I am looking for references on unitary groups in the algebraic setting: that is, given a quadratic extension $E/F$, the unitary groups (if I understand correctly) are subgroups of the Weil restriction of scalars of $GL(n)$ from $E$ down to $F$, which are fixed by some involution. More specifically, I am looking for some sort of classification, which ones are quasi-split, if they are simply-connected (or with simply-connected derived subgroup), etc. Does anyone know where to look?

  • $\begingroup$ I don't understand the issues well enough to give an answer, but Platonov/Rapinchuk "Algebraic Groups and Number Theory" has an extended discussion of unitary groups (with a very general definition) in Chapter 2. See Sections 2.3.3 and 2.3.4 especially. This should tell you which are simply connected. But I did not see anything to help tell which are quasi-split. $\endgroup$ – B R Feb 13 '12 at 22:03
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    $\begingroup$ Unitary groups are all (outer) forms of $GL(n)$ so they've all got simply-connected derived subgroups (because this, by definition, can be checked after passing to an alg closed field, where the groups become $GL(n)$). Some are quasi-split and some aren't. The story is similar to division algebras; if $F$ is local then the unitary groups can be listed explicitly; if $F$ is a global field then the global situation is almost determined by the local ones but there is sometimes a finite amount of error. I think one place to see this stuff is Clozel's IHES paper from 1990 or so when he attaches... $\endgroup$ – Kevin Buzzard Feb 13 '12 at 22:46
  • $\begingroup$ ...Galois representations to certain automorphic reps of $GL(n)$. $\endgroup$ – Kevin Buzzard Feb 13 '12 at 22:48

I give an answer to my question so that it does not go unanswered.

A good reference on unitary groups in general is (as B R suggested) the book Algebraic Groups and Number theory by Platonov and Rapinchuk, especially Chapter 2 and 6. Another good reference is The Book of Involutions by Knus et al.

Also, I found great lecture notes by Bellaiche, Automorphic forms for Unitary groups and Galois representations.

Finally, the book Automorphic representations of unitary groups in three variables by Rogawski has a nice discussion on many things, in particular Cartan subgroups of unitary groups.


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