I start with a noncompact connected semisimple Lie group with finite center $G$ and fix a maximal compact subgroup $K$ of $G$. I am considering these compact groups $K$. If $\mathbb T$ is the maximal torus in $K$, I take the quotient $K/\mathbb T$. I read that they are Kahler manifold. I am interested to know if there is a natural way to find a dense set in $K/\mathbb T$ which is parametrized by $(j\_1/2^{n\_1}, \dots, j\_m/2^{n\_m})$, where $j\_i\in \mathbb Z$ and $n\_i\in \mathbb N$. 


Something like a dense grid, as can be constructed on $\mathbb R^d$ and some other groups/spaces like in a Heisenberg group, is what I have in mind. 



Quotient of a compact Lie group by maximal Torus
The answer is yes. As Aakumadala mentioned in a comment, the quotient $K/\mathbb{T}$ is isomorphic to the flag variety of $K\_{\mathbb{C}}$. If we let $N = N\_{\mathbb{C}}$ denote the nilpotent radical of a Borel subgroup of $K\_{\mathbb{C}}$, the flag variety has a dense $N$-orbit with a simply transitive $N$ action. Note that when $K = SU(3)$, we find that $N$ is a Heisenberg group, as you anticipated.


The underlying manifold of the dense $N$-orbit is just an affine space, isomorphic to $\mathbb{C}^n = \mathbb{R}^{2n}$ for some $n$. If you make this identification, then the dense embedding $\mathbb{Z}[1/2] \subset \mathbb{R}$ gives you a dense set parametrized by $\mathbb{Z}[1/2]^{2n}$.


