Groups $GL(\infty)$ over finite fields and multiplications of double cosets

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 infinite number of copies of $\mathbb F$. Consider the direct sum $\mathbb V=V\oplus V^\diamond$ and the group $\mathbf{GL}$ of all continuous linear operators in $\mathbb V$. We reduce the theory of unitary representations of $\mathbf{GL}$ to projective representations of a category whose morphisms are linear relations in finite-dimensional linear spaces over $\mathbb F$. In fact we consider a certain family $Q_\alpha$ of subgroups in $\mathbf{GL}$ preserving two-element flags, show that there is a natural multiplication on spaces of double cosets with respect to $ Q_\alpha$, and reduce this multiplication to products of linear relations.