标题： 一种谱隙吸收原理 显示英文标题

标题： A Spectral Gap Absorption Principle

摘要： 我们证明了在特征为零的局部域上的单连通半单代数群的酉表示遵循谱间隙吸收原理：即谱间隙在张量积下保持不变。 我们通过证明简单代数群的酉对偶由矩阵系数的可积参数过滤来实现这一点。 这是一个闭理想滤链，它捕捉了不包含平凡表示的对偶的每一个闭子集。 换句话说，我们证明了一个表示具有谱间隙当且仅当存在某个$p < \infty$，使得其矩阵系数对于每个$\epsilon>0$都属于$L^{p+\epsilon}(G)$。 通过这样做，我们继续了Bader和Sauer在此领域的工作，并证明了他们提出的猜想。 我们还利用这一原理对Bekka和Valette提出的一个猜想给出了肯定解答：从半单群到格的限制映射的像在Fell拓扑中永远不是稠密的。 显示英文摘要

摘要： We show that unitary representations of simply connected, semisimple algebraic groups over local fields of characteristic zero obey a spectral gap absorption principle: that is, that spectral gap is preserved under tensor products. We do this by proving that the unitary dual of simple algebraic groups is filtered by the integrability parameter of matrix coefficients. This is a filtration of closed ideals that captures every closed subset of the dual that doesn't contain the trivial representation. In other words, we show that a representation has a spectral gap if and only if there exists some $p < \infty$ such that its matrix coefficients are in $L^{p+\epsilon}(G)$ for every $\epsilon>0$. Doing this, we continue the work of Bader and Sauer in this area and prove a conjecture they phrased. We also use this principle to give an affirmative solution to a conjecture raised by Bekka and Valette: the image of the restriction map from a semisimple group to a lattice is never dense in Fell topology.