At first sight it's unclear how much the question has to do with the ambient semisimple group $G$: consider the simplest case when the rank is 1 and $U$ is just the additive group.   It's true that the conjugation action of $G$ itself on its function algebra has a rich structure.    This was shown in characteristic 0 by Kostant and then in a more algebraic setting by Richardson, after which Steve Donkin (Invent. Math. 91, 1988) generalized their results to almost all prime characteristics; his paper is freely available online at

http://gdz.sub.uni-goettingen.de/dms/load/img/?PPN=GDZPPN002104814

The maximal unipotent subgroup $U$ plays an essential role in the study of a semisimple group, but as a variety it just has the structure of an affine space.    So its actions on its own functions won't provide direct information related to the representations of $G$.