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). 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 of "The Nil Hecke Ring and Singularity of Schubert Varieties," 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 $k[U]_L$?

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). 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," 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 $k[U]_L$?

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 $G$). In fact there is even a natural ring structure on this direct limit such that this isomorphism is a ring isomorphism.

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 $k[U]_L$?

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). 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 of "The Nil Hecke Ring and Singularity of Schubert Varieties," 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 $k[U]_L$?