Examples of exotic modules for the additive group

Question

Let $k$ be an algebraically closed field of positive characteristic $p > 0$, and let $X$ be an intedeterminate over $k$. I am interested in the additive group scheme $\mathbb{G}_a$, that is, the affine algebraic group scheme having coordinate ring $k[X]$. We can identify $\mathbb{G}_a$ with the unipotent radical of a Borel subgroup $B$ for the group scheme $SL_2$, e.g., if we identify $B$ with the subgroup of upper triangular matrices in $SL_2$, then we can identify $\mathbb{G}_a$ with the subgroup $U$ of upper triangular unipotent matrices in $SL_2$.

I am specifically interested in rational modules for $\mathbb{G}_a$, or equivalently, $k[X]$-comodules. Given a rational $B$-module $M$, I can of course restrict $M$ to $U = \mathbb{G}_a$ and obtain a rational $\mathbb{G}_a$-module. But what I'd really like are some explicit examples of rational $\mathbb{G}_a$-modules that do not arise via restriction from a rational $B$-module, and I haven't had any luck so far finding explicit examples in the literature.

Can anyone point me to explicit examples in the literature of (finite-dimensional) rational $\mathbb{G}_a$-modules that do not arise as the restriction of rational $B$-modules?

Here's the (single) example I've been able to cook up from scratch for the case $p=2$. Let $V$ be the $k$-vector space with basis vectors $v_0$ and $v_1$. I define a linear map $\Delta: V \rightarrow V \otimes k[X]$ by

$\Delta(v_0) = v_0 \otimes 1 + v_1 \otimes (X+X^2)$,

and $\Delta(v_1) = v_1 \otimes 1$. Then it's straightforward to check that this defines a $k[X]$-comodule structure on $V$ (equivalently, a rational $\mathbb{G}_a$-module structure on $V$), which cannot come from the restriction of a rational $B$-module structure on $V$ because of the non-homogeneous term $X+X^2$.

Answer 1

At first glance it seems that your $p=2$ example can be generalized to arbitrary primes by using $p$-polynomials such as $X + X^p$. In prime characteristic there are lots of exotic embeddings of the additive group involving such polynomials, which in turn translate into morphisms of (affine) algebraic groups. See for instance the discussion in Chapter 20 of my 1975 Springer GTM book, with references to older literature including Demazure-Gabriel. On the other hand, representations of the multiplicative group are pretty much the same in all characteristics, so you would typically see the same incompatibility you note for $p=2$ when the additive group is represented using as coordinate functions some nontrivial $p$-polynomials.

I'm not sure whether concrete examples of the type you want are written down explicitly, though people studying group actions and invariant theory in prime characteristic may have dealt with the situation.