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 over a finite field $\mathbb F_q$. 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. We examine first the group $\mathbb{GLB}$, a topological completion of the inductive limit group $\varinjlim GL(n, \mathbb F_q)$. As was shown by Gorin, Kerov, and Vershik [arXiv:1209.4945], the traceable factor representations of $\mathbb{GLB}$ admit a complete classification, achieved in terms of harmonic functions on the Young graph $\mathbb Y$. We show that there exists a parallel theory for ergodic coadjoint-invariant measures, which is linked with a deformed version of harmonic functions on $\mathbb Y$. Here the deformation means that the edges of $\mathbb Y$ are endowed with certain formal multiplicities coming from the simplest version of Pieri rule (multiplication by the first power sum $p_1$) for the Hall-Littlewood (HL) symmetric functions with parameter $t:=q^{-1}$. This fact serves as a prelude to our main results, which concern topological completions of two inductive limit groups built from finite unitary groups. We show that in this case, coadjoint-invariant measures are linked to some new branching graphs. The latter are still related to the HL functions, but the novelty is that now the formal edge multiplicities come from the multiplication by $p_2$ (not $p_1$) and the HL parameter $t$ turns out to be negative (as in Ennola's duality).