Homomorphisms Between Verma Modules in Characteristic P

JOURNAL OF ALGEBRA 112, 58-85 (1988)

Homomorphisms between Verma Modules in Characteristic p

JAMES FRANKLIN*-'

School of Mathematics, University of New South Wales, Kensington 2033, New South Wales, Australia

Communicated by Jacques Tits

Received April 28, 1986

INTRODUCTION

Let g be a complex semisimple Lie algebra, with a Bore1 subalgebra b c g and Cartan subalgebra h c b. In classifying the finite dimensional representations of g, Cartan showed that any simple finite dimensional g-module has a generating element u, annihilated by n = [b, b], on which h acts by a linear form I E h*. Such an element is called a primitive vector (for the module). Harish-Chandra [9] considered g-modules, not necessarily finite dimensional, with a primitive vector, in particular the “universal” modules of this kind, called “Verma modules” by Dixmier [7]. These are constructed using the universal enveloping algebra % of g: the Verma module corresponding to I is

VA=%! en+ 1 @(A-l(A)) REh

-Y; is naturally a 43- (and hence g-) module. By regarding finite dimensional modules as quotients of Verma modules, Harish-Chandra [9] was able to give a uniform proof of the existence of a finite dimensional simple g-module VA (called a Weyl module) for each dominant integral 2. Verma modules were also used in Bernstein, Gel’fand and Gel’fand’s proof of Weyl’s formula for the characters of the V, [2]. Recently they have found uses in analysis as a more calculatory alternative

* I am grateful to Professor Roger Carter for constant help and encouragement with this work. + Financial support was provided by the University of Sydney’s James King of Irrawang Travelling Scholarship. 58 0021-8693/88 $3.00 Copyright 0 1988 by Academic Press, Inc. All rights of reproduction in any form reserved. VERMA MODULES IN CHARACTERISTIC p 59 to C*-algebras, via Duflo’s study of the primitive ideals of the universal enveloping algebra [20]. Composition series of Verma modules and homomorphisms between Verma modules were studied by Verma [19] and Bernstein, Gel’fand and Gel’fand [2, 33. Writing p for half the sum of the dominant roots, the Weyl group W of g relative to b has a translated “dot” action on h* defined by w . I = w(A + p) - p (w E W, A E A*). Verma showed that any non-trivial g-homomorphism of Verma modules ^yj -+ VA is injective; that the space of g-homomorphisms “yj -+ $5 is at most one dimensional; and that, if II -s, .A is a positive integral multiple of CI(a a positive root), then there is a non-trivial homomorphism Ys, 1 --t VA. Bernstein, Gel’fand and Gel’fand showed conversely that any non-trivial homomorphism VP -+ ~j, is a composition of homomorphisms of this form. More strongly, they proved that if VP is a simple composition factor of Y;., then there exist positive roots CI,, .. . . CI, such that p=ss,,. ... .sZZ.s,,.jb and s,,~,. . ..s.,.A- S .s . “‘S,, . E.is a positive integral multiple of c(;for each i = 1, .. . . Y.The n?ult~&ities of the V,, as composition factors of the %‘j.were conjectured by Kazhdan and Lusztig [ 131 and established in the two papers [ 1,4]. In characteristic p, the situation is not so well understood. Let K be an algebraically closed field of characteristic p. % has a Z-form %YZ(that is, “ZZ is a subring and aZ 0, C = “2) described by Kostant [ 143. The K-algebra aZ 0, K, called the hyperalgebra, has the same finite dimensional representations as the simply connected algebraic group G, corresponding to g (Sullivan [ 181). If 1 is an integral weight (that is, 2”E h* and I(h,) E Z for all positive roots a) then there are Z-forms in $;. and V, (that is, Z-lattices closed under multiplication by aZ). These can be tensored with K to form %=modules %& and Vi.,K. The V,,, are not, as over Cc,irreducible. Each VA,, has, however, a unique simple quotient L,,,, and all simple finite dimensional %= (or rational G,) modules occur in this way. Further, any finite dimensional @

For a positive root r and an integral weight A, there exists a non-zero homomorphism VA _ dr + ?‘j. (d E R + ) provided that (A + p)( h,) = Mp’ + d for some MEZ, eEZt, and d< p’/N, where N is almost always 1, but may be 2 or 3 for certain high roots. This result may be interpreted geometrically in terms of the afline Weyl group: a non-zero homomorphism exists between ?.J$and 6. provided p and J. are mirror images in a p’-hyperplane of h*, and the distance of reflection d is not too large-usually, not more than pe. The main purpose of this work is to extend it to derive similar results for homomorphisms between the Weyl modules in characteristic p. A sub- sequent article will derive a result for the I/,,, which is the same as that above for the Verma modules, but with the extra condition that the weight ,4.must be dominant, and also sufliciently far from certain of the walls of the dominant region.

1. PRELIMINARIES

g is a complex semi-simple Lie algebra of rank 1, h a Cartan subalgebra of g, h* its dual. @ is the set of roots, Qf the set of positive roots (relative to some base d = {E,, .. . . a,}).

n, = c g”, n.- = 2 g-” and bk=n_+h. Ita’ It@+ W is the Weyl group of @, generated by reflections s,, c1E @+. We take a Chevalley basis of g, {e,(clE@+), f,(ccc@+)), h,(i= 1, .. . . I)}, with e,Eg’, f,EC, hi= Cc,,,f,,l. Th ere is an ordering relation < on h* defined by >+< A’ if i is a sum of positive roots (equivalently, of simple roots).

For rE @+, expressed as a sum of simple roots cf= 1n+,, the height ht r of r is defined to be If=, ni. If r, r’ E Qt and r < r’ then ht r < ht r’. C!Zis the universal enveloping algebra of g, %(n + ) that of n + , %(n ~ ) of n , @(b ) of b- and 4!(h) of h. There is a tensor product decomposition of %

and a direct sum decomposition

#: @ + %(b ~ ) is the projection corresponding to this direct sum decomposition. VERMA MODULES IN CHARACTERISTIC/J 61

Since h is commutative, e(h) is isomorphic to S(h), the symmetric algebra on h. Hence 43!(h) can be regarded as the algebra of polynomial functions on h*. Then for any 1 E h *, the map AH A(A), RE h, defines an algebra homomorphism S(h) + C, “evaluation at A.” This extends to a homomorphism %(b_)=@(n-)@%(h)+@(n_). The image of us% under this map is denoted u(A) and is called “the evaluation of u at 1.” Let m= I@+l. The m-tuples rc=((7c(a))!,ED+, with each n(a)EZ+, are called “partitions”; the set of all partitions is denoted p. For n E p, the weight of rc, wt 7c,is defined to be Clea+ n(a) a, and the degree of II, deg Z, is defined to be C,, G+ z(a). The partitions of a fixed weight are thus the different ways in which that weight can be written as a sum of positive roots. We fix some ordering (a,, .. . . a,) of the positive roots. 1%has a C-basis, the Poincare-Birkhoff-Witt (PBW) basis:

{r;,H,E,,: 71,TC’EP, A E (Z’)‘} where K=f,, n(a1).. . f,, n(%l) HA=/+...@ if A = (il, .. . . i,) E,, = e,“;‘“~’ . . ,;M.

