Connection 1-form on Lie group If we regard $S^{2n-1} \to \mathbb{CP}^{n-1}$ as a principal $S^1$ bundle, how do I show that $$A=\frac{1}{2\pi}\sum_i(x_i dx_i-y_i dy_i),$$
where $(x_1,y_1,\dotsc,x_{2n},y_{2n})$ are coordinates on $S^{2n-1}$, satisfies the following relation:
$$(R_a)^*A = \mathrm{Ad}(a^{-1}) A$$ is true for all $a \in S^1$?

Well, the action is abelian, so $Ad(a^{-1})$ is just the identity. Now, applying the right action by $a\in S^1$ to your connection, what kind of shape on $S^{2n-1}$ does it trace out? Try thinking about the case when $n=1$. 
