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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.

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this has nothing to do with local fields. You may look up Tits' article in AMS symposium on algebraic groups and discontinuous subgroups, where he gives the quasi-split forms of the unitary group. –  Venkataramana Dec 14 '12 at 21:28
4  
According to Tits, Classification of algebraic semisimple groups, Proc. Sympos. Pure Math. 9 (1966), a semisimple group $G$ over a field $F$ is quasi-split if and only if the centralizer of a maximal $F$-split torus $S$ of $G$ is a (maximal) torus. A simple calculation using this criterion shows that a special unitary group of a $2m+1$-dimensional space is quasi-split if and only if it is of $F$-rank $m$, and a special unitary group of a $2m$-dimensional space is quasi-split if and only if it is of $F$-rank $m$. This implies the desired assertion. –  Mikhail Borovoi Dec 14 '12 at 21:33
    
@Borovoi Thank you! This is most helpful! –  M Turgeon Dec 16 '12 at 2:57

2 Answers 2

up vote 5 down vote accepted

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.

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Thank you for your comment. Maybe my original question was not clear enough, but what I am looking for is a reference about being quasi-split. The decomposition I do understand. –  M Turgeon Dec 14 '12 at 21:52
    
Thank you for this addendum. It is quite enlightening. –  M Turgeon Dec 16 '12 at 2:56

As pointed out in comments, there is a detailed survey by Tits from the algebraic group viewpoint in his AMS proceedings paper, along with quite a few references to earlier literature. From the viewpoint of groups over local fields, it's worth looking closely at his table of "indices" at the end of the paper. There he shows concisely which labelled Dyhkin diagrams can occur over various kinds of fields. In particular, you are interested in the twisted type $^2 \! A_n$ with $n$ even. He indicates that in the local field case this diagram can occur only relative to a quadratic extension and corresponds then to a quasi-split special unitary group. In his set-up, "quasi-split" corresponds to the case where $n$ is twice the relative rank $r$ and the Dynkin diagram is folded accordingly. (In general, various special unitary groups over division algebras are possible.)

While Tits does not spell out all the details of how such groups are constructed, he does provide a nice overview of the basic algebraic group theory leading to such a classification list. (Some improvements are contained in his 1971 paper in the Crelle Journal.) Also, student of his at Bonn named Martin Selbach elaborated further on the theory:

Klassifikationstheorie halbeinfacher algebraischer Gruppen. Diplomarbeit, Univ. Bonn, Bonn, 1973. Bonner Mathematische Schriften, Nr. 83. Mathematisches Institut der Universitat Bonn, Bonn, 1976. v+140 pp.

P.S. For the characteristic 0 theory, presented in a different style, you might try the lecture notes by I. Satake (which I haven't looked at in a long time): Classification theory of semi-simple algebraic groups. With an appendix by M. Sugiura. Notes prepared by Doris Schattschneider. Lecture Notes in Pure and Applied Mathematics, 3. Marcel Dekker, Inc., New York, 1971. viii+149. Satake also used labelled Dynkin diagrams, but with different conventions than Tits ("Satake-Tits diagrams").

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Thank you for the references, I will look into it! –  M Turgeon Dec 16 '12 at 2:55

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