Consider the Hermitian bounded symmetric domain for $k \leq m$: $$ C_{k, m} = \{ Z \in \mathbb{C}^{m\times k} \,|\, Z^*Z < I_k \} $$ where $I_k$ is the $k\times k$ unit matrix. If I am not mistaken, this is known as AIII or Cartan domain of type I.
The goal is to construct some kind of (finite but arbitrarily large) lattice or grid in $C_{k, m}$ which is as uniform as possible with respect to the distance induced by the embedding to $\mathbb{C}^{m\times k}$.
Literature is permeated with adelic business and it's hard to find explicit examples. Moreover, I am not sure whether discrete subgroups of the isometry group are of particular interest because the application requires "good access" to the lattice points. With generators of a discrete subgroup I am not sure I will be able to efficiently enumerate the lattice points in a manner that actually respects the distance between the lattice points.
For the special case of $k=1$ we just get the unit disc in $\mathbb{C}^m$ and given $N$ equally spaced points in $P \subset [0,1]$ we can map the "hypercube grid" $P^{m} \times P^m \in \mathbb{R}^{2m}$ to the disc by $(a,b)\to z=F^{-1}(a)+\imath F^{-1}(b) \to zf(\|z\|)\in C_{1,m}.$ The function $f:\mathbb{R}\to\mathbb{R}$ satisfies certain nonlinear ODE so that the whole resulting map is actually measure preserving from the hypercube to the unit disc.
I am also interested in the following, closely related questions:
- What is the explicit form of the unbouded realization of $C_{k, m}$ and what is the formula for the Cayley map?
- Is there a measure preserving map from a hypercube to $C_{k, m}$?
