**Colloquium Publications**

Volume: 61;
2013;
590 pp;
Hardcover

MSC: Primary 22; 11; 58;
**Print ISBN: 978-0-8218-4990-3
Product Code: COLL/61**

List Price: $115.00

Individual Member Price: $92.00

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#### Supplemental Materials

# The Endoscopic Classification of Representations: Orthogonal and Symplectic Groups

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*James Arthur*

Within the Langlands program, endoscopy is a
fundamental process for relating automorphic representations of one
group with those of another. In this book, Arthur establishes an
endoscopic classification of automorphic representations of orthogonal
and symplectic groups \(G\). The representations are shown to
occur in families (known as global \(L\)-packets and
\(A\)-packets), which are parametrized by certain self-dual
automorphic representations of an associated general linear group
\(GL(N)\). The central result is a simple and explicit formula
for the multiplicity in the automorphic discrete spectrum of
\(G\) for any representation in a family.

The results of the volume have already had significant
applications: to the local Langlands correspondence, the construction
of unitary representations, the existence of Whittaker models, the
analytic behaviour of Langlands \(L\)-functions, the spectral theory of
certain locally symmetric spaces, and to new phenomena for symplectic
epsilon-factors. One can expect many more. In fact, it is likely
that both the results and the techniques of the volume will have
applications to almost all sides of the Langlands program.

The methods are by comparison of the trace formula of \(G\)
with its stabilization (and a comparison of the twisted trace formula
of \(GL(N)\) with its stabilization, which is part of work in
progress by Moeglin and Waldspurger). This approach is quite
different from methods that are based on \(L\)-functions,
converse theorems, or the theta correspondence. The comparison of
trace formulas in the volume ought to be applicable to a much larger
class of groups. Any extension at all will have further important
implications for the Langlands program.

#### Table of Contents

# Table of Contents

## The Endoscopic Classification of Representations: Orthogonal and Symplectic Groups

#### Readership

Research mathematicians interested in automorphic forms, the Langlands program, and number theory.