What is a left-invariant Vector field? Is left invariant vector field on  $\mathbb{G}$,is same as a vector field which is invariant under group transformation.
 A: I guess you need a plain english explanation.
A vector field $X$ is a function that associate smoothly to every point $p$ of $G$ an element or vector $X_p$ of the tangent space of the group $G$ (which in this case is also a manifold). So for every point $p$ you have the vector $X_p \in T_p(G)$.
Now suppose you have a function $F$ that maps every point of $G$ to another point of $G$. Let's suppose that the point $p$ goes to $q$ i.e. $F(p)=q$.
Then if you consider the vector field $X$ you already had, you can play two different games with this vector field:

*

*First of all you can calculate $X_{F(p)}$. This is the original vector field $X$ but evaluated in the new point $F(p)$ which is $q$, i.e. $X_{F(p)}=X_{q}$. In particular if you apply the vector to a function $f\in C^{\infty}(G)$, you have $X_{F(p)}(f)$;


*The other game you can do is to calculate $(F_*X_{p})(f)$ which by definition is $X_p(f\circ F)$. Here, $F_*X_{p}$ is given by taking the vector $X_p$ calculated from the old vector field and considering its image by a the linear application $F_*$ which is the differential of $F$.
The two games are in general different because in the first case you leave the vector field untouched and you change the point to which is applied, while in the second case you use the old point you had and you evaluate the vector field in the point but then you move the resulting vector by apply the function.
Now if the two games coincide then you have that the vector field is invariant i.e.
$$X_{F(p)}(f)=X_p(f\circ F)$$
You can think about this with an hypersimple physical example: Consider a river and a thermometer. The river is flowing so a particle of water is moved along the flow. If you're interested on the changing of temperature along the flow of the river, in any given time you can either measure the temperature few istants after the time given measuring the temperature in the new place where the particles will be or you can measure now the temperature of the particles and assign this temperature to the place where you guess they will be given the velocity they have.
Now your case: if the function $F$ is the function given by the left multiplication of the group, then you have a "Left Invariant Vector Field".
Actually it's easier to work out some example to get effectively the idea.
