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J ROHLFS

'^ ( xy ) = '^x'^y for xeX, уеГ, aeq. Suppose that for every cocycle с for Н^(д,Г) the с-twisted g-action has a fixpoint in X, If XqG X is a fixpoint of the g-action on X, then we have by 7 ь-> Xo • 7 a map Г —> X and an exact sequence 1 -^ Г --» X —> XIГ -^ 1. We get an exact sequence of pointed sets

1 -^ jP-> X^-> (Х/Г)«-4 H4g, Г) -^ 1,

which describes the fixpoint set {Х/ГУ completely. This can be applied to actions of g on symmetric spaces, their compactifications and to actions on Bruhat-Tits buildings.

ii ) If Г does not act freely, the above method to construct fixpoints works as well. However there may be fixpoints which do not arise from that construction.

For every cocycle (1, b), ЬеГ, we now determine X{b).

LEMMA 1.4. The natural inclusion SO„ (R) ^^S1„(R) induces a bijection ЯЧ0, SO„(R)) ^ ЯЧа, SIJR)).

Proof . If (1, g) is a cocycle for H^(g, Sl„ (R)) we have a twisted operation given by x^-^^x - g~^ on X. This operation is isometric and thus [8: I, 13.5] has a fixpoint у eX. If XqgX is the point corresponding to SO^ (R), then у = Xo • h"^ for some h € Sl„ (R). Hence y = ''y * g~^ = Xq • h'^ = Xq • ''h'^ • g~^ and h~^ • g • "^heSOn (R). This means that i is surjective.

To prove the injectivity of i we consider two cocycles (1, ki) and (1, кг), kl, k2eSO„ (R), and suppose that there exists a geSl^CR) such that k2 = g^ * fci ' "^g- From [8: III, 7.4] we have a unique Cartan decomposition Sl„ (R) = SOn (R) • F. Here F is a closed and r-stable subset, F = F~\ and for к e SO^ (R) we have k~^ • Pk = P, Write g = k'p with к € SO„ (R) and peP. Then кг = p-i. ^^-1. fc^ . Tfc . Tp = fc-i • fe^ • ^k • p' for some р'б F. From the uniqueness of the Cartan decomposition we get кг = k"^ • ki • ^k and thus i is injective. q.e.d.

Remark , The argument just given shows that an analog of Lemma 1.4 holds more generally for any finite group g acting isometrically on a symmetric space. For some other special cases see [1: 6.8] and [19: 1.6]

We now compute H^(g, Sl„ (R)). In fact we just have to reinterpret the classical theorem of Sylvester giving the classification of real quadratic forms.

We recall the notion of a signature. If 6 e Gl^ (R) is symmetric, then there exists an Ae Gl^ (R) such that 'A- b - A = %,. Here 'x • tï, , • x = X[=iX?-Xj"=r+i ^f for xeW considered as row vector. The tuple {r,s) with r + s = n is called the signature of b. By Sylvester's theorem we have a bijection H Hg, Gl^ (R)) ^ {(r, s) G N X N/r + s = n} induced by fe н^ signature (b).