On a Certain Residual Spectrum of Sp8

Pogge, James Todd

Canadian journal of mathematics, Tome 56 (2004) no. 1, pp. 168-193

Résumé

Let $G=\text{S}{{\text{p}}_{\text{2n}}}$ 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 $G\left( \mathbb{A} \right)$ acting on the Hilbert space ${{L}^{2}}\left( G\left( F \right)\backslash G\left( \mathbb{A} \right) \right)$ . Main contributions have been made by Langlands. He described, using his theory of Eisenstein series, an orthogonal decomposition of this space of the form: $L_{dis}^{2}\left( G\left( F \right)\backslash G\left( \mathbb{A} \right) \right)\,=\,{{\oplus }_{\left( M,\,\pi\right)}}L_{dis}^{2}{{\left( G\left( F \right)\backslash G\left( \mathbb{A} \right) \right)}_{\left( M,\,\pi\right)}},\,\text{where}\,\left( M,\,\pi\right)$ 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 $L_{\text{dis}}^{2}{{\left( G\left( F \right)\backslash G\left( \mathbb{A} \right) \right)}_{\left( M,\pi\right)}}$ is a space of residues of Eisenstein series associated to $\left( M,\,\pi\right)$ . In this paper, we will completely determine the space $L_{\text{dis}}^{2}{{\left( G\left( F \right)\backslash G\left( \mathbb{A} \right) \right)}_{\left( M,\pi\right)}}$ , when $M\simeq \text{G}{{\text{L}}_{2}}\times \text{G}{{\text{L}}_{2}}$ . This is the first result on the residual spectrum for non-maximal, non-Borel parabolic subgroups, other than $\text{G}{{\text{L}}_{n}}$ .

Détail

DOI : 10.4153/CJM-2004-008-0

Mots-clés : 11F70, 22E55

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