##
**Kac-Moody groups, their flag varieties and representation theory.**
*(English)*
Zbl 1026.17030

Progress in Mathematics (Boston, Mass.). 204. Boston, MA: Birkhäuser. xvi, 606 p. (2002).

The theory of Kac-Moody Lie algebras has undergone a tremendous development since they were introduced more than 40 years ago. These algebras generalize the finite-dimensional semisimple Lie algebras, and they have close and important connections to several areas in mathematics, and also in mathematical physics. Today the topic has become a standard tool in mathematics. The purpose of the present book is to present the algebro-geometric and topological aspects of Kac-Moody theory in characteristic \(0\).

A lot of different topics are treated in this monumental work. Many of the topics are brought together for the first time in book form. The emphasis is on the study of Kac-Moody groups and their flag varieties. Among the main topics are, the Weyl-Kac character formula, a detailed construction of Kac-Moody groups and their flag varieties; the Demazure character formula, a study of the geometry of Schubert varieties, including normality and Cohen-Macaulayness; the Borel-Weil-Bott theorem; the Bernstein-Gelfand-Gelfand resolution; the Kempf resolution; conjugacy theorems for the Cartan subalgebras and invariance of the generalized Cartan matrix; determination of the defining ideal of the flag varieties via the Plücker relations; a study of the \(T\)-equivariant and singular cohomology of the flag varieties via the nil-Hecke ring, including positivity results for the cup product, and various criteria for the smoothness and rational smoothness of points on Schubert varieties. A particular chapter is devoted to the realization of affine Kac-Moody algebras and the corresponding groups, as well as their flag varieties.

There are several sections introducing the necessary background material, like ind-varieties; pro-groups, pro-Lie algebras; Tits systems; the nil-Hecke ring; Coxeter groups; the Bernstein-Gelfand-Gelfand \(\mathcal O\) category; Lie algebra homology; as well as appendices on results from algebraic geometry; local cohomology; results from topology; relative homological algebra; introduction to spectral sequences.

Although the book is devoted to the general Kac-Moody theory, many of the topics of the book will be useful for those only interested in the finite-dimensional case.

The book is self contained, but is on the level of advanced graduate students. Most statements, definitions and proofs are short, and there are few examples. Moreover many results are given as exercises, some of these are on the difficult side. For the motivated reader who is willing to spend considerable time on the material, the book can be a gold mine. Chapter by chapter the contents is:

I. Kac-Moody algebras; Basic theory

This section contains the definition of Kac-Moody algebras, root space decompositions and the Weyl groups associated to Kac-Moody algebras. Moreover the dominant chamber and the Tits cone are introduced, and the invariant bilinear form and the Casimir operator are treated.

II. Representation theory of Kac-Moody algebras

Here the Bernstein-Gelfand-Gelfand \(\mathcal O\) category is introduced, the Weyl-Kac character formula is proved, and the important Shapovalov bilinear form is given.

III. Lie algebra homology and cohomology

Results of Konstant-Garland-Lepowsky are proved and the Laplacian is introduced and used in calculations.

IV. An introduction to ind-varieties and pro-groups

Ind-varieties, ind-groups and their Lie algebras are defined and the smoothness of ind-varieties discussed. Pro-groups and pro-Lie algebras are introduced.

V. Tits systems; Basic theory

The basic theory of Tits systems and refined Tits systems is presented.

VI. Kac-Moody groups; Basic theory

The definitions of Kac-Moody groups and the related parabolic subgroups are given. Moreover, the representations of Kac-Moody groups are treated.

VII. Generalized flag varieties of Kac-Moody groups

The ind-variety structure of generalized flag varieties is given, and line bundles on these varieties are studied. Moreover the group \(\mathcal G^{\text{min}}\), defined by Kac-Peterson, and the group \(\mathcal U^-\) are studied.

VIII. Demazure and Weyl-Kac character formulas

The normality of Schubert varieties is proved and the Demazure Character formula proved. Moreover extensions of the Weyl-Kac character formula and the Borel-Weil-Bott theorem are given.

IX. Bernstein-Gelfand-Gelfand and Kempf resolutions

The Bernstein-Gelfand-Gelfand resolution and the dual Kempf resolution are obtained in the general Kac-Moody situation.

X. Defining equations of \(\mathcal G/\mathcal P\) and conjugacy theorems

The quadratic relations defining \(\mathcal G/\mathcal P\) in the canonical projective embedding are given, and the conjugacy theorems for Lie algebras, and groups, are proved.

XI. Topology of Kac-Moody groups and their flag varieties

The nil-Hecke ring is introduced, and the \(T\)-equivariant cohomology of \(\mathcal G/\mathcal B\) is determined. Some positivity results of cup products are given, and the degeneracy of the Leray-Serre spectral sequence for the fibration \(\mathcal G^{\text{min}} \to \mathcal G^{\text{min}}/T\) is proved.

XII. Smoothness and rational smoothness of Schubert varieties

The singular locus and rational smoothness of Schubert varieties are studied.

XIII. An introduction to affine Kac-Moody Lie algebras and groups

The affine Kac-Moody Lie algebras and groups are introduced and studied.

