  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).
