Small representations of \(\mathrm{SL}_2\) in the finite Morley rank category. (Q2915898)

In this paper, the authors consider, in a finite Morley rank context, a faithful action of \(G=\mathrm{(P)SL}_2(K)\) on an Abelian group \(V\) with \(\mathrm{RM}(V)\leq 3\mathrm{RM}(K)\). They show that then either \(V\cong K^2\) is the natural module for \(G=\mathrm{SL}_2(K)\), or \(V\cong K^3\) is the irreducible \(3\)-dimensional representation of \(G=\mathrm{PSL}_2(K)\) in characteristic different from \(2\).NEWLINENEWLINE In characteristic zero this boils down to a theorem of \textit{J. G. Loveys} and \textit{F. O. Wagner} [Proc. Am. Math. Soc. 118, No. 1, 217-221 (1993; Zbl 0793.03041)]; in positive characteristic, the existence of bad fields, i.e.\ fields of finite Morley rank with a distinguished proper infinite connected Abelian subgroup [\textit{A. Baudisch} et al., J. Inst. Math. Jussieu 8, No. 3, 415-443 (2009; Zbl 1179.03041)] \textit{horribly complicate matters}. The proof proceeds by first considering the action of a torus \(T<G\) on a \(T\)-minimal subgroups of \(V\), and then deducing linearity of the action of \(G\) on \(V\).