Word maps of Chevalley groups over infinite fields

E. A. Egorchenkova

Zapiski Nauchnykh Seminarov POMI, Problems in the theory of representations of algebras and groups. Part 34, Tome 478 (2019), pp. 108-127

Résumé

Let $G$ be a simply connected Chevalley group over an infinite field $K$ and let $\widetilde{w}: G^n\rightarrow G$ be a word map that corresponds to a non-trivial word $w$. It has been proved in: (Israel J. Math. 210 (2015), 81-100) that if $w = w_1w_2w_3w_4$ is a product of any four words on independent variables, then every non-central element of the group $G$ is contained in the image of $\widetilde{w}$. A similar result for a word $w = w_1w_2w_3$ that is a product of three independent words was obtained in: (Archiv der Math. 112 (2019), no. 2, 113-122) under the condition that the group $G$ is not of types $B_2, G_2$. In this paper we prove that for groups of types $B_2, G_2$ all elements of big Bruhat cell $B \mathfrak{n}_{w_0} B$ are contained in the image of a word map $\widetilde{w}$ where $w = w_1w_2w_3$ is a product of three independent words. For groups of types $A_r, C_r, G_2$ (respectively, for groups of type $A_r$) or groups over a perfect field $K$ (respectively, over a perfect field $K$ such that $\mathrm{char} K$ is not a bad prime for $G$) that has $\dim K \leq 1$ (here $\dim K$ is cohomological dimension of $K$) it has been proved here that all split regular semisimple elements (respectively, all regular unipotent elements) of the group $G$ are contained in the image $\widetilde{w}$ where $w = w_1w_2$ is a product of two independent words. Also, for any isotropic (but not necessary split) simple algebraic group $\mathcal G$ over a field $K$ of characteristic zero we show that for a word map $\widetilde{w}: \mathcal{G}(K)^n\rightarrow \mathcal{G}(K)$, where $w = w_1w_2$ is a product of two independent words, all unipotent elements are contained in $\mathrm{Im}\, \widetilde{w}$.