We consider for a simple,
simply connected algebraic group over an algebraically closed field of characteristic p,
modules induced from characters on a Borel subgroup. We ask if the socle levels of
the modules induced from characters in a general position determine the socle
levels of modules induced from characters in a singular position. Technically,
the question may be phrased in terms of the infinitesimal group subscheme
determined by the Frobenius morphism of the global group. Qualitatively, we
show that the socle levels of the global induced modules are induced from
the socle levels of the infinitesimal induced modules, assuming only that
Bott’s Theorem applies. Quantitatively, we show that the multiplicities of
the composition factors of the module induced from an infinitesimal socle
layer are determined by the structure of the layer as a module for the Borel
subgroup.
