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Groups $GL(\inftty)$ over finite fields and multiplications of double cosetsстатья
- Автор: Neretin Yury
- Журнал: Journal of Algebra
- Год издания: 2021
- Издательство: Academic Press
- Местоположение издательства: United States
- DOI: 10.1016/j.jalgebra.2021.06.011
- Аннотация: 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 $W=V\oplus V'$. The object of the paper is the group $GL$ of continuous linear operators in $W$. 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 $Q_n$ of subgroups in $GL$ preserving two-element flags, show that there is a natural multiplication on spaces of double cosets with respect to $Q_n$, 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 $GL$.Prelminary version of the paper is avalable viahttps://arxiv.org/abs/2002.09969
- Добавил в систему: Неретин Юрий Александрович