The theory of discretely decomposable restriction was established by T. Kobayashi. He gave necessary/sufficient conditions for the discrete decomposability and the \({\mathfrak {g}}\) -admissibility by associated variety and asymptotic K-support. He also proposed conjectures and problems in [Adv. Stud. Pure Math. ’00, Contemp. Math. ’11, Progr. Math. ’15]. In this chapter, we consider Kobayashi’s conjectures on discretely decomposable restriction and give affirmative answers to some of them. To solve the problems, we study fundamental properties of discretely decomposable restriction, such as compactness of the action of functions, and estimate of infinitesimal characters by growth of Hilbert representations. As an analog of Kobayashi’s criterion for the \({\mathfrak {g}}\) -admissibility, we give a criterion for the discrete decomposability by wave front set.