For individual simple types, look at the planches at the end of Bourbaki, Chap; 4-6 (or English translation published by Springer). Google Scholar permits an online search.

Perhaps this should be viewed as a question about representation theory of $G$; here much but not everything is known, especially for $p$ large enough.

Concerning arXiv, it has far too many entries every day to permit easy access; but the subject labels provide a good short list if your interests are specific enough.

There seems to be some confusion about the difference between "A. Milgram" and "R.J. Milgram". The latter used a version of his middle name "James" and was a prolific algebraic topologist at Stanford (now retired).

A second edition is one possibility, but a more likely scenario is that a technical typist moved fingers too far when typing the year.

You can probably find some insights in the Math Reviews of Lusztig's papers by Bhama Srinivasan, a former student of J.A. Green; she got heavily involved with the finite groups of Lie type in particular.

@LSpice: Yes, I've now deleted the comment.

@LSpice: Sorry for the delay in responding, but I had to search awhile to find older printings of my book GTM 21. In Theorem 32.1, not having yet embraced the idea of "root datum", I gave too superficial an account of isomorphism between "simple" algebraic groups. In the revised printing I added the obvious exception, the group of type D$_\ell$ with even rank $\ell\geq 6$. (What is actually proved here is the equivalent Theorem' following Chevalley which involves extending a map. A complicated but concrete method compared to Takeuchi, later Steinberg.)

Note that the word "complex" isn't needed here, just an algebraically closed field of arbitrary characteristic (as noted by Marc Palm). Springer's book is one source.

P.S. This is one of the many questions I should have asked him when we spent a month at INI in Cambridge (UK), sharing an office. Probably it was in my first visit to IAS in 1968-69 that he posted a note from a publisher who declined to publish his lectures on "Chevrolet" groups.

@LSpice: I've always been somewhat naive about this issue, but in the case of Steinberg's Yale lectures there have always been efforts to replace the mimeographed notes by typeset and edited ones. Eventually this was done by AMS after Steinberg's death. What I don't know is how the ownership changed then.

P.S. The tag 'algebraic-groups' is probably more suitable here than the tag 'lie-groups', since the methods used are most often algebraic though suggested by Harish-Chandra's analytic methods.

The standard source for these characters is usually the paper by J.A. Green in the Transactions of AMS )1955), which is freely available online; or the version in Ian Macdonald's book (latest edition). In any case, the term principal seriesis most often used for the family of representations induced from 1-dimensjonal representaton of a Borel subgroup, here the upper (or lower) triangular matrices. (Carter's 1985 book has a more general viewpoint based on the Deligne-Lusztig construction.)