Cosets, genericity, and the Weyl group

In a connected group of finite Morley rank in which, generically, elements belong to connected nilpotent subgroups, proper normalizing cosets of definable subgroups are not generous. We explain why this is true and what consequences this has on an abstract theory of Weyl groups in groups of finite Morley rank. The only known infinite simple groups of finite Morley rank are the simple algebraic groups over algebraically closed fields and this is a motivation, among many others, for a classification project of these groups. It borrows ideas and techniques from the Classification of the Finite Simple Groups but at the same time it may provide, sometimes, a kind of simplified version of the finite case. This is mostly due to the existence of well-behaved notions of genericity and connectivity in the infinite case, which unfortunately have no direct finite analogs. The present note deals with a very specific and technical topic concerning such arguments based on genericity in the case of infinite groups of finite Morley rank, which serve here to bypass allegro potential complications of various nature, including finite combinatorics. As a result, we show similarities with algebraic groups in any case as far as a theory of Weyl groups is concerned, and naturally this applies also to non-algebraic configurations which are encountered throughout much of the current work in the area. In a connected reductive algebraic group, maximal (algebraic) tori are conjugate and cover the group generically, with the Weyl group governing essentially the structure of the entire group. In the abstract context, we use the term “generous” to speak of a subset “whose union of conjugates is generic in the group”, the typical property of tori in the classical algebraic case. There are at least two abstract versions of tori in groups of finite Morley rank, which coincide at least in the case of a reductive algebraic group, decent tori on the one hand and Carter subgroups on the other. The main caveat with these two more abstract notions for a seemingly complete analogy with algebraic groups is in both cases an unknown existence, more precisely the existence of a nontrivial decent torus on the one hand and the existence of a generous Carter subgroup on the other. Anyway, here we follow an approach resolutely adapted to the second notion. With both notions there are conjugacy theorems, the conjugacy of maximal decent tori [Che05] and of generous Carter subgroups [Jal06]. This gives a natural notion of Weyl group in each case, N(T )/C(T ) for some maximal 1 ha l-0 02 04 56 4, v er si on 3 12 S ep 2 00 8 decent torus T or N(Q)/Q for some generous Carter subgroup Q. In any case and whatever the Weyl group is, it is finite and, as with classical Weyl groups and algebraic tori in algebraic groups, its determination and its action on the underlying subgroup is fundamental in the abstract context. As an element of the Weyl group is a coset in the ambient group, it is then useful to get a description of such cosets, even though recovering from such a description the structure and the action of the Weyl group is in general a particularly delicate task. This is mostly due to the fact that, in practice, one can only get a generic, and thus weak, description of the coset. In [CJ04] such arguments were however developed intensively, and this was highly influenced by one of the most critical aspects of the early work, notably by Nesin, on the socalled “bad” groups of finite Morley rank ([BN94, Theorem 13.3]). In this paper, a pathological coset, whose representative is typically a Weyl element which should not exist, is usually shown to be both generous and nongenerous, and then the coset does not exist. This is the main protocol, sometimes refered to as “coset arguments”, for the limitation of the size of the Weyl group. Generosity is usually obtained by unexpected commutations between the Weyl elements and the underlying subgroup, and in general this may depend on the specific configuration considered. It is certainly the pathological property in any case, and we shall prove here at a reasonable level of generality that the existing cosets should be nongenerous. In particular, we rearrange as follows the protocol of [CJ04] in the light of further developments of [Jal06] concerning generosity. Theorem 1 (Generix and the Cosets) Let G be a connected group of finite Morley rank in which, generically, elements belong to connected nilpotent subgroups. Then the coset wH is not generous for any definable subgroup H and any element w normalizing H but not in H. The assumption on the generic elements of G in Theorem 1 can take several forms, and we will explain this shortly. The most typical case where Theorem 1 applies is however the case in which H = Q is a generous Carter subgroup of G. In particular, the present paper is also an appendix of [Jal06] on the structure of groups of finite Morley rank with such a generous Carter subgroup, and more precisely a follow-up to Section 3.3 in that paper. The general idea of the protocol