On natural representations of the symplectic group

Let [V.sub.k] be the Weyl module of dimension [MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII.] for the group G = Sp(2n, F) arising from the A:-th fundamental weight of the Lie algebra of G. Thus, [V.sub.k] affords the grassmann embedding of the A:-th symplectic polar grassmannian of the building associated to G. When char(F) = p > 0 and n is sufficiently large compared with the difference n - k, the G-module [V.sub.k] is reducible. In this paper we are mainly interested in the first appearance of reducibility for a given h : = n - k. It is known that, for given h and p, there exists an integer n(h, p) such that [V.sub.k] is reducible if and only if n [greater than or equal to] (h, p). Moreover, let n [greater than or equal to] n (h, p) and [R.sub.k] the largest proper non-trivial submodule of [V.sub.k]. Then dim([R.sub.k]) = 1 if n = n(h,p) while dim ([R.sub.k]) > 1 if n > n(h,p). In this paper we will show how this result can be obtained by an investigation of a certain chain of G-submodules of the exterior power [W.sub.k] := [[conjunction].sup.k]V, where V = V(2n,F). 2000 Mathematics Subject Classification : 20C33, 20E24, 51B25, 51E24. Key words and phrases : symplectic grassmannians, Weyl modules for symplectic groups.