Connected components of the orthogonal group O(2n) in characteristic 2.

Question

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

Answer 1

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

Answer 2

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.

Answer 3

This is also described in section 23 of The Book of Involutions.

Answer 4

An alternative presentation of Dickson's invariant is given in §4.1.2 of the article On the adjoint quotient of Chevalley groups over arbitrary base schemes, JIMJ 2010, by P.-E. Chaput and myself. There we show how the determinant $O(2n)\to\mathbb{Z}/2\mathbb{Z}$ in characteristic 0 (or say different from 2) extends to a morphism of group schemes over $\mathbb{Z}$, reducing to Dickson's invariant modulo the prime 2.