xα acts bijectively on weight spaces: ∗ xα : Mλ → Mλ+α is an isomorphism for all λ ∈ h . The set of weights of a simple cuspidal module thus have form λ + ZΦ. The classiﬁcation of cuspidal weight modules with ﬁnite dimensional weight spaces was completed by Mathieu in 2000 [Ma]. His approach used coherent families - another type of module important to this thesis. A coherent family of degree d is a weight module M such that ∗ • dimMλ = d for all λ ∈ h .

• The function λ 7→ tr(u · −|Mλ ) is polynomial in λ for each ﬁxed u com- muting with h.

Here u · − is the linear map M → M mapping m to u · m while u · −|Mλ is its restriction to the weight space Mλ . It can be shown that each cuspidal module occurs as a submodule of a coherent family. Since one can show that coherent families only exist for simple ﬁnite dimensional Lie algebras of type A and C, the same is true for cuspidal modules, see [Fe, Fu1] for details.

3.2 Whittaker modules 3.2.1 Kostant’s modules Whittaker modules were originally deﬁned by Kostant in 1978 [Ko]. They are deﬁned in terms of a triangular decomposition as in (1.4), and can be abstractly

26 described as modules M on which the algebra of positive roots n+ acts locally ﬁnitely. This just means that that dimU (n+) · m < ∞ for all m ∈ M. To construct such modules we let η : n+ → C be a nonsingular Lie algebra homomorphism. This means that η is determined by the nonzero scalars η(α) for α ∈ ∆. For each such η we have a corresponding one dimensional n+- module Cη where each x ∈ n+ acts by the scalar η(x). We may now deﬁne the corresponding Whittaker module Wη by

Wη = U (g) ⊗U (n+) Cη .

For each central character χ : Z(g) → C we have the corresponding central ideal Iχ generated by z − χ(z), z ∈ Z(g). We may now deﬁne

χ χ Wη = Wη /I Wη

χ A result due to Kostant in 1978 states that Wη is simple for all χ.

3.2.2 Generalized Whittaker modules

Whittaker modules are generated by eigenvectors for the subalgebra n+. Gen- eralized Whittaker modules arise by instead looking at generalized eigenvectors for a larger class of subalgebras. More precisely, let (n,g) be a pair of Lie algebras such that • n is a subalgebra of g • n is quasi-nilpotent • the adjoint n-module g/n is locally nilpotent A module W is said to be a generalized Whittaker module for the Whittaker pair (n,g) if the action of n on M is locally ﬁnite in the sense that

dimU (n) · w < ∞ for all w ∈ W.

This type of generalized Whittaker modules were originally studied in [BM].

3.3 Gelfand-Zetlin modules

The Lie algebras gln of n × n complex matrices can be sequentially embedded in each other with respect to the upper left corner:

gl1 ⊂ gl2 ⊂ ··· ⊂ gln−1 ⊂ gln.

Let Γ be the subalgebra of U (gln) generated by all the centers Z(gli) for 1 ≤ i ≤ n. Gelfand-Zetlin modules arose out of an attempt to construct bases for simple ﬁnite dimensional gln-modules that were eigenbases with respect to the algebra Γ.

27 The combinatorial data of such Gelfand-Zetlin basis elements can be en- coded as a doubly indexed complex vector t = {ti j | 1 ≤ j ≤ i ≤ n}, which also can be visualized as a tableau:

tn,1 tn,2 ··· tn,n−1 tn,n tn−1,1 ······ tn−1,n−1 [t] = ········· t2,1 t2,2 t1,1

A tableau [t] is called standard if all its entries ti, j lie in the same coset of C/Z a c and if for each sub-tableau of form we have a ≤ b < c. For example, b 1 4 7 9 1 5 8 is standard. 3 7 6

It is well known that simple ﬁnite dimensional gln-modules are parametrized by their highest weight vectors λ = (λ1,...,λn) where λi+1 −λi ∈ N. Gelfand and Zetlins original result in [GZ] states the corresponding ﬁnite dimensional module V(λ) has an eigenbasis consisting of all standard tableaux which has top row λ1 λ2 − 1 ······ λn − n + 1 and where the action of the generators of gln upon this basis is given by the formulas

 ! k−1(t −t )  k ∏ j=1 k,i k−1, j ek,k+1 · [t] = −∑i=1 k [t + εk,i]  ∏ j6=i(tk,i−tk, j)   k+1 !  k ∏ j=1(tk,i−tk+1, j) ek+1,k · [t] = ∑i=1 k [t − εk,i] (3.1) ∏ (tk,i−tk, j)  j6=i  !  k k−1 ek,k · [t] = ∑ (tk,i + i − 1) − ∑ (tk−1,i + i − 1) [t],  i=1 i=1 where εk,i is the unit vector having a 1 in position (k,i) and zeroes elsewhere. For example: The simple gl3-module V(1,4,7) is 8-dimensional with basis 1 3 5 1 3 5 1 3 5 1 3 5 1 3 , 1 3 , 1 4 , 1 4 , 1 2 1 2 1 3 5 1 3 5 1 3 5 1 3 5 1 4 , 2 3 , 2 4 , 2 4 . 3 2 2 3

28 About 40 years after Gelfand and Zetlins original paper, Drozd, Ovsienko, and Futorny had the idea that the formulas (3.1) could be applied on a wider range of tableaux to obtain many new simple inﬁnite dimensional modules, see [DFO1, DFO2, DFO3]. A tableau [t] is called generic if each row has non-integer differences, i.e. ti, j −ti,k ∈/ Z for all 1 ≤ j < k ≤ i ≤ n − 1.

