On annihilators of bounded (g, k)-modules

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Let g be a semisimple Lie algebra and k⊂ g be a reductive subalgebra. We say that a g-module M is a bounded (g, k)-module if M is a direct sum of simple finite-dimensional k-modules and the multiplicities of all simple k-modules in that direct sum are universally bounded. The goal of this article is to show that the "boundedness" property for a simple (g, 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 g*) under the assumption that the main field is algebraically closed and of characteristic 0. In particular this implies that if M₁, M₂ are simple (g, k)-modules such that M₁ is bounded and the associated varieties of the annihilators of M₁ and M₂ coincide then M₂ 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.