of [CJ04] has been used repeatedly in various contexts, most notably to get a fine description of p-torsion in terms of connected nilpotent subgroups of bounded exponent and of decent tori [BC07]. Applied to the most natural kind of Weyl groups, the protocol shows that centralizers of decent tori are connected in any connected group, implying in particular that the Weyl group N(T )/C(T ) attached to a decent torus T acts faithfully on T . This corresponds to the most typical and smooth applications of the protocol in [CJ04], generally a lemma expedited at the early stage of the analysis of each configuration considered there. With [Che05] and [Jal06], and eventually the finiteness of conjugacy classes of uniformly definable families of decent tori of [FJ08, Theorem 6.4], it became clear that, for that specific lemma, the protocol 2 ha l-0 02 04 56 4, v er si on 3 12 S ep 2 00 8 had implementations autonomous from these specific configurations. Proofs may have appeared in [AB08, Fre07b], with a conceptually better and more general implementation in the second case but, regrettably, with no connection at all to [CJ04] in both cases. A much more delicate use of the protocol can be found in [CJ04, Proposition 6.17]. It is proved there, still in a specific configuration, that the centralizer of a certain finite subgroup of a decent torus is connected, with then a much more restrictive faithful action of the Weyl group. As this special application of the protocol contains the main difficulty possibly inherent to the subject, we mostly refer to this example. As we will see below, the key point is that generosity is in general related to a finiteness property, as opposed to a uniqueness property, a delicate aspect treated “by hand” in [CJ04, Proposition 6.17] and much more conceptually here. Theorem 1 has general consequences on the action of the Weyl group on the underlying subgroup, again whatever these are. Back to the concrete example of a reductive algebraic group, the maximal algebraic torus is a divisible abelian subgroup, and the Weyl group acts faithfully on it. The main corollary of Theorem 1 is a general form of this in the abstract context of groups of finite Morley rank. Corollary 2 Let G be a connected group of finite Morley rank in which, generically, elements belong to connected nilpotent subgroups. Suppose that H is a definable connected generous subgroup, that w is an element normalizing H but not in H, of finite order n modulo H, and that {h | h ∈ H} is generic in H. Then CH(w) < H. In the case of a connected reductive algebraic group, the subgroup H in Corollary 2 is typically the maximal torus T and w a representative of a nontrivial element of order n of the Weyl group. In the finite Morley rank case, H may typically be a generous n-divisible Carter subgroup Q, and w a representative of a nontrivial element of order n of the Weyl group N(Q)/Q. One gets then, for instance if Q is a divisible abelian generous Carter subgroup as in Corollary 14 below, consequences qualitatively similar in the finite Morley rank case. As for Theorem 1, the statement adopted in Corollary 2 is far more general than what it says about this typical case. Less typical applications can be found in [DJ07, §4.2] in the context of connected locally solvable groups, the smallest class of groups of finite Morley rank containing connected solvable groups and Chevalley groups of type PSL 2 and SL 2 over algebraically closed fields. Besides, the reader can find there a form of Theorem 1, actually weaker, but which reformats uniformly and in a hopefully informative way the original arguments of [CJ04] in this context of “small” groups. 3 ha l-0 02 04 56 4, v er si on 3 12 S ep 2 00 8 1 Technicalities and environment Before passing to the proofs, we review briefly the background needed, or surrounding. Groups of finite Morley rank are equipped with a rudimentary notion of finite dimension on their definable sets, satisfying as axioms a few basic properties of the natural dimension of varieties in algebraic geometry over algebraically closed fields. By definable we mean definable by a first-order logic formula, possibly with parameters and possibly in quotients by definable equivalence relations. The dimension, or “rank”, of a definable set A is denoted by rk (A). The finiteness of the rank implies the descending chain condition on definable subgroups, and this naturally gives abstract versions of classical notions of the theory of algebraic groups: • The definable hull of an arbitrary subset of the ambient group is the smallest definable subgroup containing that set. It is contained in the Zariski closure in the case of an algebraic group. • The connected component G of a group G of finite Morley rank is the smallest (normal) definable subgroup of finite index of G, and G is connected when G = G. A fundamental property of a connected group of finite Morley rank is that it cannot be partitioned into two definable generic subsets, that is two subsets of maximal rank [Che79]. Our arguments make full use of the following simpler properties.