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Let G ⊂ GL(r) be an irreducible finite complex reflection group. We show that (apart from the exception G = S6) any automor-phism of G is the product of an automorphism induced by tensoring by a linear character, of an automorphism induced by an element of N GL(r) (G) and of what we call a " Galois " automorphism: we show that Gal(K/É), where K is the field… (More)
Dedicated to George Lusztig on his 60th birthday.
Let G be a complex reflection group and K its field of definition (the subfield of generated by the reflection character). We show that Gal(K/É) injects into the group A of outer automorphisms of G which preserve reflections, and that this injection with a few exceptions can be chosen such that it commutes with the Galois action on characters of G; further,… (More)