On characteristic p Verma modules and subalgebras of the hyperalgebra
Let G be a finite dimensional semisimple Lie algebra; we study the class of infinite dimensional representations of Gcalled characteristic p Verma modules. To obtain information about the structure of the Verma module Z(λ) we find primitive weights μ such that a non-zero homomorphism from Z(μ) to Z(λ) exists. For λ + ρ dominant, where ρ is the sum of the fundamental roots, there exist only finitely many primitive weights, and they all appear in a convex, bounded area. In the case of λ + ρ not dominant, and the characteristic p a good prime, there exist infinitely many primitive weights for the Lie algebra. For G = sl3 we explicitly present a large, but not necessarily complete, set of primitive weights. A method to obtain the Verma module as the tensor product of Steinberg modules and Frobenius twisted Z(λ1) is given for certain weights, λ = pn λ1 + (pn — 1)ρ. Furthermore, a result about exact sequences of Weyl modules is carried over to Verma modules for sl2. Finally, the connection between the subalgebra u¯1 of the hyperalgebra U for a finite dimensional semisimple Lie algebra, and a group algebra KG for some suitable p-group G is studied. No isomorphism exists, when the characteristic of the field is larger than the Coxeter number. However, in the case of p — 2 we find u¯1sl3≈ KG. Furthermore, we determine the centre ofu¯nsl3, and we obtain an alternative K-basis of U-.