# unipotent groups, their forms and representations

For simplicity fix a base field $k$ of characteristic zero, and consider smooth affine algebraic $k$-groups. (It is understood that unipotent groups in positive characteristic are more complicated, as one might have interesting non-smooth ones.)

Question 1: forms of unipotent groups

If $k$ is algebraically closed, then it is clear that every connected unipotent $k$-group is a successive extension of $\mathbb{G}_\mathrm{a}$'s. Then what about the case $k$ not algebraically closed? Is there non-trivial $K$-forms of, say, the upper trangular unipotent $k$-group of $GL_n$, and of the unipotent radicals of Levi $k$-subgroups of $GL_n$, etc?

If $K$ is a finite Galois extension of $k$ of Galois group $\Gamma$, then a $K$-form of the split $k$-torus is the same as a $\Gamma$-module structure on the group of characters $\mathbb{Z}^d$. Is there analogous results for $K$-forms of unipotent $k$-groups?

For example, with $K$ a finite extension field of $k$. the scalar restriction $$Res_{K/k}\mathbb{G}_{\mathrm{m}}$$

is not split as $k$-torus, but $$Res_{K/k}\mathbb{G}_\mathrm{a}$$

splits into a direct sum of $\mathbb{G}_\mathrm{a}$, becasue $K$ is a finite vector space over $k$ viewed additively. It is from this example that I want to know if there are interesting examples of forms of unipotent groups.

Question 2: representations of unipotent groups

If one has the one dimensional unipotent group $U=\mathbb{G}_\mathrm{a}$, then an algebraic representation of $U$ on $V$ a finite dimensional $k$-vector space is the same as a unipotent operator on $V$. One can then extend this description naturally to obtain the Tannakian category of finite dimensional algebraic representations of $U$.

And what about general unipotent $k$-group $U$? By the theorem of Lie-Engel, we know that such a representation of $U$ on $V$ is upper-triangular: it stabilizes a full flag of $V$, and acts trivially on the successive quotients (because of unipotence). Is there more precise information one can find about these representations so as the determine the Tannakian category of representations of $U$?

Again, let $K$ be a finite Galois extension of $k$ with Galois group $\Gamma$, and $U$, $W$ two connected unipotent $k$-groups that are isomorphic over $K$. Then how can one distinghuish the representations of the two groups by some "action" of $\Gamma$ one the representations spaces, in the spirit one finds in representations of the $k$-torus $$Res_{K/k}\mathbb{G}_\mathrm{m}$$

thanks!

• In characteristic zero unipotent groups and nilpotent Lie algebras correspond to each other using the Campbell-Baker-Haussdorf formulas. – Torsten Ekedahl Oct 4 '10 at 18:25