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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.

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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
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2 Answers

up vote 5 down vote accepted

Theorem 17.5 in Humphreys's Linear Algebraic Groups seems to be the result you want. (Also, doesn't the result for solvable groups only imply the corresponding result for connected unipotent groups? The proof for unipotent groups doesn't require connectedness.)

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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
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This is Kolchin's theorem, first proved in Kolchin's paper http://www.jstor.org/stable/1969399

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Thanks. Kolchin's proof seems to be the one reproduced in Humphreys's book. It's good to know the primary reference, if only to be able to include the word 'matric' somewhere. –  Keerthi Madapusi Pera Feb 3 '11 at 3:33
Yes, in my Section 17 notes I cited Kolchin's papers for this result and also referred to the more general treatment by Demazure and Gabriel in IV.2.2 of their scheme-theoretic treatise Groupes algebriques (1970). Kolchin did some pioneering work in the subject in the late 1940s before the more comprehensive work of Borel and Chevalley in the 1950s. –  Jim Humphreys Feb 3 '11 at 14:19
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