A NNALES SCIENTIFIQUES DE L ’É.N.S.
H ENNING H AAHR A NDERSEN Cohomology of line bundles on G/B
Annales scientifiques de l’É.N.S. 4e série, tome 12, no1 (1979), p. 85-100
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4s serie, t. 12, 1979, p. 85 a 100.
COHOMOLOGY OF LINE BUNDLES ON G/B
BY HENNING HAAHR ANDERSEN
Let G be a connected algebraic group over a field of characteristic p > 0. Denote by B a Borel subgroup of G. Let ^ be a character of B and consider the induced line bundle L (^) on G/B. This paper deals with the questions:
(a) when is H^G/B, L(x))^0?
(fo) what is the structure (dimension, trace, G-composition factors) of I-T(G/B, L(^))?
An important partial answer to (a) is contained in the following Theorem due to G.Kempf[9]:
If H°(G/B, L(/))^=0 (i. e. if ^ is dominant) then H^G/B, L0c))=0 for f > 0 . (For another proof see [2].)
For non-dominant weights, however, very little is known about the vanishing behaviour of the cohomology of L (7). In fact the only group for which there has been given a complete answer to (a) is SL(3) (W. L. Griffith [5]) but even in this case (b) is wide open.
Our main results in this paper are (see Section 1 for notation):
(2.3) and (2.9). - I f 0 ^ < o ^ , x > ^ P - l or <o^, 5C>=ap"-l with a<p and n e N then H^G/B, LOc^ir^G/B, L(5jx+p)-p)).
(3.1). — If there exist positive integers a, n with a<p and two distinct simple roots a, P such that <oT, 7 > + ^ " + 1 ^ 0 a n d < ^, X > + ^ " < P ^ , a > + 1 ^ 0 then H1 (G/B, L0c))=0.
(4.5). — For non-dominant characters the condition in 3.1 is also necessary for vanishing of H1 (G/B, L(^)) when G has semi-simple rank 2.
The proofs of (2.3) and (2.9) are based on M. Demazure's simple proof of Bott's Theorem [4]. As a Corollary we in fact get that Bott's Theorem holds in characteristic p for line bundles induced by "small" characters (2.4) as well as the Steinberg characters (2.10) but we show also that there always exist line bundles having at least 2 non-vanishing cohomology groups (2.7). Using (2.9) together with a couple of Lemmas about the restriction of line bundles to codimension 1 Schubert varieties in G/B we are able to obtain the sufficient condition for vanishing of H1 (G/B, L (^)) in (3.1). We point out in (3.2) exactly what we need to show in order to get that this condition is also necessary. Via a Theorem of C. S. Seshadri [10] we then use this in Section 4 to handle the semi-simple rank 2 case. This
ANNALES SCIENTIFIQUES DE L'ECOLE NORMALE SUPERIEURE. - 0012-9593/1979/85/S 1.00
Section contains also some results about the G-module structure of some of the I-T(G/B, L(^))'s, which though very far from a complete answer to question (b) above, may indicate a little about what kind of results to expect. As C. S. Seshadri's work [10] is not generally available we have included an Appendix containing a brief outline of his proof of the above mentioned Theorem.
I would like to thank J. E. Humphreys for some very helpful and stimulating discussions on the problems treated in this paper.
1. Preliminaries
NOTATION. — G will denote a connected reductive algebraic group over a field k. We will assume k is algebraically closed and of positive characteristic p . We fix a maximal torus T c: G and a Borel subgroup B containing T. R will denote the set of roots of G with respect to T, R_ the set of roots of B and R+ = - R- . S c R + will be the set of simple roots and W the Weyl group.
When a e R we let U^ denote the corresponding unipotent subgroup of G and we fix an isomorphism G^: Ga -> U^ satisfying tQ^(z)t~1 =Qy,(a(t)z), teT, z e k .
The Schubert variety in G/B associated to an element w e W is defined to be the closure of the cell B w B/B. We denote it X^.
The character group of T will be denoted X (T). If T| e X (T) or more generally if T| is any linear representation of B on a vector space E we let L(r|) [or sometimes L(E)] denote the induced locally free sheaf on G/B, i. e. the sheaf whose Sections over an open subset U c= G/B are the regular functions (p: TC'^U) -> E satisfying the relation cp(xfc)=r| (b)~1 (p(x), xen~1 (U), foeB. Here n is the canonical morphism G -> G/B.
From [4] we recall the following crucial but simple Lemma valid in all characteristics.
LEMMA 1.1. — LetaieS and let P^ denote the minimal parabolic subgroup ofG having a as only positive root. Let T| : B -> GL(E) be a linear representation ofB and neX(T). lfr[
extends to P^ and < a^, a > = — 1 then
W (G/B, L (r|) ® L (u)) = 0 for all i.
