CLASSIFICATION OF PSEUDO-REDUCTIVE GROUPS

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1 CLASSIFICATION OF PSEUDO-REDUCTIVE GROUPS BRIAN CONRAD AND GOPAL PRASAD Abstract. In an earlier work [CGP], a general theory for pseudo-reductive groups G over arbitrary fields k was developed, and a structure theorem was proved whenever char(k) 2 or when char(k) = 2 subject to the arithmetic hypothesis [k : k 2 ] 2. Many new phenomena arise when [k : k 2 ] > 2, related to exceptional isogenies, non-reduced root systems, and non-split central extensions. In particular, there are such G that vary with moduli in a manner with no analogue when char(k) 2 or when char(k) = 2 with [k : k 2 ] 2. In the pseudo-semisimple case over imperfect k with any positive characteristic we analyze the structure of the automorphism scheme and obstructions to the existence of pseudo-split and quasisplit forms (including many examples with nontrivial obstruction). We also prove a general structure theorem, up to a benign central extension, over any imperfect field k of characteristic 2 (recovering the structure theorem that was already known when [k : k 2 ] = 2). This completes the classification of pseudo-reductive groups over general fields. 1 Introduction 1.1 Motivation. A pseudo-reductive group over a field k is a smooth connected affine k-group G that does not contain any nontrivial smooth connected unipotent normal k-subgroup. A pseudoreductive k-group G that is also perfect (i.e., G = D(G)) is called pseudo-semisimple. The interest in pseudo-reductive groups is due to their role in the study of arbitrary linear algebraic groups over imperfect fields (such as function fields of positive-dimensional varieties over fields of positive characteristic). Over such fields it is often straightforward to reduce arithmetic problems to the pseudo-reductive case but not to the reductive case. The structure of pseudo-reductive groups is essential to prove general finiteness questions over local and global function fields, such as finiteness for degree-1 Tate-Shafarevich sets of all affine k-group schemes of finite type; see [C1, 1] for further applications. Recent work of Gabber on compactification theorems for arbitrary linear algebraic groups uses this structure theory over general (imperfect) fields. Weil restrictions G = R k /k(g ) for finite extensions k /k and connected reductive k -groups G are pseudo-reductive [CGP, Prop ]. If G 1 and k /k is not separable then such G are not reductive [CGP, Ex ]. If K/k is a separable extension then a smooth connected affine k-group G is pseudo-reductive if and only if G K is pseudo-reductive [CGP, Prop ], so pseudo-reductivity is inherited by smooth connected normal k-subgroups (as we may check over k s ). Example If G is pseudo-reductive then D(G) is perfect [CGP, Prop ] and consequently D(G) is pseudo-semisimple. Also, for a finite extension field k /k and connected semisimple k -group G, the pseudo-reductive k-group R k /k(g ) is pseudo-semisimple if G is simply connected [CGP, Prop ]. It is essential that G is simply connected: if k /k is a nontrivial purely inseparable finite extension in characteristic p > 0 then R k /k(pgl p ) has a nontrivial commutative quotient and hence is not pseudo-semisimple [CGP, Ex ]. Date: September 1, Mathematics Subject Classification. Primary 20G30; Secondary 20G25. Key words and phrases. pseudo-reductive, Weil restriction. 1

2 2 BRIAN CONRAD AND GOPAL PRASAD Despite the abundance of pseudo-reductive groups, pseudo-reductivity is not a robust property. For instance, over any imperfect field k there is a pseudo-semisimple group G with smooth connected normal (hence pseudo-reductive) k-subgroups N N such that G/N is not pseudo-reductive and N is not perfect [CGP, Ex ]; such N and N do not exist when G is connected reductive. Also, for any imperfect k there are pseudo-reductive k-groups G for which G/Z is not pseudo-reductive for some central k-subgroup scheme Z [CGP, Ex ]. (This is due to the fact that inseparable Weil restriction does not preserve surjectivity and does not scale dimension by the field degree away from the smooth case; e.g., for nontrivial finite purely inseparable extensions k /k in characteristic p > 0, the map R k /k(sl p ) R k /k(pgl p ) has kernel R k /k(µ p ) with positive dimension.) A solvable pseudo-reductive group is necessarily commutative [CGP, Prop ], but the structure of commutative pseudo-reductive groups appears to be intractable (see [T]). There is no convenient description of commutative pseudo-reductive groups akin to Galois lattices for tori. A structure theory is developed in [CGP] modulo the commutative case and is summarized in [C1, 2] and [R]. Cartan k-subgroups of pseudo-reductive k-groups are commutative and pseudo-reductive, and the lack of a concrete description of such groups is not an obstacle in applications (see [C1]). A general theory of pseudo-reductive groups, including the open cell, root systems, rational conjugacy theorems, and the Bruhat decomposition, is established in [CGP] The standard construction. A maximal k-torus T in a pseudo-reductive k-group G is the almost direct product of the maximal central k-torus Z in G and the maximal k-torus T := T D(G) in D(G) [CGP, Lemma 1.2.5]. We say G is pseudo-split if it contains a split maximal k-torus T, in which case any two such tori are conjugate under an element of G(k) [CGP, Thm. C.2.3] and the set Φ := Φ(G, T ) of nontrivial T -weights on Lie(G) injects into X(T ) via restriction. The pair (Φ, X(T ) Q ) is a root system, possibly non-reduced when k is imperfect of characteristic 2 [CGP, Thm ] (and it always coincides with Φ(D(G), T ) since G/D(G) is commutative). The root system attached to such a pseudo-split pair (G, T ) can be canonically enhanced to a root datum R(G, T ) = (Φ, X(T ), Φ, X (T )) [CGP, 3.2]. However, in contrast with the reductive case, the construction of coroots via algebraic group techniques is not related to an explicit description of all pseudo-split pseudo-simple k-groups of rank 1 since (i) such a description when char(k) 2 or char(k) = 2 with [k : k 2 ] 2 lies much deeper than the construction of coroots (especially for the root system BC 1 when [k : k 2 ] = 2), (ii) if char(k) = 2 with [k : k 2 ] > 2 then it is unlikely that any such description over k can be found (see 4.2.2)! If G is a pseudo-semisimple k-group then the set {G i } of its minimal nontrivial perfect smooth connected normal k-subgroups is finite, the G i s pairwise commute and generate G, and every perfect smooth connected normal k-subgroup of G is generated by the G i s that it contains (see [CGP, Prop ]). A pseudo-reductive k-group G is called pseudo-simple if it is pseudo-semisimple and nontrivial with no nontrivial smooth connected proper normal k-subgroup, and absolutely pseudosimple if G ks is pseudo-simple; the latter are analogues of connected semisimple groups that are absolutely simple. A pseudo-split pseudo-semisimple group is absolutely pseudo-simple precisely when its root system is irreducible [CGP, Prop ]. The core of the characteristic-free study of pseudo-reductive groups G in [CGP] is the absolutely pseudo-simple case. If char(k) 2, 3 then every pseudo-reductive k-group is standard in a sense reviewed in 2.1. (See [CGP, Thm ] for a formulation that also characterizes the standard cases in characteristics 1 Chapter 9 in the first edition of [CGP] is completely rewritten in the second edition, incorporating significant improvements that are used throughout this monograph. All numerical labeling in the first edition of [CGP] remains unchanged in the second edition except in Chapter 9, but some results outside Chapter 9 are improved in the second edition and we use such improvements. In this monograph, we assume familiarity with notation/terminology from [CGP], especially [CGP, Ch. 9], but we provide many cross-references to aid the reader.

