I should know what is expected to be true, and what is the story.

I should also know what is known to be true, i.e. proven. And I should know how the proof goes.

I should also know what is known to be false, and counter examples.

For our flag variety, why is it true that, $T$-equivariant homology and $G$-equivariant homology only differ by Weyl covering.

(1) First of all, why is that for any Borel $B$ of $G$, the abstract Cartan $B/[B,B]$ for different $B$ are identified? Well, we may consider conjugation action of $G$ on the set of $B$. Suppose $B_1 = g B_2 g^{-1}$ for some $g$, the ambiguity of $g$: is right multiplication by $B_2$, left multiplication by $B_1$. For any $g' \in B_1 g B_2$, maybe consider the induced map on.

What is conjugation action on $B$? Fix a $b_0 \in B$, we can send $b \mapsto b_0 b b_0^{-1} = b b^{-1} b_0 b b_0^{-1}$. So, thing in the same conjugacy class in $B$ are also in the same unipotent orbit.

A conjugacy class of $B$ is contained in the unipotent orbit. But not the other way around. For example, identity matrix is its own conjugacy class (even under $G$ conjugation). However, multiplying by $[B,B]$ gives a lot.

(2) On the Lie algebra and Lie group level. Taking $W$ invariant.