In order to apply this Lemma we will need a detailed knowledge of some representations of Pa:
LEMMA 1.2. — Let /eX(T), a e S and suppose r = < a ^ , ^ > ^ 0 . Then V^=H°(Pa/B, L(^)) has a basis VQ, v^, . . . , Vy with the following properties:
(a) Vi is a T'-semi-invariant of weight 5a(^)+ia, f = 0 , 1, . . . , r ; (b) G-,(z)i;,= (J. \ zj~iV i , z e k , j = Q , 1, . . . , r.
Proof. — Elementary exercise (see [I], example 3.6).
Finally we shall need some criteria for vanishing of H°(Xy,, L(^)|x ):
46 SERIE - TOME 12 - 1979 - N° 1
LEMMA 1.3. — Let /„: G/B-^G/P^ denote the canonical morphism. Thenf^ is a Pl-fibration if and only if l(ws^) = l(w) — 1. [Here l(w) denotes the length of w.]
Proof. - The statement follows from the Bruhat decomposition, see [9], Lemma 2.1.
LEMMA 1.4. - If H°(X^, L(^)|xJ^O then H°(X^, L(^)|xJ contains a unique ^-stable line. The weight in question is w(^).
Proof. - The Lemma is well known for Xy, = G/B (i. e. for w = WQ, the element in W with maximal length), see [8], Theorem 8.3. The same proof applies.
COROLLARY 1.5.,- Let oceS. Then H°(X^, L(/)|x ) = 0 if either:
(a) 3 p e S - { a } : < ( T , x > < O o r (b) 3 p e S - { a } : < P ' , 5 , ( x ) > < 0 . Proof. - Note that i f a , p e S , o c ^ P then
/ (WQ Sa) = I (wo) -1 and / (WQ s^ Sp) = / (wo) - 2,
Hence by Lemma 1.3/p|x is a P^fibration. The fibers are isomorphic to Pp/B and if
< P ^ , X > < 0 we have H°(Pp/B, L(^))=0. Hence/p. L(^) = 0 and the Leray spectral sequence
HWX.J.R^LOC)) => H^(X,^LOc)),
shows then that H°(X^^, L(^))=0. We are done if 7 satisfies condition (a). Suppose now that ^ satisfies condition (b) and assume H°(X^, L(^))^0. Set y = - W o ( P ) . Then J(s^WoSa) = / ( w o 5 p 5 j = ?(wo) — 2. Hence the closure of B^B is P^-stable (under multiplication on the left) [3], Proposition 1.4. Hence Py acts on H° (X^, L (7)). As WQ s^ (^) is a T-weight here so is therefore Sy WQ s^ (7). But
Sy^osJx)=^osJx)-<Y^- ^ o 5 j x ) > Y = W o s J / ) + < P , U X ) > Y .
Now in general if B -> GL (E) is any linear representation of B and e e E is a T-semi-invariant whose corresponding weight is maximal among the T-weights of E [with respect to the order of X(T) induced by B] then the line generated by e is B-stable. But by Lemma 1.4 H° (Xy,^, L (/)) has only one B-stable line and the weight in question is WQ Sy_ (^). Therefore
^ ^O^ < P^» ^(7) > y cannot be bigger than WQ s^(%). This contradicts condition (b).
2. Line bundles induced by small characters
Let a e S, / e X (T) and suppose r = < o c ^ , 5C> ^0. From Lemma 1.2 we see that we have the following sequences of B-modules
r o-K^v^^o,
l o - f c ^ K ^ V ^ O , (2.1)
^-^eo^-V^
where k^ denotes the 1-dimensional B-representation ^: B -^ GL (fe) and where V" has a basis { v ^ , v^, . . . , Vr-i} with the properties:
(i) Vi is a T-semi-invariant of weight Sa0c)+fa;
(ii) Q-^z)Vj=^(j}^-iv„zekJ=l,2, . . . , r - l . _ i = i V /
Let H^: V^ -^ V^ denote the map that takes Vi into n^-i. It is easy to see that this is a B-equivariant map. The kernel K^ of H^ has a basis { ( ? i , ^ ' " , ^ a ] ,
a=msix[n\np < [ ^ , 5C> } with the properties:
(i) €i is a T-semi-invariant of weight Sa(^)+^a;
(ii) Q^e^^^z^^e^zek.
i=l\1/
Similarly the cokernel Q^ of H^ has a basis { e ^ , e^, . . . , ^ } with the properties:
(i) ^ is a T-semi-invariant of weight 5a(/)+pfa.
(ii) e-Jz)^,= t f ^ ^ z ^ - ^ z e ^
We derive the following properties of these B-representations when <oc^, /) is "small":
If 0 < < oT, 7 > ^p then V^V,_, (via H,).
