product of torus and affine space

Question

Given the set $\mathbb{C}^*\times \mathbb{C}$ with the group structure given by $(x,y)\cdot (z,w) =(xz, zy+wx) $. How can I check that this is reductive or not? ?

I think that its maximal normal solvable subgroup is itself, how can I see the identity component of this group?

Thanks

Answer 1

Two definitions and a property :

A linear representation $\rho:G\rightarrow GL(n,\mathbb{C})$ is completely reducible if for all $G$-invariant subspace $V$ in $\mathbb{C}^n$, there exists a $G$-invariant subspace $W$ for which $\mathbb{C}^n=V\oplus W$.

A group $G$ is linearly reductive if for all $n\geq 1$ and all group morphism $\rho:G\rightarrow GL(n,\mathbb{C})$ the linear representation $\rho$ is completely reducible.

Over a field of characteristic $0$, $G$ is linearly reductive if and only if its identity component $G^0$ is connected.

Remark that :

1. $G:=\mathbb{C}^*\times \mathbb{C}$ is connected because $\mathbb{C}^*$ and $\mathbb{C}$ are connected.

2. Regarding the property given above, and remark 1, in order to show that $G:=\mathbb{C}^*\times \mathbb{C}$ is not reductive, it suffices to find a representation $\rho_0:G\rightarrow GL(n,\mathbb{C})$ such that $\rho_0$ is not completely reducible.

Sketch of solution :

Consider the function $\rho_0:\mathbb{C}^*\times \mathbb{C}\rightarrow GL(2,\mathbb{C})$ sending $(x,y)$ to $\begin{pmatrix}x&y\\0&x\end{pmatrix}$. Show that this is a group morphism (whence a linear representation) but that linear representation is not completely reducible.