Then the weight of E,, (in the usual sense of the weight of the adjoint representation of h in a) is wt rc’, and the weight of F, is - wt 7~.The degree of F,H, E,. (in the usual sense for a polynomial expression) is deg 7t+ deg H, + deg n’. (The notation here is adapted from Jantzen [ 121 and Humphreys [IO].)

LEMMA 1.1. (a) The Z-span of a PBW basis is a ring (denoted ZPBW). (b) If (ZPBW), denotes the ubeliun subgroup of ZPBW generated by the PBW basis elements of degree ar most n, then (ZPBW), (ZPBW),. G W’BW,+n,. (c) vu E ZPBW is u sum of basis elements of the same weight, and u’ is also a sum of basis elements of the same weight, then uu’ us a sum qf basis elements of weight wt u + wt u’. Proof For this lemma only, we rename the Chevalley basis 1x1 >. . . . X,} (M=2m+I). A ny product of PBW basis elements is a monomial in the Xis, and for any monomial Xi, . . Xj5 there is a per- mutation 0 such that XjO,,,... X/,,,, is a PBW basis element. So the three results follow from the fact that for any monomial Xi, . . . Xjc and any per- mutation c, X,, . . . Xjr - X,,,, . . X,,$, is a Z-sum of monomials X,, . . .X,, 62 JAMESFRANKLIN

(t < s) of the same weight as Xi, ... X,. It is sufficient to prove this fact when 0 is a transposition, but in this case it follows from the multiplication table for a Chevalley basis: XjXj - XjXi = [Xi, Xj] is a Z-multiple of some X, of weight wt Xi + wt X,.

COROLLARY 1.2. Any monomial in the Chevalley basis elements of g is a Z-sum of PBW basis elements of the same weight and equal or lower degrees.

COROLLARY 1.3. The ring ZPBW is independent of the choice of PBW basis (that is, of the choice of ordering of the positive roots). Proof An element of another PBW basis is a product of elements of the original PBW basis, so by Lemma 1.1(a) it is in the Z-span of the original PBW basis. Similar results hold for subalgebras of g. In particular the Lie subalgebra n- has basis {f,: CIE @+ }, so %!(n_) has basis {F,: 71E p}. Clearly the Z-span of the F, is the ring ZPBW n %(n _ ). The ring ZPBW is said to be a “H-form” of the algebra a’, meaning that % = ZPBW 0, C. % has another H-form %=, the ring generated by the e;/n! and f;/n!, a~@+, nEZ+. Thus ez contains ZPBW. The interplay between these two rings is important when passing to modular representations; in particular it is significant that f; is divisible by p in ez but not in ZPBW. 9Yzhas a Z-basis (the Kostant basis):

where

e+(orl) emn) e,, = L . . . r, 7c’(cll)! n’(a,)! (Kostant [ 141, Humphreys [ 10,26.4]). Note that F,, E,, are Z-multiples off,, e,. respectively, but H, is not a Z- (or C-) multiple of h,. It is clear that &(n-), %(n+), and e(h) have Z-forms a(x),, %(n+), and Q(h), with Z-bases, respectively {fn: nep}, {e,.: dip} and {h,: A+?+)‘}. @(be) h as a E-form @(be), with Z-basis {f,ha: nip, A&Z+)‘}. VERMA MODULES IN CHARACTERISTICP 63

Then

(the tensor products being over i?.), and @(b-),=%(n-),8%(h),.

Algebras over K, an algebraically closed field of characteristic p, can then be formed by tensoring these Z-forms with K. Thus

%K has K-basis {fnhAen, @ 1: rc, rr’ E p, A E (Z+ )‘). The subalgebras @(L)~, @(n+),, !&(b-), and @(ZZ)~are defined in the obvious way. Questions about the effect of multiplication by the err, CIE @+, can often be reduced to the case where a is simple by the following lemma on the structure of %(n + )z.

LEMMA 1.4. %!(n +)z is the ring generated by the e:/n!, E E A, n b 1. This result is well known; a proof can be found in [8, Proposition 9.101.

2. VERMA MODULES

For L E h*, I, is a left ideal of % defined by

I,= 1 %e,+ 2 @(h,-I(hi)). aGO+ i=l The Verma module is the left B-module %!JZ,,denoted K.. Properties (Dixmier [7, Chap. 71). (i) The L-weight space of q: is one dimensional. If u2 # 0 is in this weight space, then Z,u, = 0. (ii) VA= S!(n _ ) u),, and @(n _ ) acts freely in VA. In particular, if the p-weight space of Y’JJis non-zero, then p < 1. (iii) Dim, Horn&VA, VLS)d 1 for all ,J,;~‘E h*, and any nonzero homomorphism VA+ Vi, is injective. (iv) A “translated” action of the Weyl group W on h* is defined by

s, .A= &(A + p) - p, AEh*,crE@+.

(Here p=iC a4G+a). Then (Dixmier [7, 7.6.13]), if (2 + p)(h,) E iz +, there exists an injective homomorphism “y^,,i + “y^. Further, all %-homomor-

481/112!1-5 64 JAMES FRANKLIN phisms between Verma modules are compositions of homomorphisms of this form. The construction of Verma modules over C can be imitated over Z by using az in place of 42. This will yield the Verma modules over Z. These can then be tensored with the field K to give Verma modules in characteristic p. The lattice of integral weights, denoted hg, consists of the elements of h* which take integer values on all roots (equivalently, on all simple roots). For ,I E h,*, the left ideal Zj.,z of 4!Yzis detined by

(Note that becausee,(h’P~‘h”) E e(h) z e,. for any rc’, i, n, the @(b ~ )= in the above definition may be replaced by 9&, so that I,,, is indeed a left ideal of 4&.)