If ap < < oc , 5C) ^ (a+ l)p for some a<p then
(2-2) K^Q^^^DJ^®^^/,^^^^,
where ©a is the fundamental weight corresponding to a.
[When E is a B-representation we denote by E^ the same representation raised to the p'th power.]
Let now p=half the sum of the positive roots (= ^ coj. Combining (2.1), (2.2) and
aeS
Lemma 1.1 we get.
THEOREM 2.3. — Let aeS:
(i) y < o ^ X + P > ^ 0 then
H^G/B, LOc^H^^G/B, L(K^p) ® L(-p));
(ii) ifQ^ <a^, x + P > ^P ^^
H^G/B, L^^H^^G/B, L(5,(x+P)-P));
(iii) (/' ap < < a , ^ + p > ^ (a+l)j?/or som^ l ^ a < j ? ^n w^ have two long exact sequences:
(a) . . . ^Hi+l(G/B,L(s^+p)-p))^Hi(G/B,L(^
^^(G/B.LO^^L^p))-. . . .
4s SERIE - TOME 12 - 1979 - N° 1
(b) ...-H^^G/B^ftV^^)^))
® L ( 5 J x + P - P ( a + ^ - l ) c O a ) ) - P ) ) ^ Hl + l( G / B , L ( V ^ p )
®L(-p))^Hl(G/B,L((V^_l)J( p ))®L(5Jx+P)-P(a+^-l)coJ-P))-^ • • • COROLLARY2.4. - (i) if < o^, x + P > ^0 and H^1 (G/B, L (s»0c + P) - P + n a)) =0 for n=0, 1, . . . , <o^, / > r^nH^G/B, L(/))=0.
(ii) ifweW and 0^ <a^, 7 + p ) ^p for a H a e R + n w "1] ^ - ^n H^G/B, LCc^H^ (M;) (G/B, L(w(x+p)-p));
(iii) let Ao denote the interior of the bottom alcove in the dominant chamber. Ifw e W and
^C+pew'^Ao) t^n
0 /or i^l(w), Hl( G / B , L ( x ) ) = <
H° (G/B, L (w (x + p) - P)) for i = I {w\
Proof. - (i) follows from Theorem 2.3 (i) by considering the exact sequences arising when one takes a B-filtration of K^+p with 1-dimensional factors, (ii) and (iii) follow by repeated use of Theorem 2.3 (ii).
Let us illustrate the results in Corollary 2.4 by an.
Example 2.5. - Let G be of type B^. Then X (T) has rank 2. In the Figure below we have named the Weyl chambers in X(T) H°, H1, H2, H3 and H4 in such a way that if the characteristic is zero then ^eH1 if and only if H^G/B, L(^))=0 for j^i, i=0, 1, . . . , 4.
By 2.4 (i) it is easy to see that if ^eH1 then H3 (G/B, L(^))=0 (compare observation below) and by Serre duality we therefore get that H1 (G/B, L(/))=0 for ^eH3. On the
ANNALES SCIENTIFIQUES DE L'ECOLE NORMALE SUPERIEURE
Figure we have also indicated the bottom alcoves. By 2.4 (iii) the vanishing behaviour for line bundles induced by characters from these alcoves is as in characteristic 0.
Set N = dim G/B. It is easy to see that the canonical sheaf on G/B is L (- 2 p) and hence H^(G/B,L(^)^H°(G/B,L(-^-2p)). As H°(G/B, L0c))^0 if and only i f ^ is dominant (i.e. <oT, 7 > ^ 0 for alloceS) we find H^G/B, L(^))^0 if and only if
< a ^ X > ^ - 2 f o r a l l a e S .
Suppose now x^X(T) satisfies p ^ < o T , 5 c > ^ 2 p - 2 f o r some a e S and suppose there exists (3eS-{a} such that < P^, ^ O c + p ) - p > > -2. By the above observation H1^ (G/B, L (s^ (x + P) - p)) = 0 and from Theorem 2.3 (iii) we get
H^^G/B, LOc^H^G/B, L(V^)(x)L(-p)) and
. . . ^ HN( G / B , L ( ^ ( x + p ) - p + p o c ) ) - ^ HN( G / B , L ( V ^ p )
^(-p^H^^G/B^^Oc+^-p+pa))-^.
Now <oc , s j 5 c + p ) + j ? a > = - < o T , 5 c > - l + 2 p which by assumption lies between 0 and p. Hence by Theorem 2.3 (ii) we have
H^G/B, L{s^p)-p+p^)^Hi+l{G/B, LOc-poc)) and we find by inserting this above
(2.6) t^-^G/B, LOc^H^G/B, L(x-pa)).