Theorem 3. For each ﬁxed generic tableau [t], let V([t]) be the vector space spanned by all tableaux [l] that satisﬁes li, j − ti, j ∈ Z. Then V([t]) is a gln- module under the formulas (3.1).

The theorem gives rise to the largest known family of modules. This family of generic Gelfand-Zetlin modules is parametrized the set of generic tableaux, n(n+1) i.e. by 2 generic complex parameters. The simple generic Gelfand-Zetlin modules were recently classified in [FGR]. Although the stated theorem is for gln, it translates easily to the special Lie algebras of type A. An analogous construction for the orthogonal algebras (type B and D) were considered in [Maz1]. A more abstract approach is to define Gelfand-Zetlin modules as modules on which the subalgebra Γ ⊂ U (g) acts locally finitely. By this definition there are also many other Gelfand-Zetlin modules than the ones discussed above.

3.4 Parabolically induced modules

Let g be a (semi-)simple Lie algebra with triangular decomposition g = n− ⊕ h ⊕ n+. A subalgebra p of g is said to be parabolic if it contains the Borel subalgebra h ⊕ n+. Parabolic subalgebras admit a Levi decomposition

p = (a ⊕ ha) ⊕ n,

0 where n is nilpotent, a = a ⊕ ha is reductive, a is semisimple and ha ⊂ h is abelian and central in a0. A generalized Verma module over g is an induced module

Mp(V) = U (g) ⊗U (p) V, where V is a simple a0-module and nV = 0. A parabolically induced module is a module of form U (g) ⊗U (p) V where p is a parabolic subalgebra of g and V is a ﬁnitely generated p-module. Many general results have been proved about generalized Verma modules and parabolically induced modules, in particular, for g = sln one can give general conditions for when a parabolically induced module is simple. See [Fu2, FM, KM1, KM2, MS, SM] for details.

Part II: Results of the Thesis

4. U (h)-free modules

The majority of papers in this thesis are concerned with construction and clas- siﬁcation of so called U (h)-free modules and related objects. A weight modules V can be can be characterized as a module on which U (h) acts locally ﬁnitely:

dimU (h) · v < ∞ for all v ∈ V. By contrast, a U (h)-free module is a module on which U (h) acts freely, so in this sense U (h)-free modules are the opposite of weight modules. We deﬁne F to be the full subcategory of U (g)-Mod with objects

ob( ) = {M ∈ (g)-Mod | ResU (g)M is free of ﬁnite rank}. F U U (h)

Such modules are isomorphic to U (h)⊕k when restricted to U (h), where k is the rank of the module. The easiest case is when k = 1; then as a vector space (and as a U (h)-module) M is isomorphic to U (h) – a polynomial algebra in rank g variables. As an example, let g = sl2 with basis as in (1.5). In this case U (h) is isomorphic to C[h]. Denote by Mc the vector space C[h] equipped with the following sl2-module structure. For any polynomial f (h) ∈ C[h] we deﬁne h · f (h) = h f (h), x · f (h) = f (h − 2), 1 y · f (h) = − 4 (h + c + 2)(h − c) f (h + 2).

Then ResU (g)M is free of rank 1. Moreover, for all non-constant f (h) we U (h) c have deg f (h − 2) − f (h) = deg f (h) − 1, and we note that the element (x−1) ∈ U (g) can be used to reduce any element f (h) to a nonzero constant which shows that the module Mc is simple for all c. Moreover, since the action of x does not increases the degree of a polynomial, we have dimU (n+) · f (h) ≤ deg f (h) + 1, which shows that the modules Mc in fact all Whittaker modules in the sense of Section 3.2.

33 This is however not the typical case as the following exempliﬁes. Denote by Nc the vector space C[h] equipped with the following sl2-module structure. h · f (h) = h f , 1 x · f (h) = 2 (h + c) f (h − 2), 1 y · f (h) = − 2 (h − c) f (h + 2).

The modules Nc are still U (h)-free but are no longer Whittaker modules. Fur- thermore, the simplicity of the modules Nc is no longer obvious. For example, N0 is clearly not simple: hC[h] is a submodule. In fact Nc will be simple precisely for c ∈/ N. This last point also illustrates a problem inherent to the category F. The quotient N0/hN0 is not U (h)-free, so the category F is not closed under taking quotients. This suggests that it might be categorically more interesting to study a larger category which contains F and is abelian, i.e. closed under forming direct sums, submodules, and quotients. Here the natural choice is to deﬁne M to be the full subcategory of U (g)- Mod having objects

ob( ) = {M ∈ (g)-Mod | ResU (g)M is finitely generated}. M U U (h) This category is abelian and contains all objects of F, all finite dimensional modules, and even some non-free infinite dimensional modules. U (h)-free modules were introduced in Paper I which also classified all rank 1 U (h)-free modules for Lie algebras of type A. This classification was extended to all other types of Lie algebras (see Section 1) in Paper II. Similar modules were also investigated for the Virasoro algebra, see [CC, CG].

34 5. On Paper I

The ﬁrst paper investigates sln+1-modules whose restriction to U (h) is free of rank 1. The main result of the paper is the classiﬁcation of such modules. For sl2 we also give a formula for tensor product decomposition.

5.1 Construction and classification of U (h)-free modules As a vector space, any rank 1 U (h)-free module can be identified with the left regular U (h)-module U (h)U (h), which itself can be identified with C[h1,h2,...,hn] where {h1,...,hn} is a basis of h. In fact the paper uses the explicit choice of basis 1 n+1 hk := ek,k − ∑ ei,i. n + 1 i=1

For any root vector xα we have the following relations in U (g):

xα h = (h − α(h))xα for all h ∈ h.