A lot of different topics are treated in this monumental work. Many of the topics are brought together for the first time in book form. The emphasis is on the study of Kac-Moody groups and their flag varieties. Among the main topics are, the Weyl-Kac character formula, a detailed construction of Kac-Moody groups and their flag varieties; the Demazure character formula, a study of the geometry of Schubert varieties, including normality and Cohen-Macaulayness; the Borel-Weil-Bott theorem; the Bernstein-Gelfand-Gelfand resolution; the Kempf resolution; conjugacy theorems for the Cartan subalgebras and invariance of the generalized Cartan matrix; determination of the defining ideal of the flag varieties via the Plücker relations; a study of the \(T\)-equivariant and singular cohomology of the flag varieties via the nil-Hecke ring, including positivity results for the cup product, and various criteria for the smoothness and rational smoothness of points on Schubert varieties. A particular chapter is devoted to the realization of affine Kac-Moody algebras and the corresponding groups, as well as their flag varieties.

There are several sections introducing the necessary background material, like ind-varieties; pro-groups, pro-Lie algebras; Tits systems; the nil-Hecke ring; Coxeter groups; the Bernstein-Gelfand-Gelfand \(\mathcal O\) category; Lie algebra homology; as well as appendices on results from algebraic geometry; local cohomology; results from topology; relative homological algebra; introduction to spectral sequences.

Although the book is devoted to the general Kac-Moody theory, many of the topics of the book will be useful for those only interested in the finite-dimensional case.

The book is self contained, but is on the level of advanced graduate students. Most statements, definitions and proofs are short, and there are few examples. Moreover many results are given as exercises, some of these are on the difficult side. For the motivated reader who is willing to spend considerable time on the material, the book can be a gold mine. Chapter by chapter the contents is:

I. Kac-Moody algebras; Basic theory

This section contains the definition of Kac-Moody algebras, root space decompositions and the Weyl groups associated to Kac-Moody algebras. Moreover the dominant chamber and the Tits cone are introduced, and the invariant bilinear form and the Casimir operator are treated.

II. Representation theory of Kac-Moody algebras

Here the Bernstein-Gelfand-Gelfand \(\mathcal O\) category is introduced, the Weyl-Kac character formula is proved, and the important Shapovalov bilinear form is given.

III. Lie algebra homology and cohomology

Results of Konstant-Garland-Lepowsky are proved and the Laplacian is introduced and used in calculations.

IV. An introduction to ind-varieties and pro-groups

Ind-varieties, ind-groups and their Lie algebras are defined and the smoothness of ind-varieties discussed. Pro-groups and pro-Lie algebras are introduced.

V. Tits systems; Basic theory

The basic theory of Tits systems and refined Tits systems is presented.

VI. Kac-Moody groups; Basic theory

The definitions of Kac-Moody groups and the related parabolic subgroups are given. Moreover, the representations of Kac-Moody groups are treated.

VII. Generalized flag varieties of Kac-Moody groups

The ind-variety structure of generalized flag varieties is given, and line bundles on these varieties are studied. Moreover the group \(\mathcal G^{\text{min}}\), defined by Kac-Peterson, and the group \(\mathcal U^-\) are studied.

VIII. Demazure and Weyl-Kac character formulas

The normality of Schubert varieties is proved and the Demazure Character formula proved. Moreover extensions of the Weyl-Kac character formula and the Borel-Weil-Bott theorem are given.

IX. Bernstein-Gelfand-Gelfand and Kempf resolutions

The Bernstein-Gelfand-Gelfand resolution and the dual Kempf resolution are obtained in the general Kac-Moody situation.

X. Defining equations of \(\mathcal G/\mathcal P\) and conjugacy theorems

The quadratic relations defining \(\mathcal G/\mathcal P\) in the canonical projective embedding are given, and the conjugacy theorems for Lie algebras, and groups, are proved.

XI. Topology of Kac-Moody groups and their flag varieties

The nil-Hecke ring is introduced, and the \(T\)-equivariant cohomology of \(\mathcal G/\mathcal B\) is determined. Some positivity results of cup products are given, and the degeneracy of the Leray-Serre spectral sequence for the fibration \(\mathcal G^{\text{min}} \to \mathcal G^{\text{min}}/T\) is proved.

XII. Smoothness and rational smoothness of Schubert varieties

The singular locus and rational smoothness of Schubert varieties are studied.

XIII. An introduction to affine Kac-Moody Lie algebras and groups

The affine Kac-Moody Lie algebras and groups are introduced and studied.

Reviewer: Dan Laksov (Stockholm)

### MSC:

17B67 | Kac-Moody (super)algebras; extended affine Lie algebras; toroidal Lie algebras |

22E67 | Loop groups and related constructions, group-theoretic treatment |

22-02 | Research exposition (monographs, survey articles) pertaining to topological groups |

17-02 | Research exposition (monographs, survey articles) pertaining to nonassociative rings and algebras |

14M15 | Grassmannians, Schubert varieties, flag manifolds |

14M20 | Rational and unirational varieties |