The average size of the kernel of a matrix and orbits of linear groups

Let O be a compact discrete valuation ring of characteristic 0. Given a module M of matrices over O, we study the generating function encoding the average sizes of the kernels of the elements of M over finite quotients of O. We prove rationality and establish fundamental properties of these generating functions and determine them explicitly for various natural families of modules M. Using p-adic Lie theory, we then show that special cases of these generating functions enumerate orbits and conjugacy classes of suitable linear pro-p groups.