Bounded reductive subalgebras of sl(n)

Let $\mathfrak{g}$ be a semisimple Lie algebra and $\mathfrak k\subset\mathfrak g$beareductivein$\mathfrak g$subalgebra\mathrm{.}A$(\mathfrak g, \mathfrak k)$-moduleisa$\mathfrak g$-modulewhichafterrestrictionto$\mathfrak k$becomesadirectsumoffinite-dimensional$\mathfrak k$-modules. I.Penkov and V.Serganova introduce definition of bounded $(\mathfrak g, \mathfrak k)$-modulesforreductivesubalgebras$\mathfrak k\subset\mathfrak g$,i\mathrm{.}e\mathrm{.}$(\mathfrak g, \mathfrak k)$-moduleswhose$\mathfrak k$-multiplicities are uniformly bounded. A question arising in this context is, given $\mathfrak{g}$, to describe all reductive in $\mathfrak{g}$ bounded subalgebras, i.e. reductive in $\mathfrak{g}$ subalgebras $\mathfrak{k}$ for which at least one infinite-dimensional bounded $(\mathfrak g, \mathfrak k)$-module exists. In the present paper we describe explicitly all reductive in ${\mathfrak{s}\mathfrak{l}}_n$ bounded subalgebras. 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.