LEMMA 2.1. Zj.,L= ZAn 4?$.

Proof: Clearly I,,, c Zj.n 9&. For the reverse inclusion,

(since

is the algebra of all polynomial functions taking integral values on hf, it contains

Z [(hlmiq(hl’), .. . . (h’-lf(h”):ijEiZ+]; by symmetry the reverse inclusion also holds). So ee has a Z-basis {fnAAezr: TC,TC’ E p, A E (Z + )‘}, where

& ,,...,i,)=(“’ -iyl)) . ..(“yq. VERMA MODULES IN CHARACTERISTIC p 65

Then an element of 9&, Cz,A,a,in,A,n, fnAAen, (in,A,n,E b) is in IA iff when n = (0, ...) 0) and A = (0, .. . . 0) then in,A,ns= 0. But if rr’ # (0, .. . . 0) then

while if A # (0, .. . . 0) then

fJAE ;=l1 ,.,,,, @(b-J z (hi-;(hi)) = hz.

So an element in both az and I, is in IA,+, as required. For 1 E h,*, the Verma module over Z, denoted Vi,=, is defined to be the left +&module G&/Z,,, . V& can be regarded as a i2-submodule of Vj.‘,,by identifying u + I, z (u E 4$) with u + I, E VA. (This is an embedding, since if u+Z;,=u’+Zj.,\;ithu,u’E~~,thenu-u’EZ~n~~=Zi,Z.Henceu+Zj,,= U’ + Z,,,). Hence %(n-)L acts freely in Vj.,z, and if uA= l’+ II.,, then uj. is in the A weight space of Y!&, and +& = @(n _ )z Uj, (because 3,~ c ?ZUAn Bi(n ~ ) uA = (az n %!(n_ )) vi. since %!(n_ ) acts freely =@(n-), VA).

These modules can be tensored with K. For 1 E hi, Zj.,Kis defined to be the left ideal Z, H@ K of aK. Then the Verma module ouer K, denoted V&, is defined to be the left 4!YKmodule42KIZi,K.

LEMMA 2.2. 9’jK = Y& 0 K as %=module. (Since 4& acts on Y&, C&K=%z@K acts on Y&OK.) Proof There is a surjective az-module homomorphism 4& --* “y;:z given by UHUU~, UE%$. Hence there is a surjective homomorphism of %=modules %K+ Y&@ K. Clearly Zn,Kis in the kernel of this map, so there is a surjective %rhomomorphism VAK = 4YK/ZA,K+ VAz 0 K. It remains to show that the map is injective. Now ’ a sum of free Z-modules since “y^,== %(n _ )H uAand @(n -)z acts freely in Y&. so as K-space, 66 JAMFSFRANKLIN

and so +C,K= %A,K g vA,z 0 K as K-space. Hence the %rsurjection Y’& -+ Y& 0 K is an isomorphism, as required. Remark. Zi,K = Z,,, Q K, so it is generated as erideal by the ez/n! @ 1 (cr~@+,nal) and the (“~-~@J)@i (i=l,...,l,n>l). Hence if a 4?Kmodule is generated by an element which is annihilated by the e;/n! @ 1 and the (hl-L(hJ) @ 1, then it is a homomorphic image of “I’& (and Y’& is universal wih this property).

3. CONSTRUCTION OF S,,,

From the properties of Verma modules over C (Section 2), it follows that for any r E @+, d> 0 and 1 E h* such that (A + p)(h,) = d there is a one dimensional space of homomorphisms ^y^s,A + ^y^. Since @(n- ) acts freely on any Verma module, for each of these homomorphisms there is a unique element u of %(n-) such that the homomorphism is given by u’u,,.~ H U’UU~(u’ E %). The condition that uuA should be annihilated by the generators of ZAPdr, namely the e, and the hi- (A - dr)(h,), determines u up to a scalar. S,,(A) is then defined to be the unique u E %(n_) satisfying this condition and also the condition (3) below, which determines the scalar. S,,(A) thus satisfies (1) e,S,,(A)v,=O for all ME@+. (2) (hi - (A - dr)(hi)) S,,(A) ui. = 0 for i = 1, .. . . 1. Condition (2) means that S,,(A) is of weight -dr, and so can be written as s,,(n)=c FIAT(~)some c,(A) E C x wtn=dr The condition that determines S,,,(A) uniquely is then (3) The term F,c,(l) for which F, has highest degree is

where r = c nici. i Note that it is not yet clear that any element satisfying (1) and (2) has a non-zero term in ni f ;;; this will become clear when S,J(n) is constructed below. Note also that the statement (3) is independent of the order in which the roots are taken, since if the f ;; are commuted to a different order, the commutators are terms of lower degree (as in Lemma 1.1). VERMA MODULES IN CHARACTERISTIC p 67

S,,(n) is also characterisable as the least element ZJ of the lattice %(n_) n ZPBW such that u’u,,.~ H U’UV~ (u’ E 9) is a non-zero homomorphism Ys,.I -+ VA. (It is not yet clear that Sr,d(3L)is even in ZPBW; this also will appear in the construction below.) We will construct the element S,,(n) satisfying the conditions (1) to (3). To do so, we use the fact (Proposition 4.2, below) that S,,(n) depends polynomially on I (for fixed r, d) to define an element of 9(k) whose evaluation at all relevant 2 (that is, all 1 such that (A + p)(h,) = d) is S,,(n). This element, when evaluated at J’s other than those such that (,X+p)(h,) =d, will give the elements needed to define homomorphisms between Verma modules over K. Since the Iti’s such that (A + p)(h,) = d lie in a hyperplane of h*, the condition that an element of @(b_) should evaluate to S,,(n) for all 1 such that (A+ p)(h,) = d does not determine the element completely. There are different ways of resolving this ambiguity; the one chosen here is to define an element S,,, in &(b- ), depending on r, d and also on a simple root v], which satisfies (0’) S,,, = C, F,c, with c, E C[h,, .. . . h,, .. . . h,]. (1’) e,Sr,d,qE CgE o+ @e, + e’(b - )(hr + PVC) - 4 for BE@+. (2’) (hi - (A - dr)(M) S,,, E 1, for i = 1, .. . . 1, for all 1 E h*. (3’) The term F,c, of (0’) for which F, has highest degree is ni f;;, Notes. (a) Although the definition of S,,,, is meaningful for any simple root q, the element will be constructed only for certain q, to be described below. (b) It is the omission of h, in (0’) that causes S,,, to be determined by its values on the hyperplane (J. + p)(h,) = d (in fact we must have q < Y, so that r =C, n,E with n, #O, whence h,=C, m,c with mp #O and so @Chi, .. . . &, .. . . h,] restricted to the hyperplane is the algebra of polynomial functions on the hyperplane). (c) From (2’) the sum in (0’) is in fact over those rc such that wt rr = dr. When an S,,, satisfying the conditions (O’k(3’) is evaluated at a 1 E h* such that (A+p)(h,) =d, the result satisfies the conditions (l)-(3) for S,,(A). For, (3) follows directly from (3’), (2) follows from (2’) because u and u(1) have the same weight, for any u E @(b ~ ) and 1 E h*, while (1) follows from (1’) since, if (1’) holds, &Lsr,d,,(~))(~) = (P(eSlsr,d,,)w)> from Lemma 3.1 below = UP, + p(k) -d)(l), some uE%(&), from (1’) = 0 since (A + p)(h,) -d = 0. 68 JAMESFRANKLIN Thatis, e,S,,,(l)E Zj.9 as required for (1). Hence S,,,(L) is equal to S,,(n), for all q for which such an element S,,, exists.

