12
T. A. SPRINGER
these cocycles for the relation: (cs) ~ (c's) if and only if there is c e C such that
c's =
c~l
cs (sc), form the 1-cohomology set
Hl(k,
C). It has a privileged ele-
ment 1, coming from the constant function cs = e.
3.2. Reductive k-groups. Now let G be a connected reductive fc-group. It is said to
be quasi-split if it contains a Borel subgroup which is defined over k (this is a very
restrictive property). G is split (over /:) if it has a maximal torus which is defined
over k and /c-split. In this case G is quasi-split.
EXAMPLE.
G = 50(F) (see [2, pp. 15-16]). This is quasi-split but not split if and
only if the dimension n of the underlying vector space is even and the index equals
in - 1.
From the splitting of 2.14 one concludes that G is an inner form of a quasi-split
group.
Now let B be a Borel subgroup of G and T c B a maximal torus, both defined
over k5. Let cjjQ(G) = (X, A, Xv, Ay) be the based root datum defined by (G, B, T).
If s G rk there is gs e G(ks) such that
intfoX^O = B, int(gs)(sT)=T.
Then int(gs)o5- defines an automorphism of T depending only on s (since the coset
Tgs is uniquely determined). This automorphism determines an automorphism
fj,G(s) of X, permuting the elements of A (since int(gs) o s fixes B). It is easy to check
that juG defines a homomorphism juG: Tk - Aut (J0(G). Let G' be a £-form of G.
Then ^G = ^ if and only if G and G' are inner forms of each other.
3.3. Restriction of the base field [22, 1.3]. Let / c ks be a finite separable exten-
sion of A:. Let Gbean /-group. Then there exists a &-group H = i*//*G charac-
terized by the following property [2, 1.4]: for any /t-algebra A we have H(A) =
G(A ®k I). In particular, H(k) = G(/). Let 2 be the set of ^-isomorphisms / -* ks.
We then have H(ks) =
G(Ar5)2'.
The action of rk = Gal(£5/A:) on H(k5) is as follows.
If ^ e G(ksY is a function on 2 with values in (j(fc5), and s e rk, then, for a e 2",
(.y $(j) = 0C? a).
Rl/kG is obtained from G by restriction of the ground field from / to /:. If G is
connected or reductive then so is Rl/k G.
Now let G be connected and reductive. Fix B and T (defined over ks) as in 3.2
and let (poiG) be the based root datum defined by (G, B, T). Then H = i?//*G con-
tains the Borel subgroup Bx = J?- and the maximal torus Tx =
J2".
The based
root datum p0(Ri/kG) (relative to Bx and T{) is then
^(G)1.
The action of V^ on the
lattice Jf2, is like before: if s e rh (j e X2 then (s fi)(a) = ^(^ (7).
3.4. Anisotropic reductive groups. A connected reductive A:-group G is called
anisotropic (over /:) if it has no nontrivial /c-split /r-subtorus.
EXAMPLES,
(i) Let F be a nondegenerate quadratic form on a Avector space
(char k ^ 2). Let G = SO(F) be the special orthogonal group of F (the identity
component of the orthogonal group 0(F)). It is anisotropic over k if and only if F
does not represent 0 over k (the proof is given in [2, p. 13]).
(ii) If A: is a locally compact (nondiscrete) field then G is anisotropic over k if and
only if G(k) is compact.
(iii) If k is any field then G is anisotropic if and only if G(k) has no unipotent
elements 7* e and the group of its Arational characters Hom*(G, GL^ is trivial.
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