The Endoscopic Classification of Representations Orthogonal and Symplectic Groups

James Arthur

2013年10月 · Colloquium Publications 61 巻 · American Mathematical Soc.

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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

. The representations are shown to occur in families (known as global

-packets and

-packets), which are parametrized by certain self-dual automorphic representations of an associated general linear group

. The central result is a simple and explicit formula for the multiplicity in the automorphic discrete spectrum of

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 -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 with its stabilization (and a comparison of the twisted trace formula of
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 -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.