Iftherefore - 5C+j? a - 2 p is dominant we can conclude that H^'^G/B, L(^))^0. Butwe assumed above that s^ (^ + p) — p does not belong to the Weyl chamber which in characteristic zero contains the characters whose line bundles have H^^O. I f ^ therefore is non- singular and satisfies the above conditions then L (7) will have at least two non-vanishing cohomology groups. To be precise what we have proved is the following.
COROLLARY 2.7. — I// satisfies the conditions:
(a) there exist two distinct simple roots a, P such that p ^ < a ^ , ^ > ^ 2 p - 2 and
< l O j X + P ) - P > ^ - l ;
(b) < y ^ x - p a > ^ - 2 / o r a ^ y e S ;
then H^^G/B, L(/)) and H^^G/B, LQc)) are both ^0.
Remark!.^. — From the Dynkin diagrams of the various root systems one can easily see that the conditions in Corollary 2.7 will always be satisfied for some ^ e X (T) unless G is of type AI . In all other cases there will therefore exist line bundles on G/B with at least 2 (in case G2 at least 3) non-vanishing cohomology groups.
Let now S^ denote the unique simple submodule of V^. With notation as in Lemma 1.2 we have S^=span^ v, ( r ^0 ^ Suppose now that <oT, ^ > = a p " - l for some a<p,
46 SERIE - TOME 12 - 1979 - N° 1
n e N. Then S^+p = span { VQ, Vpn, . . . , v^n] and it is easy to see that we have exact sequences O-c^S^/c^O,
0 —> k —> r^ —> S" —> 0" ^(x+p) ' 4+p ^X+P-P^ ' u-
From the corresponding exact sequences of locally free sheaves we get by tensoring with L ( — p) and using Lemma 1.1:
THEOREM 2.9. - Z/<oT, 7 > =ap"-l with a<p then
H^G/B, LOc^H^^G/B, L(5,(x+P)-P)).
COROLLARY 2.10. - Let q=pn and set ^q=(q-\)p. Ifp> <oT, p ) for alldeR+ then H,0/B,L(»(^p)-p,).{^,^^^'^,_,^
for all w e W .
Proof. - Apply Theorem 2.9.
Remark. — The ^'s are known as the Steinberg weights and H°(G/B, L(/^)) as the Steinberg modules. It is a fact that the Steinberg modules are irreducible [6] and so are therefore H^G/B, L(w(^+p)-p)) under the assumption of Corollary 2.10.
3. Vanishing of H1
THEOREM 3.1. — Let /eX(T) and suppose there exist a, R e S , a 7^?, a, n e N with a<p such that
<o^, x > + a p " + 1 ^ 0 and < P^, 5 0 ) + a < P^, a > p " + l ^0.
r^nH^G/B,!^))^.
Proof. — We first note that ifoc is any simple root then the divisor X^^ in G/B is defined by the invertible sheaf L(o)J, i.e. we have the exact sequence
O^L(-CO,)^OG/B^OX^O.
Tensoring this sequence with L(X) and looking at the associated cohomology sequence we see that the induced map H^O/B, L(X-coJ) -^ H^G/B, L(k) is injective if H°(X^^, L(X))=0. To prove the Theorem it will therefore be enough to find two non- negative integers r and s with the properties:
(i) H^G/B, L(x+rco,+scop))=0 and
(ii) H°(X^, L(^+nco,+m(Op))=H°(X^, L(x+^co,+mo)p))=0
ANNALES SCIENTIFIQUES DE L'ECOLE NORMALE SUPERIEURE
for all (n, m) with O ^ ^ r and O ^ m ^ s . We claim that
( r , . s ) = ( - a ^ - < o c ' , x > - l , - a < ?', a > p " - < P', 7 > - 1) will do the job: Choose ^i eX(T) such that
UZl+P^P^+^a+SCOp.
Pasv computations show that
^Xi)^"-! and < p ' , ^ > = - l .
By Theorem 2.9 we have H^G/B, LOc+ro^+5(Op))^H°(G/B, L(^i)) and the latter is zero by Lemma 1.1. Hence (r, s) satisfies (i). To see that (ii) is also satisfied note first that when n ^ r we have
<o^, /+^o\+mcop> =<oc^, 5C> 4-^<oT, ^ > +r= -a^-.^O.
According to Corollary 1.5 H°(X^, L(7+nco^+m(0p)) therefore vanishes. The same Corollary shows that we are done if we also show that
<P^ Ux+^+mcop)><0 for O^n^r, O^m^s.
