On annihilators of bounded $(\mathfrak{g},\mathfrak{k})$-modules

Let $\mathfrak{g}$ be a semisimple Lie algebra and $\mathfrak{k}\subset \mathfrak{g}$ be a reductive subalgebra. We say that a $\mathfrak{g}$-module $M$ is a bounded $(\frak g, \frak k)$-moduleif$M$isadirectsumofsimplefinite-dimensional$\frak k$-modulesandthemultiplicitiesofallsimple$\frak k$-modules in that direct sum are universally bounded. The goal of this article is to show that the “boundedness” property for a simple $(\mathfrak{g},\mathfrak{k})$-module $M$ is equivalent to a property of the associated variety of the annihilator of $M$ (this is the closure of a nilpotent coadjoint orbit inside ${\mathfrak{g}}^{\ast }$) under the assumption that the main field is algebraically closed and of characteristic 0. In particular this implies that if $M_1,M_2$ are simple $(\mathfrak{g},\mathfrak{k})$-modules such that $M_1$ is bounded and the associated varieties of the annihilators of $M_1$ and $M_2$ coincide then $M_2$ is also bounded. This statement is a geometric analogue of a purely algebraic fact due to I. Penkov and V. Serganova and it was posed as a conjecture in my Ph.D. thesis.