On annihilators of bounded $(\frak g, \frak k)$-modules

Abstract

Let $\frak g$ be a semisimple Lie algebra and $\frak k\subset\frak g$ be a reductive subalgebra. We say that a $\frak g$-module $M$ is a bounded $(\frak g, \frak k)$-module if $M$ is a direct sum of simple finite-dimensional $\frak k$-modules and the multiplicities of all simple $\frak k$-modules in that direct sum are universally bounded. The goal of this article is to show that the "boundedness" property for a simple $(\frak g, \frak k)$-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 $\frak g^*$) 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 $(\frak g, \frak k)$-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.