But
^ (X + "c^ + /"cop) = / + nw^ + mcop -« o^, ^ > + n) a and so
< P ^ s j 7 + n o ^ + m c o p ) > = < P ^ 7 > + m - ( < o r , x > + ^ ) <P^, oc>
^ < P ^ X > + 5 - « a ' , x > + ^ ) < P ^ a >
= - a < P ^ , a > ^ - l + ( a pn+ l ) < p ^ , a > = - l + < p ^ , o c > < 0 . Remark 3.2. - If ^ is dominant then all the higher cohomology groups of L (^) are zero (PL [2]). It seems very likely that for non-dominant weights the condition in Theorem 3.1 is also necessary for vanishing of H1. In fact this will be true if we have non-vanishing at the edge points - (a +1) p" co^ + ^ a < - P^, a > p" cop: Let namely 7 be a non-dominant weight
P^a
and choose a eS such that < a ^ , 5 c > < 0 . L e t a < p , n e N be determined by - (a +1) p" ^ < oc^, % > ^ - ap" -1. If we assume that the condition in Theorem 3.1 is not satisfied then < p^, ^ > ^ - a < p^, a > p" for all P e S - {a}. We want to show H1 (G/B, L(^))^0. This follows by semi-continuity if 7 belongs to one of the chambers where H1 ^ 0 in characteristic zero, i.e. if there exists y e S: s^ (7 + p) — p is dominant. So assume that ^ does not belong to any of these chambers. We claim that then H°(X^, L(^ np(0p))=0 for all (n?) with -(a+l)^n^<oT, x > and
— a < P ^ , a > p " ^ n p ^ < P ^ , ^ > : P^oc. For y^oc this follows via Corollary 1.5 from the fact < a , ^ Up cop > < 0, and for y = a it follows^from the assumption that Sp (7 + p) — p is not dominant. This assumption implies namely that <(oc , S p ( ^ + p ) — p ) <0 and hence
<oT, sp(^ n^)> = <a^, sp0c+p+(^ ny(o^-5c-p)>
Y Y
= < ^ ^ p ( ^ + p ) - p > + l + < a> <, ^ ^ o ) , - 7 - p - ( n p - < a ^ , 7 > - l ) P >
< l + n , - < a ' , 5 c > - l - ( n p - < a ' , x > - l ) < ^ , P > ^ 0 .
4e SERIE - TOME 12 - 1979 - N° 1
As in the proof of Theorem 3.1 we conclude that
H^G/B, L(^ np (Dp - co,))->H1 (G/B, L(^ np(0p)) P P is injective for all yeS. Repeated use of this shows that
Hl( G / B , L ( - ( a + l ) pn( o , - ^ a < P \ a > pnG ) p ) ) is injected into H1 (G/B, LQc)).
4. Semi-simple rank 2
We first recall a Theorem due to Seshadri (valid without any assumption on the semi- simple rank).
THEOREM 4.1 (C. S. Seshadri [10]). - There exists a locally free sheaf M on G/B with the properties:
(a) for any vector bundle V on G/B there is a long exact sequence F
. . . -^ H^G/B, V) -> H^G/B, V^) -> PTfG/B, V®M) -^ . . . (b) M^^I^Q) w^r^ Q 15 a ^-representation whose set of ^-weights is
{- ^ n^\0^n^p-l}-{0}.
a e R +
(For construction and further details about M see the Appendix^) We shall need the following easy consequence of Seshadri's Theorem.
COROLLARY 4.2. - Let%eX(T):
(a) the Frobenius H1 (G/B. L(x)) -^ H1 (G/B, L(p^)) is injective if H°(G/B,L(Q)(g)L(px))=0;
(b) H°(G/B, L(Q)®L(^5c))=0 if the weights p^- ^ n<,a, O^n^p-1 aJJ are non-
a e R +
^omfnant.
F
Proq/: - By Theorem 4.1 (a) H^G/B, LQc)) -^ H^G/B, L(^5c)) is injective if H°(G/B, M®Lte))=0. Applying Theorem 4.1(a)with V=M(x)L(x) we get (a) while (b) follows by taking a full filtration of Q.
Remark 4.3. - With notation as in Remark 3.2 observe that i f a e N then 5j-(fl+l)o^-^ ^ P ' . o O c o p + r t - p
(3^bc
is dominant, i.e. -(fl+l)c0a-^ f l < PV, a>cop belongs to the H^chamber. By
(3^a
semi-continuity H^G/B, L(-(^+l)o^- ^ a < P \ a>cop))^0. One way of
P^a
showing non-vanishing at the edge points mentioned in Remark 3.2 would therefore be to show that the Frobenius
•^
H ^ G / B . L ^ a + l ) ^ - ^ aO^oQfflp))
P?'a
^(G/B.M-^+l);/'®,-^ f l < P ' , a > p " ( O p ) )
P^a
is injective. By Corollary 4.2 a sufficient condition for this is:
(4.4)
^ ^ a - ^ + l ) ^ ^ - ^ a < P ^ , a > p " c o p
a e R + p ^ a
is non-dominant for all n^, a, n with
O^n^p-1, 0^a<j? and n^O.