LEMMA 3.1. Zf UE@(&), no@, ,l~h*, then #(m(A))(A)= @(nu))(n). Proof: It is sufficient to prove the result when u = u’c (u’ E @(n- ), c E e(h)) and n = ecr (CIE @‘), since fi, left multiplication by e, and evaluation at A are linear, and the algebra 42(n+) is generated by the e,, and fi(esebu) = fi(eBp(e,u)) for all u E ?Z!(b-). Decomposing erxU’according to the direct sum 42= CBS@+42ea @ @(b _ ), e, u’ = de, + fi(e, u’). so

g;(wW) = fi( u’e,c + fi(e,u’) c)(A) = (fi(e,u’) cHJ.1 since #( u’e, c) = 0 = #(e,u’)(;l) c(A) = fi(e,u’c(A))(i), proving the lemma. (d) The conditions (O’)-(3’) for S,,,, do not include, as would be natural, any integrality condition. Unfortunately, for a few roots r (to be called “exceptional”) it is impossible to find an S,,, in ZPBW (or even in 4!&) which evaluates to S,,(n). The non-exceptional roots will therefore be dealt with first, and the modifications necessary to deal with the exceptional roots will be explained later (Section 6). Exceptional and non-exceptional roots are now distinguished.

DEFINITION. A root r is exceptional if it is the highest root or the highest short root of G,, E, or F4. (There are thus live exceptional roots-in E, the highest root and the highest short root are the same.) The significant difference between the exceptional and the non-exceptional roots appears in the next two lemmas.

LEMMA 3.2. Zfr=CEEd n,E is not exceptional, then there is a simple root v] such that n, = 1 and q has the same length as r.

(We will say q is a “simple root of the same length as r occurring once in r”. ) Proof: If some n, = 0, then r is in the root system generated by the sim- VERMA MODULES IN CHARACTERISTIC p 69 ple roots E’ such that nEz# 0. This root system has rank less than 1, so by induction we may assume the result holds (it clearly holds for rank one). So we may assume that all n, > 0. If r is long, r < p, where p is the highest root; if r is short, r < CI,where o! is the highest short root (Humphreys [ 10, Lemma 10.4A, Exercise 10.111). The highest roots /I are (Humphreys [ 10, p. 66, Table 21) A, E, + ... +E, B, E,+~E~+~E~+ ... +2~, (cl long) c, 2&,+2&*+ ... +2&,.-l+&, (E, long)

D, E,+2E2+ ... +2&,-z + E/p, + E/

E, ~,+2E,+2&,+3~,+2E,+c,

E, 2E, + 2E2+ 3E3 +4&z, + 3E5 + 2~~ + E,.

In each case there is at least one long simple root q such that m, = 1 in P=CEEd m,E. The condition r < B implies n, < m, for all E E A, and since by assumption n, > 1, we have n, = 1. In B, and C, the highest short roots t( are respectively Ed + c2 + . . + E, and &i + 2~~ + ... + Lb-, + I+ (E, short) so in each case there is a short simple root q such that m, = 1 in CI= x,, d rn,:E. Again this implies n, = 1 for a short root r < c(. In E, and F4 there is a unique submaximal long root (Tables III and IV in Springer [ 16, p. 1401) and the same argument applies to all long roots except the highest. If Fz, the short rOOtS art-. E3, Ed, E2 + E3, E3 + Ez,, E, + E2 + E3, E2 + E3 + Ez,, E2 + 2E3 + E4, El+E2+Ej+Eq, El +E2+2E3+Eq, E,+2E2+2E,+Eq, E1+2E2+3~3+Eq, E,+2E2+3E3+2E4. All of these except the last have nE,= 1 or ns4= 1. In G2 the short root a + b has n, = 1 and the long root 3a + b has nh = 1. This completes the proof.

LEMMA 3.3. rfr =CEEd n,E and ‘1 is a simple root of the same length as r such that n, = 1, then there exists a sequence of reflections s,, .. . . s,_ , (with eachsi=s,,forsome~iEA)suchthatr>s,r>s,s,r> ...>s,~,...s~Y=~.

(Note that in such a sequence none of the simple roots E, are equal to 9, since n, = 1.)

ProoJ: r=rj+CEZv w, so r(b) = u(k) + EeZn w(h,). r(b) = 2, and, since q and r are of the same length, q(h,) < 1, so that CE+q n,&(h,) 2 1. So El(h,)>O for some E~#I]. Therefore r>s,r and s,r=q+CEZvn:& (where n:, = n,, - r(h,, ) and n: = n, for E # a,). Then s, r and q satisfy the conditions 70 JAMES FRANKLIN of the lemma, so by induction on the height of r we may assume there is a sequence of reflections s2, .. . . s, _ i such that

so r>s1r>s2s,r> ... >s,-,...s,r=q, as required. The importance of these two lemmas is that for non-exceptional roots we can perform inductions on the height of r which “avoid” q. Attention is now restricted to the non-exceptional roots; this restriction holds until Section 6. For these roots, the elements S,,, satisfying (Of)-(3’) above will now be constructed (for those 7 of the same length as r occurring once in r). S,,, will also be characterisable as the least element of the lattice ZPBW n %(a_) 0 C[h,, .. . . h,, .. . . h,] such that UU,~.~H US,,,,(I) ul, u E 42, is a @-module homomorphism <,.), -+ Vj:,, for all ,l such that (2 + p)(h,) = d. (This characterisation does not hold for the exceptional roots.) It will in fact be shown that (for non-exceptional roots) S,,, satisfies integral versions of the conditions (O’t(3’):

