Assume that $G\leq GL_n$ is a symmetric group (i.e. if $g\in G$ then $g^*\in G$ and $G$ is Zariski closed) and $K$ be a maximal compact subgroup of $G$. Say $\rho:G\rightarrow GL(V)$ is a representation of the group $G$. I denote the action of $g\in G$ on $v\in V$ by $g\cdot v$. Say $\langle \_,\_\rangle$ is a $K$-invariant Hermitian inner product on $V$. Does it hold that $$ \langle v,g\cdot w\rangle=\langle g^*\cdot v,w\rangle? $$ Or equivalently, do we have $\rho(g)^*=\rho(g^*)$?
I assume that since $\langle\_,\_\rangle$ is $K$-invariant, that should be the case but I could not prove it. Thanks for your help.