On the Image of a Word Map with Constants of a Simple Algebraic Group

Abstract

Let G be an algebraic group, $$ \tilde{w}:\kern0.5em {G}^n\to G $$ a word map with constants, T a fixed maximal torus of G, W the Weil group of G, and π : G → T/W the factor morphism. Some properties of the maps $$ \tilde{w} $$ and π ∘ $$ \tilde{w} $$ are studied. In particular, it is proved that for the adjoint group of G of type Ar, Dr, or Er, the map π ∘ $$ \tilde{w} $$ is a constant map only for the words vgv−1, where g ∈ G and v is a word with constants. As a corollary, one can generalize a result by T. Bandman and Yu. G. Zarhin (2016) as follows: the image of a word map $$ \tilde{w}:\kern0.5em {\mathrm{PGL}}_2^n\to {\mathrm{PGL}}_2 $$ with constants contains a representation of every semisimple conjugacy class ≠ = 1, or w = vgv−1 for some g, v.