A Geometric Interpretation of the Characteristic Polynomial of Reflection Arrangements

We consider projections of points onto fundamental chambers of finite real reflection groups. Our main result shows that for groups of type $A_n$, $B_n$, and $D_n$, the coefficients of the characteristic polynomial of the reflection arrangement are proportional to the spherical volumes of the sets of points that are projected onto faces of a given dimension. We also provide strong evidence that the same connection holds for the exceptional, and thus all, reflection groups. These results naturally extend those of De Concini and Procesi, Stembridge, and Denham which establish the relationship for 0-dimensional projections. This work is also of interest for the field of order-restricted statistical inference, where projections of random points play an important role.