Inverse Galois problem for simple Lie type groups

Progress towards the Inverse Galois problem over $\mathbb{Q}$ is very well documented for sporadic groups ($M_{23}$ is the only case open) and for $PSL_n(q)$ (a lot of cases known, but wide open in general).

I'm interested in progress for families of simple Lie type groups other than $PSL_n(q)$.

Important note (edit). I'm looking for results for complete families, not individual groups.

I haven't been able to find almost any information, and that suggests that the answer is "nothing is known". But it would be nice to have a reference for that, if it is the case.

On a side note, as an example of progress for those groups (so that they are not completely intractable), Belyi proved that the 6 families of classical simple Lie groups are realizable over $\mathbb{Q}^{ab}$.

• where have you been looking? typing Inverse Galois in arxiv's search box produces many articles dealing with projective linear, orthogonal and symplectic groups. to name just a few, look at works by Sara Arias-de-Reyna, Luis Dieulefait, Sug Woo Shin, Gabor Wiese, and David Zywina. Oct 3 '15 at 17:19
• @DrorSpeiser There's plenty of articles with partial results, but that's not quite what I'm looking for. Oct 3 '15 at 17:40

There is Thompson and Volklein, who prove that the symplectic groups are Galois groups:

Thompson, J. G.(1-FL); Völklein, H.(1-FL)
Symplectic groups as Galois groups.
J. Group Theory 1 (1998), no. 1, 1–58.
12F12


More information on realization of simple groups can be found in Volklein's book:

MR1405612 (98b:12003) Reviewed
Völklein, Helmut(1-FL)
Groups as Galois groups. (English summary)
An introduction. Cambridge Studies in Advanced Mathematics, 53.
Cambridge University Press, Cambridge, 1996. xviii+248 pp. ISBN: 0-521-56280-5
12F12


Finally, a zoo of low-order (not that low) is discussed by David Zywina.

• That's nice, I hadn't noticed the result of Thompson-Volklein. Oct 3 '15 at 17:30

It's hard to keep track of all relevant literature on the inverse Galois problem for finite groups of Lie type, but at this point many special cases have been worked out while others remain open. One rather recent paper here indicates how subtle the approaches have become. (For an arXiv preprint, see here.)

Work by Gunter Malle, including his joint book with B.H. Matzat, is an important source; Malle might also have advice to give by email. (One obvious question is what kind of search tools you are using and what kind of library access you have?)

As you realize, finite special linear groups and their relatives pose extra problems---especially in low ranks. The study of finite simple or almost-simple groups is natural but has so far been successful only in special cases rather than through some unified argument.

• I'm interested on the realizability of a whole family of Lie type groups, not as much on individual cases. Perhaps I should have made that clearer. I have Volklein and Serre's textbooks, but not Malle-Matzat. I'll definitely check it out. Oct 3 '15 at 17:38
• As far as I can tell, there is very little literature which treats an entire family of groups of Lie type over arbitrary finite fields (but I'm not a specialist). Anyway, the book by Malle-Matzat has a reputation of being rather difficult due to reliance on Matzat's framework. Malle himself has been close to the finite groups of Lie type for a long time and can probably provide updated information. Oct 3 '15 at 18:20