Let $U$ be a unitary group defined with respect to an extension $E/F$ of non-archimedean local fields, and assume it is realised with respect to a pair $(V,q)$, where $V$ is an $n$-dimensional vector space over $E$ and $q$ is a hermitian form on $V$. By a decomposition theorem, $V$ decomposes as a sum of hyperbolic planes and another (possibly trivial) hermitian space of dimension at most 2. My question is as follows:
If $n$ is odd, this other hermitian space is in fact a line, and in that case $U$ is quasi-split. If $n$ is even, then $U$ is quasi-split if and only if this other space is trivial (that is, $V$ is really a sum of hyperbolic planes). Where could I find a reference for this characterisation of quasi-splitness?
This is often use in many papers, but I have never seen a reference for it. Any help would be greatly appreciated.
Best Answer
There is a text by Scharlau about "Hermitian...". Also the older book by O'Meara.
The point is that, first, over non-archimedean local fields a quadratic form in five or more variables has an isotropic vector. In case the residue characteristic is not two, this has a reasonably elementary direct proof. Then note that a hermitian form is a (special type of) quadratic form in twice as many variables. Thus, there is no anisotropic hermitian form in more than two variables.
Edit: in response to questioner's comment, "quasi-split" means (reductive and) a "Borel subgroup" defined over the field. Then "Borel subgroup" means parabolic subgroup that remains minimal under extending scalars. If the whole space were decomposable as hyperbolic planes and an anisotropic two-dimensional space, any quadratic extension would produce a "smaller" parabolic (the Borel, here, because the minimal parabolic is still next-to-Borel).
About algebraic groups, J. Tits' article in Corvallis is good, also the book Platonov and Rapinchuk, "Algebraic groups and number theory". Both these pay attention to such rationality properties, while many classics (such as Borel's "Linear algebra groups") emphasize the algebraically closed groundfield case.