Small representations of \(\mathrm{SL}_2\) in the finite Morley rank category. (English) Zbl 1267.20053

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 \(2\).
In characteristic zero this boils down to a theorem of J. G. Loveys and 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 [A. Baudisch et al., J. Inst. Math. Jussieu 8, No. 3, 415-443 (2009; Zbl 1179.03041)] 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 \(V\).


20F11 Groups of finite Morley rank
03C45 Classification theory, stability, and related concepts in model theory
03C60 Model-theoretic algebra
20G05 Representation theory for linear algebraic groups
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