3 CLASSIFICATION OF PSEUDO-REDUCTIVE GROUPS 3 2 and 3, assuming [k : k 2 ] 2 when char(k) = 2. The degree restriction in characteristic 2 can be eliminated by using the main theorem of this monograph; see Corollary ) Informally, a standard pseudo-reductive group is obtained from the Weil restriction of a connected reductive group by replacing a Cartan subgroup with a more general commutative pseudo-reductive group. The significance of the ubiquity of the standard construction is that many questions about standard pseudo-reductive k-groups can be reduced to analogues for connected semisimple groups (over finite extensions of k) and for commutative smooth connected affine k-groups. 1.3 Exotic groups and degenerate quadratic forms. If k is imperfect with char(k) = 3 then a structure theorem is provided by [CGP, Thm (1)]: every non-standard pseudo-reductive k-group has the unique form G 1 G 2 where G 1 is a standard pseudo-reductive k-group and G 2 is an exotic pseudo-reductive k-group in a sense reviewed in 2.3 (resting on exceptional isogenies in characteristic 3 arising from a nontrivial factorization of the Frobenius isogeny for connected semisimple groups of type G 2 ). For arithmetic problems over such k, we can often replace G 2 with an associated semisimple group of type G 2 (see [CGP, 7.3]). Overall, if char(k) 2 then a general pseudo-reductive k-group is understood in terms of connected semisimple groups over finite extensions of k and auxiliary commutative smooth connected affine k-groups. If k is imperfect with char(k) = 2 and [k : k 2 ] = 2 then there is a structure theorem [CGP, Thm ] using exotic constructions for types B n (n 2), C n (n 2), and F 4 (see 2.3) and a birational group law construction for type BC n with any n 1 (see [CGP, ]). The condition [k : k 2 ] = 2 for imperfect k of characteristic 2 is harmless for proving arithmetic finiteness theorems in [C1], but we want to go further since the case [k : k 2 ] > 2 arises for generic fibers of smooth affine groups over varieties of dimension > 1 over fields of characteristic 2. Moreover, the proof of Gabber s compactification theorem encounters additional complications in characteristic 2 that are most efficiently overcome using our main theorem and techniques in its proof. Our construction of new classes of absolutely pseudo-simple k-groups when [k : k 2 ] > 2 ultimately rests on the following examples that arise from degenerate quadratic spaces: Example Let (V, q) be a quadratic space over a field k with char(k) = 2, d = dim V 3, and q 0. Let B q : (v, w) q(v + w) q(v) q(w) be the associated symmetric bilinear form and let V be the defect space of v V such that the linear form B q (v, ) on V vanishes. The restriction q V is 2-linear (i.e., additive and intertwines with squaring on k), and dim(v/v ) = 2n for some integer n 0 since V/V admits a non-degenerate symplectic form (arising from B q ). Assume 0 < dim V < dim V and (V, q) is regular in the sense that ker(q V ) = 0; regularity is preserved by any separable extension on k (Lemma 7.2.1). Under these hypotheses, in order that q be non-degenerate (i.e., the projective hypersurface (q = 0) P(V ) is k-smooth) it is necessary and sufficient that dim V = 1, which is to say d = 2n + 1, and it is well-known that in such cases SO(q) is adjoint semisimple and absolutely simple of type B n with O(q) = µ 2 SO(q). Since char(k) = 2, SO(q) is the maximal smooth closed k-subgroup of O(q). Motivated by this, in general we define SO(q) to be the maximal smooth closed k-subgroup of the k-group scheme O(q); i.e., SO(q) is the Zariski-closure of O(q)(k s ) in O(q). The properties of SO(q) for regular (V, q) with 0 < dim V < dim V are studied in , where we show that SO(q) is absolutely pseudo-simple with root system B n over k s with 2n = dim(v/v ), and that over k s the dimension of its root groups is 1 for long roots and dim V for short roots. Moreover, the minimal field of definition over k for the geometric unipotent radical of SO(q) is the finite extension K k 1/2 of k generated by the square roots (q(v )/q(v)) 1/2 for nonzero v, v V.

