This question is motivated partly by a recent question on Chevalley groups over arbitrary commutative rings (and see also this older question). The answers to that question point to a large and daunting literature which I am unfamiliar with, so I am curious about understanding a particular case which hopefully is easy. Namely, consider a Chevalley group $G$ and fix an algebraically closed field $k$ of arbitrary characteristic. For $n > 0$ set $R_n := k[t]/t^n$. Then we can consider the Chevalley group $G(R_n)$. I'm curious what is known about the structure of this group - for example, can we describe the Hopf algebra of regular functions on this group? Dually, is there a nice description of the hyperalgebra of this group? Since $R_n$ is finite over $k$ and is local, one would hope the answers are not too wild in this case. (I've looked at the papers of Lusztig that were mentioned in the answers to the linked question, but unless I'm missing something it seems that he only explicitly constructs the ring of regular functions on $G(A)$ in the case that $A$ is an integral domain.)

This question is motivated partly by a recent question on Chevalley groups over arbitrary commutative rings (and see also this older question). The answers to that question point to a large and daunting literature which I am unfamiliar with, so I am curious about understanding a particular case which hopefully is easy. Namely, consider a Chevalley group $G$ and fix an algebraically closed field $k$ of arbitrary characteristic. For $n > 0$ set $R_n := k[t]/t^n$. Then we can consider the Chevalley group $G(R_n)$. I'm curious what is known about the structure of this group - for example, can we describe the Hopf algebra of regular functions on this group? Dually, is there a nice description of the hyperalgebra of this group? Since $R_n$ is finite over $k$ and is local, one would hope the answers are not too wild in this case. (I've looked at the papers of Lusztig that were mentioned in the answers to the linked question, but unless I'm missing something it seems that he only explicitly constructs the ring of regular functions on $G(A)$ in the case that $A$ is an integral domain.)

This question is motivated partly by a recent question on Chevalley groups over arbitrary commutative rings. The answers to that question point to a large and daunting literature which I am unfamiliar with, so I am curious about understanding a particular case which hopefully is easy. Namely, consider a Chevalley group $G$ and fix an algebraically closed field $k$ of arbitrary characteristic. For $n > 0$ set $R_n := k[t]/t^n$. Then we can consider the Chevalley group $G(R_n)$. I'm curious what is known about the structure of this group - for example, can we describe the Hopf algebra of regular functions on this group? Dually, is there a nice description of the hyperalgebra of this group? Since $R_n$ is finite over $k$ and is local, one would hope the answers are not too wild in this case. (I've looked at the papers of Lusztig that were mentioned in the answers to the linked question, but unless I'm missing something it seems that he only explicitly constructs the ring of regular functions on $G(A)$ in the case that $A$ is an integral domain.)

This question is motivated partly by a recent question on Chevalley groups over arbitrary commutative rings (and see also this older question). The answers to that question point to a large and daunting literature which I am unfamiliar with, so I am curious about understanding a particular case which hopefully is easy. Namely, consider a Chevalley group $G$ and fix an algebraically closed field $k$ of arbitrary characteristic. For $n > 0$ set $R_n := k[t]/t^n$. Then we can consider the Chevalley group $G(R_n)$. I'm curious what is known about the structure of this group - for example, can we describe the Hopf algebra of regular functions on this group? Dually, is there a nice description of the hyperalgebra of this group? Since $R_n$ is finite over $k$ and is local, one would hope the answers are not too wild in this case. (I've looked at the papers of Lusztig that were mentioned in the answers to the linked question, but unless I'm missing something it seems that he only explicitly constructs the ring of regular functions on $G(A)$ in the case that $A$ is an integral domain.)