(0”) S,,, = 1 F,cn with c, E Z[h,, .. . . I;,,, .. . . h,]. TI wtrr=dr

(3”) The term F,c, of (0”) for which F, has highest degree is ni f;;. These obviously imply the conditions (O’k(3’); the stronger conditions involving integrality are needed because tensoring with the field K of characteristic p (Section 5) is possible only when applied to the elements of some lattice. To construct S,,, , for any positive root r, any positive integer d and any simple root rl of the same length as r occurring once in r, we proceed by induction on the height of r. A root of height one is a simple root E, and the only possible choice for q is v = E. We take SE,dE= ff for all EE d, d > 0. It is necessary to show this satisfies (0”)-(3”): VERMA MODULES IN CHARACTERISTIC /I 71

Condition (0”) is clear: When F,=ff, c, = 1, otherwise c, ==O. (1”) By Lemma 1.4, it is suflicent to consider only the case when o! is a simple root q, Then

e;-’ (n-j)!

where k = min(n, d), if q = e

(Kostant [ 141; Humphreys [ 10, Lemma 26.21). So if TV# E or if n > d, then

$f:to$ % f$Ji.z E + m>l while if da n

+- ~~~(h.+n-d)‘..(12,il-d) - .

c+ %(b-)z (h, + p(hJ - d) Ellf@+ c s,, . VI>1 as required. Condition (2”) is true since

EIAZ. Condition (3”) is clear. ’ This completes the tirst step in the inductive definition of Sr,d,,,(and of Lu)). TO define S,& for a (non-exceptional) root r of height greater than one, and a simple root 7 of the same length as r occurring once in r, we use a sequence of reflections which “reflects r down to $’ (Lemma 3.3 above):

Temporarily writing I’ for sir, and s, for s,, we have s,, = s,s,s; ‘, giving the folIowing commutative diagram of reflections: 72 JAMES FRANKLIN

h* h*

Hence we have the following diagram of inclusions of Verma modules:

provided that p = s, . il < 2, 2’ = s, .2 < I, s, . p < ,LLand s,, . ,I’ < I.‘. Since dim, Hom,(KP.,, %‘J = 1, any two such composite inclusions Y”s,.~ + Yj, are scalar multiples of each other. Take 1 E h* such that (2 + p)(h,) = d and (2 +p)(h,) > dr(h,) (the set of such I’s is Zariski dense in the hyperplane (2 +p)(h,)= d). Write q= (l+p)(h,). Then the inclusions of Verma modules are as pictured above, since s;tl=A-(l+p)(h,)r=i-dr

since q > dr(h,), which is positive because s,r < r,

S, .p = s, . (A - dr) = A- dr - (A - dr + p)(h,) = 1’ - dr’

since q > Wh,). As was proved above, f,“vl is in the highest weight space of VA,, since f; satisfies the conditions (1 )-( 3). Similarly, f: ~ dr~h~)vPis in the highest weight space of KR.P. The induction hypothesis is that there exists an S,,,,, satisfying (0”-(3”), and therefore an S,,,,(,I’) = S,,,,,(I.‘) satisfying (l)-(3). Hence S,.,,,(ll’) vi., is in the highest weight space of KE.P. The condition that SE %(n_ ) should be such that Sv, is in the highest weight space of a VU in YL determines S VERMA MODULES IN CHARACTERISTICP 73

up to a scalar. Further, fz- dr(hc)Su,is in the highest weight space of eC .p, so that f”- dr(h&)Sis a scalar multiple of S,,,,,(A.‘) f;. We choose the scalar in S by sitting the two expressions equal: fi - d+,)S = S,,,,,( A’) f8. It is now clear that S satisfies the conditions for S,,,(A): (1) and (2) follow from the fact that Su, is in the highest weight space of “y,, while (3) follows from the last equation, since the term of highest degree (in thef,‘s) on the right is f~l . . . f~‘“’ - r(h,)+ Y). . .f? and hence the term of highest degree in sisf:l...f?...f:, which is the term of highest degree of S,,(A) of condition (3). (Lemma 1.1 has been used to commute the factors in the terms of highest degree; the commutators produced by doing so are of lower degree.) So the last equation may be rewritten as f;-"'hr's,,d(n) = S,.,&&') f$

The equation is true for all 1, such that (A+p)(h,) =d and q = (A+p)(h,)> dr(h,). This equation is essentially that in Shapovalov [15, p. 3101 and will be called Shapovalov’s equation. To complete the inductive step, it is necessary to show that there exists an S,,, in %(b _ ) which evaluates to the elements S,,(A) for all relevant I (namely, those for which (A + I) = d and (I. + p)(h,) > dr(h,)). The main step in doing this is to use Shapovalov’s equation to show that S,,(E.) depends polynomially on I. (that is, if S,,(A) =C, F,c,(A), the c,(A) are polynomial functions of A).

4. CONSTRUCTION AND PROPERTIES OF Sr,d,v

Here the construction of S,,, begun in Section 3 is completed, and it is shown that the element satisfies the properties (O”t(3”). The first lemma shows how commuting ff with elements of @(n ) produces coefficients which are polynomials in q (indeed, more strongly, that commuting f ,” with elements of @(n _ ) n ZPBW produces coefficients in Z[q] ). This will be useful for commuting f; with S,,,,,(A’) in Shapovalov’s equation.

