This is the first of two Astérisque volumes dedicated to a well-known conjecture of B. H. Gross and D. Prasad. (The second volume is here.) Formulated some twenty years ago, the conjecture has to do with representations of classical groups; specifically, the original conjecture described what should happen when one took an irreducible admissible representation of SO(n) and restricted it to a subgroup isomorphic to SO(n–1). The conjecture was recently proved by Waldspurger and Mœglin.
The two Astérisque volumes include articles by Gross and Prasad explaining (and generalizing) their conjecture and giving many examples, followed by a series of articles (mostly by Waldspurger) that culminate in the complete proof. Together, the two volumes make up a high-level expository account of the conjecture and proof.
I really like the idea of presenting a major new result in this form. It does not, of course, render it any less technical and difficult, but it does gather all the crucial ideas in one place. Kudos to the authors for having done it.
Fernando Q. Gouvéa is Carter Professor of Mathematics at Colby College and the Editor of MAA Reviews.