4 4 BRIAN CONRAD AND GOPAL PRASAD For any nonzero v 0 V, the map v (q(v)/q(v 0 )) 1/2 is a k-linear injection of V into k 1/2 with image V containing 1 and generating K as a k-algebra. If we replace v 0 with a nonzero v 1 V then the associated k-subspace of K is λv where λ = (q(v 0 )/q(v 1 )) 1/2 K. In particular, the case V k occurs precisely when dim V 2, which is precisely when the regular q is degenerate, and always [k : k 2 ] = [k 1/2 : k] dim V. If [k : k 2 ] = 2 then SO(q) turns out to be the quotient of a basic exotic k-group [CGP, 7.2] modulo its center, and If [k : k 2 ] > 2 then the SO(q) s are a new class of absolutely pseudo-simple k-groups of type B n (with trivial center), though for n = 1 these are a special case of the pseudo-split non-standard absolutely pseudo-simple groups of type A 1 with trivial center built in [CGP, Ch. 9]. In 7.3 we relate isomorphisms SO(q ) SO(q) to conformal isometries q q and use this to construct additional absolutely pseudo-simple k-groups of type B with trivial center via Brauer Severi varieties corresponding to 2-torsion classes in Br(k). In particular, the homothety class of q V is an invariant that varies with moduli. Via the exceptional isogeny Sp 2n SO 2n+1 in characteristic 2, a fiber product construction involving the groups SO(q) and their Brauer Severi variants is used in 8.2 to construct new absolutely pseudo-simple groups of type C n when n 2 and [k : k 2 ] 2. The equality B 2 = C 2 underlies a refinement with fiber products in 8.3 leading to even more such k-groups for n = 2 when [k : k 2 ] 16. Over general imperfect fields k of characteristic 2, a structure theorem must account for many new phenomena. In 4.2 we review how the case of characteristic 2 presents special challenges, and how the case [k : k 2 ] > 2 exhibits novel behavior; some of these difficulties will also be addressed later in this Introduction. The main result of this monograph is a structure theorem for pseudoreductive groups over arbitrary imperfect fields k of characteristic 2 (see ). We also give results on automorphism schemes and on pseudo-split and quasi-split k-forms for imperfect k of any positive characteristic (see 1.7); the latter rests on our main theorem. 1.4 Tame central extensions. A key ingredient in our work is a generalization of the standard construction, better-suited to the peculiar demands of characteristic 2. Before we address that, it is instructive to review the principle underlying the proof of ubiquity of the standardness property for pseudo-reductive groups away from the case char(k) = 2 with [k : k 2 ] > 2. In [CGP], standardness of pseudo-reductive groups is proved via splitting certain central extensions. Let s recall the most basic instance, to see why it breaks down completely if char(k) = 2 with [k : k 2 ] > 2 (especially if [k : k 2 ] > 8; see 1.4.3(i)). Let G be an absolutely pseudo-simple k-group. If K/k is the minimal field of definition for the k-subgroup R u (G k ) G k and if G := G ss K is the maximal semisimple quotient of G K over K then for the simply connected central cover q : G G and µ := ker q Z G, as in [CGP, Def ] there is a canonically associated k-homomorphism ξ G : G D(R K/k (G )) = R K/k ( G )/R K/k (µ) arising from the natural map i G : G R K/k (G ). By pseudo-reductivity of G over k, the kernel ker ξ G = ker i G contains no nontrivial smooth connected k-subgroup. This kernel is central when the root system of G ks is reduced [CGP, Prop ], a property that holds whenever char(k) 2 [CGP, Thm ]. For any imperfect k of characteristic 2 and n 1 there exists a pseudo-split pseudo-simple k-group G with root system BC n [CGP, ], and in such cases ker ξ G is not central (see 1.4.2) Assume the root system of G ks is reduced, and that ξ G is surjective (as holds when char(k) 2, 3, but can fail over any imperfect field of characteristic 2 or 3 due to the existence of exceptional

5 CLASSIFICATION OF PSEUDO-REDUCTIVE GROUPS 5 isogenies in these characteristics). In such cases, G is standard if and only if there exists a k- homomorphism f giving a commutative diagram R K/k ( G ) ( ) f G ξg R K/k ( G )/R K/k (µ) Finding such an f is equivalent to splitting the q-pullback central extension 1 ker ξ G E R K/k ( G ) 1. Such extensions split because of a general fact (see [CGP, Ex ]): if k /k is a finite extension of fields and G is a connected semisimple k -group that is simply connected then for any commutative affine k-group scheme Z of finite type containing no nontrivial smooth connected k-subgroup (such as Z = ker ξ G above), every central extension of k-group schemes is (uniquely) split over k. q 1 Z E R k /k(g ) 1 ( ) Assume G ks has a non-reduced root system over k s (so char(k) = 2), so ker ξ G is non-central by [CGP, Prop ]. If [k : k 2 ] > 2 then ξ G is generally not surjective [CGP, Thm (4)]. If [k : k 2 ] = 2 (so k K 2 K) then a miracle happens (see [CGP, Rem , Thm ]): the maximal pseudo-reductive quotient G := G pred over K 2 is pseudo-split with a non-reduced root K 2 system, Z G = 1, G(F ) = Aut F (G F ) for all separable extension fields F/K 2, G is uniquely determined up to K 2 -isomorphism by its rank, and G = R K 2 /k(g). Hence, G is pseudo-split (!) with trivial center and has only inner automorphisms (over separable extensions of k). The absence of outer automorphisms in absolutely pseudo-simple cases with a non-reduced root system (over k s ) when [k : k 2 ] = 2 has the striking consequence that for such k a general pseudoreductive k-group G (perhaps not pseudo-semisimple) with a non-reduced root system over k s is uniquely a direct product G 1 G 2, where G 1 has a reduced root system over k s and G 2 = R k i /k(g i ) for a canonically determined collection {k i } of finite separable extensions of k and pseudo-split absolutely pseudo-simple k i -groups G i with a non-reduced root system (so G i is determined up to k i -isomorphism by its rank and the minimal field of definition K i /k i for its geometric unipotent radical) [CGP, Prop , Thm (1)]. Thus, if [k : k 2 ] = 2 then the contributions from the reduced and non-reduced irreducible components of the root system over k s can be uniquely separated from each other over k, with the non-reduced part well-understood Two substantial difficulties arise if ξ G is not surjective (so char(k) = 2, 3) or G ks has a non-reduced root system: (i) Assume G ks has a reduced root system (so ker ξ G is central) but ξ G is not surjective (so char(k) = 2, 3). The additional possibilities for ξ G (G) force us to go beyond the simply connected semisimple K-group G and consider members of a wider class of absolutely pseudo-simple groups G over finite extensions k of k, called generalized basic exotic and basic exceptional; see Chapter 8. (Such G have simply connected maximal geometric semisimple quotient, and the basic exceptional case which occurs if and only if [k : k 2 ] 16 rests on the equality of root systems B 2 = C 2.) For such G, the k-groups D(R k /k(g )) have a splitting property for central extensions as in ( ) if either char(k) = 3 or char(k) = 2 with [k : k 2 ] 8 (see [CGP, Prop ], 1.5, and Corollary B.3.4

