Using an old idea of Hunt (P.A.M.S. 1956) this book reachs a beautiful simplicity in its treatement of conjugacy of maximal tori in a compact connected Lie group G and in the proof that the exponential map of G is onto. As far as I know, most previous texts (including Dieudonné's "éléments d'analyse" vol. 5, ISBN 2040009329) used either Hunt's idea mixed with Hopf-Rinow theorem Introduction to Compact Lie Groups (Series in Pure Mathematics), The structure of Lie groups (Holden-Day series in mathematics), either A. Weil's proof using the Lefschetz fixed point formula, to achieve the same objectives Lie Groups (Graduate Texts in Mathematics), Harmonic analysis on homogeneous spaces (Pure and applied mathematics, 19). The method used in Representations of Finite and Compact Groups (Graduate Studies in Mathematics ; V. 10) are similar to Sepansky's, but I failed to understand some Simons' arguments and I had to switch into Wallach's. The rest of Sepansky's work is fine, but more standard, treating roots and weights as simultaneous eigenvalues of representations of the algebra of a maximal torus (instead of thinking of them as simultaneous eigenvalues of representations of the torus itself, as in Lie Groups: Beyond an Introduction, or Representations of Compact Lie Groups.Of course,the book includes a detailed proof of H.Weyl's character formula and it ends with Borel-Weil construction of all irreducible representations of compact semi-simple Lie groups. I think that the theory of representations of compact semi-simple Lie groups is much simpler and easy to understand than the (equivalent) theory of representations of complex semi-simple Lie algebras. For its study, Sepansky's book is brief, simple and self-contained; therefore I fully recommende it. However, the cited books by Knapp and Bump add a rather extensive content on real Lie algebras and their classification. On the other hand, Fegan's little book is still valuable, but it should need a second edition to fill many details and gaps. Finally I will cite a similar book Lie Groups (Universitext) which contains Borel-Weil construction; it's scope is not much broader than Sepansky's but it relies heavily on topological group actions, orbits, etc. I think that Sepansky's book can also be compared with the content of "éléments d'analyse" vol. 5 of Jean Dieudonné, for its combination of originality, transparency and (relative) brevity.

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