有限域上无限矩阵群的有限迹与表示
Finite traces and representations of the group of infinite matrices over a finite field
摘要：
本文研究了具有$q$元的有限域上几乎上三角无限矩阵的局部紧无限维群$\mathbb{GLB}$的表示理论。这个群体是由S.K.定义的，AV，和Andrei Zelevinsky在1982年作为一般线性群$\mathbb{GL}(n,q)$的一个适当的$n=\infty$模拟。它作为$\mathbb{GL}(\infty,q)$的替代方案，后者的表示理论很差。我们最重要的结果是通过Borel子群$\mathbb{B}$上的某些概率测度描述了群$\mathbb{ GLB}$的半有限幂幺迹（特征标），并构造了相应的$II_\infty$型von Neumann因子表示.作为主要工具，我们使用群代数$L_1(\mathbb{GLB})$中光滑函数的子代数$\mathcal A(\mathbb{ GLB})$。这个子代数是有限维群代数${\mathbb C}(\mathbb{GL}(n,q))$在抛物嵌入下的归纳极限。正如在其他例子的渐近表示理论，我们发现显着的性质，无限的情况下，不发生有限群，如乘法不可分解的字符或连接到概率概念。无限维岩堀-赫克代数$\mathcal H_q(\infty)$起着特殊的作用，在我们的考虑，并允许理解深类比的发展理论与代表理论的无限对称群$S(\infty)$已深入研究了许多以前的论文。
英文摘要：
The article is devoted to the representation theory of locally compact infinite-dimensional group $\mathbb{GLB}$ of almost upper-triangular infinite matrices over the finite field with $q$ elements. This group was defined by S.K., A.V., and Andrei Zelevinsky in 1982 as an adequate $n=\infty$ analogue of general linear groups $\mathbb{GL}(n,q)$. It serves as an alternative to $\mathbb{GL}(\infty,q)$, whose representation theory is poor. Our most important results are the description of semi-finite unipotent traces (characters) of the group $\mathbb{ GLB}$ via certain probability measures on the Borel subgroup $\mathbb{B}$ and the construction of the corresponding von Neumann factor representations of type $II_\infty$. As a main tool we use the subalgebra $\mathcal A(\mathbb{ GLB})$ of smooth functions in the group algebra $L_1(\mathbb{GLB})$. This subalgebra is an inductive limit of the finite--dimensional group algebras ${\mathbb C}(\mathbb{GL}(n,q))$ under parabolic embeddings. As in other examples of the asymptotic representation theory we discover remarkable properties of the infinite case which does not take place for finite groups, like multiplicativity of indecomposable characters or connections to probabilistic concepts. The infinite dimensional Iwahori-Hecke algebra $\mathcal H_q(\infty)$ plays a special role in our considerations and allows to understand the deep analogy of the developed theory with the representation theory of infinite symmetric group $S(\infty)$ which had been intensively studied in numerous previous papers.
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