Thus we observe that any module structure on U (h) is determined up to isomorphism by the action of generators of sln+1 on the generator 1 of C[h1,h2,...,hn]. We now choose the generating set {e1,n+1,...,en,n+1,en+1,1,...,en+1,n} of sln+1 and deﬁne pi := ei,n+1 · 1 and qi := ei,n+1 · 1. Any rank 1 U (h)-free module is then determined by the the 2n polynomials p1, p2,..., pn,q1,q2,...,qn. Introducing algebra homomorphisms σi : U (h) → U (h) deﬁned by σi f (h1,...,hn) = f (h1,...,hi − 1,...,hn), we are able to write down the action of sln+1 on U (h) explicitly as

hi · f = hi f , ei,n+1 · f = piσi( f ), −1 (5.1) en+1,i · f = qiσi ( f ), −1 −1 ei, j · f = piσi(q j) − q jσ j (pi) σiσ j f , for 1 ≤ i, j ≤ n, i 6= j.

35 Conversely however, it is not immediately clear which 2n-tuples

(p1, p2,..., pn,q1,q2,...,qn) give rise to a sln+1 module under the formulas (5.1). This is the main question answered in the paper. The paper proceeds by constructing a set of modules. S For all b ∈ C and for all S ⊂ {1,...,n} we denote by Mb the rank 1 U (h)- free module determined by the 2n-tuple (p1, p2,..., pn,q1,q2,...,qn) where ( (h1 + h2 + ···hn + b) if i ∈ S pi = (h1 + h2 + ···hn + b)(hi − b − 1) if i ∈/ S and ( −(hi − b) if i ∈ S qi = −1 if i ∈/ S

The main result of the paper is that any rank 1 U (h)-free module can be S obtained from a module Mb by twisting it by an automorphism. ∗ n+1 More precisely, for each (n + 1)-tuple a = (a1,a2,...,an+1) ∈ (C ) we let ϕa be the automorphism of sln+1 deﬁned by

ai ϕa(ei, j) = ei, j for i 6= j and ϕa(h) = h for all h ∈ h. a j

Furthermore, let τ be the automorphism of sln+1 deﬁned by

τ(ei, j) = e j,i for i 6= j and τ(h) = −h for all h ∈ h.

Theorem 30 then states that a complete set of isomorphism classes of rank 1 U (h)-free modules are given by

S ∗ {Fa(Mb )|a ∈ (C ) × {1},S ⊂ {1,...,n},b ∈ C} S ∗ ∪{Fa ◦ τ(Mb )|a ∈ (C ) × {1},S ⊂ {1,...,n},b ∈ C}. Moreover, this list of modules is irreduntant for n > 1. See Theorem 29 and 30 of Paper I for details.

5.2 Submodules and properties Having classiﬁed the rank 1 U (h)-free modules, we turn to the problem of S determining which of them are simple. The conclusion is that Mb is simple unless both (n+1)b ∈ N and S = n := {1,...,n}. This determines the simples since the functors τ and Fa are equivalences.

36 When (n + 1)b ∈ N and S = n we show that there is a short exact sequence n 0 → W → Mb → Q → 0 where W is simple and finitely generated (but not free) over U (h), and Q is (n+1)b+n a simple finite dimensional module of dimension n . This shows that all rank 1 U (h)-free modules has length ≤ 2. Worth noting is that when g = sl2, and 2b ∈ N the above sequence becomes n 0 → M−b−1 → Mb → L(2b) → 0, showing that all simple finite dimensional modules can be constructed as quotients of U (h)-free modules. This is analogous to the known fact that any such simple finite dimensional module can be expressed as a quotient of Verma modules: with notation as in Section 3.1.1 we have

0 → M(−2b − 2) → M(2b) → L(2b) → 0.

5.3 Clebsch-Gordan problem for sl2 The classical Clebsch-Gordan theorem gives a formula for the decomposition of the tensor product simple ﬁnite dimensional sl2 modules. Namely, for non- negative integers m ≥ n we have

L(m) ⊗C L(n) ' L(m + n) ⊕ L(m + n − 2) ⊕ ··· ⊕ L(m − n + 2) ⊕ L(m − n). In the paper, it is established that an analogous formula holds for the decomposition of then tensor product of a simple ﬁnite dimensional module and a U (h)-free module. We show that for all 2b ∈ C \ N, we have

k MS ⊗ L(k) ' MMS . 2b C 2b+ k−2i i=0 2

37 6. On Paper II

The second paper continues the classiﬁcation started in Paper I, but it takes a different approach. The idea for the ﬁrst part of this paper was suggested by Olivier Mathieu.

6.1 Weighting functor and coherent families ∗ Let M be any U (g)-module and for each λ ∈ h let mλ be the corresponding maximal ideal in U (h) i.e. the ideal generated by all elements of form h − λ(h). Extending λ : h → C to an algebra homomorphism λ : U (h) → C, we have mλ = ker(λ). We may now deﬁne M M = Mλ , λ∈h∗ where Mλ = M/mλ M. M then has the natural structure of a U (h)-module. However, we may further extend this to a g-module structure as follow: for each root vector xα deﬁne

xα · (m + mλ M) = (xα · m) + mλ+α M.

This lets us deﬁne a functor W : U (g)-Mod → U (g)-Mod called the weighting functor. It maps M to M , and maps a morphism f : M → N to W ( f ) deﬁned by W ( f ) m + mλ M = f (m) + mλ N. It can now be shown that

• W (M) is always a weight module. • W (M) ' M if M is already a weight module. • W maps U (h)-free modules of rank d to coherent families of degree d.

