Representations of semisimple Lie algebras in the BGG category \(\mathcal O\).

*(English)*Zbl 1177.17001
Graduate Studies in Mathematics 94. Providence, RI: American Mathematical Society (AMS) (ISBN 978-0-8218-4678-0/hbk). xvi, 289 p. (2008).

The book is focused on algebraic methods in the representation theory of a semisimple Lie algebra \(\mathfrak g\) over \(\mathbb C\). The exposition is on the level of a graduate course. The emphasis is on highest weight modules, starting with Verma modules and cumulating in the determination of formal characters of simple highest weight modules in the setting of the Kazhdan-Lustig Conjecture, the proof of which (1979) is beyond of the algebraic framework of the book. The book contains exercises, examples and a good list of references for a further study and research work.

Let \(U(\mathfrak g)\) be the universal enveloping algebra. The category \(\text{Mod}\, U(\mathfrak g)\) of all \(U(\mathfrak g)\)-modules is a very large one. Many interesting Lie group representations can be effectively studied in terms of a more little subcategory, namely BGG category \({\mathcal O}\) introduced in early 1970s by Joseph Bernstein, Israel Gelfand and Sergei Gelfand.

Let \(\mathfrak g=\mathfrak n^-\oplus \mathfrak h\oplus \mathfrak n\) denote the Cartan decomposition. The category \(\mathcal O\) is defined to be the full subcategory of \(\roman{Mod}\, U(\mathfrak g)\) whose objects are the modules \(M\) satisfying the following three conditions. (1) \(M\) is a finitely generated \(U(\mathfrak g)\)-module. (2) \(M\) is \(\mathfrak h\)-semisimple, that is, \(M\) is a weight module: \(M=\oplus_{\lambda\in \mathfrak h^*} M_\lambda\). (3) \(M\) is locally \(\mathfrak n\)-finite: for each \(v\in M\) the subspace \(U(\mathfrak n) v\) of \(M\) is finite dimensional.

Chapter 0 summarizes the basic facts on the structure of semisimple Lie algebras over \(\mathbb C\). The reader is also required to be familiar with elementary homological algebra.

Part I (Highest Weight Modules, Chapters 1–8) is written in a textbook style. Chapter 1 (Category \(\mathcal O\): Basics) starts with axioms of \(\mathcal O\) and their immediate consequences. Chapter 2 deals with characters of finite dimensional modules. To derive the classical formulas of Weyl and Kostant the author follows the approach of BGG by studying characters as formal \(\mathbb Z\)-linear combinations of Verma modules. Chapter 3 is about homological methods in category \(\mathcal O\): functors \(\roman{Hom}\) and \(\roman {Ext}\), the duality functor on \(\mathcal O\), dominant and antidominant weights, tensoring Verma modules with finite dimensional modules, projective objects in \(\mathcal O\) and BGG reciprocity, contravariant forms on modules. Chapter 4 (Highest weight modules 1) focuses on Verma modules and their simple quotients. Chapter 5 (Highest weight modules 2) is about further development due to Bernstein-Gelfand-Gelfand and Shapovalov. A central result is BGG Theorem on composition factors of Verma modules which can be reformulated for regular weights in terms of Bruhat ordering. The author emphasizes the alternative proof of the theorem of Jantzen in the framework of Jantzen filtrations. Chapter 6 (Extensions and Resolutions) develops a BGG-resolution of a finite dimensional module simple \(L(\lambda)\), its terms are direct sums of Verma modules. This realizes the Weyl-Kostant character formula as an Euler characteristic. The reminder of the chapter treats higher \(\roman{Ext}\) functors. Chapter 7 (Translations functors). The translation functors are introduced and their effects on Verma modules and simple modules are examined. As a result, a general proof is obtained that self-dual projectives are those corresponding to antidominant weights. Chapter 8 (Kazhdan-Lustig theory). This chapter marks a shift from a textbook to a survey mode. The construction of Kazhdan-Lustig polynomials is outlined, and Kazhdan-Lustig Conjecture is formally stated. Independent proofs of the conjecture were published by Beilinson-Bernstein and Brylinski-Kashiwara; the respective literature is introduced.

Part II (Further Developments). Chapter 9. (Parabolic Versions of Category \(\mathcal O\)). The chapter is about subcategory \(\mathcal O^{\mathfrak p}\) determined by a parabolic subalgebra \(\mathfrak p \supset\mathfrak b \) of \(\mathfrak g\). Chapter 10 (Projective Functors and Principal Series) This chapter deals with the projective functors defined by Joseph Bernstein and Sergei Gelfand together with the related category of principal series Harish-Chandra modules for a complex semisimple Lie group having \(\mathfrak g\) as a Lie algebra. Chapter 11 (Tilting modules). Tilting modules are characterized by the condition that both the module and its dual have standard filtrations. Motivation arises indirectly from the study of fusion rules in parallel module categories for semisimple algebraic groups, quantum enveloping algebras at root of unity, or affine Kac-Moody algebras. Chapter 12 (Twisting and Completion Functors). The chapter deals with shuffled Verma modules, twisting functors, and completion functors. Chapter 13 (Complements). The author sketches some interactions between category \(\mathcal O\) and other parts of representation theory. The following notions are discussed: Universal enveloping algebras and classification of primitive ideals; parallels to category \(\mathcal O\), Kac-Moody algebras, analogues of highest weight modules in various categories; blocks, their quivers an representation types.

