Let G be a semisimple algebraic group scheme. defined and split over the prime subfield of an algebraically closed field k of characteristic P > 0. if F: G + G is the Frobenius morphism, and T is a maximal split torus in G, then TG,. is the inverse image of T under F’. Thus TG’, is a non-reduced subgroup scheme of G. Let B be a Bore1 subgroup of Gz containing 7. whose opposite B” determines the positive system of roots in the root system of G relative to T. Let TB, = B xc; TG, = B n TG,, the scheme theoretic intersection of B with 7-G‘,. It is well known [J2, Sect. 3.1 that injective ‘TG,-mods( as a factor of Q,(p), coincides with the multiplicity of L,(p) as a composition factor in lnd$;(i,). At the same time, if i is generic relative to an alcove in the sense defined in [CPS 1, Appendix A] the multiplicities of L,(p) in Tndyi;(/I) coincide with the multiplicities [IndE(/i): L(p)]. where L.(/f) denotes the rational irreducible G-module of high weight /i relative to B”. Thus, in the generic case, one can translate the problem of computing multiplicities of irreducibles in Indg(/l) to 2 question about the structure of rational injectivc TG,-modules. We begin, here, a study of the structure of rational injectivc 7%,-modules from the point of view of the parabolic subgroup schemes of 7’G,. In this, we proceed by analogy with the early work [H] of Harish-Chandra, [ Sp] of Springer, and with the more recent work [RS] of Ronan and Smith. If P is a parabolic subgroup scheme of G, containing T. let TP, = P xc; TG, = P n TG,. Then TP, has a Levi factorization: TP, = TL, x lip,, where C:,,, denotes the unipotcnt radical of TP,, and TL, its Levi factor