On Annihilators of Bounded (g, k)-Modules

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 kmodules and the multiplicities of all simple k -modules in this 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 M 1 , M 2 are simple (g, 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.