These notes were written to be distributed to the audience of the first author’s Takagi Lectures delivered June 23, 2018. These are based on a work-in-progress that is part of a collaborative project that also involves Akshay Venkatesh.

In this work-in-progress we give a new construction of some Eisenstein classes for GL_N (Z) that were first considered by Nori [41] and Sczech [44]. The starting point of this construction is a theorem of Sullivan on the vanishing of the Euler class of SL_N (Z) vector bundles and the explicit transgression of this Euler class by Bismut and Cheeger. Their proof indeed produces a universal form that can be thought of as a kernel for a regularized theta lift for the reductive dual pair (GL_N, GL₁). This suggests looking to reductive dual pairs (GL_N, GL_k) with k ≥ 1 for possible generalizations of the Eisenstein cocycle. This leads to fascinating lifts that relate the geometry/topology world of real arithmetic locally symmetric spaces to the arithmetic world of modular forms.

In these notes we do not deal with the most general cases and put a lot of emphasis on various examples that are often classical.

这些笔记是为了分发给第一作者于 2018 年 6 月 23 日发表的 Takagi 讲座的听众而编写的。这些笔记基于一项正在进行的工作，该工作是一个合作项目的一部分，该项目也涉及 Akshay Venkatesh。

在这项正在进行的工作中，我们给出了 GL _N ( Z ) 的一些爱森斯坦类的新构造，这些构造首先由 Nori [41] 和 Sczech [44] 考虑。这个构造的起点是沙利文关于 SL _N ( Z ) 向量丛的欧拉类消失的定理以及 Bismut 和 Cheeger 对该欧拉类的显式违反。他们的证明确实产生了一种通用形式，可以将其视为还原对偶对（GL _N， GL _{1 ）的}正则化 theta 升力的内核。这表明寻找k ≥ 1 的还原对偶对 (GL _N , GL _k )来实现爱森斯坦余循环的可能推广。这导致了令人着迷的提升，将实算术局部对称空间的几何/拓扑世界与模形式的算术世界联系起来。

在这些注释中，我们不处理最一般的情况，而是重点关注各种通常经典的示例。