Since we know that coherent families only exists when the Lie algebra is of type A or type C (see the discussion in Section 3.1.4), it now easily follows that U (h)-free modules also only exist in type A and C. So to complete the classification of rank 1 U (h)-free modules for all simple complex finite dimensional Lie algebras as started in Paper I, only the classification in type C remains.

38 6.2 Classiﬁcation in type C

We proceed by explicitly constructing an object M0 which is U (h)-free or rank 1 in U (sp2n)-mod. We use the matrix realization of sp2n from Section 1.4. Letting hi := ei,i −en+i,n+i we have U (h) ' C[h1,...,hn]. We let M0 be this vector space equipped with the sp2n-action

hi · f = hi f 2 8ei,n+i · f = (2hi − 1)(2hi − 3)σi ( f ) −2 −2en+i,i · f = σi ( f ) 4(ei,n+ j + e j,n+i) · f = (2hi − 1)(2h j − 1)σiσ j( f ) −1 −1 −(en+i, j + en+ j,i) · f = σi σ j ( f ) −1 2(ei, j − en+ j,n+i) · f = (2hi − 1)σiσ j ( f ) for all 1 ≤ i, j ≤ n, i 6= j. Here the matrix elements on the left side together form a basis for sp2n. The algebra-homomorphisms σi are defined by σi(hk) = hk − δi,k as before. Now if M is U (h)-free of rank 1 we know that W (M) is a coherent family of degree 1. Since there is a unique such coherent family we know that W (M) is isomorphic to W (M0) and this allows us to prove that any rank 1 U (h)-free module can be obtained by twisting M0 by an (explicitly given) automorphism. See Section (3.4) of Paper II for details. This completes the classification of rank 1 U (h)-free modules for all simple complex finite dimensional Lie algebras.

6.3 Applying translation functors In the ﬁnal section of the paper it is shown that there exist simple U (h)-free modules of rank higher than 1. This is done by using the translation functors T(µ,λ) discussed in Section 2.4.6. It is known that when λ − µ is dominant and integral, T(µ,λ) is an equivalence. In the paper we show that the restriction of T(µ,λ) to the category of U (h)-free modules is still an equivalence. This is used to show that if M is rank 1 U (h)-free with character χµ , then for λ 6= µ, the module T(µ,λ)(M) is a simple U (h)-free of rank higher than 1.

39 7. On Paper III

Paper III deals with construction of a new large family of simple gl2n-modules. This family of modules is parametrized by the set of nonsingular complex n × n-matrices i.e. by n2 generic parameters. The only larger known family of gl2n-modules are the Gelfand-Zetlin modules which for comparison are parametrized by n(2n + 1) generic complex parameters (see Section 3.3).

7.1 Decomposition of gl2n The Lie algebra gl2n has a natural decomposition into subalgebras as indicated in the diagram AB . CD Explicitly, we have

A = {ei, j|1 ≤ i, j ≤ n}, B = {ei,n+ j|1 ≤ i, j ≤ n},

C = {en+i, j|1 ≤ i, j ≤ n}, D = {en+i,n+ j|1 ≤ i, j ≤ n}, which gives gl2n = A ⊕B ⊕C ⊕D. Note that A ' D ' sln, while B and C are commutative.

7.2 Module construction We proceed to construct our modules in steps. Let Q = (qi, j) be any complex n × n-matrix. We denote by LQ the one dimensional B-module where the action is given by Q:

ei,n+ j · v = qi, jv for all 1 ≤ i, j ≤ n, and v ∈ LQ. Since B is commutative, this is indeed a Lie algebra module. We also note that this action can be reformulated using the trace function: 0 B · v = tr(QBT )v. 0 0 Next we deﬁne

A ⊕B MQ := IndB LQ = U (A ⊕ B) ⊗U (B) LQ.

40 MQ is then an A ⊕ B-module which is isomorphic to U (A ) as a vector space (and A -module). We now consider the simplicity of the module MQ. The ﬁrst results of the paper can be summarized as follows.

Theorem 4. In the above setting the following are equivalent: 1. The matrix Q is nonsingular. 2. The module MQ is injective in B-Mod. 3. The module MQ is simple.

The remainder of the paper uses techniques of induction and taking quotients to extend the module MQ to a gl2n-module. Finally an explicit formula for the action of gl2n on this module is given. The results can be summarized as follows:

Theorem 5. Let A.B denote the matrix product while AB means the product in U (A ). Let ϕ and ψ be the algebra homomorphisms U (A ) → U (A ) ⊗ A acting on generators by ϕ(A) = A ⊗ I + I ⊗ A and ψ(A) = A ⊗ I − I ⊗ AT respectively, and let F := (e ji)i j ∈ U (A ) ⊗ A . Deﬁne an action of gl2n on MQ ' U (A ) as follows: for any a ∈ U (A ), let

AB · a = Aa − aD + tr(ψ(a).Q.BT ) CD − tr(ϕ(a).F2.Q−T .C) − tr(ϕ(a).Q−T .C)tr(F).

This is a simple gl2n-module structure.

This formula was obtained by ﬁrst determining the formula for Q = I, and then twisting the module by the automorphisms ϕS : gl2n → gl2n deﬁned by AB AB.S−1 ϕ : 7→ . S CD S.CS.D.S−1 for all nonsingular S. See Theorem 17 and Theorem 18 of Paper III for all the details.

41 8. On Paper IV

About 30 years ago, McDowell used parabolic induction to construct new simple modules from Whittaker modules, see [McD1, McD2]. This paper com- bines the techniques used by McDowell with the results of Paper I to obtain a new family of simple generalized Verma modules.

