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# On a Certain Residual Spectrum of $\Sp_8$

Published:2004-02-01
Printed: Feb 2004
• James Todd Pogge
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## Abstract

Let $G=\Sp_{2n}$ be the symplectic group defined over a number field $F$. Let $\mathbb{A}$ be the ring of adeles. A fundamental problem in the theory of automorphic forms is to decompose the right regular representation of $G(\mathbb{A})$ acting on the Hilbert space $L^2\bigl(G(F)\setminus G(\mathbb{A})\bigr)$. Main contributions have been made by Langlands. He described, using his theory of Eisenstein series, an orthogonal decomposition of this space of the form: $L_{\dis}^2 \bigl( G(F)\setminus G(\mathbb{A}) \bigr)=\bigoplus_{(M,\pi)} L_{\dis}^2(G(F) \setminus G(\mathbb{A}) \bigr)_{(M,\pi)}$, where $(M,\pi)$ is a Levi subgroup with a cuspidal automorphic representation $\pi$ taken modulo conjugacy (Here we normalize $\pi$ so that the action of the maximal split torus in the center of $G$ at the archimedean places is trivial.) and $L_{\dis}^2\bigl(G(F)\setminus G(\mathbb{A})\bigr)_{(M,\pi)}$ is a space of residues of Eisenstein series associated to $(M,\pi)$. In this paper, we will completely determine the space $L_{\dis}^2\bigl(G(F)\setminus G(\mathbb{A})\bigr)_{(M,\pi)}$, when $M\simeq\GL_2\times\GL_2$. This is the first result on the residual spectrum for non-maximal, non-Borel parabolic subgroups, other than $\GL_n$.
 MSC Classifications: 11F70 - Representation-theoretic methods; automorphic representations over local and global fields 22E55 - Representations of Lie and linear algebraic groups over global fields and adele rings [See also 20G05]