论文标题
组$ gl（\ infty）$在有限字段和双cosets的乘以
Groups $GL(\infty)$ over finite fields and multiplications of double cosets
论文作者
论文摘要
令$ \ mathbb f $为有限字段。考虑无限数量的$ \ mathbb f $的直接总和$ v $，请考虑双空间$ v^\ diamond $，i。考虑直接sum $ {\ mathbb v} = v \ oplus v^\ diamond $。本文的对象是$ \ Mathbb v $中连续线性运算符的$ \ mathbf {gl} $。我们将$ \ mathbf {gl} $的单一表示形式的理论减少到某个类别的投影表示形式，其形态是$ \ mathbb f $的有限维线性空间中的线性关系。实际上，我们考虑了一个$ \ mathbb v $保留两元素标志的子组的某个家庭$q_α$，这表明，相对于$q_α$，在双倍库的空间上存在自然乘法，并将此乘法减少到线性关系的产物中。我们证明该组具有$ \ mathrm {i} $，并获得了$ \ mathbf {gl} $的所有不可约合统一表示的“上限”。
Let $\mathbb F$ be a finite field. Consider a direct sum $V$ of an infinite number of copies of $\mathbb F$, consider the dual space $V^\diamond$, i.~e., the direct product of an infinite number of copies of $\mathbb F$. Consider the direct sum ${\mathbb V}=V\oplus V^\diamond$. The object of the paper is the group $\mathbf{GL}$ of continuous linear operators in $\mathbb V$. We reduce the theory of unitary representations of $\mathbf{GL}$ to projective representations of a certain category whose morphisms are linear relations in finite-dimensional linear spaces over $\mathbb F$. In fact we consider a certain family $ Q_α$ of subgroups in $\mathbb V$ preserving two-element flags, show that there is a natural multiplication on spaces of double cosets with respect to $ Q_α$, and reduce this multiplication to products of linear relations. We show that this group has type $\mathrm{I}$ and obtain an 'upper estimate' of the set of all irreducible unitary representations of $\mathbf{GL}$.
