超矩阵上的不变函数

簇集合上的不变函数

Invariant Functions On Cluster Ensembles

本文定义了簇集合上关于簇模群作用的不变函数，将其应用于已知与新例子，证明其具有几何与数论解释，并对仿射Dynkin图相关的簇集合进行了分类。

单李代数上不变函数的内在构造

Intrinsic construction of invariant functions on simple Lie algebras

给出了复单李代数上本原伴随不变函数的一种构造算法。这种结构是内在的，因为它不诉诸于任何表征。通过提升李代数的Kostant切片上的坐标函数，得到了整个李代数上的一个本原不变函数。这种不变函数的内在构造对于较大的例外李代数（如E）是最有用的。最后概述了该算法的Maple实现，并将在未来的工作中应用于这些特殊的李代数。

On vector-valued twisted conjugate invariant functions on a group

We study the space of vector-valued (twisted) conjugate invariant functions on a connected reductive group.

The b-functions of semi-invariants of quivers

In this paper we compute b-functions (or Bernstein-Sato polynomials) of various semi-invariants of quivers. The main tool is an explicit relation for the b-functions between semi-invariants that correspond to each other under reflection functors (or castling transforms). This enables us to compute recursively the b-functions for all Dynkin quivers, and extended Dynkin quivers with prehomogeneous dimension vectors.

On invariant random positive definite functions

We give the definition of an invariant random positive definite function on a discrete group, generalizing both the notion of an invariant random subgroup and a character. We use von Neumann algebras to show that all invariant random positive definite functions on groups with infinite conjugacy classes which integrate to the regular character are constant.

Invariant generalized functions supported on an orbit

We study the space of invariant generalized functions supported on an orbit of the action of a real algebraic group on a real algebraic manifold. This space is equipped with the Bruhat filtration. We study the generating function of the dimensions of the filtras, and give some methods to compute it. To illustrate our methods we compute those generating functions for the adjoint action of $\mathrm{GL}_3(\mathbb{C})$. Our main tool is the notion of generalized functions on a real algebraic stack, introduced recently by Sakellaridis.