Invariant measures of infinite-dimensional groups over finite fields

We study the problem of characterizing the set of $G$-invariant measures on a space of infinite-dimensional matrices over a finite field. The groups $G$ in consideration are inductive limits of the general linear groups $GL(n, q)$ and the even unitary groups $U(2n,q^2)$ over a finite field. The problem is equivalent to characterizing the nonnegative harmonic functions on branching graphs that are Hall-Littlewood deformations of the Young graph. The talk is based on joint work with Grigori Olshanski.