  For $G(\mathbb{R})$ a split, simply connected, semisimple Lie group of rank
$n$ and $K$ the maximal compact subgroup of $G$, we give a method for computing
Iwasawa coordinates of $G/K$ using the Chevalley generators and the Steinberg
presentation. When $G/K$ is a scalar coset for a supergravity theory in
dimensions $\geq 3$, we determine the action of the integral form
$G(\mathbb{Z})$ on $G/K$. We give explicit results for the action of the
discrete $U$--duality groups $SL_2(\mathbb{Z})$ and $E_7(\mathbb{Z})$ on the
scalar cosets $SL_2(\mathbb{R})/SO_2(\mathbb{R})$ and
$E_{7(+7)}(\mathbb{R})/[SU(8,\mathbb{R})/\{\pm Id\}]$ for type IIB supergravity
in ten dimensions and 11--dimensional supergravity in $D=4$ dimensions,
respectively. For the former, we use this to determine the discrete U--duality
transformations on the scalar sector in the Borel gauge and we describe the
discrete symmetries of the dyonic charge lattice. We determine the
spectrum--generating symmetry group for fundamental BPS solitons of type IIB
supergravity in $D=10$ dimensions at the classical level and we propose an
analog of this symmetry at the quantum level. We indicate how our methods can
be used to study the orbits of discrete U--duality groups in general.
