  Let $\mathfrak g$ be a semisimple Lie algebra and $\mathfrak
k\subset\mathfrak g$ be a reductive in $\mathfrak g$ subalgebra. A $(\mathfrak
g, \mathfrak k)$-module is a $\mathfrak g$-module which after restriction to
$\mathfrak k$ becomes a direct sum of finite-dimensional $\mathfrak k$-modules.
I.Penkov and V.Serganova introduce definition of bounded $(\mathfrak g,
\mathfrak k)$-modules for reductive subalgebras $\mathfrak k\subset\mathfrak
g$, i.e. $(\mathfrak g, \mathfrak k)$-modules whose $\mathfrak
k$-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
$\mathfrak{sl}_n$ 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.
