Groups GL(∞) over finite fields and multiplications of double cosets

Let $\mathbb F$ be a finite field. Consider direct sum $V$ of an infinite number copies F$, consider the dual space $V^\diamond$, i.~e., product F$. ${\mathbb V}=V\oplus V^\diamond$. The object paper is group $\mathbf{GL}$ continuous linear operators in V$. We reduce theory unitary representations to projective certain category whose morphisms are relations finite-dimensional spaces over In fact we family $ Q_\alpha$ subgroups V$ preserving two-element flags, show that there natural multiplication on double cosets with respect Q_\alpha$, and this products relations. has type $\mathrm{I}$ obtain 'upper estimate' set all irreducible $\mathbf{GL}$.