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

Let F be a finite field. Consider a direct sum V of an infinite number of copies of F , consider the dual space V ⋄ , i.e., the direct product of an infinite number of copies of F. Consider the direct sum V = V ⋄ ⊕ V. The object of the paper is the group GL ‾ of continuous linear operators in V. We reduce the theory of unitary representations of GL ‾ to projective representations of a certain category whose morphisms are linear relations in finite-dimensional linear spaces over F. In fact we consider a certain family Q ‾ α of subgroups in GL ‾ 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 I and obtain an 'upper estimate' of the set of all irreducible unitary representations of GL ‾. [ABSTRACT FROM AUTHOR]