There is a Frobenius morphism 6 : Y Y such that 8{Vat(d)) = Vat(dp) for all
fundamental roots a, and such that 6 induces the p-power map on A p(T). Set
7 = (5_10. Then 7 : X —• Y is a well-defined homomorphism of abstract groups.
Similarly, let F be the Frobenius morphism of X which induces the p-power map on
T and on all T-root groups corresponding to fundamental roots and their negatives.
Then cf) and jF agree on a generating set for X . Hence j 7 F . It remains to show
that 7 is a morphism.
Write X [jBwB, the Bruhat decomposition (where each w £ W(X), the
Weyl group, is identified with a coset representative in X). For each w we have
BwB ww~1BwB C wU~B. Thus X is covered by a finite number of translates of
the open dense set C = U~B.
We claim that it suffices to show that 7 j C is a morphism. To see this, suppose
7 J C is a morphism and fix # £ X . Then 7 j #C = A ^ ) ° {4 I C) 0 A^-i, where
A^-i : X X is left multiplication by
and X^g) : Y Y is left multiplication
by f(g). Each term in the composition is a morphism, hence so is 7 j #C. It follows
that there are principal open sets U^, •, Uhn covering X , such that 7 [ Uht is a
morphism for each i. Let / be in the coordinate ring of Y, and consider 7*(/). For
each i there is an element gi in the coordinate ring of X , and an integer n;, such
that 7*(/) I Uhi gi/h™*. As Uhnt = J/^., and since the open sets cover X , we
conclude that the ideal generated by the functions h™1 (1 i n) must be the full
coordinate ring of X . Hence there are elements / 1 , . . . , ln of the coordinate ring such
that 1 = E W - Set tf = fl Uhi. Then we have
* ( / ) | U = E 7 * ( / ) W = E^'ft-
Now [7 is a nonempty open set, so is dense in each Uht. It follows that 7*(/) = E ^ff«
on each {7^.. Hence 7*(/) = Y^hgi o n -^- This shows that 7 is a morphism. Therefore
it suffices to show that 7 j C is a morphism.
Now C = U~B = U~TU, and as a variety, E f T i / = J] U-a X n ^ X n Ua, where
the first and last products are over all positive roots and the middle product is the
direct product of 1-dimensional tori, the number of them being equal to the rank of
X . The discussion of the first paragraph shows that 7 restricted to each root group
is a morphism. It remains to show that 7 { T is a morphism, where T YIT{.
Write K[T] = K[xf,...,x±], where K[Tt] = K[xf] for each i. Similarly, let
f(T) A = n A;, with K[A] K[yf,..., yt]- For each i, let ^-, \i{ be isomorphisms
from if* to T{,Ai, respectively. Then cf(6i(a)) =
for integers czj. So
for each k, we have f*{yk)(6i(a)) =
Hence £*(yfc) = IT?'*.
But then the hypothesis d(p 0 implies that p divides Cik for all i,fc. Writing
c^- = pr^j, we have (f)(Si(a)) =
and so 7(^(0)) = nMj(
a r
a n
d as
= II^?* - Therefore 7 j T is a morphism, as required.
L e m m a 1.3 Let X be a simple connected algebraic group in characteristic p 0,
let q be a power of p, and let X be a dominant weight for X. Suppose that there is
a rational indecomposable extension of the irreducible module Vx(qX) by the trivial
X-module. Then one of the following holds:
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