Infinite-dimensional groups over finite fields and Hall-Littlewood symmetric functions

Abstract

The groups mentioned in the title are certain matrix groups of infinite size with elements in a finite field Fq. They are built from finite classical groups and at the same time they are similar to reductive p-adic Lie groups. In the present paper, we initiate the study of invariant measures for the coadjoint action of these infinite-dimensional groups. Of special interest for us are ergodic invariant measures, which are a substitute of orbital measures.We examine first the group GLB, a topological completion of the inductive limit group lim→GL(n,Fq). The traceable factor representations of GLB were studied by Gorin, Kerov, and Vershik (2014) [12]. We show that there exists a parallel theory for ergodic coadjoint-invariant measures, which is linked with harmonic functions on the “HL-deformed Young graph” YHL(t). Here the deformation means that the edges of the Young graph Y are endowed with certain formal multiplicities depending on the Hall-Littlewood (HL) parameter t specialized to q−1.This fact serves as a prelude to our main results, which concern topological completions of two inductive limit groups built from finite unitary groups, of even and odd dimension. We show that in this case, coadjoint-invariant measures are linked to some new branching graphs. They are still related to the HL symmetric functions, but in a nonstandard way, and the HL parameter t takes now the negative value −q−1. As an application, we find several families of unitarily invariant measures, including analogues of the Plancherel measure.