This is the relation of measures which appears in the Theorem. We shall see below that ZcitO)
takes the form T*° (up to isomorphism; T*e = #-fixed points in T*), and the measure used
in the integration over H is pulled back from the measure dT*e on
via the isomorphism
T j ^ T V V ^ T * * . The factor [T*e{R) : (1 + 0)(T*(R))]/[T%(R) : N(T*(R))] which relates
drH with dTe o (1 + 0) o AT-1, will be computed for each torus considered in the course of the
proof below.
B. Stable conjugacy.
Let us recall the structure of the set of (F-rational) conjugacy classes within the stable
(F-) conjugacy class of a regular element t in H. By definition, the centralizer Zu(t) of t in
H is a maximal F-torus T # . The elements t,t' of H are conjugate if there is g in H with
t' Int(g - 1 )£(= g~lig). They are stably conjugate if there is such g in H ( = H(F)) . Then
ga ga{g~l) lies in T # for every a in the Galois group T = Gal(F/F) , and g \-+ {a i— » ga}
defines an isomorphism from the set of conjugacy classes within the stable conjugacy class of
t to the pointed set D(TH/F) = k e r ^ ^ F , T ^ ) - Hl(F,H)]. In our case Hl(F,H) is trivial,
hence D(TH/F) is a group.
1. Lemma. The set of stable conjugacy classes of F-tori in H injects naturally in the image
in Hl(F,W) ofkev[Hl{F^) - i / ^ H ) ] , where N = Norm(T^,H), and W is the Weyl
group ofT*H in H . This map is an isomorphism when H is quasi-split. Note that the image is
Hl{F, W) when Hl(F, H) is trivial, and Hl(F, W) is the group of continuous homomorphisms
p : T W, when T acts trivially on W.
Proof. Indeed, the tori T and T ^ are conjugate in H, thus T = g~lT*Hg for some g in H .
For any t in T there is i* in T ^ with t = g~lt*g. For t in T, erg-1 at*erg = at = t = g~1t*g,
thus at* = g~lt*ga £ T ^ , and ga G Norm(T|
,H) . Since t (and so t*) is regular, gG
is uniquely determined modulo T ^ , namely in W. For a general t* in T ^ we then have
a(g~lt*g) g~1(ga(g~1))a(t*)(a(g)g~1)g, so that the induced action on T ^ is given by
a*(t*) = lnt{ga)(j(t*)). The cocycle p = p(T) : T - W is given by p(a) = gamodT*H.
It determines T up to stable conjugacy. Conversely, a {ga} in kev[H1(F, N ) Hl(F, H)]
determines an action cr*(t*) = Int(pcr)(cr(t*)) on T ^ . By a well-known theorem of Steinberg,
when H is quasi split over F , an F-conjugacy class in H of a regular t* contains a rational
element h_1t*h (in H), whose centralizer is an F-torus which defines ga.
In our case of H = GSp(2), the Weyl group W is the dihedral group D±, generated by
the reflections sx = (12)(34) and s2 = (23). Its other elements are 1, (12)(34)(23) = (3421)
(which takes 1 to 2, 2 to 4, 4 to 3, 3 to 1), (23)(12)(34) = (2431), (23)(3421) = (42)(31),
(3421)2 = (23)(41), (23)(23)(41) = (41). Let us list the F-tori T according to the subgroups
of W, the split torus corresponding to {1}, and conclude the following.
2. Lemma. We have that Hl(F,T) is trivial except when p(T) is the subgroup ofW of the
form ((14)(23)) or ((14)(23), (12)(34), (13)(24)), where HX(F,T) =1/2.
Proof. Recall that if T # splits over the Galois extension E of F then Hl(F, T # ) =
7 7 1 ( G a l ( F / F ) , T ^ ( F ) ) , where T*H(E) = {diag(a, 6, \/b, A/a); a, 6, A e F x } , and Gal(F/F )
Previous Page Next Page