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Sur Les Conjectures de Gross et Prasad I
Wee Teck Gan, University of California at San Diego, CA, Benedict H. Gross, Harvard University, Cambridge, MA, Dipendra Prasad, Tata Institute of Fundamental Research, Mumbai, India, and Jean-Loup Waldspurger, Institut de Mathématiques de Jussieu-CNRS, Paris, France
A publication of the Société Mathématique de France.
 Astérisque 2012; 318 pp; softcover Number: 346 ISBN-13: 978-2-85629-348-5 List Price: US$105 Member Price: US$86.40 Order Code: AST/346 See also: Sur Les Conjectures de Gross et Prasad II - Colette Moeglin and Jean-Loup Waldspurger A note to readers: Half of this book is in English and half is in French. About 20 years ago Gross and Prasad formulated a conjecture determining the restriction of an irreducible admissible representation of the group $$G = SO(n)$$ over a local field to a subgroup of the form $$G' = SO(n-1)$$. The conjecture stated that for a given pair of generic $$L$$-packets of $$G$$ and $$G'$$, there is a unique non-trivial pairing, up to scalars, between precisely one member of each packet, where $$G$$ and $$G'$$ are allowed to vary among inner forms; moreover, the relevant members of the $$L$$-packets are determined by an explicit formula involving local root numbers. For non-archimedean local fields this conjecture has now been proved by Waldspurger and Mœglin, using a variety of methods of local representation theory; the Plancherel formula plays an important role in the proof. There is also a global conjecture for automorphic representations, which involves the central critical value of $$L$$-functions. This volume is the first of two volumes devoted to the conjecture and its proof for non-archimedean local fields. It contains two long articles by Gan, Gross, and Prasad, formulating extensions of the original Gross-Prasad conjecture to more general pairs of classical groups including metaplectic groups, and providing examples for low rank unitary groups and for representations with restricted ramification. It also includes two articles by Waldspurger: a short article deriving the local multiplicity one conjecture for special orthogonal groups from the results of Aizenbud-Gourevitch-Rallis-Schiffmann on orthogonal groups and a long article (which appeared in Compositio Mathematica in 2010) completing the first part of the proof of the Gross-Prasad conjecture by extending an integral formula relating multiplicities in the restriction problem to harmonic analysis from supercuspidal representations to general tempered representations here. A publication of the Société Mathématique de France, Marseilles (SMF), distributed by the AMS in the U.S., Canada, and Mexico. Orders from other countries should be sent to the SMF. Members of the SMF receive a 30% discount from list. Readership Graduate students and research mathematicians interested in classical groups, metaplectic groups, branching laws, Gross-Prasad conjectures, local root numbers, and central critical $$L$$-value. Table of Contents W. T. Gan, B. H. Gross, and D. Prasad -- Symplectic local root numbers, central critical L-values, and restriction problems in the representation theory of classical groups W. T. Gan, B. H. Gross, and D. Prasad -- Restrictions of representations of classical groups: Examples J.-L. Waldspurger -- Une formule intégrale reliée à la conjecture locale de Gross-Prasad, $$2^e$$ partie : Extension aux représentations tempérées J.-L. Waldspurger -- Une variante d'un résultat de Aizenbud, Gourevitch, Rallis et Schiffmann