8.1 Generalized Verma modules Parabolic induction was discussed in Section 3.4. Recall a parabolic subalgebra p of g has a Levi-decomposition p = (a ⊕ ha) ⊕ n, where n is nilpotent, 0 a = a ⊕ ha is reductive, a is semisimple and ha ⊂ h is abelian and central in a0, and that a generalized Verma module over g is an induced module

Mp(V) = U (g) ⊗U (p) V, where V is a simple a0-module and nV = 0.

8.2 Our construction

Consider sln+1 as a subalgebra of sln+2 with respect to the upper left corner and let p := sln+1 + h + n+. p The paper studies the modules Mp(V) where Res V is a rank 1 U (h)-free sln+1 sln+1-module as in Paper I. The main result is the determination of necessary and sufﬁcient conditions for simplicity of Mp(V). 1 n+1 Let hi = ek,k − n+1 ∑i=1 ei,i for 1 ≤ i ≤ n+2 so that {h1,...,hn} is a basis for the Cartan subalgebra of sln+1 and {h1,...,hn,hn+2} is a basis for the Cartan subalgebra of sln+1. Each p-module V as above can be parametrized as V(a,S,b,λ) for a = ∗ n+1 (a1,...,an,1) ∈ (C ) , S ⊂ {1,...,n+1}, b ∈ C and λ ∈ C. Here a,S, and b corresponds to the classiﬁcation parameters in Paper I and λ is the eigenvalue of hn+2 acting on V. As vector spaces we now have

Mp(V(a,S,b,λ)) ' C[en+2,1,en+2,2,...,en+2,n+1] ⊗ C[h1,...,hn],

42 and we consider its elements as polynomials in en+2,1,en+2,2,...,en+2,n+1 with coefficients in U (h) = C[h1,...,hn]. n+1 m m1 m1 mn+1 For m = (m1,...,mn+1) ∈ N we write E = en+2,1en+2,2 ···en+2,n+1. This lets us express every element of v of Mp(V(a,S,b,λ)) uniquely as m v = ∑ E Pm, m∈Nn+1 where finitely many coefficients Pm ∈ U (h) are nonzero. This can be refined into homogeneous components: with |m| := m1 + ··· + mn+1 we may write

N m v = ∑ ∑ E Pm. k=0 |m|=k We deﬁne the degree of v by degv = N.

8.3 Simplicity

It is shown in the paper that degei,n+2 · v < degv for all 1 ≤ i ≤ n + 1. Thus if v is a homogeneous nonzero element of a proper submodule S ⊂ Mp(V) of minimal degree, v must be annihilated by ei,n+2 for all 1 ≤ i ≤ n + 1. It turns out that this implies that for each m with |m| = N − 1, the vector

−1 −1 −1 −1 (a1 (m1 + 1)σ1 (Pm+ε1 ),...,an (mn + 1)σn (Pm+εn ),(mn+1 + 1)Pm+εn+1 ) is a solution to the linear system A(λ,b,S,N)x = 0 where A(λ,b,S,N) is a (n + 1) × (n + 1)-matrix with coefﬁcients in U (h) explicitly deﬁned in the paper. We proceed by proving that det A(λ,b,S,N) = (−nb − λ − N + 1)(b − λ − N + 2)n.

This immediately shows that Mp(V(a,S,b,λ)) is simple whenever (−nb−λ − N + 1) and (b − λ − N + 2) both are nonzero. We also prove the converse statement that Mp(V(a,S,b,λ)) is reducible whenever either (−nb − λ − N + 1) or (b − λ − N + 2) are zero by explicitly constructing proper nontrivial submodules in both cases. We have thus proved the following theorem.

Theorem 6. Let V(a,S,b,λ) be a simple p-module as above. Then the corresponding generalized Verma module Mp(V(a,S,b,λ)) is simple if and only if (−nb − λ − N + 1) 6= 0 and (b − λ − N + 2) 6= 0.

43 9. Summary in Swedish - Sammanfattning på svenska

9.1 Bakgrund Inom algebra studerar man algebraiska objekt så som grupper, ringar, krop- par, och algebror. Ett sätt att analysera dessa objekt är att representera deras element som linjära avbildningar i ett vektorrum, det vill säga att till varje element i den algebraiska strukturen tillskriva en linjär avbildning på så vis att multiplikationen i den algebraiska strukturen motsvaras av sammansättning av linjära avbildningar i vektorrummet. Vektorrummet tillsammans med denna extra struktur kallas för en modul. Representationsteori kan därmed ses som ett sätt att förstå komplicerade algebraiska strukturer genom att översätta dem till linjär algebra - ett välkänt matematiskt område som är enkelt att arbeta med. Genom att undersöka delmoduler och kvotmoduler kan en godtycklig modul brytas ned i sina enklaste komponenter. Dessa komponenter kallas enkla moduler och de kan karakteriseras som moduler vilka saknar äkta delmoduler. För vissa algebraiska strukturer – till exempel för ändliga grupper – är den grundläggande representationsteorin redan väl utredd. Här har vi för varje ändlig grupp en ändlig lista med alla enkla moduler, och vi vet att varje modul kan brytas ned som en direkt summa vars komponenter tillhör denna lista. Representationsteori för Lie algebror är speciellt tillämpbart inom fysiken. Här kan till exempel olika partikeltillstånd beskrivas som element i moduler. För Lie algebror ﬁnns det oändligt många enkla moduler, vilket gör det svårare att få en komplett bild från ett representationsteoretiskt perspektiv. Trots detta är vissa klasser av enkla moduler välutforskade.

