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I am looking for a reference for the following fact: The orthogonal group $O_{2n}$ over an algebraically closed field of characteristic 2 has exactly two connected components.

To be more precise, let $O_q$ denote the orthogonal group of the quadratic form $q(x)=x_1 x_2 +x_3 x_4+\cdots +x_{2n-1}x_{2n}$ over an algebraically closed field $k$. In characteristic $p\neq 2$ the determinant takes two values on $O_q$, 1 and $-1$, and therefore the subgroup $SO_q:=O_q\cap SL_{2n}$ is of index 2 in $O_q$; it is known that $O_q\cap SL_{2n}$ is connected.

In characteristic 2 the determinant takes only one value 1 on $O_q$ (because $-1=1$), and therefore $O_q\cap SL_{2n}=O_q$. Still there is a homomorphism $D\colon O_q\to \mathbf{Z}/2\mathbf{Z}$ given by a polynomial $D$ called the Dickson invariant, see J.A.~Dieudonn\'e, Pseudo-discriminant and Dickson invariant, Pacific. J. Math. 5 (1955), 907--910. This homomorphism $D$ indeed takes both values 0 and 1 on $O_q$, and therefore its kernel ker $D$ is a closed subgroup of index 2 in $O_q$. I would like to know that ker $D$ is connected. In other words, I am looking for a reference to the assertion that the orthogonal group $O_q$ has at most two connected components. This is proved in Brian Conrad's handout "Properties of orthogonal groups" to his course Math 252 "Algebraic groups", see http://math.stanford.edu/~conrad/252Page/handouts/O(q).pdf . Is there any other reference for this fact?

I will be grateful to any references, comments, etc.

Mikhail Borovoi

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3 Answers

up vote 6 down vote accepted

Presumably this is treated in detail in chapter 7 of the book The Classical Groups and K-Theory, by A.J.Hahn and O.T.O'Meara.

On page 424 it says in theorem 7.2.23 that the elementary subgroup has index two. And elementary matrices are in the connected component of 1.

Wilberd

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I concur with Wilberd's "presumably", since this large book by Hahn and O'Meara takes real work to get into due to its treatment of classical groups over very general rings. The slender earlier survey by Dieudonne on classical groups does not use algebraic groups or algebraic geometry. But his concise treatment in II.10 (2nd edition) is an alternative for fields of characteristic 2: see MR0310083 (46 #9186) Dieudonne, Jean A., La geometrie des groupes classiques.Ergebnisse der Mathematik und ihrer Grenzgebiete, Band 5. Springer-Verlag, Berlin-New York, 1971. –  Jim Humphreys Mar 25 '10 at 23:09
    
From looking at the table of contents (I don't have a copy of Hahn and O'Meara nearby at the moment), it appears that Chapter 7.2 is based on the work of Bass that I referred to in my post. –  Marty Mar 26 '10 at 0:49
    
True, but the authors point out that their treatment is a little different. It is certainly long. –  Wilberd van der Kallen Mar 26 '10 at 14:54
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When working with orthogonal groups and Spin groups, in arbitrary characteristic (over an arbitrary commutative base ring, in fact), I like the article "Clifford Algebras and Spinor Norms over a Commutative Ring", by Hyman Bass.

In particular, Bass describes a short exact (in the fppf topology over a base ring $R$) sequence: $$1 \rightarrow \mu_2 \rightarrow Spin \rightarrow SO \rightarrow 1,$$ of group schemes, arising from a projective $R$-module equipped with a nonsingular $R$-valued quadratic form. Here, the group $SO$ is defined as the kernel of the "degree" homomorphism from $O$ to the locally constant sheaf $Z / 2 Z$, which generalizes the determinant appropriately to arbitrary base rings (in particular, $det = (-1)^{deg}$, when $2$ is invertible).

So I think that connectedness follows from Bass's paper, given that the well-known group $Spin$ is connected.

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The article itself is archived by JSTOR, which means it is freely available to many but not all university users: Hyman Bass, American Journal of Mathematics, Vol. 96, No. 1 (1974), pp. 156-206. The slightly earlier book by Dieudonne in French may not be easily found in libraries, but it treats these groups over fields efficiently. –  Jim Humphreys Mar 29 '10 at 17:37
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This is also described in section 23 of The Book of Involutions.

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In the Book of Involutions it is written: The associated classical algebraic group is known to be connected (Borel, 23.6)... –  Mikhail Borovoi Apr 15 '10 at 11:29
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The same fact ($O(2n) = SO(2n) \rtimes Z/2$) is Proposition 5.2.2 in Knus's "Quadratic and hermitian forms over rings", with full proof. $SO(2n)$ is connected because it is almost simple. Is this a more satisfactory reference? –  Skip Jun 2 '10 at 14:58
    
(cont'd) That should be "Proposition 5.2.2 in Chapter IV" (p.225) –  Skip Jun 2 '10 at 14:59
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