# References needed for representation theory of certain unipotent algebraic groups in characteristic zero

Due to my preoccupation with characteristic p > 0 representation theory of algebraic groups, I am quite ignorant of some basic facts pertaining to their characteristic zero theory, mostly concerning Lie theory. I am virtually certain that the following results are true, and well known, but have been unable to locate a proper reference for them. One of them I in fact proved myself in my dissertation, but of course most journals will not accept an non peer-reviewed dissertation as a primary reference. The other I have never seen a proof of but have good reason to believe is true. The theorems can be found in the following pdf file:

http://homepages.utoledo.edu/mcrumle/references%20needed.pdf

The two theorems are given as theorems 1.1 and 1.2 of section 1. Please ignore the rest, except perhaps for the introduction which gives notation.

Thank you very much for any help.

$\text{Question, version 2.0}$

Here is the full statement of the theorem I am looking for. $U_n$ denotes the algebraic group of all $n \times n$ unipotent upper triangular matrices over a field $k$ (so in particular, $U_2$ is the additive group $G_a$ and $U_3$ is the Heisenberg group). The Lie algebra of $U_n$ is denoted $\mathfrak{u}_n$, identified as the space of all $n \times n$ strictly upper triangular matrices over $k$, and likewise $\mathfrak{gl}_d$ denotes the Lie algebra of $GL_n$, the general Linear group over $k$. A Lie algebra homomorphism $\phi: \mathfrak{u}_n \rightarrow \mathfrak{gl}_d$ is of course a $k$-linear map preserving the bracket operator $[X,Y] = XY - YX$. The theorem:

Let $k$ be a field of characteristic zero, let $n$ be arbitrary, and let $\phi: \mathfrak{u}_n \rightarrow \mathfrak{gl}_d$ be a Lie algebra homomorphism such that $\phi(X)$ is a nilpotent matrix for all $X \in \mathfrak{u}_n$. Then the formula

$\Phi(g) = e^{\phi(\log(g))}, \qquad g \in U_n$

defines a $d$-dimensional representation of $U_n$. Further, any $d$-dimensional representation of $U_n$ over $k$ is of this form.

For clarity, let me state this theorem for the case $n = 3$ (i.e. the Heisenberg group) using slightly different terminology:

Let $k$ be a field of characteristic zero. Every $d$-dimensional representation of the Heisenberg group $U_3$ over $k$ is of the form

$e^{xX+yY+(z-xy/2)Z}$

where $X,Y$ and $Z$ are $d \times d$ nilpotent matrices over $k$ satisfying $Z = [X,Y]$ and $[Z,X] = [Z,Y] = 0$. Further, any such collection of matrices $X,Y,Z$ satisfying the above gives a representation of $U_3$ according to this formula.

As you can see I am quite ignorant concerning Lie theory applied to algebraic groups in characteristic zero. I really need a specific reference to this theorem, or something very close to it. It is not terribly important that I understand the proof so I would prefer to bypass as much background material as possible.

Thanks again for any help.

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So you need to learn about Jordan decomposition and about the Lie-Kolchin theorem. Try the book by T A Springer on Linear algebraic groups. – Wilberd van der Kallen Nov 10 '11 at 9:13
Please copy the statements of the theorems into the question. – S. Carnahan Nov 10 '11 at 9:20