from PART III - General classification and applications
Published online by Cambridge University Press: 05 June 2015
Calculations in characteristics 2 and 3
Over imperfect fields k of characteristics 2 and 3 there are pseudo-reductive groups that are not standard. Interesting classes of such groups (to be called exotic) were discovered by Tits, resting on special properties of the group G2 in characteristic 3 and the group F4 in characteristic 2, as well as groups in characteristic 2 of type Bn and Cn with n ≥ 2. A basic common feature of these types is that the unique pair of non-orthogonal positive simple roots {a, b} with distinct root lengths, say with a long and b short, satisfies. In other words, the Dynkin diagram has an edge with multiplicity p.
The definition of exotic pseudo-reductive k-groups requires a lot of preparation (essentially all of §§7.1 and 7.2), due to the intervention of subtleties related to the field of definition over k of a Levi ks-subgroup (see Example 7.2.2). The case of G2 provides everything we need for a complete result in characteristic 3.
Tits' method for constructing exotic examples in characteristics 2 and 3 uses general axiomatic arguments with root systems that he sketched in [Ti2, §5] over a separably closed field (and applied to the root systems of a split G2 in characteristic 3 and a split F4 in characteristic 2, as well as variants for types B and C in rank ≥ 2 in characteristic 2). We will use an alternative approach that is well suited to working with arbitrary k-forms of type G2 (resp. types F4 and Bn and Cn with n ≥ 2) for any imperfect k of characteristic 3 (resp. 2). This section largely focuses on the calculations with (forms of) these types that underlie the constructions to be given in §7.2.
Let k be an arbitrary field of characteristic p ∈ {2, 3}, and let G be a connected semisimple k-group that is absolutely simple and simply connected with Dynkin diagram having an edge with multiplicity p. We do not assume that G is k-split.
To save this book to your Kindle, first ensure no-reply@cambridge.org is added to your Approved Personal Document E-mail List under your Personal Document Settings on the Manage Your Content and Devices page of your Amazon account. Then enter the ‘name’ part of your Kindle email address below. Find out more about saving to your Kindle.
Note you can select to save to either the @free.kindle.com or @kindle.com variations. ‘@free.kindle.com’ emails are free but can only be saved to your device when it is connected to wi-fi. ‘@kindle.com’ emails can be delivered even when you are not connected to wi-fi, but note that service fees apply.
Find out more about the Kindle Personal Document Service.
To save content items to your account, please confirm that you agree to abide by our usage policies. If this is the first time you use this feature, you will be asked to authorise Cambridge Core to connect with your account. Find out more about saving content to Dropbox.
To save content items to your account, please confirm that you agree to abide by our usage policies. If this is the first time you use this feature, you will be asked to authorise Cambridge Core to connect with your account. Find out more about saving content to Google Drive.