6 6 BRIAN CONRAD AND GOPAL PRASAD below) but this fails more generally: if char(k) = 2 and [k : k 2 ] > 8 then D(R k /k(g )) can admit pseudo-simple central extensions by α 2 and by Z/(2) (see for pseudo-split examples with root system A 1, and see B.1 B.2 for pseudo-split examples with root system of type B or C of any rank 2). (ii) If char(k) = 2 and [k : k 2 ] > 2 then there are pseudo-split absolutely pseudo-simple k-groups G with root system BC n (any n 1) such that Z G = 1 but Aut F (G F ) is strictly larger than G(F ) for every separable extension field F/k (see [CGP, Ex ]). Consequently, in contrast with the case [k : k 2 ] = 2 in 1.4.2, it is generally impossible to split off (as a direct factor of G over k) the contribution from non-reduced irreducible components of the root system over k s ; see Example Moreover, as is explained in [CGP, ], the classification of pseudo-split absolutely pseudo-simple groups with a non-reduced root system over k s rests on invariants of linear algebraic nature that do not arise (in nontrivial ways) when [k : k 2 ] = To overcome 1.4.3(i) and 1.4.3(ii) when proving an exhaustive classification theorem with char(k) = 2 and [k : k 2 ] > 2, we need an entirely different technique to realize an absolutely pseudosimple k-group G as a central quotient D(R k /k(g ))/Z for a canonically determined member G of a well-understood class of absolutely pseudo-simple groups over a canonically associated purely inseparable finite extension k /k, with G ss k simply connected. (The pairs (G, k /k) replace ( G, K/k) from the standard absolutely pseudo-simple case above.) The construction of such central quotient maps D(R k /k(g )) G is not achieved by splitting a general class of central extensions, in contrast with ( ) in the standard case. For the pairs (G, k /k) that arise from our constructions, the associated perfect smooth connected k-groups D(R k /k(g )) have two special properties: the maximal semisimple quotient over k is simply connected, and the scheme-theoretic center contains no nontrivial unipotent k-subgroup scheme. This latter condition replaces the condition imposed on Z in ( ) that it does not contain any nontrivial smooth connected k-subgroup. For any perfect smooth connected affine group G over a field k, a central extension 1 Z E G 1 with affine E of finite type is called k-tame if Z contains no nontrivial unipotent k-subgroup scheme. A key example of such Z is R k /k(µ ) for an extension field k /k of finite degree and a k -group scheme µ of multiplicative type. In Chapter 5 we show that for any such G, if K/k is the minimal field of definiton over k for R u (G k ) G k then the set of k-tame central extensions E of G that are perfect, smooth, and connected is in natural bijection with the set of connected semisimple central extensions E of G := G ss K over K. Thus, there is a unique perfect smooth connected k-tame central extension G of G for which the associated connected semisimple central extension of G is simply connected; it is called the universal smooth k-tame central extension of G. This has nothing to do with pseudo-reductivity, and it is elementary to check that G is pseudo-reductive when G is. Our strategy is to pass from G to G (which has better properties) and then use a pseudoreductive variant of the Isomorphism Theorem to describe G in terms of a known list of constructions (when G is of minimal type in a sense discussed in 1.6). In an initial version of our work, the universal smooth k-tame central extension was not used; we argued differently via the Pseudo- Isogeny Theorem in Appendix A. As an example, if G := D(R k /k(g )) for a finite extension k /k and connected semisimple k - group G then G = R k /k( G ), where G G is the simply connected central cover. address a wider class of pseudo-semisimple possibilities for G. We next

7 CLASSIFICATION OF PSEUDO-REDUCTIVE GROUPS Generalized standard groups. In [CGP, Ch. 7 8], we constructed some non-standard absolutely pseudo-simple groups G called basic exotic over arbitrary imperfect fields k of characteristic 2 or 3 such that G ss k is simply connected and the root system Φ of G k is reduced. The s possibilities for Φ are F 4, B n, and C n in characteristic 2 (with any n 2) and G 2 in characteristic 3. Over any imperfect k of characteristic 2, in [CGP, Ch. 9] we constructed a class of pseudo-split absolutely pseudo-simple groups G with root system BC n (for any n 1); if [k : k 2 ] = 2 then the isomorphism classes of such k -groups are classified by (n, K /k ) where n is the rank and K is the minimal field of definition over k for the geometric unipotent radical of G (which can be any nontrivial purely inseparable finite extension of k ). For imperfect k of characteristic 2 or 3, with [k : k 2 ] = 2 in the BC n -cases, Weil restrictions to k of groups G as above over finite-degree extensions k /k are perfect and satisfy the splitting result for central extensions as in ( ) [CGP, Prop , Thm (3)]. (That splitting result for central extensions fails in some BC n -cases with k = k whenever [k : k 2 ] > 2; see Examples B.4.1 and B.4.3.) Thus, for imperfect k of characteristic 3 or of characteristic 2 with [k : k 2 ] = 2, it is reasonable to define a notion of generalized standard pseudo-reductive group by using Weil restrictions to k of both basic exotic groups and the known pseudo-split pseudo-simple groups with a non-reduced root system over finite-degree extensions k /k. This captures all deviations from standardness over such k (see [CGP, Thm ]), but such exhaustiveness fails when char(k) = 2 with [k : k 2 ] > 2. Consider imperfect k of characteristic 2. If [k : k 2 ] = 2 and G is a pseudo-split absolutely pseudosimple k-group of rank 1 with a reduced root system and G ss SL k 2 then G R K/k (SL 2 ) for a purely inseparable finite extension K/k (see [CGP, Prop ]). As we review in 3.1 (and analyzed in [CGP, ]), whenever [k : k 2 ] > 2 there are many more possibilities for such G: in addition to the field invariant K/k, there are linear algebra invariants (such as certain K -homothey classes of nonzero kk 2 -subspaces of K). These new pseudo-split pseudo-semisimple groups with root system A 1 can be used to shrink short root groups (for type B) or fatten long root groups (for type C) in pseudo-split basic exotic k-groups with rank n 2. Over any imperfect field k of characteristic 2 satisfying [k : k 2 ] > 2, in Chapters 7 and 8 we use this technique in conjunction with Example to construct many new absolutely pseudo-simple groups G for which G ss is simply connected (of type B or C) with rank k n 2; such k-groups have no analogue when [k : k 2 ] = 2. These refined constructions are called generalized basic exotic groups; see Definitions and In 8.3 we combine the equality of root systems B 2 = C 2 with the type-b basic exotic constructions in the rank-2 case to build another family of absolutely pseudo-simple groups called basic exceptional. The generalized basic exotic and basic exceptional groups (and the derived groups of their Weil restrictions through finite extensions of the ground field) satisfy a splitting result for central extensions as in ( ) when [k : k 2 ] 8 (see Corollary B.3.4), but this fails whenever [k : k 2 ] > 8 (see B.1 B.2). In [CGP, Ch. 9] we constructed pseudo-split absolutely pseudo-simple groups with root system BC n over any imperfect k of characteristic 2 (for any n 1). The data classifying the possibilities for such k-groups up to isomorphism is much more intricate when [k : k 2 ] > 2, and one encounters new behavior in these examples that never occurs when [k : k 2 ] = 2 (e.g., if K/k is the minimal field of definition of R u (G k ) G k then when [k : k 2 ] > 2 there can be proper subfields F K over k such that the non-reductive maximal pseudo-reductive quotient G pred F over F has a reduced root system; see [CGP, Ex ]). Weil restrictions to k of generalized basic exotic groups, basic exceptional groups, and the constructions with non-reduced root systems are used to define a generalized standard construction over an arbitrary field k; see Definitions and

