We give a survey on Mœglin's construction of representations in the Arthur packets for p-adic quasisplit symplectic and orthogonal groups. The emphasis is on comparing Mœglin's parametrization of elements in the Arthur packets with that of Arthur.
Zelevinsky’s classification theory of discrete series of p-adic general linear groups has been well known. Mœglin and Tadić gave the same kind of theory for p-adic classical groups, which is more complicated due to the occurrence of nontrivial structure of L-packets. Nonetheless, their work is independent of the endoscopic classification theory of Arthur (also Mok in the unitary case), which concerns the structure of L-packets in these cases. So our goal in this paper is to make more explicit the connection between these two very different types of theories. To do so, we reprove the results of Mœglin and Tadić in the case of quasisplit symplectic groups and orthogonal groups by using Arthur’s theory.
In his monograph Arthur (The endoscopic classification of representations: orthogonal and symplectic groups, Colloquium Publications, American Mathematical Society, Providence, 2013) characterizes the L-packets of quasisplit symplectic groups and orthogonal groups. By extending his work, we characterize the L-packets for the corresponding similitude groups with desired properties. In particular, we show these packets satisfy the conjectural endoscopic character identities.
We develop a general procedure to study the combinatorial structure of Arthur packets for p-adic quasisplit Sp(N) and O(N) following the works of Mœglin. This allows us to answer many delicate questions concerning the Arthur packets of these groups, for example the size of the packets.
Arthur classified the discrete automorphic representations of symplectic and orthogonal groups over a number field by that of the general linear groups. In this classification, those that are not from endoscopic lifting correspond to pairs (ϕ,b), where ϕ is an irreducible unitary cuspidal automorphic representation of some general linear group and b is an integer. In this paper, we study the local components of these automorphic representations at a nonarchimedean place, and we give a complete description of them in terms of their Langlands parameters.