9.2 Kända klasser av enkla moduler I det här avsnittet låter vi g vara en enkel ändligtdimensionell Lie algebra över de komplexa talen. Det är välkänt att g har en triangulär uppdelning

g = n− ⊕ h ⊕ n+.

9.2.1 Viktmoduler Viktmoduler är en fundamental klass av g-moduler som har studerats sedan början av 1900-talet. En viktmodul är en modul i vilken element från Cartan- algebran h verkar genom skalär multiplikation på basvektorer i modulen.

44 Mer exakt; om M är en modul kan vi för varje funktional λ : h → C deﬁniera motsvarande viktrum

Mλ = {v ∈ M|h · v = λ(h)v för alla h ∈ h}.

Om M = ⊕λ∈h∗ Mλ så är M en viktmodul, och de λ för vilka Mλ 6= 0 utgör vikter för M. Det visar sig att det finns en naturlig partiell ordning på h∗ vilket gör att vi kan vi kan introducera så kallade högstaviktmoduler. Dessa är moduler som genereras av en maximal viktvektor med avseende på den partiella ordningen. Ett fundamentalt resultat är att det för varje λ ∈ h∗ finns en unik enkel högstaviktmodul L(λ) med högsta vikt λ. Enkla högstaviktmoduler kan således parametriseras av h∗. År 2000 färdigställde Olivier Mathieu klassifikationen av enkla viktmoduler vars viktrum är ändligtdimensionella genom att klassificera så kallade ”cuspidal modules“ – moduler vars mängd vikter är tät i h∗, se [Ma].

9.2.2 Ändligtdimensionella moduler Man kan visa att varje enkel modul av ändlig dimension är en högstavikt- modul, och därmed av form L(λ). Omvänt kan man vidare visa att L(λ) är ändligt dimensionell endast om λ kan skrivas som en summa av så kallade fundamentala vikter. Geometriskt kan detta ses som att enkla ändligtdimen- sionella moduler klassiﬁceras av ett koniskt gitter i h∗.

9.2.3 Whittakermoduler

Moduler där verkan av n+ är lokalt ändlig, det vill säga dimU (n+) · v < ∞ för alla vektorer v i modulen kallas för Whittakermoduler. Whittakermoduler beskrevs ursprungligen av Kostant år 1978, se [Ko]. Det enklaste sättet att konstuera Whittakermoduler är att välja en ickesingulär Lie algebra homo- morﬁ η : n+ → C med vilken man deﬁnierar en endimensionell modul Cη genom x · v = η(x)v för alla x ∈ n+ och v ∈ Cη .

Genom att inducera till g erhålls en Whittakermodul Wη = U (g) ⊗U (n+) Cη . När man slutligen tar en kvot med en verkan av Z(g) erhålls en enkel Whit- takermodul. Generaliserade Whittakermoduler kan också deﬁnieras för godtyckliga så kallade Whittakerpar (n,g) där n ⊂ g uppfyller vissa ändlighets-villkor, se [BM].

9.2.4 Gelfand-Zetlinmoduler Gelfand-Zetlinmoduler är en stor klass moduler som existerar för Lie algebror av typ A, B, och D. I Gelfand och Zetlins ursprungliga konstruktion [GZ]

45 studerades serien av inbäddningar

gl1 ⊂ gl2 ⊂ ··· ⊂ gln, med avseende på över vänster hörn. Låt Γ vara delalgebran i U (gln) som genereras av Z(gl1),...,Z(gln). Vi kan nu deﬁniera Gelfand-Zetlinmoduler som de moduler för vilka verkan av Γ är lokalt ändlig. Man kan visa att man i enkla ändligtdimensionella gln-moduler kan välja en Gelfand-Zetlin-bas bestående triangulära scheman av tal, så kallade generiska Gelfand-Zetlin-tablåer. Med avseende på denna bas är det enkelt att skriva ner en explicit verkan av generatorer i gln. Genom att använda samma formler på en utökad klass av tablåer erhålls en stor klass av enkla gln-moduler. Dessa n(n+1) moduler kan parametriseras av 2 komplexa tal.

9.3 Avhandlingens resultat Avhandlingen bidrar till representationsteori för Lie algebror genom att konstruera och klassiﬁcera nya klasser av moduler över Lie algebror. Modulerna som konstrueras är alla oändligtdimensionella icke-vikt moduler. Speciellt fokus ligger på så kallade U (h)-fria moduler som behandlas i Paper I, Paper II och Paper IV. Dessutom konstrueras en stor klass enkla gl2n-moduler i Paper III.

9.3.1 U (h)-fria moduler Låt g vara en enkel ändligtdimensionell Lie algebra. De g-moduler M som uppfyller ResU (g) ' (h)⊕k, U (h) U det vill säga moduler vars restriktioner till U (h) är fria av ändlig rang kallas för U (h)-fria moduler. Dessa är i en mening motsatsen till viktmoduler - Cartanalgebran verkar fritt istället för diagonalt. U (h)-fria moduler av rang 1 undersöks i Paper I och Paper II. Som h- moduler är dessa isomorfa med U (h), vilket innebär att deras element kan identifieras med polynom i rank g variabler. Deras modulstrukturer bestäms entydigt av de ändligt många polynomen {xα · 1|α är en rot till g} vilket gör direkta beräkningar relativt enkla att genomföra. Paper I fokuserar på typ A. Dess huvudresultat är en klassifikation av moduler som är U (h)-fria av av rang 1 för Lie algebran sln+1. Vi visar att denna klass av moduler kan parametriseras av delmängder av {1,...,n} tillsammans med n + 2 generiska komplexa tal. I artikeln ges även explicita formler för sln+1-verkan på dessa moduler. Dessutom visar vi att de flesta moduler i klassifikationen är enkla, och för resterande moduler bestämmer vi deras komposi- tionsserier. Speciellt ger detta upphov till exempel på nya enkla moduler som

