论文标题
单位组的自动形式的对称时期
Symmetric periods for automorphic forms on unipotent groups
论文作者
论文摘要
令$ k $为一个数字字段，$ \ mathbb {a} $是其Adeles的戒指。让$ u $是一个单体群体定义的$ k $，$σ$ a $ k $ - 理性的$ u $带有固定点$ u^+$的差额。由于C. Moore的结果，空间$ l^2（u（k）\ backslash u _ {\ mathbb {a}}）$是$ _ {\ mathbb {a}} $的表示。设置$ p^+：ϕ \ mapsto \ int_ {u^+（k）\ backslash {u} _ {\ mathbb {a}}}^+} ϕ（u）du $，是$ l^2（k）u（k）\ l^2（k）\ l^2（k） u _ {\ Mathbb {a}}）$，我们证明，如果$π$是$ l^2（u（k）\ backslash u _ {\ mathbb {a}}）$ l^2（u（k）\ backslash u _ { $π^\ vee =π^σ$。这是由于Y. Benoist和作者所依赖的作者而导致的本地结果的全球类似物。
Let $k$ be a number field and $\mathbb{A}$ be its ring of adeles. Let $U$ be a unipotent group defined over $k$, and $σ$ a $k$-rational involution of $U$ with fixed points $U^+$. As a consequence of the results of C. Moore, the space $L^2(U(k)\backslash U_{\mathbb{A}})$ is multiplicity free as a representation of $U_{\mathbb{A}}$. Setting $p^+:ϕ\mapsto \int_{U^+(k)\backslash {U}_{\mathbb{A}}^+} ϕ(u)du$ to be the period integral attached to $σ$ on the space of smooth vectors of $L^2(U(k)\backslash U_{\mathbb{A}})$, we prove that if $Π$ is a topologically irreducible subspace of $L^2(U(k)\backslash U_{\mathbb{A}})$, then $p^+$ is nonvanishing on the subspace $Π^\infty$ of smooth vectors in $Π$ if and only if $Π^\vee=Π^σ$. This is a global analogue of local results due to Y. Benoist and the author, on which the proof relies.