8 8 BRIAN CONRAD AND GOPAL PRASAD In 10.2 it is shown that the generalized standard construction satisfies many nice properties. Moreover, by Proposition , the information required to make a generalized standard presentation of a pseudo-reductive k-group G consists of data that is uniquely functorial with respect to isomorphisms in the pair (G, T ) where T is a maximal k-torus of G (any T may be used). 1.6 Minimal type and general structure theorem. To formulate the exhaustiveness of the generalized standard construction, we need to consider the property of a pseudo-reductive k-group G being of minimal type: this notion was introduced in [CGP, Def ] and is reviewed in 2.4. Every pseudo-reductive k-group G admits a canonically associated pseudo-reductive central quotient G of minimal type with the same root datum as G (over k s ) [CGP, Prop (iii)]. To prove theorems about general pseudo-reductive groups, it is often harmless and genuinely useful to pass to the minimal-type case (e.g., see the proofs of Propositions and , Corollary , Propositions B.3.1 and C.1.1, and Theorem C.2.15). Generalized basic exotic and basic exceptional k-groups are of minimal type and admit an intrinsic characterization via this condition; see Definition and Theorem However, the minimal type property of a pseudo-reductive group is not inherited by pseudo-reductive central quotients in general; absolutely pseudo-simple counterexamples that are standard exist over every imperfect field (see Example 2.4.4). Hence, a general structure theorem for pseudo-reductive groups must go beyond the minimal type case. There is a weaker condition on a pseudo-reductive k-group G that we call locally of minimal type: for all roots a of (G ks, T ks ), the pseudo-simple k s -group (G ks ) a of rank 1 generated by the ±aroot groups admits a pseudo-simple central extension of minimal type. This property is inherited by pseudo-reductive central quotients of G, and it is easy to check that all generalized standard pseudo-reductive groups are locally of minimal type. In general, a pseudo-semisimple k-group is locally of minimal type if and only if its universal smooth k-tame central extension G is of minimal type (Proposition 5.2.3); passage from G to G is technically very useful in proofs. The locally minimal type property can only fail if char(k) = 2 and [k : k 2 ] > 2. In B.4 we show that if char(k) = 2 with [k : k 2 ] > 2 then for any n 1 there are pseudo-split absolutely pseudo-simple k-groups with root system BC n that are not locally of minimal type, and likewise (see 4.2.2, B.1 B.2) with root systems B n and C n for any n 1 when [k : k 2 ] > 8 (best possible, by Proposition B.3.1); here B 1 and C 1 mean A 1. These examples suggest that there is no general structure theorem (akin to the standard construction ) beyond the locally minimal type class. The condition locally of minimal type is more robust than minimal type, so we seek to describe pseudo-reductive groups locally of minimal type. Our main result (Theorem ) addresses this: Structure Theorem. A pseudo-reductive group locally of minimal type is generalized standard. The novelty is that we settle the case char(k) = 2 with [k : k 2 ] > 2 (the only k for which a pseudoreductive k-group can fail to be locally of minimal type). The cases char(k) 2 or char(k) = 2 with [k : k 2 ] 2 constitute the main content of [CGP, Thm ]. As an illustration, if G is an arbitrary pseudo-reductive group over a field k (we do not assume G to be locally of minimal type) then G/Z G is pseudo-reductive of minimal type (see Proposition 4.1.2), so it is generalized standard. However, even for standard G, the k-group G/Z G can fail to be perfect when G D(G). Indeed, if (R k /k(g ) C)/R k /k(t ) is the standard presentation associated to a maximal k-torus T G (with C = Z G (T )) then by arguing similarly to [CGP, Rem ] we see that G/Z G = (R k /k(g ) C)/R k /k(t ) where C R k /k(t /Z G ) is the (pseudo-reductive) image of C. This is a standard pseudo-reductive group, and it is pseudo-semisimple if and only if R k /k(t ) C is surjective. Thus, we get standard pseudo-reductive G satisfying Z G = 1 and

