Automorphisms and generalized involution models of finite complex reflection groups

Abstract

We prove that a finite complex reflection group has a generalized involution model, as defined by Bump and Ginzburg, if and only if each of its irreducible factors is either G ( r , p , n ) with gcd ( p , n ) = 1 ; G ( r , p , 2 ) with r / p odd; or G 23 , the Coxeter group of type H 3 . We additionally provide explicit formulas for all automorphisms of G ( r , p , n ) , and construct new Gelfand models for the groups G ( r , p , n ) with gcd ( p , n ) = 1 .