**Background**


Let $G$ be a semisimple linear algebraic group over an algebraically closed field $k$ of arbitrary characteristic. Let $B \subseteq G$ be a Borel subgroup of $G$ and let $U \subseteq B$ be its unipotent radical. Consider $k[U]$ as a $U$-module under left multiplication in $U$; let's call this module $k[U]\_L$. By, say, identifying $U$ with the big cell in the flag variety $G/B^-$, it is not hard to see that $k[U]\_L$ is isomorphic as a $U$-module to a direct limit of standard modules $H^0(\lambda)$ for $G$. (In characteristic 0 one can also see this by identifying $k[U]$ with the dual zero Verma module for the enveloping algebra of $G$, cf [this paper](http://www.ams.org/journals/jams/2000-13-04/S0894-0347-00-00347-7/home.html)). In fact there is even a natural ring structure on this direct limit such that this isomorphism is a ring isomorphism.


EDIT: A reference for this fact is Lemma 2.5 and the discussion following it in "[The Nil Hecke Ring and Singularity of Schubert Varieties](http://www.kryakin.com/files/Invent_mat_%282_8%29/123/123_06.pdf)," by Shrawan Kumar. Although the construction is done there over $\mathbb C$, it works over any algebraically closed field, and although Kumar only proves that there is a $T$-equivariant morphism, it is easy to check that the morphism he constructs is $U$-equivariant.


**Question**


Now consider $k[U]$ as a $U$-module under the *conjugation* action of $U$ on itself; let's call this module $k[U]\_C$. Is there a nice description of $k[U]\_C$ as a $U$-module in terms of $G$-modules in a way analogous to the description of $k[U]\_L$?



Decomposition of the ring of functions on the unipotent radical of a Borel
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).


