Pseudo reductive group

Examples of pseudo reductive groups that are not reductive

Suppose that k is a non-perfect field of characteristic 2, and a is an element of k that is not a square. Let G be the group of nonzero elements x + y√a in k[√a]. There is a morphism from G to the multiplicative group G_m taking x + y√a to its norm x² – ay², and the kernel is the subgroup of elements of norm 1. The underlying reduced scheme of the geometric kernel is isomorphic to the additive group G_a and is the unipotent radical of the geometric fiber of G, but this reduced subgroup scheme of the geometric fiber is not defined over k (i.e., it does not arise from a closed subscheme of G over the ground field) and the unipotent k-radical is trivial. So G is a k-reductive group but is not a reductive k-group. A similar construction works using a primitive nontrivial purely inseparable finite extension of an imperfect field in any positive characteristic, the only difference being that the formula for the norm map is a bit more complicated than in the preceding quadratic examples.

More generally, if K is a non-trivial purely inseparable finite extension of k and G is any non-trivial connected reductive group defined over K then the Weil restriction H=R_K/k(G) is a smooth connected affine algebraic group defined over k for which there is a homomorphism from H_K onto G. The kernel of this K-homomorphism descends the unipotent radical of the geometric fiber of H and is not defined over k (i.e., does not arise from a closed subgroup scheme of H), so R_K/k(G) is pseudo-reductive but not reductive. The previous example is the special case using the multiplicative group and the extension K=k[√a].

Classification and exotic phenomena

Over fields of characteristic greater than 3, all pseudo-reductive groups can be obtained from reductive groups by the "standard construction", a generalization of the construction above. The standard construction involves an auxiliary choice of a commutative pseudo-reductive group, which turns out to be a Cartan subgroup of the output of the construction, and the main complication for a general pseudo-reductive group is that the structure of Cartan subgroups (which are always commutative and pseudo-reductive) is mysterious. The commutative pseudo-reductive groups admit no useful classification (in contrast with the reductive case, for which they are tori and hence are accessible via Galois lattices), but modulo this one has a useful description of the situation away from characteristics 2 and 3 in terms of reductive groups over some finite extensions of the ground field.

Over imperfect fields of characteristic 2 and 3 there are some extra exotic pseudo-reductive groups coming from the existence of exceptional isogenies between groups of types B and C in characteristic 2, between groups of type F₄ in characteristic 2, and between groups of type G₂ in characteristic 3, using a construction analogous to that of the Ree groups. Moreover, in characteristic 2 there are additional possibilities arising not from exceptional isogenies but rather from the fact that for simply connected type C (I.e., symplectic groups) there are roots that are divisible (by 2) in the weight lattice; this gives rise to examples whose root system (over a separable closure of the ground field) is non-reduced; such examples exist with a split maximal torus and an irreducible non-reduced root system of any positive rank. The classification in characteristic 3 is as complete as in larger characteristics, but in characteristic 2 the classification provided in the published literature is complete only when [k:k^2]=2 (due to complications caused by the examples with a non-reduced root system, as well as phenomena related to certain regular degenerate quadratic forms that can only exist when [k:k^2]>2). Subsequent work of Conrad & Prasad (2016), building on additional material included in the second edition of Conrad, Gabber & Prasad (2015), completes the classification in characteristic 2 (up to a controlled central extension) by providing an exhaustive array of additional constructions that only exist when [k:k^2]>2 , ultimately resting on a notion of special orthogonal group attached to regular but degenerate and not fully defective quadratic spaces in characteristic 2.