This is a review of Collected Works of John Tate. Parts I, II, edited by Barry Mazur and Jean-Pierre Serre. American Mathematical Society, Providence, Rhode Island, 2016. For several decades it has been clear to the friends and colleagues of John Tate that a “Collected Works” was merited. The award of the Abel Prize to Tate in 2010 added impetus, and finally, in Tate’s ninety-second year we have these two magnificent volumes, edited by Barry Mazur and Jean-Pierre Serre. Beyond Tate’s published articles, they include five unpublished articles and a selection of his letters, most accompanied by Tate’s comments, and a collection of photographs of Tate. For an overview of Tate’s work, the editors refer the reader to [4]. Before discussing the volumes, I describe some of Tate’s work. 1. Hecke L-series and Tate’s thesis Like many budding number theorists, Tate’s favorite theorem when young was Gauss’s law of quadratic reciprocity. When he arrived at Princeton as a graduate student in 1946, he was fortunate to find there the person, Emil Artin, who had discovered the most general reciprocity law, so solving Hilbert’s ninth problem. By 1920, the German school of algebraic number theorists (Hilbert, Weber, . . .) together with its brilliant student Takagi had succeeded in classifying the abelian extensions of a number field K: to each group I of ideal classes in K, there is attached an extension L of K (the class field of I); the group I determines the arithmetic of the extension L/K, and the Galois group of L/K is isomorphic to I. Artin’s contribution was to prove (in 1927) that there is a natural isomorphism from I to the Galois group of L/K. When the base field contains an appropriate root of 1, Artin’s isomorphism gives a reciprocity law, and all possible reciprocity laws arise this way. In the 1930s, Chevalley reworked abelian class field theory. In particular, he replaced “ideals” with his “idèles” which greatly clarified the relation between the local and global aspects of the theory. For his thesis, Artin suggested that Tate do the same for Hecke L-series. When Hecke proved that the abelian L-functions of number fields (generalizations of Dirichlet’s L-functions) have an analytic continuation throughout the plane with a functional equation of the expected type, he saw that his methods applied even to a new kind of L-function, now named after him. Once Tate had developed his harmonic analysis of local fields and of the idèle group, he was able prove analytic continuation and functional equations for all the relevant L-series without Hecke’s complicated theta-formulas. Received by the editors September 5, 2016. 2010 Mathematics Subject Classification. Primary 01A75, 11-06, 14-06. c ©2017 American Mathematical Society

The collected works

Chiral rings in N = 2 superconformal theories

Groupes et algébres de lie

Introduction to lie algebras and representation theory

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A diagrammatic approach to categorification of quantum groups II

Introduction * Kac--Moody Algebras -- Basic Theory * Representation Theory of Kac--Moody Algebras * Lie Algebra Homology and Cohomology * An Introduction to ind-Varieties and pro-Groups * Tits Systems -- Basic Theory * Kac--Moody Groups -- Basic Theory * Generalized Flag Varieties of Kac--Moody Groups * Demazure and Weyl--Kac Character Formulas * BGG and Kempf Resolutions * Defining Equations of G/P and Conjugacy Theorems * Topology of Kac-Moody Groups and Their Flag Varieties * Smoothness and Rational Smoothness of Schubert Varieties * An Introduction to Affine Kac-Moody Lie Algebras and Groups * Appendix A. Results from Algebraic Geometry * Appendix B. Local Cohomology * Appendix C. Results from Topology * Appendix D. Relative Homological Algebra * Appendix E. An Introduction to Spectral Sequences * Bibliography * Index of Notation * Index

Kac-Moody Groups, their Flag Varieties and Representation Theory

* Preface * Frobenius Splitting: General Theory * Frobenius Splitting * Schubert Varieties * Splitting and Filtration * Cotangent Bundles of Flag Varieties * Group Embeddings * Hilbert Schemes of Points * Bibliography * Index

Frobenius splitting methods in geometry and representation theory

The nil Hecke ring and cohomology of GP for a Kac-Moody group G

Let G be a Kac-Moody group with Borel subgroup B and compact maximal torus T. Analogous to Kostant and Kumar [Kostant, B. & Kumar, S. (1986) Proc. Nati. Acad. Sci. USA 83, 1543-1545], we define a certain ring Y, purely in terms of the Weyl group W (associated to G) and its action on T. By dualizing Y we get another ring 1, which, we prove, is "canonically" isomorphic with the T-equivariant Ktheory KT(G/B) of GIB. Now KT(G/B), apart from being an algebra over KT(pt.) A(T), also has a Weyl group action and, moreover, KT(G/B) admits certain operators {DW},,w similar to the Demazure operators defined on A(T). We prove that these structures on KT(G/B) come naturally from the ring Y. By "evaluating" the A(T)-module W at 1, we recover K(G/B) together with the above-mentioned structures. We believe that many of the results of this paper are new in the finite case (i.e., G is a finite-dimensional semisimple group over C)

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Shrawan Kumar

Papers

Kac-Moody Groups, their Flag Varieties and Representation Theory

Frobenius splitting methods in geometry and representation theory

The nil Hecke ring and cohomology of GP for a Kac-Moody group G

$T$-equivariant $K$-theory of generalized flag varieties

Demazure character formula in arbitrary Kac-Moody setting