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

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