Unfortunately (4.4) does not hold for all groups and so this method is not fine enough for proving that the condition stated in Theorem 3.1 is both necessary and sufficient for vanishing of H1. However we shall now see that it does work for groups of semi-simple rank 2:
THEOREM 4 . 5 . — (i) Let G be of type A 2 and denote by a and |3 the two simple roots. If 7eX(T) 15 non-dominant and does not belong to any of the ^-chambers then H^G/B, L(/))^0 if and only if there exist a, neN with a<p such that either
< oT, 5c > ^ -(a+ l)p" and < ^, ^ > ^" or < (T, ^ > ^ -(a+ 1)^ an^ < oT, x > ^n; (ii) Let G be of type B2 and denote by a and (3 ^ ^wo sfmp^ roots with < oT, P the two simple roots with < a , (3 > = - 2. J/^ e X (T) is non-dominant and does not belong to any of the ^-chambers then H^G/B, \.{^))^if and only if there exist a, ne^with a <p such that
H -chamber
H -chamber ^^chamber^^A^^
/\ / \ /\ 7\ / \ y \
AAAAA^ AAAA,
T¥¥¥¥VpV¥¥TT¥¥
H ^chamber \ / \ / \ / \ / H2 chamber \/\/\
yyyvyvvv\A/\/\A/
A/\/\/\/H^,niVV\/\/\A/^
Fig. 1. - Type A;,.
4e SERIE - TOME 12 - 1979 - N° 1
either <oT, ^ > ^ -(a+l)p" ^nd < p ^ , ^ > ^ f l p " or < P^, ^>^-(a+l)p" an^
<cx',x>^2aj/1;
(hi) Z^r G be of type G^ and denote by a and P r^ 2 simple roots with < a^, P > = — 3. J/
7eX(T) f5 non-dominant and does not belong to any of the ^-chambers then H^G/B, L(^))^0 if and only if there exist a, n e N with a<p such that either
<^,^^-(a^l)pnand(^^)^apnor(^,^^-(a+l)pnand(^,^^3apn. Remark 4.6. — Via Serre-duality we obtain corresponding necessary and sufficient conditions for vanishing of H^^G/B, L(^)), N = dim G/B. The exact formulation of these conditions is left to the reader. On Figure 1-3 below we have illustrated the results of 4.5 by shading the alcoves with the property that if 7 belongs to the interior of such an alcove then H^G/B, L0c))^0 [resp. H^^G/B, L(/))^0] in positive characteristic but =0 in characteristic zero.
Fig. 2. - Type B^.
[On this figure we have also indicated the expected vanishing behaviour of H2.]
Proof'of Theorem 4.5. — Theorem 3.1 shows that the conditions are necessary and by Remark 4.3 we are done if we show that (4.4) holds in each of the 3 cases. For type A^ this was already done by Seshadri in [10]. The method is the same in the other cases. Let us treat the first of the two type G2-cases and leave the others to the reader: For type G2 we have R + = { o c , P, oc+P, 2 o c + P , 3oc+P, 3 o c + 2 p } and some easy computations show that nia+n2P+^3(a+P)+^4(2a+P)+^5(3a+P)+n6(3a+2p)
= (2 HI - 3 U2 - ^3 + ^4 + 3 Us) CO^ + ( - HI + 2 U2 + ^3 - ^5 + ^>) 0)p •
Fig. 3. - Type €2.
What we have to check is therefore that the inequalities
— ( 2 n i — 3 ^ 2 — ^3+^4 + 3 ^ 5 ) — (a -(-l)jm^O,
— ( — HI + 2 ^2 + ^3 — ^5 + ^3) + a?" ^ 0.
cannot both be satisfied if O ^ n ^ p — 1 , i = l , 2 , . . . , 6 . Assume that they are satisfied. From the second we get 2 n 2 +n3 ^nl +n5— n6 +f lJ?" an(! by inserting in the first we conclude that the left hand side of this is
^—n^-\-n^—n^—ln^—nQ—pn^n^—pn<Q. We have reached a contradiction.
4° SERIE - TOME 12 - 1979 - N° 1
Remark. - The interested reader may check that the above method can be used to prove that condition (4.4) is satisfied also for all groups of semi-simple rank 3 but not e. g. for groups of type A^ if n ^ 6 !