46 inte är fria, men ändligt genererade över U (h). För sl2 bevisas dessutom en version av Clebsch-Gordans formel: vi ger en formel för uppdelning av ten- sorprodukten mellan enkla U (h)-fria moduler och enkla ändligtdimensionella moduler. Paper II fortsätter med att utvidga klassifikationen av U (h)-fria moduler av rang 1 till typ C. Dessutom etableras ett samband mellan U (h)-fria moduler och koherenta familjer. Vi konstruerar en viktningsfunktor W på g-Mod och visar att om M är U (h)-fri av rang n så är W (M) en koherent familj av grad n. Vi kan därigenom använda den sedan tidigare kända klassifikationen av koherenta familjer för att analysera U (h)-fria moduler. Med dessa metoder visar vi att U (h)-fria moduler endast kan existera i typ A och C, vilket därmed färdigställer klassifikationen av U (h)-fria moduler av rank 1 för enkla ändligt- dimensionell Lie algebror. Vi använder även translationsfunktorer för att visa att det existerar enkla U (h)-fria moduler av rang högre än 1.

9.3.2 Generaliserade Whittakermoduler för gl2n I paper III konstrueras en stor familj av enkla moduler för gl2n. Konstruktionen bygger på följande blockuppdelning av gl2n i fyra n × n-matrisalgebror: AB . CD

Modulerna konstrueras genom att börja med en endimensionell B-modul och därefter stegvis inducera först till A ⊕ B, sedan till A ⊕ B ⊕ D, och till sist till hela gl2n. De resulterande modulerna kan karakteriseras som enkla moduler vilka är isomorfa med U (A ) som A -moduler och där verkan av B är lokalt ändlig. Modulerna är därmed exempel på generaliserade Whittakermoduler för Whit- takerparet (B,gl2n). Modulerna från konstruktionen parametriseras av invert- erbara n × n-matriser, det vill säga av n2 generiska parametrar. Således ger detta den näst största kända klassen av gl2n-moduler efter Gelfand-Zetlinmodulerna.

9.3.3 Generaliserade Vermamoduler från U (h)-fria moduler I Paper IV använder vi parabolisk induktion för att konstruera en ny klass av sln+2-moduler från U (h)-fria sln+1-moduler. Vi låter p vara summan av standard Borel-algebran i sln+2 och sln+1 inbäddad i övre vänster hörn. Modulerna vi undersöker är av form U (g) ⊗U (p) V, där ResU (p) V är en av modulerna klassiﬁcerade i Paper I. Vi ger nöd- U (sln+1) vändiga och tillräckliga villkor för att en den resulterande modulen ska vara enkel.

47 Acknowledgements

During my years as a doctoral student I have come to think that doing mathematics is less about writing proofs and more about developing ideas and ways of thinking. I’m thankful to the many people who helped inspire the ideas underlying this work. First I want to thank my doctoral advisor Volodymyr Mazorchuk whose advice and support made it possible for me to write my dissertation. Throughout our many discussions I’ve been impressed with his encouragement, optimism, and excitement for mathematics. I’m also grateful for the work he’s doing with keeping the algebra group in Uppsala active by organizing speeches, seminars, and conferences. Indeed, it is through these meetings with other algebraists from around the world that I have made con- nections with people that were important to writing the papers of the thesis. I’m especially grateful to Olivier Mathieu for his friendliness and hospitality during my visit to Institut Camille Jordan in Lyon 2015. The second paper in this thesis is heavily inspired by his ideas. The mathematics in Paper IV were essentially done in one intense week of work during my visit to Wilfrid Laurier in Canada. Here I want to thank Kaiming Zhao as well as our co-authors Gen- qiang and Yan-an for their hospitality and friendliness. Additionally I want to thank Arne Meurman in Lund who got me interested in the subjects of algebra and representation theory in the first place. Thanks also to my family for the continual encouragement over the almost 10 years that I’ve been studying mathematics. I didn’t know anybody in Uppsala when I first moved here in 2011, so I’m particularly grateful to the friends that I’ve made since then. Among my friends at the mathematics department I’m especially thankful to Katja, Sei- don, and Djalal for our many interesting discussions about mathematics and life. I also appreciate to have had so many friendly office mates over the years - thank you Martin, Marta, Reza, Viktoria, and Anna. With risk of forget- ting people I intend to keep the list short so I will finish by just thanking all my other friends – thank you for making life as a doctoral student in Uppsala more enjoyable!

