# Sur Les Conjectures de Gross et Prasad I

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*Wee Teck Gan; Benedict H. Gross; Dipendra Prasad; Jean-Loup Waldspurger*

A publication of the Société Mathématique de France

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.