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 Bore1subalgebra 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.