p
1/1,... , z/p of t, n and */ serve as coordinates on M(BOQ). The assumption that
UJM is G-invariant and non-vanishing forces LUM to have the form
t^1 .. .t^dti .. .dtidxi... dxpdv\ .. .di/p.
Since we are only interested in the form near the identity, we may assume \t{\ 1
and take UJM to be
dti... dtidxi ... dxpdui... dvv
Consequently, we may take the forms UJX and toy on and to be given by
dt\ . . . dtidx\ ... dxpdvi .. . dvv
and
dXdxi . . . dxpdi/i ... di/p
respectively. The form w j is not defined over F in general but this is not a
problem because there is always a constant c £ Fx such that cux is defined over
F.
3. T h e Variety 5 ° .
It was mentioned in section 2, that there is a relation among the variables
z(W,a) for every closed path W+ WQ, .. . , Wp, Wp+\ = Wo selected through
the Weyl chambers. These relations are not all independent. In fact, this section
shows that all the relations are consequences of the relations that arise for rank
two groups. This result is closely related to the fact that the Weyl group is a
Coxeter group. Chapter III studies the rank two situation carefully. Building
on lemma 3.1 and the results of Chapter III, Chapter IV will draw some general
conclusions about the vanishing of principal values on divisors.
The rank two root systems occurring at the codimension two intersections of
walls will be called nodes.
L E M MA 3.1. Every relation among the coordinates (z(W,a)) on a patch
S°(J9oo, Bo) of the variety of regular stars is a consequence of
i) z(W, a) + z(W a) = 0 where W and W are adjacent walls separated by a
wall of type a, and
ii) exp(zpX-ap) -exp(ziX_
a i
) = 1 where Wi, Wi,. . . , Wp is the path around
a node (so that p = 4, 6, 8,12 according as the node is of type A\ x A\, A2,
&2, G2) and zi,.. . , zp are the corresponding wall variables.
PROOF 1. Consider any closed path W\) . . . , Wq+i = W\. The chambers are
separated by walls
(Wlyai) = (W2,ai),... ,(Wq,aq) = (Wuag).
Reflection in these walls corresponds respectively to elements LO\) ... ,10q of the
Weyl group, and U{W% W{+\ or
UJq . . .UJlWl = W\.
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