The Burger-Sarnak Method and Operations on the Unitary Duals of Classical Groups

多量子比特黄金门

Multi-Qubit Golden Gates

本文为紧致酉李群构造最优拓扑生成元，推广黄金门至多量子比特，并证明Sarnak-Xue密度假设变体，从而获得高效通用门集，显著减少T型门和CS门数量。

Parabolically induced unitary representations of the universal group U(F)^+ are C_0

By employing a new strategy we prove that all parabolically induced unitary representations of the Burger-Mozes universal group U(F)^+, with F being primitive, have all their matrix coefficients vanishing at infinity. This generalizes the same well-known result for the universal group U(F)^+, when F is 2-transitive.

The universal group of Burger--Mozes and the Howe--Moore property

By constructing a new unitary representation we prove the universal group $U(F)^+$ of Burger--Mozes does not have the Howe--Moore property when $F$ is primitive but not $2$-transitive. It is well known $U(F)^+$ does have this property when $F$ is $2$-transitive. Along the way, we give a characterization of the universal group, when $F$ is primitive, to have the Howe--Moore property, and also prove $U(F)^+$ has the relative Howe--Moore property. These two results are a consequence of a strengthening of Mautner's phenomenon for locally compact groups acting on d-regular trees and having Tits' in…

通过内窥分类研究酉群上同调自守表示的统计

Statistics of Cohomological Automorphic Representations on Unitary Groups via the Endoscopic Classification

本文利用内窥分类计算酉群上具有给定无穷远处上同调因子的自守表示族在水平趋于无穷时的精确统计量，得到表示计数和 Satake 参数平均值的首项及误差界，并由此推出上同调增长渐近、平均 Sato-Tate 等分布律和 Sarnak-Xue 密度假设等结果。

带边曲面上平坦酉丛的符号差

Signature for flat unitary bundles over surfaces with boundary

本文研究带边紧曲面基本群在Hermitian型经典单Lie群中的表示，通过扩展Atiyah-Patodi-Singer的rho不变量，将符号差、Toledo与rho不变量以统一方式呈现，给出带边流体经典符号差公式的一个版本。

半单群中格的渐近上同调与一致稳定性

Asymptotic Cohomology and Uniform Stability for Lattices in Semisimple Groups

本文为高阶半单群中的格建立渐近上同调理论，证明第二上同调消失可推出关于有限维酉几乎表示的一致稳定性，从而推广了 Kazhdan 及 Burger–Ozawa–Thom 的刚性结果。