Let $G=\text{SL}(n,\mathbb R)$, a unimodular Lie group and let $\mu$ be a (the) bi-invariant Haar measure on it. Let $\alpha$ be any automorphism on this group. I wonder if $\alpha_*\mu=\mu$
By the unimodularity, if $\alpha$ is an inner automorphism (defined by conjugation of a group element) then $\alpha_*\mu=\mu$ by the bi-invariance. But what happens for general automophisms?