论文标题
AX+B组和DIRICHLET系列的表示
Representation of ax+b group and Dirichlet Series
论文作者
论文摘要
令$ g $为$ ax+b $组。本质上有两个$ g $，$，$，l^2（\ Mathbb r^+））$和$（μ^*，l^2（\ Mathbb r^+））$的两个不可约合的无限尺寸单一表示。在本文中，我们给出了有关$μ$的光滑矢量及其梅林变换的各种特征。令$ \ f d $为正整数$ \ mathbb z^+$上支持的delta发行版的线性总和。我们研究了矩阵系数的梅林变换$μ_{\ f d，f}（a）$，$ f $ smooth。我们以Dirichlet系列$ L（S，\ f D）$来表达这些Mellin变换。我们确定了足够的条件，使得广泛的矩阵系数$μ_ {\ f d，f} $是本地集成的功能，并估计siegel集上$ l^2 $ -norms $μ_ {\ f f} $。我们进一步得出了一种不平等，该不平等可能有可能用于研究Dirichlet系列$ L（S，\ f d）$。
Let $G$ be the $ax+b$ group. There are essentially two irreducible infinite dimensional unitary representations of $G$, $(μ, L^2(\mathbb R^+))$ and $(μ^*, L^2(\mathbb R^+))$. In this paper, we give various characterizations about smooth vectors of $μ$ and their Mellin transforms. Let $\f d$ be a linear sum of delta distributions supported on the the positive integers $\mathbb Z^+$. We study the Mellin transform of the matrix coefficients $μ_{ \f d, f}(a)$ with $f$ smooth. We express these Mellin transforms in terms of the Dirichlet series $L(s, \f d)$. We determine a sufficient condition such that the generalized matrix coefficient $μ_{\f d, f}$ is a locally integrable function and estimate the $L^2$-norms of $μ_{\f d, f}$ over the Siegel set. We further derive an inequality which may potentially be used to study the Dirichlet series $L(s, \f d)$.
