Mathematics Stack Exchange is a question and answer site for people studying math at any level and professionals in related fields. Join them; it only takes a minute:

Sign up
Here's how it works:
  1. Anybody can ask a question
  2. Anybody can answer
  3. The best answers are voted up and rise to the top

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?

share|cite|improve this question
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. – B R Feb 13 '12 at 22:03
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... – Kevin Buzzard Feb 13 '12 at 22:46
...Galois representations to certain automorphic reps of $GL(n)$. – Kevin Buzzard Feb 13 '12 at 22:48
up vote 4 down vote accepted

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.

share|cite|improve this answer

Your Answer


By posting your answer, you agree to the privacy policy and terms of service.

Not the answer you're looking for? Browse other questions tagged or ask your own question.