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Oct 17, 2012 at 18:09 comment added Angelo To Jesko: yes, definitely. There is a whole lot of work on homogeneous spaces, toric varieties, wonderful compactifications, spherical varieties, and so on, in which one exploits the group action to get results on the intersection theory.
Oct 17, 2012 at 16:43 comment added Jesko Hüttenhain How unfortunate. Let me ask more broadly then: Are there any particular techniques or results in intersection theory on $G$-varieties that make use of the $G$-action? In fact, let me add that to my original question.
Oct 17, 2012 at 16:23 history answered Angelo CC BY-SA 3.0