We will now take a closer look at some of the H^G/B, L(^))'s which "are zero in characteristic 0 but non-zero in characteristic p'\
Let G be of type A^. Suppose ^eX(T) is such that ap^, <oT, ^ > ^ ( a + l ) p - 2 and
— ( a + l ) j ? ^ < p , 7 > ^ — < a , 5 C > — 2 for some a <p [i. e. ^ belongs to one of the bottom j^-alcoves in H2 where H1 (G/B, L (x)) ^0, Theorem 4.5 (i)]. Then
PROPOSITION 4 . 7 . — Wi^n ^ fs as afoor^ H1 (G/B, L(^)) fs irreducible with highest weight
%+apft. (Here highest weight is with respect to B = the opposite Borel-subgroup ofB= the Borel subgroup corresponding to the set of positive roots.)
Proof. — For a = 1 this follows via Serre-duality from (2.6) and the fact that ifk belongs to the bottom alcove in the dominant chamber then H°(G/B, L(X)) is irreducible ([6], 4.1). We now use induction on a: SeU == - ^ - 2 p. Note first that 5p (k + p) - p does
not belong to the I-^-chamber. and that < P^, S p ( ^ + p — p ( P + ( a — l ) c o p ) ) — p > ^0. Hence H^G/B, Usp(?i+p)-p))=0
and
Hl(Pp/B,L(V(t^)( p )®L(5p(?l+p-p(P+(a-l)o)p))-p))=0.
[L(V^_^) is constant on Pp/B.] From the Leray spectral sequence relative to the canonical morphism G/B -> G/Pp we conclude that also
H3(G/B,L(V£_^)^®L(sp(X+p-p(P+(a-l)cop))-p))=0 and Theorem 2.3 (iii) gives therefore
H^G/B, Lpi^H^G/B, L(V^p) ® L(-p))
^H2(G/B,L(V(t^)^®L(5p(^+p-p(P+(a-l)o)p))-p)).
Let now ¥„ denote the B-submodule offV^^)^ spanned by the first (n + l)-basis vectors (see Lemma 1.2). From the exact sequences of locally free sheaves associated to the sequences of B-modules
(
4-
8) o-^_^v,-.^_,^^p-o,
we get for n = a — 1 using the above observations
. . . ^ H ^ G / B . - r ^ ^ H ^ G / B . L ^ - ^ H ^ G / B ^ L C s p a + r t + a p p - p ) )
^ H ^ G / B . '/ ,_;)-^0, where we for convenience have set
^-2=L(V^)®L(5p(?i+p-p(p+(a-l)(Dp))-p).
N
For any neX(T) the alternating sum ^ (-l^TrH1 (G/B, L(u)) is independent of the
1 = 0
characteristic. Using this fact and observing that the induction hypothesis tells us that the highest weight of H1 (G/B, L (s? (X + p) + ap |3 - p)) is 5p (k + p) + ap P - p + {a -1) p a we find that the highest weight of H^G/B, L (sp (?i + p) + ap P - p)) is the same as it would be in characteristic zero, namely equal to SpS^ ( s p ( X - + p ) + a p P ) — p = W o ( ^ — a p p ) — 2 p = — w o Oc + ap P). We claim now:
(i) - ^o (X + ^ P) is not a weight of H3 (G/B, (t\_^);
(ii) dim H2 (G/B, (^.^-dimH^G/B, ^)
+ dim H2 (G/B, L(5p(?i+p)+flpP-p))=dimS(-u;o(x+app)).
[Here S (— WQ (^ + ap ?)) denotes the irreducible G-module with highest weight -WoOc+^P).]
Let us first see that this claim implies the proposition: The long exact sequence above together with (i) shows that - WQ (/ + ap P) is a weight ofH2 (G/B, L (?i)) and together with (ii) that dim H2 (G/B, L (?i)) ^ dim S (- u;o (x + ap P)). Hence
H2 (G/B, L(X))^S (-Wo(x+^P)).
By Serre-duality H^G/B, L(X)) is isomorphic to the dual of H^L^)) and hence H ^ L ^ ^ - W o a + ^ P ^ S a + ^ P ) .
To prove the claim we will use the filtration o^a-i given by (4.8): (i) follows easily just by looking at the weights that occur. To prove (ii) we first note that the weights Sp (X + p) — p + kp P, k = 1, 2, . . . , a — 1 all lie outside the region where H1 (and H°) is non- zero. Hence the filtration of i^^ given by (4.8) shows that H1 (G/B, (^.2)) = 0, and we conclude that dimH2 (G/B, ^.2) -dim H3 (G/B, ^.2)= the Euler characteristic of
^_2=the sum of the Euler characteristics of L ( 5 p ( ^ + p ) — p + ^ c p p ) , k=l, 2, . . . , a— 1. These we compute by WeyPs character formula: The Euler characteristic of L(a) is
n <oT,^+p>/<or, p>.
aeR+
By induction hypothesis dim H2 (G/B, L (5p (X + p) + ap P - p)) = Euler characteristic of L(sp(?i+p)+aj?P-p)-dimS(sp(?i+p)+(2j?P-p+(^-l)pa).
