Cohomological properties of the smooth globalization of a Harish-Chandra module

PU(1,1) 有限覆盖群的 Harish-Chandra 模的整模型

Integral models of Harish-Chandra modules of the finite covering groups of PU(1,1)

本文针对 PU(1,1) 的有限覆盖群，计算其实诱导与上同调诱导对应的 Harish-Chandra 模整模型。

扭曲 $\mathcal D$-模族与 Harish-Chandra 模的算术模型

Families of twisted $\mathcal D$-modules and arithmetic models of Harish-Chandra modules

本文发展了一般基概形上的扭曲 $\mathcal D$-模理论及闭 $K$-轨道形式理论，并利用它们几何构造了上同调诱导模的半整模型，进而证明了其投射性和无挠相对李代数上同调等算术良好性质。

A construction of generalized Harish-Chandra modules for locally reductive Lie algebras

We study cohomological induction for a pair $(\frak g,\frak k)$, $\frak g$ being an infinite dimensional locally reductive Lie algebra and $\frak k \subset\frak g$ being of the form $\frak k_0 + C_\gg(\frak k_0)$, where $\frak k_0\subset\frak g$ is a finite dimensional reductive in $\frak g$ subalgebra and $C_{\gg} (\frak k_0)$ is the centralizer of $\frak k_0$ in $\frak g$. We prove a general non-vanishing and $\frak k$-finiteness theorem for the output. This yields in particular simple $(\frak g,\frak k)$-modules of finite type over $\frak k$ which are analogs of the fundamental series of ge…

Generalized Harish-Chandra modules with generic minimal $\frak k$-type

We make a first step towards a classification of simple generalized Harish-Chandra modules which are not Harish-Chandra modules or weight modules of finite type. For an arbitrary algebraic reductive pair of complex Lie algebras $(\g,\k)$, we construct, via cohomological induction, the fundamental series $F^\cdot (\p,E)$ of generalized Harish-Chandra modules. We then use $F^\cdot (\p,E)$ to characterize any simple generalized Harish-Chandra module with generic minimal $\k$-type. More precisely, we prove that any such simple $(\g,\k)$-module of finite type arises as the unique simple submodule o…

Algebraic methods in the theory of generalized Harish-Chandra modules

This paper is a review of results on generalized Harish-Chandra modules in the framework of cohomological induction. The main results, obtained during the last 10 years, concern the structure of the fundamental series of $(\mathfrak{g},\mathfrak{k})-$modules, where $\mathfrak{g}$ is a semisimple Lie algebra and $\mathfrak{k}$ is an arbitrary algebraic reductive in $\mathfrak{g}$ subalgebra. These results lead to a classification of simple $(\mathfrak{g},\mathfrak{k})-$modules of finite type with generic minimal $\mathfrak{k}-$types, which we state. We establish a new result about the Fernando-…

Cohomology of Standard Modules on Partial Flag Varieties

Cohomological induction gives an algebraic method for constructing representations of a real reductive Lie group $G$ from irreducible representations of reductive subgroups. Beilinson-Bernstein localization alternatively gives a geometric method for constructing Harish-Chandra modules for $G$ from certain representations of a Cartan subgroup. The duality theorem of Hecht, Mili\vcić, Schmid and Wolf establishes a relationship between modules cohomologically induced from minimal parabolics and the cohomology of the $\ms{D}$-modules on the complex flag variety for $G$ determined by the Beilinson-…

A Geometric Embedding for Standard Analytic Modules

In this manuscript we make a general study of the representations realized, for a reductive Lie group of Harish-Chandra class, on the compactly supported sheaf cohomology groups of an irreducible finite-rank polarized homogeneous vector bundle defined in a generalized complex flag space. In particular, we show that the representations obtained are minimal globalizations of Harish-Chandra modules and that there exists a whole number q, depending only on the orbit, such that all cohomologies vanish in degree less than q. The representation realized on the q-th cohomology group is called a standa…

理性模式

Rationality patterns

本文通过建立通用范畴框架，扩展了Loewy分类方案和Borel-Tits准则，用于研究Harish-Chandra模块的分类和定义域问题。

Rational Structures on Automorphic Representations

This paper proves the existence of global rational structures on spaces of cusp forms of general reductive groups. We identify cases where the constructed rational structures are optimal, which includes the case of GL($n$). As an application, we deduce the existence of a natural set of periods attached to cuspidal automorphic representations of GL($n$). This has consequences for the arithmetic of special values of $L$-functions that we discuss in subsequent articles. In the course of proving our results, we lay the foundations for a general theory of Harish-Chandra modules over arbitrary field…

The Borel-Weil theorem for reductive Lie groups

In this manuscript we consider the extent to which an irreducible representation for a reductive Lie group can be realized as the sheaf cohomolgy of an equivariant holomorphic line bundle defined on an open invariant submanifold of a complex flag space. Our main result is the following: suppose $G_{0}$ is a real reductive group of Harish-Chandra class and let $X$ be the associated full complex flag space. Suppose $\mathcal{O}_λ$ is the sheaf of sections of a $G_{0}$-equivariant holomorphic line bundle on $X$ whose parameter $λ$ (in the usual twisted $\mathcal{D}% $-module context) is antidomin…