  The Mackey-type identity mentioned in the title relates the operations of
parabolic induction and restriction for invariant functions on the Lie algebras
of the finite unitary groups $U(N, q^2)$. This result is applied to
constructing positive harmonic functions on a new branching graph with a
negative Hall-Littlewood parameter, as introduced in the authors' paper [Adv.
Math. vol. 395 (2022), 108087; arXiv:2102.01947].This in turn implies the
existence of an infinite-parameter family of invariant measures for the
coadjoint action of an infinite-dimensional analogue of the groups $U(N, q^2)$.