LEMMA 4.1. FornEp,qeZ+,

FnfZ' = c f:-'F,,c,c,i,,,(q) i,n' with Cnjn’ E Z[x]. 74 JAMES FRANKLIN

Proof: The lemma is proved by induction on deg K. It is true for degree zero. We assume it is true for F,, and prove it true for F,f,, ct E @+.

f’nfafl = Fn i .fX-‘fx+ ieci(q), i=o where ci(x) = ( - 1)i N, E. . . N, + + ijE,J;), (This commutation formula for f,f” is easily proved by’induction on q; the N,, are the structure constants of ;he Lie algebra: [fi, f,] = -N,,f,+ s, with k,, = 0 if r + s is not a root.) Now because %z is a ring,

and so i! divides N,,e ...N,+iE,E. Thus c~[x]EZ[X]. Then

Fnfctfi= i (FJlei)fa+~ECi(q) i=O

= i (1 ff-‘p’Fz~cn,j,z~(q - i)> fcx+ ieCi(q)y i=O j,x’ where the caj,nSare in Z[x], by induction hypothesis. Each F,,f,+. is a Z-sum of the F,.,, since ZPBW is a ring, so, on collecting terms, the lemma is proved. (Note: It has not been required in the statement of the lemma that rc’(s) = 0, that is, that all they,% have been collected in thefz-’ at the left. However, it is clear that we can require this, after some further com- mutations which do not affect the result. If Z(E) # 0, there may be non-zero c,,~,~,for negative values of i.) It can now be shown that S,+(k) depends polynomially on A.

PROPOSITION 4.2. $,A~)= c F,44h),... . W,)), wttr=drI where c, E Z[h,, .. . . &,, .. . . h,]. Proof. Shapovalov’s equation is VERMA MODULES IN CHARACTERISTIC fI 75

where (A+p)(h,)=d, q=(1+p)(h,)>dr(h,), il’=s;i=l-q& and r’ = s,r. By the induction hypothesis on Srs,d,tl,

sr,,d,,(n’) = 1 F,cxn’(h,), . . . . A’(h)) x wt n=dr with ck E Z[h,, .. . . &, .. . . h,]. So

f~-dr’hq&l) = 1 F,f;c;(A’(h,), ...) /I’(/?,)) n

= c (1 fl- ‘&c~,;,.Gd) ch(n’(h), .. . . A’(h)) n i,d with c,,~,~,E Z[x], by the last lemma. It is convenient to assume, as in the note after the last lemma, that all the rc’(s) are zero. Then the last expression is equal to

Here c,(A) = Cn,i,n,c,,,,.(q) ch(A’(h,), .. .. l’(h,)), where the sum is over those X, i, rr’ such that f;-iF,V = F,,,. (The sum over rc has only a finite number of non-zero terms since ck is non-zero only if wt rc= dr.) Since q = I(h,) + 1 and A’(hJ = A(hi) - (1(/z,) + 1) s(hi), c,.(A) is the evaluation at 1 of the polynomial

1, Cn,i,d(k+ 1) CXh, - WC+ 1) Glh ...Y h- vb + 1) E(W), 7z.i.n which is in Z[h,, .. . . &,, .. . . h,], as required. The restriction of the sum in s,,,(A) to those 7c such that wt rc= dr follows directly from the weights of the terms in Shapovalov’s equation. Note. The c,‘s in the expression for s,,(A) are unique, since if S,,(1)= c FA(~) = c f’,zc;(~) 77 n then each pair c,(A) and c;(A) agree on a set of A’s dense in the hyperplane (A + p)(h,) = d. So each c, - ck is divisible in @[hi, .. . . h,] by h, + p(h,) -d. But C[hi, .. . . h,](h, + p(h,) - d) n C[hl, .. . . h,, .. . . h,] is zero, since h, = m,b,+L,mA m, # 0. (In fact, under our assumptions, m, = 1.) So c, = CLfor all 7~.It is also clear that the c,‘s are independent of the choice of the sequence of reflections from r down to q (in particular, of the choice of E): there can be only one element of %(~)nC[hr, .. . . k,,, .. . . h,] whose evaluation at a set of A’s dense in (A + p)(h,) = d determines a 76 JAMES FRANKLIN homomorphism V*, 1 + 6.) and whose term of highest degree in the Fz is as in condition (3). The inductive definition of S,,, is now completed by defining

Sr.+, = 1 F, c, where the c,‘s are the elements of Z[h,, .. . . h,, .. . . h,] of Proposition 4.2. It can now be shown that S,,, satisfies the conditions (O”)( 3”). Con- dition (0”) is already clear from the definition, and (2”) follows from the fact that all the F,c, have weight -dr. Condition (3”) is true because (3) is already known to be true for S,,(A). Condition (1”) can easily be proved at once from the fact that S,,,,(A) = S,,(A) defines a homomorphism K,. ). + VA, but in fact we prove a stronger version of ( 1”) which gives more precise information on the integrality of (e”,/n!) S,,,. First, there is a lemma on commutation.

LEMMA 4.3. For n 2 0, EE A, YIE p,

with c,, E H[h,] and deg c,, d X(E). (Zn particular, although (e;/n!) F, is not in ZPBW, its “%(bP) part” is.)

Proof: The obvious rule fi(f,uh,)=f, #Z(U)hi, for C(E@+, i= 1, .. . . 1, is used a number of times. The result is proved by induction on the degree of F,. It is true for degree zero. The result is assumed true for F,, and proved for f, F, = FRO.

Case 1. If a=&, ~,(E)=z(E)+ 1, and

where N=n-1-(wtn)(h,)EZ. But by induction assumption, &(ez/n!) F,) and /((ez-‘/(n- l)!) Fz) are both sums of terms F,,,c,,~, with c,,, EZ[~,] of degree at most rc(.s). So #((ez/n!) f,F,) =&F,,c,*, with c,, E Z[h,] of degree at most q,(e) =X(E) + 1. WRMA MODULES IN CHARACTERISTIC p 77

Case 2. If o!#E, q,(s) = n(s), and

(Since eJ-jE=forphep+ NEiE--afOL--(i+ljE,e:f, can be written as a Z-sum of terms fordiEef-’ by successivecommutations. The Ni are integral because (e:/n!) f, E %&and the f,- ,,(e; ~ i/(n - i)!) are elements of a Kostant basis of 3$.) so

But by the induction assumption, each fi(efei/(n-- i)!) F,) is a sum of terms F,., c,-, with c,,.E Z[h,] of degree at most rc(s) = rrO(s). The result follows immediately. The next lemma relates divisibility by h,.+ p(h,) - d to the ring of polynomials with integral coeflicients.

