On a Certain Residual Spectrum of $Sp_8$

[摘要] Let $G=Sp_{2n}$ be the symplectic group defined over a numberfield $F$. Let $mathbb{A}$ be the ring of adeles. A fundamentalproblem in the theory of automorphic forms is to decompose theright regular representation of $G(mathbb{A})$ acting on theHilbert space $L^2igl(G(F)setminus G(mathbb{A})igr)$. Main contributions have been made by Langlands. He described, using histheory of Eisenstein series, an orthogonal decomposition of thisspace of the form: $L_{dis}^2 igl( G(F)setminus G(mathbb{A})igr)=igoplus_{(M,pi)} L_{dis}^2(G(F) setminus G(mathbb{A})igr)_{(M,pi)}$, where $(M,pi)$ is a Levi subgroup with acuspidal automorphic representation $pi$ taken modulo conjugacy(Here we normalize $pi$ so that the action of the maximal splittorus in the center of $G$ at the archimedean places is trivial.) and $L_{dis}^2igl(G(F)setminus G(mathbb{A})igr)_{(M,pi)}$is a space of residues of Eisenstein series associated to$(M,pi)$. In this paper, we will completely determine the space$L_{dis}^2igl(G(F)setminus G(mathbb{A})igr)_{(M,pi)}$, when$MsimeqGL_2imesGL_2$. This is the first result on theresidual spectrum for non-maximal, non-Borel parabolic subgroups, other than $GL_n$.

[学科分类] 数学（综合）