Residual Spectra of Split Classical Groups and their Inner Forms

Abstract

Abstract. This paper is concerned with the residual spectrum of the hermitian quaternionic classical groups G ′ n and H ′ n defined as algebraic groups for a quaternion algebra over an algebraic number field. Groups G ′ n and H ′ n are not quasi-split. They are inner forms of the split groups SO 4n and Sp 4n . Hence, the parts of the residual spectrum of G ′ n and H ′ n obtained in this paper are compared to the corresponding parts for the split groups SO 4n and Sp 4n .

Global Jacquet–Langlands correspondence, multiplicity one and classification of automorphic representations

“…[MS]) to define the normalization using L-functions for the split group. For the classical hermitian quaternionic groups we used this technique to obtain the parts of the residual spectra in [Gr1], [Gr2], [Gr3], [Gr4]. …”

“…However, its analogue for inner forms could have been obtained using first the transfer of the Plancherel measure for discrete series representations (see [MS]) to define the normalization using L-functions for the split group. For the classical hermitian quaternionic groups we used this technique to obtain the parts of the residual spectra in [Gr1], [Gr2], [Gr3], [Gr4].…”

On the residual spectrum of Hermitian quaternionic inner form of SO_8

“…Other parts are papers [11,12,13]. I would like to thank my advisor G. Muić for many useful discussions and constant help during the preparation of this paper.…”

“…The method was already applied by the author to obtain the residual spectra of inner forms of SO 4 and Sp 4 in [11], an inner form of Sp 8 in [13], and some simple parts of the residual spectra of inner forms of SO 4n and Sp 4n in general in [12]. Also, the residual spectrum of the general linear group over a division algebra was obtained in [14], but the calculation relies on the results in the body of the paper which were obtained using the comparison of trace formulas.…”

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

“…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.…”