9 CLASSIFICATION OF PSEUDO-REDUCTIVE GROUPS 9 G D(G) by taking C = R k /k(t /Z G ) for a field k that is not separable over k and G such that Z G is not k -smooth (e.g., G = SL p with p = char(k) > 0). 1.7 Galois-twisted forms. A k-form of a pseudo-reductive group G over a field k is a pseudoreductive k-group H such that H ks G ks. If G is reductive then it admits a unique pseudo-split k-form. We prove such uniqueness in the the pseudo-reductive case (Proposition C.1.1) via a pseudo-reductive version of the Isomorphism Theorem (Theorem 6.2.1). Existence is harder; the commutative case seems hopeless, so we focus on the pseudo-semisimple case (in Appendix C). Over many imperfect k (with positive characteristic p) there are pseudo-semisimple G without a pseudo-split k-form, due to a field-theoretic obstruction that cannot arise if G is absolutely pseudosimple or if [k : k p ] = p; see Example C.1.2. Additional examples allowing [k : k p ] = p are given in Example C.1.3, but those are also not absolutely pseudo-simple. By using the full force of [CGP], for absolutely pseudo-simple G we show that a pseudo-split k-form exists when char(k) 2 and also when char(k) = 2 with [k : k 2 ] = 2 except possibly if G is standard of type D 2n with n 2 and k admits a quadratic Galois extension (or a cubic Galois extension when n = 2). The same conclusion holds in the standard absolutely pseudo-simple case when char(k) = 2 without restriction on [k : k 2 ] (subject to the same exceptions for type D 2n ). This is optimal because for every n 2 and every imperfect field k of characteristic 2 that admits a quadratic (or cubic when n = 2) Galois extension, we construct a standard absolutely pseudo-simple k-group G of type D 2n that does not admit a pseudo-split k-form. These matters are discussed in C.1. Suppose char(k) = 2 with [k : k 2 ] > 2, and consider non-standard absolutely pseudo-simple k- groups G. If [k : k 2 ] = 4 then a pseudo-split k-form exists (Corollary C.2.16), so assume [k : k 2 ] 8. If k has sufficiently rich Galois theory then in Example C.3.1 we make (non-standard) A 1 -examples over k without a pseudo-split k-form. These are used in C.4 to make higher-rank non-standard examples without a pseudo-split form over such k: generalized basic exotic k-groups whose root system over k s is B n or C n for any n 2, and minimal type absolutely pseudo-simple k-groups whose root system over k s is BC n for any n 1. In 6.2 we prove that if G = D(G) then the automorphism functor of G on k-schemes is represented by a finite type affine k-group scheme Aut G/k (it is often not representable for commutative G; see Example 6.2.5). If G = D(G) then Aut G/k is generally not smooth (Example 6.2.7) but its maximal smooth closed k-subgroup Aut sm G/k has structure analogous to the semisimple case. In C.2 the notion of pseudo-inner form is defined in terms of (Aut sm D(G)/k )0, and we use the Structure Theorem from 1.6 and the structure of (Aut sm D(G)/k )0 to prove uniqueness of pseudo-inner forms that are quasi-split (i.e., minimal pseudo-parabolic k-subgroups are k s -minimal), as well as the existence of such k-forms provided [k : k 2 ] 4 if char(k) = 2. If char(k) = 2 and [k : k 2 ] 8 then the above non-standard k-groups of types B, C, or BC over k s without a pseudo-split form also have no quasi-split form (due to Lemma C.2.2). Notation. For a scheme X of finite type over a field k and a closed subscheme Z of X K for an extension field K/k, the intersection of all subfields k K over k such that Z descends (necessarily uniquely) to a closed subscheme of X k is also such a subfield, called the minimal field of definition of Z X K relative to k; see [EGA, IV 2, 4.8] for a detailed discussion of the existence of such a field. The interaction of this minimal field with respect to extension of the ground field is addressed in [CGP, Lemma 1.1.8]. (In [CGP] the terminology field of definition is used, with minimality always understood to be required. In this monograph we explicitly keep minimality in the terminology.)

10 10 BRIAN CONRAD AND GOPAL PRASAD For a smooth connected affine group G over a field k and the (minimal) field of definition K/k for R u (G k ) G k, we define G red K to be the quotient G K/R u,k (G K ) that is a K-descent of the maximal reductive quotient G red of G k k. Taking K/k instead to be the minimal field of definition for R(G k ) G k yields the quotient G ss K := G K/R K (G K ) of G K as a K-descent of the maximal semisimple quotient of G k. The quotient G/R u,k (G) is maximal among all pseudo-reductive quotients of G over k, so it is denoted G pred. Using terminology reviewed in 2.4, the quotient G pred /C G pred of G is pseudoreductive of minimal type and is maximal among all such quotients of G; it is denoted G prmt. Acknowledgements We thank Ofer Gabber for his very illuminating advice and suggestions, and Kęstutis Česnavičius and Teruhisa Koshikawa for helpful comments. G.P. would like to thank the Institute for Advanced Study (Princeton), and his host Peter Sarnak, for hospitality and support during Much of this work was done while he visited the Institute. He would also like to thank the Mathematics Research Center at Stanford University for support during a visit in the summer of 2013 and thank RIMS (Kyoto) for its hospitality during July B.C. is grateful to IAS for its hospitality during several visits while working on this monograph. B.C. was supported by NSF grant DMS and G.P. was supported by NSF grants DMS and DMS

11 CLASSIFICATION OF PSEUDO-REDUCTIVE GROUPS 11 Contents 1. Introduction Motivation The standard construction Exotic groups and degenerate quadratic forms Tame central extensions Generalized standard groups Minimal type and general structure theorem Galois-twisted forms 9 2. Background review Standard groups Pseudo-semisimplicity and root systems Exotic constructions Minimal type Groups of rank-1 and the minimal field of definition for R u (G k ) A non-standard rank-1 construction Minimal field of definition for R u (G k ) Central extensions and groups locally of minimal type Central extensions Beyond the quadratic case Groups locally of minimal type Universal smooth k-tame central extension Construction of central extensions Properties of G Root fields, automorphisms, and isomorphisms Root field Isomorphism Theorem and automorphism schemes Groups with a non-reduced root system Constructions with regular degenerate quadratic forms Type F Regular degenerate quadratic spaces Conformal isometries and Brauer Severi varieties Constructions when Φ has a double bond Additional constructions for type B Constructions for type C Exceptional construction for rank Generalized exotic groups Definitions and elementary properties Functoriality Intrinsic characterization Automorphisms and main theorem Center and automorphism schemes Generalized standard groups Structure theorem 112 Appendix A. Pseudo-Isogenies 117 A.1. Main result 117

