Let $ G $ a Lie group. Consider a short exact sequence of finite dimensional $ \mathbb{C}[G] $ modules.
$$
0 \to W \to V \to \mathbb{C} \to 0
$$
What is a sufficient condition on $ G $ to imply that all sequences of this form must split?
For example if $ G $ is a compact group then every irrep of $ G $ is completely reducible so if $ V/W $ is a trivial 1d irrep then some complement $ W^\perp $ exists and is a trivial 1d subrep of $ V $ and the sequence splits.
I'm especially interested in what situations this would fail. So it would be especially interesting to have an example of a group $ G $ and a SES of continuous finite dimensional $ \mathbb{C}[G] $ modules
$$
0 \to W \to V \to \mathbb{C} \to 0
$$
that doesn't split.
Best Answer
The question is:
Let $ G $ a Lie group. Consider a short exact sequence of finite dimensional $ \mathbb{C}[G] $ modules. $$ 0 \to W \to V \to \mathbb{C} \to 0 $$ What is a sufficient condition on $ G $ to imply that all sequences of this form must split?
Some sufficient conditions are:
In summary, I believe that for a Lie group $ G $, all short exact sequences of the form described above split if and only if $ G $ is linearly reductive (i.e. the category of finite dimensional complex representation of $ G $ is semisimple) if and only if $ G $ is semisimple or compact or the complexification of a compact group or isogeneous to a direct product of groups of these three types.
As for examples of short exact sequence of this form that do not split:
Aphelli points out that unipotent groups are an easy source of examples of Lie groups with complex representations that are reducible but indecomposable, and such representation are never semisimple.
Consider the abelian unipotent group $$ G:=\begin{bmatrix} 1 & b \\ 0 & 1 \end{bmatrix} $$ and its natural two dimensional representation $ V $ $$ \begin{bmatrix} 1 & b \\ 0 & 1 \end{bmatrix} \begin{bmatrix} x \\ y \end{bmatrix} =\begin{bmatrix} x+by \\ y \end{bmatrix} $$ There is a one dimensional trivial subrepresentation, call it $ W $, $$ \begin{bmatrix} 1 & b \\ 0 & 1 \end{bmatrix} \begin{bmatrix} x \\ 0 \end{bmatrix} =\begin{bmatrix} x \\ 0 \end{bmatrix} $$ which is the kernel of the homomorphism $ \pi: V \to \mathbb{C} $ given by $ \begin{bmatrix} x \\ y \end{bmatrix} \mapsto y $. (The representation $ V $ is not semisimple, see for example the proof given here Show that the only subrepresentation of $A$ is $W= \{(t,0) \mid t \in \Bbb R\}$. So $V$ is not reducible, but it is indecomposable. or here Indecomposable but not irreducible representation and direct sums)
So we have a short exact sequence $$ 0 \to W \to V \to \mathbb{C} \to 0 $$ which does not split because any embedding $ s: \mathbb{C} \to V $ would have to have image $ W $, since that is the only 1d subrepresentation, but then it would not be a splitting because $ \pi(s(\mathbb{C}))=\pi(W)=0 $.