LEMMA 4.4. Z[h,, .. . . h,] = Z[h,, .. . . hq, .. . . h,, h, + p(h,) - d]. Prooj It will be shown that in the expression h,=CEEd mE.s,m, = 1. Since q is a simple root occurring once in r,

r=rj+ C n,E &frl so 2r (identifying h and h* via the Killing form) hr = (r, r)

=&+ c 2n,E 9 Efq (r, r)

since q is the same length as r

=h,+ 1 mEhE with me E Z. EZO Therefore ZCh ,, .. . . h,] = Z[h,, .. . . h,,, .. . . h,, h,] = Z[h,, .. . . iis, . .. . 4, hr + 0,) - 4. The strengthened form of condition (3”) for S,,,, can now be proved. 78 JAMES FRANKLIN

PROPOSITION 4.5. For n > 1, EE A, fi((e:/n!) S,,,) = C,, F,,c’,., where c;r E Z[h,, ...) h,] (=Z[h,, ...) I(, ...) h,, h, + p(h,) - d] and the degree of the ch, in h,+ p(h,) -d is at most d. Further, the I$ are divisible by hr + p(k) - de Proof From the definition of S,,, above,

s r,d,q =C%T with c, E Z[h,, .. .. fi,, .. .. h,] A

=cc Fn,c,,,, cn n R’ with c,,,, ~i?[h,], of degree at most n(s), by Lemma 4.3

where ck = C, c,,,,c,. If E#Q Z[h,] c Z[h,, .. . . hV, .. . . h,], so C;,E b[h 1, ***,h,, .. . . h,]; that is, the degree of ch, in h, is zero. If E= q, x(q) 6 d, since wt rc= dr, so the degree of the c~,~~,and hence of ca,, in h, is at most d. But because h, = h, + CE+,, mEhE, the degree of chs in h,+p(h,)- d (in Z[h 1, .**,h,,, .. . . h,]) is equal to its degree in h, (in Z[h,, .. . . h,]). Hence the degree of CL,in h, + p(h,) - d is at most d, as required. The divisibility of the CL,by h, + p(h,) - d follows from the definition of s r,d,ll. For a set of A.% dense in the hyperplane (A + p)(h,) = d, S,,,(A) = S,,(A), which satisfies condition (1). So for such I,

(/i($d,q)) (Ah=~s,,,v,=o-

So #((e;/n!) S,,,,)(A) =0 for these A. Therefore fi((eg/n!) S,,,) is divisible by h,+p(h,)-d (in @(b&)), and hence the ck, are all divisible by h,+p(h,)-d (in Z[h,, .. . . h,, .. . . h,]). This completes the proof.

EXAMPLES. When r = a + b in type A*, v may be either a or 6.

S u+b,d,b= f fl-i(Nllbfa+b)ift-’ (h,+l)h;..(h,-i+2). i=O Here N,, may be chosen to be 1. VERMA MODULESIN CHARACTERISTICP 79

A formula for any root in type A, can be obtained by multiplying the formula of Carter and Lusztig [S, p. 2401 by d! and replacing h, with -CEfl m,h, - p(h,) - d (where h, = CEEdmehE). Carter and Lusztig also give some formulas for type Gz. When r = 2a + b in type B,, b is the only possible choice for q. Then

SZa+b,d,b= c f:d-rp2S(Nabfo+b)r f2o+ b)‘i ;r-‘+’ 0 < r,s r+s

5. HOMOMORPHISMS BETWEEN VERMA MODULES IN CHARACTERISTIC p

The elements of %(n_), obtained by evaluating s,,, at weights ;1 such that (J.+ p)(h,) is (not d but) congruent to d (mod p) are used to define homomorphisms between the Y&, the Verma modules in characteristic p. It is easy to show at once that the map UU,,@ 1 H us,,d,, @ 1 v1@ 1, u E eK, is a @=module homomorphism “yj,, + “#‘&, provided p = I - dr is a reflection of 1 in a p-hyperplane, and d < p. But to obtain the full result, which applies to reflections across p’-hyperplanes as well as across p-hyperplanes, it is necessary to show that the result is not altered by dividing S,,,(n) by the highest power of p which divides it. The main step in showing that the map to be defined is a homomorphism is:

PROPOSITION 5.1. Let ,IE h,* and let N be an integer which divides S,,,(A) in fj?&. Then for n 3 1, EE A,

Proof (It is clear from Proposition 4.5 that

b($ s,,,(n) ) (A)= i u, (r:+p;r)-d), u,E@!(n-1,; m=l ( the point of the proof is to show that N divides the u,, not just the

u (l+p)(h,)- .) m m 3

481.‘112’1-6 80 JAMFSFRANKLIN

The proof is based on Carter and Payne [6, pp. 9-l 11. From the definition of S,,, , s r,+,=~F,rc,, c,EIZCh,,...,h~,...,h,l

= ;fxm where C,= fl (~(a))! c,. K XE@+ Each C,(A) is divisible by IV, since they, are Kostant basis elements. Then

Now since Z[h,, .. . . h,] = H[h,, .. . . &, .. . . hi, h, + p(h,) - d], by Lemma 4.4, any element of 43(h), can be written as a finite sum

with H,E%(h),nQ[h, ,..., &,, .. . . h,].

Writing the H,,. in this way,

with Hz,,;,, ~%(h),nQ[h,, .. . . I$, .. . . h,]. Since H,,.,, and C, are both in Q[h,, .. . . h,, . .. . h,], the sum over m may be restricted to the range 1 to d, by Proposition 4.5. Thus

(Lemma 3.1)

as required. The main theorem can now be proved.

THEOREM 5.2. Let r be a non-exceptional root (that is, a root other than the highest root and the highest short root in Eg, F4 and G2), and let r] be a simple root such that n, = 1 in r = I,, d n,~. Let d be a positive integer; A an integral weight such that (A + p)(h,) = Mp’+ dfor some ME Z and such that 81 VERMA MODULES IN CHARACTERISTIC p d < pe; p = A - dr; and p/’ the highest power of p dividing S,,,(I) in sZ. Then the map

rv,~*~u~@* v,@l for uESK is a non-zero homomorphism of aK modules Vp,K + Vi,K. Proof V& = %KIIfl,K, and Ip,~ is generated by the (et/n!)@ 1, rzE Z, uE@+, and the

(hidi(hi’)@ 1, nEZ, i= 1, .. . . 1.

But by Lemma 1.4, it is sufficient to consider only the e;/n! 0 1, EE A, rather than all the e;/n! 0 1. These generators annihilate S,,,(L)/pfC3 1 vi 0 1 since

by Proposition 5.1. Since by the assumptions of the theorem pe divides (A + p)(h,) - d, and d < p’, each (A + p)(k) - 9 m=l , .. . . d, ( m 9 is a multiple of p. Hence

Also 82 JAMES FRANKLIN since S,,,,(n) has weight -dr,

=O since (“i-i(hi))@l EZA,K.

