  The site and bond percolation problems are conventionally studied on
(hyper)cubic lattices, which afford straightforward numerical treatments. The
recent implementation of efficient simulation algorithms for high-dimensional
systems now also facilitates the study of $D_n$ root lattices in $n$ dimension
as well as $E_8$-related dense packing lattices. Here, we consider the
percolation problem on $D_n$ for $n=3$ to $13$ and on $E_8$ relatives for $n=6$
to 9. Precise estimates for both site and bond percolation thresholds obtained
from invasion percolation simulations are compared with dimensional series
expansion on $D_n$ lattices based on lattice animal enumeration. As expected,
the bond percolation threshold rapidly approaches the Bethe lattice limit as
$n$ increases for these high-connectivity lattices. Corrections, however,
exhibit clear yet unexplained trends. Interestingly, the finite-size scaling
exponent for invasion percolation is found to be lattice and percolation-type
specific.