12 12 BRIAN CONRAD AND GOPAL PRASAD A.2. Proof of Pseudo-Isogeny Theorem 118 A.3. Relation with semisimple case 120 Appendix B. Clifford constructions 121 B.1. Type B 121 B.2. Type C 122 B.3. Cases with [k : k 2 ] B.4. Type BC 125 Appendix C. Pseudo-split and quasi-split forms 131 C.1. General characteristic 131 C.2. Quasi-split forms 138 C.3. Rank-1 cases 150 C.4. Higher-rank and non-reduced cases 151 Appendix D. Basic exotic groups of type F 4 and relative rank References 154

13 CLASSIFICATION OF PSEUDO-REDUCTIVE GROUPS 13 2 Background review 2.1 Standard groups. Let k be a field. The standard construction of pseudo-reductive groups provides both a general structure theorem for pseudo-reductive k-groups when char(k) 2 as well as a guide for the main results we shall prove in this monograph when char(k) = 2. As motivation, consider a pseudo-reductive k-group G with (minimal) field of definition k /k for its geometric unipotent radical. The extension k /k is purely inseparable of finite degree and R u (G k ) descends to the maximal smooth connected unipotent normal k -subgroup R u,k (G k ) G k. Consider the maximal reductive quotient G := G k /R u,k (G k ) over k. The natural map i G : G R k /k(g ) (2.1.1) is a first attempt to relate G to a canonically associated Weil restriction of a connected reductive group. This construction is problematic for two reasons: (i) the field k is a coarser invariant than the collection of minimal fields of definition k j /k for the kernels of projections of G k onto the simple factors H j of the adjoint central quotient of the maximal reductive quotient Gred G ad k k, (ii) i G might have nontrivial kernel. The following construction bypasses both of these problems. Let k be a nonzero finite reduced k-algebra. Let G Spec(k ) be a smooth affine group scheme with connected reductive fibers. Writing k = k i for fields k i, and letting G i denote the k i -fiber of G, R k /k(g ) = R k i /k(g i ). Let T be a maximal k -torus in G, and define T := T /Z G to be the associated maximal k -torus in G := G /Z G, where Z G denotes the scheme-theoretic center of G. Suppose there is given a commutative pseudo-reductive k-group C and a k-group factorization R k /k(t ) φ C R k /k(t ) (2.1.2) of the Weil restriction to k of the canonical quotient map q : T T. (Beware that R k /k(q) may not be surjective when k is not k-étale, and we do not require φ to be surjective.) The natural G -action on G over k defines a natural R k /k(g )-action on R k /k(g ) over k, and hence a natural action of C on R k /k(g ) via composition with C R k /k(t ) R k /k(g ). The anti-diagonal map R k /k(t ) R k /k(g ) C (2.1.3) is an inclusion with central image, and the associated central quotient G = (R k /k(g ) C)/R k /k(t ) is always pseudo-reductive [CGP, Prop ]. Informally, G is obtained from R k /k(g ) by replacing the Cartan k-subgroup R k /k(t ) with the commutative pseudo-reductive k-group C (whose structure we treat as a black box); G is a kind of pushout of R k /k(g ) along φ. Such G are called standard, and C is a Cartan k-subgroup of G. Every commutative pseudo-reductive k-group is standard, by using k = k, T = G = 1, and C = G. By [CGP, Thm ] any non-commutative standard pseudo-reductive k-group G admits a standard presentation : a description as in (2.1.3) using a 4-tuple (G, k /k, T, C) as above (including a specified factorization (2.1.2) that we suppress from the notation) for which the fibers of G Spec(k ) are semisimple, absolutely simple, and simply connected. If j : R k /k(g ) G denotes the map arising from a standard presentation then the triple (G, k /k, j) is uniquely determined by G up to unique isomorphism [CGP, Prop ] and there is a natural bijection between the set of maximal k -tori T in G and the set of Cartan k-subgroups C of G via the requirement that j(r k /k(t )) C [CGP, Prop (2),(3)]. In this sense, a standard presentation

