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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 Individual Members: US$94.50 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
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