Is anyone aware of any books/papers that discuss the details of the indefinite (special) orthogonal groups $SO(n,m)$, their universal covers, representation theory, etc. (possibly some connections with physics, if any)? My searches thus far have hardly come up with much...

I suppose that I should add that I would be particularly interested in the case where $n,m>1$ (of course there is an enormous litany of references for the case $SO(3,1)$, given its connections to special relativity).

There is some mention of it in a few places that I've come across (e.g. here), I guess I'm just looking for something more focused on these groups themselves (though that's not necessary -- all suggestions are welcome!).

In regards to the potential connections with physics, one is typically interested in the irreducible representations of the universal cover of a given group, and in the standard case of $SO(3,1)$ that would be the double cover $Spin(3,1)$, but $Spin(n,m)$ is not, in general as I understand it, simply connected, so... any help with understanding what it is in more general cases, and its irreps, would be nice. :)


In my opinion, one of the best introductory books for Lie groups is Lie Groups - An introduction trough linear groups by Wulf Rossman.

In that book there are many examples and, I am not wrong, $SO(n,m)$ is explained (at least for some $n$'s and $m=1$).

It is a very good book anyway.

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    $\begingroup$ I don't think that is quite what I'm looking for (see my updated question, as I realized I should maybe be slightly more specific ;) ), but thanks nonetheless! I'm always happy to check out another potentially good book. $\endgroup$ – OperaticDreamland Oct 14 '16 at 14:20
  • $\begingroup$ I wouldn't do a downvote for that, but ok... $\endgroup$ – Edu Oct 14 '16 at 15:31
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    $\begingroup$ I wouldn't either, which is why I didn't. (In fact, I upvoted it, but my reputation was under 15, so it doesn't show up. Just for good measure, I upvoted it now that I've gone above 15, so at least that downvote is cancelled out.) $\endgroup$ – OperaticDreamland Oct 15 '16 at 2:03
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    $\begingroup$ @OperaticDreamland Oh, no worries! I didn't look at your reputation, I simply assumed that it was you because usually the downvote is followed by a comment regarding it and the only comment was yours so... $\endgroup$ – Edu Oct 15 '16 at 11:06
  • $\begingroup$ Sure, I understand. No worries! :) $\endgroup$ – OperaticDreamland Oct 15 '16 at 15:06

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