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 $\mathbb{V}=V^{\diamond }\oplus V$. The object of the paper is the group $\overline{\mathrm{GL}}$ of continuous linear operators in $\mathbb{V}$. We reduce the theory of unitary representations of $\overline{\mathrm{GL}}$ 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 ${\bar{Q}}_{\mathit{\alpha }}$ of subgroups in $\overline{\mathrm{GL}}$ preserving two-element flags, show that there is a natural multiplication on spaces of double cosets with respect to ${\bar{Q}}_{\mathit{\alpha }}$, and reduce this multiplication to products of linear relations. We show that this group has type I and obtain an ‘upper estimate’ of the set of all irreducible unitary representations of $\overline{\mathrm{GL}}$.

让 $\mathbb{F}$是一个有限域。考虑无限数量的副本的直接和V$\mathbb{F}$, 考虑对偶空间 ${伏}^{\diamond }$, 即无限数量副本的直接乘积 $\mathbb{F}$. 考虑直接和$\mathbb{伏}={伏}^{\diamond }\oplus 伏$. 论文的对象是群$\overline{\mathrm{GL}}$ 的连续线性算子 $\mathbb{伏}$. 我们减少了单一表示的理论$\overline{\mathrm{GL}}$ 到某个范畴的射影表示，其态射是有限维线性空间中的线性关系 $\mathbb{F}$. 事实上我们考虑某个家庭${\overline{问}}_{\mathit{\alpha }}$ 中的子组 $\overline{\mathrm{GL}}$ 保留双元素标志，表明在双陪集空间上存在自然乘法 ${\overline{问}}_{\mathit{\alpha }}$，并将这种乘法减少到线性关系的乘积。我们证明该组具有类型 I 并获得所有不可约幺正表示的集合的“上估计”$\overline{\mathrm{GL}}$.