Complete reducibility, Külshammer’s question, conjugacy classes: A D₄ example

Abstract

Let k be a nonperfect separably closed field. Let G be a connected reductive algebraic group defined over k. We study rationality problems for Serre's notion of complete reducibility of subgroups of G. In particular, we present a new example of subgroup H of G of type D4 in characteristic 2 such that H is G-completely reducible but not Gcompletely reducible over k (or vice versa). This is new: all known such examples are for G of exceptional type. We also find a new counterexample for Külshammer's question on representations of finite groups for G of type D4. A problem concerning the number of conjugacy classes is also considered. The notion of nonseparable subgroups plays a crucial role in all our constructions.of G that are G-cr over k but not G-cr (or vice versa). All these examples are for G of exceptional type (E 6 , E 7 , E 8 , G 2 ) in p = 2 and constructions are very intricate. The first main result in this paper is the following: Theorem 1.2. Let k be a nonperfect separably closed field of characteristic 2. Let G be a simple k-group of type D 4 . Then there exists a k-subgroup H of G that is G-cr over k but not G-cr (or vice versa).A few comments are in order. First, one can embed D 4 inside E 6 , E 7 or E 8 as a Levi subgroup. Since a subgroup contained in a k-Levi subgroup L of G is G-cr over k if and only if it is L-cr over k (Proposition 2.3), one might argue that our "new example" is not really new. However we have checked that our example is different from any example in [35, Thm.

Complete reducibility of subgroups of reductive algebraic groups over nonperfect fields IV: An $F_4$ example

“…Second, the third author had already found several examples of a subgroup H satisfying Theorem 1.3 for G of type D 4 , G 2 , E 6 , and E 7 and for k of characteristic 2 in [28], [29], [31]. We stress that Theorem 1.3 not only extends our collection of such examples, but it is new: we give the first connected such subgroup.…”

“…Forth, our method to construct H (via group theoretic argument and geometric invariant theory) is almost identical to the constructions in D 4 , G 2 , E 6 , and E 7 examples mentioned above [28], [29], [31]. Since the same method works for many examples (for finite and connected H) we believe that there should be some general phenomenon underlying these constructions (cf.…”

“…Then, in Section 4, we prove Theorem 1.3. Note that since our method is almost identical to that in [28], [29], [31], our proof is just a minimum skeleton. Finally, in Section 5, we discuss a rationality problem related to complete reducibility and the centre conjecture.…”

“…This suggests that we need to work with small p. Such proper nonseparable subgroups are hard to find. Only a handful of such examples are known [5,Sec.7], [28], [29], [31] and all of them are finite subgroups. In our example, H (or the nonseparable part of H) is connected (and very close to G 2 ).…”

Complete reducibility of subgroups of reductive algebraic groups over non-perfect fields IV: An 𝐹4 example

“…Second, the third author had already found several examples of a subgroup H (or H ′ ) satisfying Theorem 1.3 for G of type D 4 , G 2 , E 6 , and E 7 and for k of characteristic 2 in [29], [30], [32]. We stress that Theorem 1.…”