It seems appropriate now to link to the recent post of Richard Stanley, at mathoverflow.net/questions/254782/…

Not too bad. The adjoint representation of GL(5), restricted to SO(5), contains a copy of the adjoint representation together with other representations. So the condition would be genericity for the rep. of SO(5), plus regularity of some other L-functions at s=1. I haven't done the branching problem... but it shouldn't be too hard.

In many cases they are called Picard modular forms. They go back to the 1880s. See Picard, E.: Sur des fontions de deux variables independantes analogues aux fonctions modulaires. Acta Math. 2, 114-135 (1883).

Happy to see the expert commentary! I saw the preprint and other beautiful work on Moufang quadrangles and such, and thought first to refer to them. But I picked my way back in time to Allison's work in the 80s and ended up just giving references to his work. Sorry for the omission!

They're usually not phrased in terms of an order. Rather, irreducible polynomials are clumped by degree, and the number of irred polys of a given degree is well understood. So I'm basically asking whether the distribution within each fixed degree can be understood in lex-order.

But would one expect any sort of uniform geometric construction of these 238 groups? The only uniform constructions I know of Lie groups of type A-G are by Chevalley/Steinberg -- by generators and relations, like Tits for Kac-Moody groups. Otherwise, one finds these groups in a non-uniform manner by geometry -- e.g. type G_2 from octonions, etc.. So wouldn't one expect a large zoo of geometric constructions for the hyperbolic Kac-Moody groups?

Yep - the kernel of $W \rightarrow O(\bar \Omega, N)$ is $\{ \pm 1 \}$.

Incidentally, the notation issues with finite simple groups of Lie type in type $D$ are bemoaned at en.wikipedia.org/wiki/Group_of_Lie_type#Notation_issues.