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 represen- tations 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 construc- ted 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 com- position 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 $;. and V, (that is, Z-lat- tices closed under multiplication by aZ). These can be tensored with K to form %=modules %& and Vi.,K. The V,,, are not, as over Cc,irreducible. Each VA,, has, however, a unique simple quotient L,,,, and all simple finite dimensional %= (or rational G,) modules occur in this way. Further, any finite dimensional @<module with a primitive vector is a quotient of a ?‘A,K (Jantzen [ 111). TO investigate the Lj.,K it has proved most useful to con- sider the structure of the V,,,. For example, Jantzen [ 111 obtains results on the composition multiplicities of the L,,, in the P’,,,. This article studies homomorphisms between the ?‘&, the Verma modules in characteristic p. Since every non-trivial homomorphism VP -+ Yi is injective, “y; is regarded as a submodule of ?‘i. (when such a homomorphism exists). Then if v1 is a primitive vector of <., there is an element u of @ such that uvj, is a primitive vector of “yj. Such an element is described by Shapovalov [15]. His treatment is extended here by showing that u is in %YZand so can be used to define a homomorphism Yti:,K-+ Y&. By modifying this element, homomorphisms “yj,, + <,,K are defined for certain pairs p, A for which there are no non-trivial homomorphisms “yj -+ q,. The final result (Theorems 5.2 and 6.3) shows: 60 JAMES FRANKLIN For a positive root r and an integral weight A, there exists a non-zero homomorphism VA _ dr + ?‘j. (d E R + ) provided that (A + p)( h,) = Mp’ + d for some MEZ, eEZt, and d< p’/N, where N is almost always 1, but may be 2 or 3 for certain high roots. This result may be interpreted geometrically in terms of the afline Weyl group: a non-zero homomorphism exists between ?.J$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= 1 n+,, 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 decom- position. 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’.