# Unitary (algebraic) groups

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?

• 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