48 References

[BG] J. N. Bernstein, S. I. Gelfand. Tensor products of finite- and infinite-dimensional representations of semisimple Lie algebras. Compositio Math. 41 (1980), 245–285. [BGG1] I. N. Bernstein, I. M. Gelfand, S. I. Gelfand; Structure of representations that are generated by vectors of highest weight. (Russian) Funckcional. Anal. i Priložen. 5 (1971), no. 1, 1–9. [BGG2] I. N. Bernstein, I. M. Gelfand, S. I. Gelfand; A certain category of g-modules. Funkcional. Anal. i Priložen. 10, (1976) 1–8. [Bl] R. Block; The irreducible representations of the Lie algebra sl(2) and of the Weyl algebra. Adv. in Math. 139 (1981), no. 1, 69–110. [BM] P. Batra, V. Mazorchuk; Blocks and modules for Whittaker pairs. J. Pure Appl. Algebra 215 (2011), no. 7, 1552–1568. [Ca] E. Cartan; Les groupes projectifs qui ne laissent invariante aucune multiplicité planet. Bull. Soc. Math. France vol. 41 (1913), 53–96. [CC] Q. Chen, Y. Cai; Modules over algebras related to the Virasoro algebra. International Journal of Mathematics 26.09 (2015): 1550070. [CG] H. Chen, X. Guo; A new family of modules over the Virasoro algebra. Journal of Algebra, 457 (2016), 73–105. [DFO1] Yu. Drozd, S. Ovsienko, V. Futorny. Irreducible weighted sl3-modules. Funktsional. Anal. i Prilozhen 23 (1989), 57–58. [DFO2] Yu. Drozd, S. Ovsienko, V. Futorny. On Gelfand-Zetlin modules. Proceedings of the Winter School on Geometry and Physics (Srni, 1990). Rend. Circ. Mat. Palermo (2) Suppl. No. 26 (1991), 143–147. [DFO3] Yu. Drozd, S. Ovsienko, V. Futorny. Harish-Chandra subalgebras and Gel’fand-Zetlin modules. In: Finite-Dimensional Algebras and Related Topics (Ottawa, ON, 1992), NATO Adv. Sci. Inst. Ser. C Math. Phys. Sci., Vol. 424, Kluwer Acad. Publ., Dordrecht, 1994, 79–93. [Di] J. Dixmier; Enveloping Algebras. American Mathematical Society, 1977. [DMO] I. Dimitrov, O. Mathieu, I. Penkov; On the structure of weight modules. Trans. Amer. Math. Soc. 352 (2000), no. 6, 2857–2869. [Fe] S. Fernando; Lie algebra modules with finite dimensional weight spaces, I. Trans. Amer. Mas. Soc., 322 (1990), 757–781. [FGR] V. Futorny, D. Grantcharov, L.E. Ramirez; Irreducible Generic Gelfand–Tsetlin Modules of gl(n). SIGMA Symmetry Integrability Geom. Methods Appl. 11 (2015), Paper 018, 13 pp. [Fu1] V. Futorny; Weight representations of semisimple finite dimensional Lie algebras. Ph. D. Thesis, Kiev University, 1987. [Fu2] V. Futorny; Weighted sl(3)-modules generated by semiprimitive elements. (Russian) Ukrain. Mat. Zh. 43 (1991), no. 2, 281–285; translation in Ukrainian Math. J. 43 (1991), no. 2, 250–254.

49 [FM] V: Futorny, V. Mazorchuk; Structure of α-stratified modules for finite-dimensional Lie algebras. I. J. Algebra 183 (1996), no. 2, 456–482. [GZ] I.M. Gelfand, M.L. Zetlin; Finite-dimensional representations of the group of unimodular matrices. Doklady Akad. Nauk SSSR (N.S.), 71 (1950), 825–828. [Ha] T. Hawkins; Emergence of the theory of Lie groups, An essay in the history of mathematics 1869–1926. Sources and Studies in the History of Mathematics and Physical Sciences. Springer-Verlag, New York, 2000. [Hu] J. E. Humphreys; Representations of Semisimple Lie Algebras in the BGG Category O. American Mathematical Society, 2008. [Ja1] J. C. Jantzen; Zur Charakterformel gewisser Darstellungen halbeinfacher Gruppen und Lie-Algebren. Math. Z. 140 (1974), 127–149. [Ja2] J. C. Jantzen; Moduln mit einem höchsten Gewicht. Lecture Notes in Mathematics, 750. Springer, Berlin, 1979. ii+195 pp. [KM1] A. Khomenko, V. Mazorchuk; On the determinant of Shapovalov form for generalized Verma modules. J. Algebra 215 (1999), no. 1, 318–329. [KM2] A. Khomenko, V. Mazorchuk; Structure of modules induced from simple modules with minimal annihilator. Canad. J. Math. 56 (2004), no. 2, 293–309. [Ko] B. Kostant. On Whittaker vectors and representation theory. Invent. Math. 48 1978, no. 2, 101–184. [Ma] O. Mathieu. Classification of irreducible weight modules, Ann. Inst. Fourier (Grenoble) 50 (2000), no. 2, 537–592. [Maz1] V. Mazorchuk; On Gelfand-Zetlin modules over orthogonal Lie algebras. Algebra Colloq. 8 (2001), no. 3, 345–360. [Maz2] V. Mazorchuk. Lectures on sl2(C)-modules. Imperial College Press, London, 2010. [McD1] E. McDowell; On modules induced from Whittaker modules. J. Algebra 96 (1985), no. 1, 161–177. [McD2] E. McDowell; A module induced from a Whittaker module. Proc. Amer. Math. Soc. 118 (1993), no. 2, 349–354. [MS] V. Mazorchuk, C. Stroppel; Categorification of (induced) cell modules and the rough structure of generalised Verma modules. Adv. Math. 219 (2008), no. 4, 1363–1426. [SM] D. Miliciˇ c,´ W. Soergel; The composition series of modules induced from Whittaker modules. Comment. Math. Helv. 72 (1997), no. 4, 503–520. [Ve] N. Verma; Structure of certain induced representations of complex semisimple Lie algebras. Bull. Amer. Math. Soc. 74 (1968), 160–166. [Zu] G. Zuckerman; Tensor products of finite and infinite dimensional representations of semisimple Lie groups. Ann. of Math. (2) 106 (1977), no. 2, 295–308.