In Chriss and Ginzburg's "Representation Theory and Complex Geometry", they describe a geometric construction of representations of the affine Hecke algebra, using the Borel-Moore homology of generalized Springer fibers.

Briefly, let $G$ be a sufficiently nice algebraic group, and choose a semisimple $s \in G$ and some nilpotent $x \in \mathfrak{g} = $ Lie$(G)$, such that $x$ is an eigenvector for the adjoint action of $s$. Consider the variety $B_{x}^{s} $ of Borel subgroups of $G$ containing $s$ and exp$(x)$. Then $C_{G}(s,x)$ acts on this variety by conjugation, and thus acts on its Borel-Moore homology.

Now, the claim is (p.415 of C & G) that this action descends to the component group because the identity component of $C_{G}(s,x)$ acts trivially. Normally, I would expect this to be true since a path in the group from an element $g$ to the identity gives a homotopy between the action of $g$ and the identity map. But Borel-Moore homology is not a homotopy invariant, so I don't understand why an element in the identity component should necessarily act trivially on the homology.

They make a similar claim earlier in the book (p. 170) without any justification there either. If anyone can point out what I'm missing, I'd really appreciate it.