Uniformly bounded multiplicities, polynomial identities and coisotropic actions

Let $G_{\mathbb{R}}$ be a real reductive Lie group and $G'_{\mathbb{R}}$ a reductive subgroup of $G_{\mathbb{R}}$ such that $\mathfrak{g'}$ is algebraic in $\mathfrak{g}$. In this paper, we consider restrictions of irreducible representations of $G_{\mathbb{R}}$ to $G'_{\mathbb{R}}$ and induced representations of irreducible representations of $G'_{\mathbb{R}}$ to $G_{\mathbb{R}}$. Our main concern is when such a representation has uniformly bounded multiplicities, i.e. the multiplicities in the representation are (essentially) bounded. We give characterizations of the uniform boundedness by polynomial identities and coisotropic actions. For the restriction of (cohomologically) parabolically induced representations, we find a sufficient condition for the uniform boundedness by spherical actions and some fiber condition. This result gives an affirmative answer to a conjecture by T. Kobayashi. Our results can be applied to $(\mathfrak{g}, K)$-modules, Casselman--Wallach representations, unitary representations and objects in the BGG category $\mathcal{O}$. We also treat with an upper bound of cohomological multiplicities.