Small representations of SL2 in the finite Morley rank category

We study definable irreducible actions of SL2(K) on an abelian group of Morley rank ≤ 3rk(K) and prove they are rational representations of the group. In this article we consider representations of SL2 which are interpretable in finite Morley rank theories, meaning that inside a universe of finite Morley rank we shall study the following definable objects: a group G isomorphic to SL2, an abelian group V , and an action of G on V ; V is thus a definable G-module on which G acts definably. Our goal will be to identify V with a standard G-module, under an assumption on its Morley rank. (A word on this notion will be said shortly, after we have stated the results.) It will be convenient to work with a faithful representation, possibly replacing SL2 by the quotient PSL2, and we shall write G ≃ (P)SL2 to cover both cases. Theorem. In a universe of finite Morley rank, consider the following definable objects: a field K, a groupG ≃ (P)SL2(K), an abelian groupV , and a faithful action of G on V for which V is G-minimal. Assume rkV ≤ 3 rkK. Then V bears a structure of K-vector space such that: • either V ≃ K is the natural module for G ≃ SL2(K), or • V ≃ K is the irreducible 3-dimensional representation of G ≃ PSL2(K) with charK 6= 2. The characteristic 0 case essentially reduces to a theorem of Loveys and Wagner (Fact 1.2 below), or the following consequence of it: Lemma 1.4. In a universe of finite Morley rank, consider the following definable objects: a field K, a quasi-simple algebraic group G over K, a torsion-free abelian group V , and a faithful action of G on V for which V is G-minimal. Then V ⋊G is algebraic. In earlier versions of this article we relied on the following proposition, which the reader will now find in an appendix (the notion of unipotence there is not quite the algebraic one). Received November 17, 2010. First author supported by NSF Grants DMS-0600940 and DMS-1101597. The contents of this paper were found while the second author was a Hill Assistant Professor at Rutgers University. c © 2012, Association for Symbolic Logic 0022-4812/12/7703-0008/$2.50