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

Cherlin, Gregory; Deloro, Adrien

J. Symb. Log. 77, No. 3, 919-933 (2012).

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 \(V\).

Reviewer: Frank Wagner (Villeurbanne)

MSC:

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

Keywords:

fields of finite Morley rank; \(SL_2(K)\); \(PSL_2(K)\); faithful actions; irreducible representations

Citations:

Zbl 0793.03041; Zbl 1179.03041

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