Let $H$ be a semisimple Lie group contained in $SL(V)$ for some finite dimensional normed vector space $V$). Let $\|\cdot\|$ denote some norm in some finite dimensional representation of $H$.
Let $v$ be a vector in $V$ of norm $1$,
then there exists $k>0$ which only depends on $dim(V)$ such that for all $h \in H$
$$\|hv \|^{-1} \le \|h^{-1}\| \le \|h\|^k.$$
$hv$ here should be intepreted as the $\rho(h)v$ where $\rho:H\to GL(V)$ is the inclusion representation.
This notion of norm is very new to me and I don't know how to prove the inequality above (I am afraid that there might be some implicit conditions.) Please see that pages 2 and 13 (equation (41)) of the following paper
recurrence properties of random walks on finite volume homogeneous manifolds by Eskin and Margulis 2002
for the source of the notion mentioned above.
Update: That $\|hv \|^{-1} \le \|h^{-1}\|$ should be easy. That $h\in SL(V)$ implies that the maximum eigenvalue is no less than $1$. Noticing that the operator norm is no less than the spectral norm, we have
$$\|h^{-1}\| \ge 1 \ge \|hv \|^{-1}.$$