# Reference request: representations of unipotent groups have a fixed point.

I'm looking for a reference for the following standard result:

Let $U$ be a unipotent algebraic group over an algebraically closed field $k$ (of any characteristic); then any algebraic representation of $U$ has a fixed point.

Statements of Engel's theorem for the analogous statement about Lie algebras seem to be ubiquitous. I can also find the statement that connected solvable groups always preserve a line in many places (for example, Borel, Theorem III.10.4). Combining this with the fact that unipotent groups have no non-trivial characters gives me the result I need. But it would be nice to have a place to which I could refer for the precise statement about unipotent groups.

• In asking a question like this, it would help to tell us what definition of a unipotent algebraic group (scheme) you are using. There are several different definitions in the literature, most of which are rather immediately equivalent to the existence of a nonzero fixed vector in any nonzero representation (this, in fact, works as a definition over any field). – mephisto Feb 3 '11 at 5:06
• A valid point. The fact is: I don't really know what definition I'm using! In the case I'm interested in, $U$ is the unipotent radical of a parabolic sub-group $P$ of a reductive group, and the fact I need is that $U$ acts trivially on every irreducible representation of $P$. – Keerthi Madapusi Pera Feb 3 '11 at 5:20

• Ah, thank you! Also, I guess I was only imagining unipotent groups as successive extensions of $\mathbb{G}_a$, but it's nice to know that the result is valid in general. – Keerthi Madapusi Pera Feb 3 '11 at 3:22
• Here's an argument that might finish things in general using Borel's result: if a unipotent group is not connected (which can only happen in characteristic $p$), then its group of connected components $\pi_0$ has $p$-power order and the sub-space of fixed points of the connected component containing the identity is stable under $\pi_0$. So it suffices to show that every representation of a finite $p$-group on a characteristic $p$ vector space has a fixed point. One can now reduce to the case of a finite field, and use the orbit-stabilizer theorem. – Keerthi Madapusi Pera Feb 3 '11 at 3:36