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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 computingIwasawa coordinates of $G/K$ using the Chevalley generators and the Steinbergpresentation. When $G/K$ is a scalar coset for a supergravity theory indimensions $\geq 3$, we determine the action of the integral form$G(\mathbb{Z})$ on $G/K$. We give explicit results for the action of thediscrete $U$--duality groups $SL_2(\mathbb{Z})$ and $E_7(\mathbb{Z})$ on thescalar cosets $SL_2(\mathbb{R})/SO_2(\mathbb{R})$ and$E_{7(+7)}(\mathbb{R})/[SU(8,\mathbb{R})/\{\pm Id\}]$ for type IIB supergravityin ten dimensions and 11--dimensional supergravity in $D=4$ dimensions,respectively. For the former, we use this to determine the discrete U--dualitytransformations on the scalar sector in the Borel gauge and we describe thediscrete symmetries of the dyonic charge lattice. We determine thespectrum--generating symmetry group for fundamental BPS solitons of type IIBsupergravity in $D=10$ dimensions at the classical level and we propose ananalog of this symmetry at the quantum level. We indicate how our methods canbe used to study the orbits of discrete U--duality groups in general.

Integral group actions on symmetric spaces and discrete duality symmetries of supergravity theories