14 14 BRIAN CONRAD AND GOPAL PRASAD of a non-commutative standard pseudo-reductive k-group G amounts to a choice of a Cartan k- subgroup of G (or equivalently, a choice of maximal k-torus of G). 2.2 Pseudo-semisimplicity and root systems. Before we discuss phenomena specific to characteristic 2, let us recall additional general terminology and results over any field k. If G is a smooth connected affine k-group then its k-unipotent radical R u,k (G) is the maximal smooth connected unipotent normal k-subgroup of G, so G/R u,k (G) is the maximal pseudo-reductive quotient of G. The k-radical R k (G) is the maximal smooth connected solvable normal k-subgroup of G. If K/k is an extension of fields then R u,k (G) K R u,k (G K ). This inclusion is an equality if K is separable over k [CGP, Prop ] but generally not otherwise (e.g., equality fails with K = k for any imperfect k and non-reductive pseudo-reductive G). Taking K = k recovers the fact that if k is perfect then pseudo-reductive k-groups are precisely connected reductive k-groups. The triviality of R k (G) is a consequence of pseudo-semisimplicity, but there are pseudo-reductive k-groups G such that R k (G) = 1 yet G D(G); see [CGP, 11.2]. In particular, if G is a smooth connected affine k-group then G/R k (G) might not be pseudo-semisimple (though it is pseudo-reductive). As for any smooth connected affine k-group, every pseudo-reductive k-group is generated by its derived group and a single Cartan k-subgroup. Thus, the main work in describing pseudo-reductive groups lies in the pseudo-semisimple case. Assume G is pseudo-split, and let T be a split maximal k-torus in G, so S := T D(G) is a split maximal k-torus of D(G) and T is an almost direct product of S and the maximal central k-torus Z of G [CGP, Lemma 1.2.5]. Consider the T -action and S-action on Lie(G). As is explained in the proof of [CGP, Thm ], the weight spaces in Lie(G) for nontrivial T -weights are supported inside Lie(D(G)) and coincide with the weight spaces for nontrivial S-weights. These weight spaces can have very high dimension, but nonetheless the pair (X(T ), Φ(G, T )) can be naturally enhanced to be a root datum for which the Q-span of Φ(G, T ) maps isomorphically onto the quotient X(S) Q of X(T ) Q [CGP, 3.2]. The underlying root system of this root datum is reduced whenever char(k) 2 and also when char(k) = 2 provided that G red has no connected semisimple normal subgroup that k is simple and simply connected of type C (where C 1 = A 1 ). In general the set of non-multipliable roots in Φ(G, T ) is equal to Φ(G red, T k k ), and this maps isomorphically onto a reduced root system in X(S) Q [CGP, Thm ]. Lemma Let H H be an inclusion of pseudo-reductive groups over a field F, and assume that their maximal tori have the same dimension. Let T be a maximal F -torus of H, and assume that Φ(H Fs, T Fs ) = Φ(H F s, T Fs ). A connected reductive F -subgroup L H is a Levi F -subgroup of H if and only if it is a Levi F -subgroup of H. Proof. We may and do assume F = F s. By [CGP, Lemma 7.2.4], if L is a Levi F -subgroup of H then it is a Levi F -subgroup of H (without needing to assuming the equality of root systems, which could fail when Φ(H, T ) is non-reduced). Assume instead that L is a Levi F -subgroup of H, and let Φ denote the common root system for (H, T ) and (H, T ). Fix a positive system of roots Φ + and let Φ denote the set of non-multipliable roots in Φ, so Φ = Φ(L, T ). Then Φ + := Φ + Φ is a positive system of roots in Φ ; let be the basis of simple roots in Φ +. By [CGP, Thm ], Levi F -subgroups of H containing T are uniquely determined by their root groups E a for a, and each E a may be chosen arbitrarily among the 1-dimensional T -stable smooth connected F -subgroups of the a-root group of (H, T ) (defined as in [CGP, Def ]: its Lie algebra is the a-weight space since a is not multipliable). Hence, there is a unique Levi F -subgroup L of H containing T such that its a-root group E a

15 CLASSIFICATION OF PSEUDO-REDUCTIVE GROUPS 15 coincides with the a-root group E a of (L, T ) for all a. Our task is to prove L = L as F -subgroups of H. It suffices to prove E a = E a for all a because any connected reductive group equipped with a chosen maximal torus is generated by the maximal torus and its root groups for the simple positive and negative roots relative to a choice of positive system of roots in the root system. Let U a be the a-root group for (H, T ), and let U a be the a-root group for (H, T ). By the dynamic construction of such root groups, U a H = U a. Choose a nontrivial element h E a (F ) = E a(f ). By [CGP, Prop ], there are unique elements u, v U a(f ) {1} such that u hv N H (T )(F ) and unique u, v U a (F ) {1} such that uhv N H (T )(F ). From the uniqueness we conclude that u = u and v = v, so u, v U a (F ) {1}. The same reasoning applies to (L, T ) and (L, T ), so in fact u, v E a(f ) E a (F ). The T -orbits of these nontrivial elements under conjugation exhaust E a {1} and E a {1}, so E a = E a as desired. 2.3 Exotic constructions. The only nontrivial multiplicities that occur for the edges of Dynkin diagrams of reduced irreducible root systems are 2 and 3, and this underlies the existence of exceptional isogenies that only exist in characteristics 2 and 3. These exceptional isogenies are used to construct certain non-standard absolutely pseudo-simple groups, called basic exotic, whose definition we now recall. One of the main tasks of this monograph (see Chapters 7 9) is to find a generalization of the basic exotic construction that is well-suited to proving a structure theorem over any imperfect k of characteristic 2, without restriction on [k : k 2 ]. Let k be imperfect with characteristic p {2, 3}, and let k /k be a nontrivial finite extension field satisfying k p k. Let G be a connected semisimple k -group that is absolutely simple and simply connected with an edge of multiplicity p in its Dynkin diagram (so G has type G 2 if p = 3 and type F 4 or B n or C n with n 2 if p = 2). By [CGP, Lemma 7.1.2], there is a unique minimal non-central normal k -subgroup scheme N G with vanishing relative Frobenius morphism, and we call the map π : G G := G /N the very special isogeny of G. Example If G k has root system B s n (n 2), so p = 2, then G = Spin(q ) for a nondegenerate quadratic space (V, q ) over k of dimension 2n + 1 and if V denotes the defect line of the symmetric bilinear form B q (x, y) = q (x + y) q (x) q (y) on V then π is the composition of G G /Z G = SO(q ) with the unipotent isogeny SO(q ) Sp(B q ) Sp 2n, where B q is the symplectic form on V /V arising from B q (so in this case G is k -split). Let T G be a maximal k-torus, and T = π(t ). By [CGP, Prop ], π carries long root groups for (G, T ) isomorphically onto short root groups for (G, T ) and carries short root groups for (G, T ) onto long root groups of (G, T ) via a Frobenius isogeny. Moreover, G is simply connected with root system dual to that of G, and the factorization of the Frobenius isogeny F G /k : G G (p) through π is via an isogeny π : G G (p) that is the very special isogeny of G. For types F 4 in characteristic 2 and G 2 in characteristic 3, this provides the unique nontrivial factorization of F G /k. Consider the Weil restriction f = R k /k(π ) : R k /k(g ) R k /k(g ); this is not surjective. For any Levi k-subgroup j : G R k /k(g ) (if one exists) the associated map G k G is an isomorphism (see [CGP, Lemma 7.2.1] for a precise link between such Levi k-subgroups and k-descents of G ), and by [CGP, Prop ] the following are equivalent: G lies inside the image of f, f 1 (G) is smooth, and f 1 (G) ks contains a Levi k s -subgroup of R k /k(g ) ks. When these equivalent conditions hold,