The title under review is a thorough survey of the modern study of the representation theory of finite groups of Lie type in the defining characteristic p. The main objects of study are the finite groups of Lie type — one of the infinite families of finite simple groups (the other two being the alternating groups and cyclic groups of prime order). Most of the major results in this subject stem from geometry, including the construction of these groups, which we now describe.
The classification theorem for complex simple Lie algebras asserts that any simple Lie algebra g over C belongs to one of four infinite families (the classical algebras), or is one of five exceptional Lie algebras. To each such algebra is associated a Dynkin diagram which encodes much of its structure. In 1955, Chevalley showed how to construct a basis for g such that the structure constants arising from the Lie bracket operation are all integral. The Z -span of this "Chevalley" basis has the structure of a Lie algebra over Z. The exponential map can then be applied to produce a group of matrices with integral entries, which can be reduced modulo any prime p. What remains is a finite matrix group over the finite field Fp.
This construction can be generalized to arbitrary finite fields, and can be made to incorporate the twisted Lie algebras (those whose Dynkin diagrams possess a nontrivial graph automorphism). Altogether, this construction produces the finite groups of Lie type — finite groups of matrices, with entries in a finite field k of characteristic p. With very few exceptions, these groups are simple.
A representation of a finite group G over a field k is called modular if the characteristic of k divides the order of G. Many of the basic results in ordinary representation theory are no longer valid in the modular setting (such as complete reducibility), and special techniques are almost certainly required to obtain interesting results, the most fruitful of which exploit the geometric nature of the groups involved. In this respect, one views a finite group of Lie type G as the fixed points of the Frobenius automorphism corresponding to k on an algebraic group G.
The classical analog is the Borel-Weil-Bott theorem, which gives a one-to-one correspondence between the irreducible representations V with weight λ of a complex simple algebraic group G and ample line bundles L(λ) on the homogeneous spaces G/P, where P is a parabolic subgroup of G. In particular, if λ is dominant, then the space of global holomorphic sections of L(λ) is exactly the representation V corresponding to λ. This cohomological approach carries over to the context of algebraic groups in characteristic p. In essence, this is where the book Modular Representations of Finite Groups of Lie Type begins.
After a brief introduction, Humphreys guides the reader through the some of the major theorems and conjectures in the subject. The geometric approach provides beautiful descriptions of the representations of these groups, but gives almost no specific information on the character data (including the dimensions of the irreducible representations). For the latter, Brauer's theory of modular characters is needed, as well as the general theory of groups with "split BN -pairs". In low rank, and for small primes, the two approaches combine to give fairly detailed information on decomposition matrices, character values, and principal indecomposable modules (projective covers of simple modules). Humphreys goes through several intricate computations and provides the reader with vast amounts of numerical data. However, such data cannot realistically be used to formulate general conjectures since the ranks and primes involved are so small, thus underscoring the difficulty of the subject.
In addition to being a leader in the field of modular representation theory, Humphreys' clarity of exposition is almost universally known. The book is expertly written, though the reader should be warned that it is not meant to be self-contained; J.C. Jantzen's Representations of Algebraic Groups is the standard reference for this material. As usual, his bibliography is compiled with great care and contains references that "may not turn up readily in a MathSciNet search". Humphreys has done a great service to the representation-theoretic community by writing this book.
John Cullinan is Visiting Assistant Professor of Mathematics at Bard College.
1. Finite groups of Lie type; 2. Simple modules; 3. Weyl modules and Lusztig’s conjecture; 4. Computation of weight multiplicities; 5. Other aspects of simple modules; 6. Tensor products; 7. BN-pairs and induced modules; 8. Blocks; 9. Projective modules; 10. Comparison with Frobenius kernels; 11. Cartan invariants; 12. Extensions of simple modules; 13. Loewy series; 14. Cohomology; 15. Complexity and support varieties; 16. Ordinary and modular representations; 17. Deligne-Lusztig characters; 18. The groups G2; 19. General and special linear groups; 20. Suzuki and Ree groups; Bibliography; Frequently used symbols; Index.