Hence the map described is a homomorphism of eK modules. By condition (4”) for Sr,d,rt, S,,, and hence S,,,,(I) have a term n,, d f,“t, where r = C n,~. Thus S,,,(I) # 0, so S,,,,(J)/#@ 1 is not zero, since p/ is the highest power of p dividing S,,, (A). So the homomorphism is non-zero, as required.

6. THE EXCEPTIONAL RENTS The previous work can be modified to obtain weaker results for the five exceptional roots, namely the highest root and the highest short root in E,, F4 and Gz. For any root and for any prime p, there is an q E A such that m, is prime to p (since in any root system, or dual root system, a root is never a multiple of an element of the root lattice). We fix such an q. Since h, = c EEdmA h, = (l/m,)(h, - CEZ tl m,h,). Thus h,, though not in Z[h 1, .. . . h,,, .. . . h,, h,], is in h[h,, .. . . h,, .. . . hl, h,] Oh,, where 7, is the localisation of h at the prime ideal pZ, that is, (n/q E Q: (p, q) = 1 }. We now follow the proofs in Sections 3-5 above, indicating the few modifications necessary to accommodate the fact that S,,, will no longer be in EPBW but in ZPBW OZ,. As before S,,, will satisfy conditions (O/)-(3’); recall that there was no mention of Z in these. When r is exceptional and r’ = s, . r < r, then r’ is not exceptional (since the exceptionals are either the highest or the highest short roots in their root systems). So, for d > 0, there exists an Sr,d,rlf,for some q’, satisfying the conditions (0”)-(3”). Then everything in Section 3 still holds, and Shapovalov’s equation is where Sr,,d,V,=Cn F,c;, with c:,~Z[h,, .. . . h,,, .. . . h,]. Then in place of Proposition 4.2 we have S,,(n) = En F,c,(L(h,), .. . . i(h,), with c, E ZCh 1, .. . . h,] (instead of Z[h,, .. . . h,, .. . . h,]). The proof is identical to that of Proposition 4.2, except that we can no longer assume that E# q’. Now since h, = ( l/m,) (h, - C, + ,, meh,), ZCh 1, .. . . h,l = ZCh,, .. . . h,, .. . . h,, h,l 0 z, becausem, is prime to p = Z[h,, .. . . fi,, .. . . h,, h, + Ah,) - 40 h, = (Z[h,, .. . . h,, .. . . h,] 0 h,)O ZCh, + p(k) - 4. VERMA MODULES IN CHARACTERISTIC p 83

The equation S,,(A) = C, F,c,(l(h,), .. . . I(h,)) holds for all 3, in a set dense in the hyperplane (A + p)(h,) =d, so for each K, there is exactly one d, E Z[h 1> ..‘, I$, ...) h,] 0 Z, whose evaluation at all such I is c,(l(h,), .. . . A(h,)). Sr,d,rl is then defined to be C, F, d,; it is in (ZPBW n S!(b-))@ H,. In place of Proposition 4.5, we have the weaker

PROPOSITION 6.1. Let r be exceptional and n be defined as above. For n2 1, EE A, ;I;( (ef/n! ) S,,,) = C,, F,, ch, where c$ E Z [h 1, .. . . h,] @ Z, ( = Z[h, ) .. . . h,, ...) ht, h,+p(h,)-d] @Z,), and the degree of c’,, in h,+ p(h,)-d is at most dn,. The ch, are divisible by h, + p(h,) - d.

Proof: The proof is identical to that of Proposition 4.5, except that the d,Eh[h ,,..., hq, .. . . h,] @iI, replace the c, E Z[h,, .. . . &, .. . . h,], and the condition z(q) < dn, replaces X(V) < d (since now q occurs more than once in r). Proposition 5.1 is replaced by

PROPOSITION 6.2. Let A E hi and let N be an integer which divides S,,,(n) in sz 0 z,. Then for n> 1, EEA,

The proof is the same as that of Proposition 5.1, except that all lattices are tensored with Z,,, and “the range 1 to d” becomes “the range 1 to dn,.” Now for any E-lattice L, any prime p, and any field K of characteristic p, L Q Z, Q K E L 0 K as K-vector spaces, via k u&kt+ux@- for uEL,xEZ,yEZ-pZ,kEK. Y Y

Clearly this isomorphism, in the present case where L = %(n ~ )L, is com- patible with the %K action. The main theorem, Theorem 5.2 on the existence of homomorphisms between Verma modules, is replaced by

THEOREM 6.3. Let r be an exceptional root (that is, the highest root or the highest short root of E,, F4 or G2). Let r] be a simple root such that the integer m, in h, = CEEdmEhE is prime to p. Let d be a positive integer, 1 an integral weight such that (A + p)(h,) = Mp’ + d for some ME H and such that 84 JAMES FRANKLIN d-c p’/n, (where r = CEEd n,c); p = ,I - dr; and pf the highest power of p dividing S,,,(A) in +2ZQZ,. Then the map

is a non-zero homomorphism of $YKmodules VP,K + V&.

Proof: The proof is identical to that of Theorem 5.2, but with d replaced by dn, and %(n-), by %(n_), 0 h,.

It remains to describe, for each exceptional root r and each prime p, a suitable choice of r,~The above proofs required that m, be prime to p. So for each r, p, the best choice of q is the simple root such that n, is minimal subject to the constraint that m, be prime to p. This choice is listed in Table I. In summary, a homomorphism exists as described in Theorem 6.3, for d -c p’/n,, where

forr=3a+2binGz,p=2,andr=highest rootinE,,p=2 for the live caseslisted in the table above for all other roots and primes, exceptional and non-exceptional.

TABLE I

root r hr P v m7 n,

E, 28, + 3~ + 48, + fez, + 5~ 2h,+3h2+4h,+6h4+5hS +4&G + 3E, + 2&* + 4h, + 3h, + 2h, 2 82 3 3 p#2 8, 2 2

F4 E, + 2~~ + 3tz, + 2~ 2h, + 3h2 + 4h, + 2h4 2 E2 3 2 P#2 E, 2 1 26, + 3&z+ 4Ej + 2Eq hl f 2h2 f 3h, f 2h, allp E, 1 2

Gz 2a+b 2h, + 3hh 3 2 2 P743 ; 3 1 3a+26 h, + 2h, 2 1 3 PZ2 K 2 2 VERMA MODULES IN CHARACTERISTIC p 85

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