# Chevalley groups over $k[t]/t^n$

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.)

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A further comment is that you deal with both a group scheme and an abstact group (which over arithmetic local rings may be just a finite group). The language such as Hopf algebras of regular functions or hyperalgbras belongs with the group scheme, whereas the structure and automorphisms of the group of points over a ring has been explored a lot. This goes back at least as far as a 1969 paper by E. Abe (one of 32 returned by MathSciNet using a search for "Chevalley group" and "local ring"). – Jim Humphreys Nov 16 '13 at 22:51
What does $G(R_n)$ as an "algebraic group" mean? The $R_n$-group scheme $G_{R_n}$ (coordinate ring $R_n\otimes_k k[G]$), or "$G(R_n)$ as a $k$-group" -- rigorously the smooth connected Weil restriction of scalars ${\rm{Res}}_{R_n/k}(G_{R_n})$ whose group of $k$-points is $G(R_n)$ and has a massive unipotent radical? (For $n=2$ it is $G\ltimes \mathfrak{g}$ via adjoint action.) What is your definition of "regular function" on $G(A)$? Chevalley groups $G$ are over $\mathbf{Z}$; the $A$-group scheme $G_A$ has coordinate ring $A\otimes_{\mathbf{Z}} \mathbf{Z}[G]$ and Lusztig computes that. – Marguax Nov 17 '13 at 1:10
@Chuck: Every linear algebraic group is a Zariski-closed subgroup of some SL$_N$ (e.g., GL$_n$ inside SL$_{2n}$ via $n \times n$ blocks $g$ and $g^{-1}$). You are using the Weil restriction viewpoint. It is very much non-reductive (I described it for $n=2$). You write the group of points as proxy for a linear algebraic group, but these are really not the same thing; it is hard to discuss things precisely that way. For example, SL$_n(k)$ is just an abstract group; the $k$-variety SL$_n$ is an algebraic group. You can make sense of "points" of the latter in a $k$-algebra, but not for the former. – Marguax Nov 17 '13 at 2:53
@Chuck, another fact (that may help in searches) is that this "$G(R_{n+1})$" (via Weil restriction, as Marguax says) is also known as the $n$th jet scheme of $G$, sometimes written $G_n$. There is a sequence $1 \to K \to G_n \to G \to 1$, realizing $G_n$ as a unipotent extension of $G$. – Dave Anderson Nov 18 '13 at 22:07
I don't know about standard, but there is the appendix to Mustata's "Jet schemes of lci canonical singularities", and some papers of Bérczi and Szenes might have good references. All this may or may not be far from what you need. (Alan Stapledon and I also wrote a paper dealing with jet schemes of groups but that's certainly not a standard reference!) – Dave Anderson Nov 20 '13 at 12:23