Let \(U(\mathfrak g)\) be the universal enveloping algebra. The category \(\text{Mod}\, U(\mathfrak g)\) of all \(U(\mathfrak g)\)-modules is a very large one. Many interesting Lie group representations can be effectively studied in terms of a more little subcategory, namely BGG category \({\mathcal O}\) introduced in early 1970s by Joseph Bernstein, Israel Gelfand and Sergei Gelfand.

Let \(\mathfrak g=\mathfrak n^-\oplus \mathfrak h\oplus \mathfrak n\) denote the Cartan decomposition. The category \(\mathcal O\) is defined to be the full subcategory of \(\roman{Mod}\, U(\mathfrak g)\) whose objects are the modules \(M\) satisfying the following three conditions. (1) \(M\) is a finitely generated \(U(\mathfrak g)\)-module. (2) \(M\) is \(\mathfrak h\)-semisimple, that is, \(M\) is a weight module: \(M=\oplus_{\lambda\in \mathfrak h^*} M_\lambda\). (3) \(M\) is locally \(\mathfrak n\)-finite: for each \(v\in M\) the subspace \(U(\mathfrak n) v\) of \(M\) is finite dimensional.

Chapter 0 summarizes the basic facts on the structure of semisimple Lie algebras over \(\mathbb C\). The reader is also required to be familiar with elementary homological algebra.

Part I (Highest Weight Modules, Chapters 1–8) is written in a textbook style. Chapter 1 (Category \(\mathcal O\): Basics) starts with axioms of \(\mathcal O\) and their immediate consequences. Chapter 2 deals with characters of finite dimensional modules. To derive the classical formulas of Weyl and Kostant the author follows the approach of BGG by studying characters as formal \(\mathbb Z\)-linear combinations of Verma modules. Chapter 3 is about homological methods in category \(\mathcal O\): functors \(\roman{Hom}\) and \(\roman {Ext}\), the duality functor on \(\mathcal O\), dominant and antidominant weights, tensoring Verma modules with finite dimensional modules, projective objects in \(\mathcal O\) and BGG reciprocity, contravariant forms on modules. Chapter 4 (Highest weight modules 1) focuses on Verma modules and their simple quotients. Chapter 5 (Highest weight modules 2) is about further development due to Bernstein-Gelfand-Gelfand and Shapovalov. A central result is BGG Theorem on composition factors of Verma modules which can be reformulated for regular weights in terms of Bruhat ordering. The author emphasizes the alternative proof of the theorem of Jantzen in the framework of Jantzen filtrations. Chapter 6 (Extensions and Resolutions) develops a BGG-resolution of a finite dimensional module simple \(L(\lambda)\), its terms are direct sums of Verma modules. This realizes the Weyl-Kostant character formula as an Euler characteristic. The reminder of the chapter treats higher \(\roman{Ext}\) functors. Chapter 7 (Translations functors). The translation functors are introduced and their effects on Verma modules and simple modules are examined. As a result, a general proof is obtained that self-dual projectives are those corresponding to antidominant weights. Chapter 8 (Kazhdan-Lustig theory). This chapter marks a shift from a textbook to a survey mode. The construction of Kazhdan-Lustig polynomials is outlined, and Kazhdan-Lustig Conjecture is formally stated. Independent proofs of the conjecture were published by Beilinson-Bernstein and Brylinski-Kashiwara; the respective literature is introduced.

Part II (Further Developments). Chapter 9. (Parabolic Versions of Category \(\mathcal O\)). The chapter is about subcategory \(\mathcal O^{\mathfrak p}\) determined by a parabolic subalgebra \(\mathfrak p \supset\mathfrak b \) of \(\mathfrak g\). Chapter 10 (Projective Functors and Principal Series) This chapter deals with the projective functors defined by Joseph Bernstein and Sergei Gelfand together with the related category of principal series Harish-Chandra modules for a complex semisimple Lie group having \(\mathfrak g\) as a Lie algebra. Chapter 11 (Tilting modules). Tilting modules are characterized by the condition that both the module and its dual have standard filtrations. Motivation arises indirectly from the study of fusion rules in parallel module categories for semisimple algebraic groups, quantum enveloping algebras at root of unity, or affine Kac-Moody algebras. Chapter 12 (Twisting and Completion Functors). The chapter deals with shuffled Verma modules, twisting functors, and completion functors. Chapter 13 (Complements). The author sketches some interactions between category \(\mathcal O\) and other parts of representation theory. The following notions are discussed: Universal enveloping algebras and classification of primitive ideals; parallels to category \(\mathcal O\), Kac-Moody algebras, analogues of highest weight modules in various categories; blocks, their quivers an representation types.

Reviewer: Victor Petrogradsky (Ulyanovsk)