Representations of Finite and Lie Groups is a brief introduction (141 pages) to the representation theory of finite and compact groups in characteristic zero. The goal of the author is to present the material in a way such that the results generalize to more general compact topological groups, by replacing averages over the elements of the group with a suitable invariant integral.
The book covers the following topics: the basics of representation theory over the complex numbers (with a page dedicated to representations over the reals, and a section dedicated to the representations of SL(2,Fp) in characteristic p); induced representations and their characters; tensor product constructions, symmetric and alternating products, exterior powers; the complex representation ring and its lambda-structure; representations of compact groups; Lie groups and Lie algebras and, finally, a chapter on SL(2,R), where the author compares its principal and discrete series with those of SL(2,Fp). Throughout the book, the study of the important examples SL(2,R), SU_2 and SL(2,Fp) is emphasized. The book ends with three appendices: integration over topological groups, rings with minimal condition and semi-simplicity, and modular representations. Each chapter includes a reasonable number of problems which are solved at the very end of the book (some in detail, some just hints).
Although the choice of topics is good, the reviewer was not impressed with the final product. The book reads like a set of lecture notes which has not been polished enough; the text could be improved to be more reader-friendly. In particular, there are too many typographical errors (typos, concatenation of mathematical symbols in sentences and an alarming lack of commas). Although some of the typos may be obvious (the book defines the complimentary series instead of the complementary series), other errors will make the book hard to read by those who are trying to learn the subject.
Álvaro Lozano-Robledo is H. C. Wang Assistant Professor at Cornell University.