Steinberg?s fixed point theorem for crystallographic complex reflection groups

Steinberg's fixed point theorem states that given a finite com-plex reflection group the stabilizer subgroup of a point is gen-erated by reflections that fix this point. This statement is also true for affine Weyl groups. Of the infinite discrete complex reflection groups, it was shown that there are some infinite complex reflections groups that have non-trivial stabilizers that do not contain a single reflection, and therefore, these groups cannot satisfy the fixed point theorem. We thus clas-sify the infinite discrete irreducible complex reflection groups of the infinite family which satisfy the statement of the fixed point theorem.Published by Elsevier Inc.