$SL(n, \mathbb C)$ Critical Point in a representation Let V be a finite dimensional representation of $SL(n, \mathbb C)$. Suppose $V$ is endowed with a Hermitian metric and that we have  $v \in V$ such that $|g \cdot v| \geq |v|$ for every $g \in SL(n, \mathbb C)$. This is equivalent to the assertion that $l(g):=\langle g\cdot v, v \rangle + \langle v, g \cdot v \rangle = 0$ for every $g \in \mathfrak{sl_n}$.
Why? The fact that $g=1$ is a critical point for the map $g \to |g\cdot v|$ suggests me that something could happen in the tangent space to it, namely $\mathfrak{sl_n}$. That's all I can think of.
Thanks!
 A: A quick-and-dirty way to see it (the argument is sound, however -- if you have time, you could flesh it out into a proper argument using limits) would be the following, I guess: 
Consider a group element close to the identity and expand it as
$$ M\approx\mathbb{1}+ m\,,$$
where $m \in \mathfrak{sl_n}$. (Note that using $g$ both for a group and an algebra element is somewhat misleading.) Then 
$$\left\|M\cdot v\right\|^2\approx \left\|\left(1+m\right)v\right\|^2=\left<\left(1+m\right)v,\left(1+m\right)v\right>\\
=\left<v,v\right>+\underbrace{\left<mv,v\right>+\left<v,mv\right>}_{l(m)}+\left<mv,mv\right>\\
\stackrel{!}{\geq} \left<v,v\right>\,.$$
Now assume $l(m)$ is nonzero. If it is positive you can switch $m\to-m$ to make it negative. If still $l(m)+\left<mv,mv\right>>0$, you rescale $m\to\lambda m$ to make the second term smaller, and you end up with 
$$\left\|M\cdot v\right\|^2\approx \left\|\left(1+m\right)v\right\|^2
=\left<v,v\right>+\underbrace{l(m)+\left<mv,mv\right>}_{\text{negative}}\\
< \left<v,v\right>\,,$$
i.e. a violation of your condition.
The key idea is that $l(m)$ as the dominant contribution to $\left\|M\cdot v\right\|^2$ is linear in $m$, so it can be positive or negative (and it is certainly negative somewhere if it is not zero everywhere).
If you want to use derivatives, it's even simpler: Write $M=e^{t g}$ with a parameter $t$ and $g \in \mathfrak{sl_n}$. Then 
$$ \frac{\text{d}}{\text{d}t} \left\|Mv\right\|^2\bigg|_{t=0}= l(g)\,,$$
i.e. there are directions where $\left\|Mv\right\|^2$ decreases if $l(g)\neq 0$.