The dimension of S (u) can be computed by Steinberg's twisted tensor product Theorem ([6],2.1). Explicity we get (setting r = < o ^ , ? i > and s^P^)) that the Euler characteristic of L (.s'p (X, + P ) - p + kp P):
. ( r + s + 2 - ^ ) ( - s - l + 2 ^ ) ( r + l + ^ ) for k=\,l, . . . , a - l , equals z _(r+s-\-2-ap)(-s-l+2ap)(r-\-l-\-ap)
+ . a ( a - l ) ( r + s + 2 ) ( - s - l + ( ^ + l ) p ) ( r + l + ( a + l ) p ) for k=a.
46 SERIE - TOME 12 - 1979 - N° 1
The equality we have to check then reads 1 a
- ^ ( r + s + 2 - ^ ) ( - 5 - l + 2 ^ ) ( r + l + ^ p )
2 k:= 1
+la ( a - l ) ( r + s + 2 ) ( - s - l + ( a + l ) p ) ( r + l + ( a + l ) p )
^ ( f l + l K - r - l - a p K - s - l + ^ + l ^ M - r - s - ^ + p ) . 4
This equality can be checked e. g. by comparing the coefficients to r2, r1 and r° on the two sides. We leave the details to the reader.
Remark 4.9. — In [7] J. E. Humphreys points out that there seems to be a correlation between the non-vanishing of the cohomology groups of certain L (^) 's and the composition behaviour of Weyl modules. Theorem 4.5 and Proposition 4.7 support the existence of such a correlation.
APPENDIX
In this Appendix we briefly outline C. S. SeshadrFs proof of Theorem 4.1.
Let X be a smooth variety over k with functionfield k (X). By k (X)1^ we denote the field extension of k(X) obtained by adjoining all p'th roots and we l e t / i X - ^ X be the normalisation of X in this extension. The sheaf M is then defined as the quotient of Ox in/* Ox, i. e.:
0 -> Ox -> f^ Ox -^ M -^ 0.
It is easy to see that M is a locally free sheaf of rank p61^ — l. One then checks that if V is a vectorbundle on X then the Frobenius homomorphism H'(X, V^-^H^X, V^) can be identified with the map occurring in the long exact cohomology sequence associated with the above short exact sequence tensored by V [note that since/is affine we have H1 (X, V^) ^ H1 (X, /* V) ^ H1 (X, /J'* V)]. In the case where X = G/B the sheaf /„ 0^ is homogeneous. The B-module by which it is induced is OQ^^/O^^^. A good description of the actual B-module structure of this representation seems to be hard to obtain (although such a description certainly would be very useful). However, the action of T is easy to find: As the big cell WQ B WQ B is an open T-stable neighbourhood of the identity in G we see that the set of weights in OQ/B^/O^/B^ consists of all weights of the form
—Y^n^cn, where the sum is over all positive roots and 0<n^<p. The statements in Theorem 4.1 follow.
REFERENCES
[1] H. H. ANDERSEN, On Schubert Varieties in G/B and Botfs Theorem {Thesis, M.I.T., 1977).
[2] H. H. ANDERSEN, Schubert Varieties in G/B (to appear).
[3] L. BAI, C. MUSILI and C. S. SESHADRI, Cohomology of Line Bundles on G/B (Ann. sclent. EC. Norm. Sup., t. 7, 1974, pp. 89-138).
[4] M. DEMAZURE, A Very Simple Proof of Botfs Theorem, (Invent. Math., Vol. 33. 1976, pp. 271-272).
[5] W. L. Jr. GRIFFITH, Cohomology of Line Bundles in Characteristic p (to appear).
[6] J. E. HUMPHREYS, Ordinary and Modular Representations ofChevalley Groups (Lect. Notes in Math., No. 528, 1976).
[7] J. E. HUMPHREYS, Weyl Modules and Botfs Theorem in Characteristic p (Seminar on Lie Theories, Queen's Papers in Pure and Applied Math., 1978).
[8] B. IVERSEN, The Geometry of Algebraic Groups (Adv. in Math., Vol. 20, 1976, pp. 57-58).
[9] G. KEMPF, Linear Systems on Homogeneous Spaces (Ann. of Math., Vol. 103, 1976, pp. 557-591).
[10] C. S. SESHADRI, Cohomology of Line Bundles on SLs/B (Notes of a lecture at I.A.S., 1976).
(Manuscrit recu Ie 2 mai 1978, revise Ie 19 octobre 1978.) H. H. ANDERSEN
School of Mathematics, Institut for Advanced Study,
Prmceton NJ 08540 U.S.A.
ANNALES SCIENTIFIQUES DE L'ECOLE NORMALE SUPERIEURE
