论文标题
半神经谎言组统一表示基质系数的衰减估计值
Decay estimates for matrix coefficients of unitary representations of semisimple Lie groups
论文作者
论文摘要
让$ g $为有限中心的连接的半完整谎言组，而$ k $是最大的紧凑型亚组。给定$ g $上的函数$ u $，我们将$ \ mathcal {a} u $定义为超过$ k $的均方根平均值，在左右的$ k $上，$ u $。我们表明，对于所有统一表示$ g $的$π$，存在于$ g $上的独特最小值的正面球形函数$ ϕ_λ $，以至于$ \ mathcal {a} \langleπ（\ cdot）ξ，η\ rangle \ leq \ leq \ leq \ leq \ vert \ vert} \ vertη\ vert _ {\ Mathcal {h}_π} ϕ_λ $。该估计值具有渐近方估计值和Lebesgue空间估计值的良好特征。实际上，这是相当于估计$ \ vert \langleπ（\ cdot）ξ，η\ rangle \ vert \ vert \ leq c（ξ，η）\，ϕ_λ $ for $ k $ finite或spooth finite或平滑的vectors $ξ$ and $η$，以及Inf infintion in Infride in Infride in Infride inftions $ g $。此外，如果我们假设有任意$ c（ξ，η）$的后一种不平等，我们可以证明以前的不等式，然后以$ c（ξ，η）$的明确知识返回后者不等式。另一方面，它在$ g $中占有任何范围，与不是全球的渐近估计相反。我们还提供一些应用程序。
Let $G$ be a connected semisimple Lie group with finite centre and $K$ be a maximal compact subgroup thereof. Given a function $u$ on $G$, we define $\mathcal{A} u$ to be the root mean square average over $K$, acting both on the left and the right, of $u$. We show that for all unitary representations $π$ of $G$, there exists a unique minimal positive-real-valued spherical function $ϕ_λ$ on $G$ such that $\mathcal{A} \langle π(\cdot) ξ, η\rangle \leq \Vert ξ\Vert_{\mathcal{H}_π} \Vert η\Vert_{\mathcal{H}_π} ϕ_λ$. This estimate has nice features of both asymptotic pointwise estimates and Lebesgue space estimates; indeed it is equivalent to pointwise estimates $\vert \langle π(\cdot) ξ, η\rangle \vert \leq C(ξ, η) \,ϕ_λ$ for $K$-finite or smooth vectors $ξ$ and $η$, and it exhibits different decay rates in different directions at infinity in $G$. Further, if we assume the latter inequality with arbitrary $C( ξ, η)$, we can prove the former inequality and then return to the latter inequality with explicit knowledge of $C( ξ, η)$. On the other hand, it holds everywhere in $G$, in contrast to asymptotic estimates which are not global. We also provide some applications.
