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.
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Translated by I. Ponomarenko.
Translated from Zapiski Nauchnykh Seminarov POMI, Vol. 478, 2019, pp. 78–99.
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Gnutov, F.A., Gordeev, N.L. On the Image of a Word Map with Constants of a Simple Algebraic Group. J Math Sci 247, 550–564 (2020). https://doi.org/10.1007/s10958-020-04819-9
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DOI: https://doi.org/10.1007/s10958-020-04819-9