EXCEPTIONAL ALGEBRAIC GROUPS

9

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

g_1

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

7

* ( / ) | 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)) =

Yl/Jj(aCii)

for integers czj. So

for each k, we have f*{yk)(6i(a)) =

ykiUl^ji^13))

=

aCt/e-

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)) =

8{Y\jij(ar^))^

and so 7(^(0)) = nMj(

a r

'

J

)

a n

d as

above

7*(^AJ)

= 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: