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

Abstract

Abstract. The part of the residual spectrum of an inner form of the split group Sp 8 supported in the minimal parabolic subgroup is decomposed. Since the considered inner form is not quasi-split, the normalization of the standard intertwining operators, required for the calculation of the poles of the Eisenstein series, is out of the reach of the Langlands-Shahidi method. Hence, a normalization technique, based on the transfer of the Plancherel measure between the split group and its inner form, is applied. The obtained decomposition reveals certain features of the residual spectrum of the inner form which do not appear for the split group.

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

“…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.…”

“…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.…”

Automorphic Forms, Cohomology and CAP Representations. The Case $GL_2$ over a definite quaternion algebra

“…As before, π = σ ⊗ τ is a cuspidal automorphic representation of L = D * × D * . To overcome the second problem, namely to actually normalize it at non-generic split places, we use the recent work of N. Grbac, [13]. With this achievement at hand, we are able to classify the cuspidal automorphic representations π of L, which give rise to residual Eisenstein series and show where their poles are located, see Proposition 3.4.…”