On the residual spectrum of Hermitian quaternionic inner form of SO_8

Abstract

Abstract. In this paper we decompose the residual spectrum supported in the minimal parabolic subgroup of an inner form of the split group SO 8 . The approach is the Langlands spectral theory. However, since the group is non-quasi-split, it is out of the scope of the Langlands-Shahidi method and the new technique for the normalization of standard intertwining operators is developed. The decomposition shows interesting parts of the residual spectrum not appearing in the case of quasi-split groups.

The residual spectrum of an inner form of 𝑆𝑝₈ supported in the minimal parabolic subgroup

“…Hence, H n (k v ) ∼ = Sp 4n (k v ) for every split place v ∈ S D . In this paper we consider the residual spectrum of the group H 2 which is an inner form of the split group Sp 8 . Let T be the maximal split torus in H 2 .…”

“…Let T ∼ = GL 1 × GL 1 × GL 1 × GL 1 be the maximal split torus of the split Sp 8 . Fix the positive roots of Sp 8 with respect to T in such a way that the split form P 0 = M 0 N 0 of the parabolic subgroup P 0 = M 0 N 0 is the standard parabolic subgroup of the split Sp 8 with the Levi factor M 0 ∼ = GL 2 × GL 2 .…”

“…Therefore, we had to develop a new technique of the normalization of the intertwining operators based on the JacquetLanglands correspondence explained in [5] and the transfer of the Plancherel measure based on the global idea explained in [29]. This technique was already used by the author of this paper in [6], [8] and [9], where we considered the residual spectrum of a non-quasi-split inner forms of SO 4 , Sp 4 , SO 8 and the parts of the residual spectra of non-quasi-split inner forms of SO 4n and Sp 4n . See also [7] where the residual spectrum of GL n over a division algebra is obtained.…”

“…In this paper we decompose the part of the residual spectrum of H 2 coming from the residues of the Eisenstein series attached to cuspidal automorphic representations of the minimal standard parabolic subgroup of H 2 defined over k. 8 , in addition to the interesting parity conditions (which appear for split groups as well) in Theorems 3.2.1, 3.3.3, and 3.3.13, the results show certain features of hermitian quaternionic groups such as the local conditions on the non-triviality of the one-dimensional representation at non-split places in Theorems 3.2.2, 3.3.2, 3.3.13, 3.3.15, and the condition on the number of non-quasi-split places of a global quaternion algebra used to define H 2 in Theorem 3.3.15. The reason for occurrence of such conditions lies in the different local normalization factors at split and nonquasi-split places which give local L-functions in the global normalizing factors.…”

“…It is a non-quasi-split inner form of the split group Sp 8 . Although, in principle, the results of this paper could be obtained using the Arthur trace formula explained in [1], our strategy of the calculation is a more direct approach of the Langlands spectral theory explained in [20] and [25].…”