Bounded reductive subalgebras of $\mathfrak{s}{\mathfrak{l}_n}$

Abstract

Let $\mathfrak{g}$ be a reductive Lie algebra and $\mathfrak{k} \subset \mathfrak{g}$ be a reductive in $\mathfrak{g}$ subalgebra. A ($\mathfrak{g},\mathfrak{k}$)-module M is a $\mathfrak{g}$-module for which any element m ∈ M is contained in a finite-dimensional $\mathfrak{k}$-submodule of M. We say that a ($\mathfrak{g},\mathfrak{k}$)-module M is bounded if there exists a constant C _M such that the Jordan-Hölder multiplicities of any simple finite-dimensional $\mathfrak{k}$-module in every finite-dimensional $\mathfrak{k}$-submodule of M are bounded by C _M . In the present paper we describe explicitly all reductive in $\mathfrak{s}{\mathfrak{l}_n}$ subalgebras $\mathfrak{k}$ which admit a bounded simple infinite-dimensional ($\mathfrak{s}{\mathfrak{l}_n},\mathfrak{k}$)-module. Our technique is based on symplectic geometry and the notion of spherical variety. We also characterize the irreducible components of the associated varieties of simple bounded ($\mathfrak{g},\mathfrak{k}$)-modules.