 
Summary: CHAPTER 4
AUTOMORPHIC REPRESENTATIONS OF GSp(4)
By James Arthur
1. In this note, we shall describe a classification for automorphic representations
of GSp(4), the group of similitudes of fourdimensional symplectic space. The
results are part of a project [A3] on the automorphic representations of general
classical groups. The monograph [A3] is still in preparation. When complete, it
will contain a larger classification of representations, subject to a general condition
on the fundamental lemma.
In the case of GSp(4), the standard fundamental lemma for invariant orbital
integrals has been established [Ha], [W]. However, a natural variant of the standard
fundamental lemma is also needed. To be specific, the theorem we will announce
here is contingent upon a fundamental lemma for twisted, weighted, orbital integrals
on the group GL(4) × GL(1) (relative to a certain outer automorphism). This has not
been established. However, it seems likely that by methods of descent, perhaps in
combination with other means, one could reduce the problem to known cases of
the standard fundamental lemma. (The papers [BWW], [F], and [Sc] apply such
methodstothetwistedanalogueofthefundamentallemma,butnotitsgeneralization
to weighted orbital integrals.)
The general results of [A3] are proved by a comparison of spectral terms in
