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

JOURNAL OF ALGEBRA

卷:585

Article

Neretin, Yury A.^1,2,3,4 

[1] Univ Vienna, Math Dept, Wolfgang Pauli Inst, Vienna, Austria

[2] Inst Theoret & Expt Phys, Moscow, Russia

[3] Moscow MV Lomonosov State Univ, MechMath Dept, Moscow, Russia

[4] Inst Informat Transmiss Problems, Moscow, Russia

关键词: Double cosets;    Unitary representation;    Chevalley groups;    Linear relations;   

DOI  :  10.1016/j.jalgebra.2021.06.011

来源: Elsevier

【 摘 要 】

Let F be a finite field. Consider a direct sum V of an infinite number of copies of F, consider the dual space V-lozenge, i.e., the direct product of an infinite number of copies of F. Consider the direct sum V = V-lozenge circle plus V. The object of the paper is the group (GL) over bar of continuous linear operators in V. We reduce the theory of unitary representations of (GL) over bar 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) over bar (alpha) of subgroups in (GL) over bar preserving two-element flags, show that there is a natural multiplication on spaces of double cosets with respect to (Q) over bar (alpha), 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) over bar. (C) 2021 Elsevier Inc. All rights reserved.

【 授权许可】

Free