In question 1,2, $G=GL_n(\mathbb C)$, let $$ X:G\rightarrow M_n(\mathbb C) $$ is a tangent vector field, define the left action on tangent vector field as $$ l(a)X(b)=a_*(X(a^{-1}b)) ~~~~\forall a,b\in G $$ $X$ is left invariant ,if $$ l(a)X(b)=X(ab) ~~~~\forall a,b\in G $$ Assume $X$ is left invariant and $X(1)=x$, then why is $$ X(a)=a\cdot x $$ the right is matrix multiplication ?
PS: Maybe ,there are some errors in my question ,I just begin Lie algebra and am not familiar with it. And I use the language of Riemannian geometry, so I use its notations.