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It is probably hard to give a complete description. A lot of partial information about this is contained in this paper of Kostant https://arxiv.org/abs/1201.4494 (in particular there is a description of the subring of invariant elements in $k[U]$; in fact Kostant works with the Lie algebra of $U$ instead of $U$ itself, but clearly they are isomorphic as $U$-varieties with respect to the adjoint action).