The main scope of this question is Question 1 of this post, in which the only answer does so through a geometric approach that I do not understand. An algebraic explanation of the following is very much desired.
The setup has $G$ a Lie group, $P \subset G$ a Lie subgroup, with corresponding Lie algebras $\mathfrak{g}$ and $\mathfrak{p}$. I understand that $T_e(G/P) = \mathfrak{g}/\mathfrak{p}$, as well as the corresponding dual statement. The proof of Lemma 1.4.9 of Chriss-Ginzburg continues that for any $g \in G$, we have $$T^*_{g\cdot e} (G/P) = g \mathfrak{p}^\perp g^{-1}.$$ Informally, I think that since all tangent spaces of $G$ are conjugate to each other (and thus conjugate to the tangent space at the identity), then the dual of this statement is clear, hence the statement itself. However I suspect this is wrong, but cannot figure out why.