Automorphic Representations of Gsp(4)

1. In this note, we shall describe a classification for automorphic representations of GSp(4), the group of similitudes of four-dimensional 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 methods to the twisted analogue of the fundamental lemma, but not its generalization to weighted orbital integrals.) The general results of [A3] are proved by a comparison of spectral terms in the stabilized trace formula. It is for the existence of the stabilized trace formula [A2] (and its twisted analogues) that the fundamental lemma is required. However, any discussion of such methods would be outside the scope of this paper. We shall be content simply to state the classification for GSp(4) in reasonably elementary terms. The paper will in fact be somewhat expository. We shall try to motivate the classification by examining the relevant mappings from a Galois group (or some extension thereof ) to the appropriate L-groups. Representations of the group GSp(4) have been widely studied. The papers [HP], [Ku], [Y], [So], and [Ro] contain results that were established directly for GSp(4). Results for groups of higher rank in [CKPS] and [GRS] could also be applied (either now or in the near future) to the special case of GSp(4).