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3
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Are there cases in which the Weyl group _does_ act on the flag variety/springer fiber?
Here is a general fact: Given a group $G$ and a subgroup $H$, there is an isomorphism between
The space $\text{End}(G/H)$, consisting of $G$-equivariant maps $G/H\to G/H$.
The group $N/H$, where $N$ …