As this is an answer to the question, I post it as one: As far as I know there is still no conceptual explanation of this observation. We were looking at this situation for unitary reflection groups in Proposition 2.1 in our preprint http://arxiv.org/abs/1404.5522, tried to give an explanation and discussed possible approaches with David Bessis. However, this lack of understanding still seems to be the state of the art.

As it is not clearly stated here so far: As far as I know there is still no conceptual explanation of this observation. We were looking at this situation for unitary reflection groups in Proposition 2.1 in our last year's preprint http://arxiv.org/abs/1404.5522, tried to give an explanation and discussed possible approaches with David Bessis. However, this lack of understanding still seems to be the state of the art, even for finite Coxeter groups.

As this is an answer to the question, I post it as one: As far as I know there is still no conceptual explanation of this observation. We were looking at this situation for unitary reflection groups in Proposition 2.1 in our preprint arxiv.org/abs/1404.5522, tried to give an explanation and discussed possible approaches with David Bessis. However, this lack of understanding still seems to be the state of the art.

As this is an answer to the question, I post it as one: As far as I know there is still no conceptual explanation of this observation. We were looking at this situation for unitary reflection groups in Proposition 2.1 in our preprint http://arxiv.org/abs/1404.5522, tried to give an explanation and discussed possible approaches with David Bessis. However, this lack of understanding still seems to be the state of the art.