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

Abstract

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 $Q_{\alpha }$ of subgroups in $\mathbb{V}$ preserving two-element flags, show that there is a natural multiplication on spaces of double cosets with respect to $Q_{\alpha }$, 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 $\mathbf{GL}$.

48pp, a revised version

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