A characterization of invariant subspaces for isometric representations of product system over $\mathbb{N}_0^{k}$

Using the Wold-von Neumann decomposition for the isometric covariant representations due to Muhly and Solel, we prove an explicit representation of the commutant of a doubly commuting pure isometric representation of the product system over $\mathbb{N}_0^{k}.$ As an application, we study a complete characterization of invariant subspaces for a doubly commuting pure isometric representation of the product system. This provides us a complete set of isomorphic invariants. Finally, we classify a large class of commuting isometric representations of the product system.