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edited Feb 16, 2012 at 13:26

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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 $G$.

ADDED: The answer to the question as stated is no, based on the rank 1 case where there is no useful connection between the (trivial) conjugation action of $U$ and the representation theory of $G$. Even with the added references, I haven't been able to figure out what is really being asked in this smallest case. It would help to start with a more explicit formulation when $G = SL_2$.

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 $G$.

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 $G$.

ADDED: The answer to the question as stated is no, based on the rank 1 case where there is no useful connection between the (trivial) conjugation action of $U$ and the representation theory of $G$. Even with the added references, I haven't been able to figure out what is really being asked in this smallest case. It would help to start with a more explicit formulation when $G = SL_2$.

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created Feb 15, 2012 at 23:12

52.9k
4
120
240

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 $G$.