Generic Splitting of Reductive Groups

(to appear in the Tôhoku Mathematical Journal)

Ina Kersten* and Ulf Rehmann

In memory of Ernst Witt

Abstract. Generic conditions for the occurrence of a parabolic subgroup of given type in a reductive algebraic group are described. Especially the notion of a generic splitting field of a reductive algebraic group is investigated. The given theory generalizes and unifies other investigations of various authors for special algebraic structures such as Azumaya algebras and quadratic forms.

Introduction

The “degree of splitting” of a connected semisimple algebraic group G over a field k is essentially determined by the types of parabolic k-subgroups of G. For example, G is anisotropic if G itself is the only parabolic k-subgroup, G is quasi-split if it contains a Borel subgroup, and G is split if it contains parabolic subgroups of every possible type.

We now assume that G is a connected reductive linear algebraic group over k. One of our main goals is to describe generic conditions for a field extension K of k, which guarantee the existence of a parabolic subgroup of G_{K} of a prescribed type, where G_{K} = G ×_{k} K denotes the algebraic group over K obtained from G by scalar extension.

Let k̄ denote an algebraic closure of k. It is known that Gk̄ splits and that the conjugacy classes of parabolic subgroups of Gk̄ are in one-to-one correspondence with the subsets of the vertices Δ of the Dynkin diagram of Gk̄. The subset Θ ⊆ Δ corresponding to the class of a parabolic subgroup P of Gk̄ is called the type of P. The set Δ itself is the type of Gk̄ and the empty set ∅ is the type of a Borel subgroup of Gk̄.

In §3 we show that the occurrence of parabolic subgroups of given type is preserved under kspecializations. More precisely, in 3.9 we prove, for any field extension L of k: If there is a parabolic subgroup of type Θ in G_{L}, then there is a parabolic subgroup of type Θ in G_{k′} for every k-specialization k′ of L, that is, for every field extension k′ such that there is a k-place L → k′ ∪ {∞}.

This leads us to the definition of a generic Θ-splitting field of G for any subset Θ ⊆ Δ. A field K is called a Θ-splitting field of G if G_{K} contains a parabolic K-subgroup of type Θ, and a Θ-splitting field F of G is called generic if every Θ-splitting field of G is a k-specialization of F. Especially a (generic) quasi-splitting field of G is a (generic) ∅-splitting field of G.

In order to find a generic Θ-splitting field of G, we study, in §3, the quotient variety V_{Θ} := Gk̄/P for a parabolic subgroup P of Gk̄ of any type Θ. Since P is self-normalizing, V_{Θ} can be identified with the conjugacy class of P in Gk̄. It is known that the variety V_{∅} is always defined over k, and we will see in 3.11 that its function field k(V_{∅}) is a generic quasi-splitting field of G. We show, more generally, that V_{Θ} is defined over a “small” finite and separable field extension k_{Θ} of k, which is the smallest extension of k such that the so called ∗-action of the Galois group Gal (k_{s}/k_{Θ}) on Δ leaves Θ invariant. (By k_{s} we denote the separable closure of k contained in k̄.) Hence k_{Θ} = k in most cases. Especially this is always true for groups of inner type. We will see in 3.16 that, for any Θ, the function field k_{Θ}(V_{Θ}) is a generic Θ-splitting field of G. Any Θ-splitting field of G contains a copy of k_{Θ}, and the Θ-splitting fields K of G are – as field extensions of k_{Θ} – characterized by the condition that K(V_{Θ}) is a purely transcendental extension of K (cf. 3.10, 3.16).

A Θ-splitting field K of G splits G “partially” in the sense that the rank of G_{K} is greater than or equal to the rank of G, but is not necessarily equal to the maximal possible value, in which case K would be a splitting field of G (cf. 1.7 and 1.10 below). If K is a splitting field of G, then the semisimple part of G_{K} is a group of Chevalley type.

Another main goal of this paper is to exhibit subsets Θ of Δ such that a corresponding generic Θ-splitting field is a generic splitting field of G. Similarly as above, a splitting field F of G is called generic if every splitting field of G is a k-specialization of F.

* Supported by the Sonderforschungsbereich 343 “Diskrete Strukturen in der Mathematik” at Universität Bielefeld sponsored by the Deutsche Forschungsgemeinschaft.

1991 Mathematics Subject Classification. Primary 20G15; Secondary 16H05, 11E04, 14L35, 14M15.

Our theory of generic Θ-splitting fields of reductive groups unifies several other investigations of similar kind for different special algebraic structures.

The earliest example of a generic splitting field has been given by Witt [37, 1935], who constructed a generic splitting field for a quaternion skew field D over k. Here splitting of course means the splitting of D into a full 2 × 2 matrix ring. We will show that the generic splitting field constructed by Witt is precisely the function field k(V_{∅}) for the algebraic k-group G = SL_{1}(D) (cf. 3.20). Witt’s result has been generalized to central simple k-algebras by Amitsur [2, 1955]. The varieties which occur in this context are the Severi-Brauer varieties over k which can be described as the k-forms of projective space. A different approach to this construction making use of non-abelian Galois cohomology has been given by Roquette [23, 1963], [24, 1964]. These results occur as particular cases in our discussion of the partial generic splitting of the algebraic group G = SL_{r+1}(D) for a finite dimensional central skew field D over k and r ≥ 0 (cf. 4.9 below). Moreover, Roquette proved [23, Th. 4, p. 413] that the function field of the Severi-Brauer variety related to the full matrix ring M_{r+1}(D) over D is a purely transcendental extension of that of the Severi-Brauer variety of D. Translated into our theory, this becomes a particular case of the fact (cf. 3.18, 3.19 below) that the generic Θ-splitting field k(V_{Θ}) of G is a purely transcendental extension of a certain corresponding Θ_{an}-splitting field of the semisimple anisotropic kernel G_{an} of G (cf. 1.8 below).

To the best of our knowledge the first who had the idea of studying partial generic splitting instead of just total generic splitting was Knebusch in the 70’s. He investigated partial generic splitting of quadratic forms [16, 1976], [17, 1977], thereby introducing his generic splitting towers. Then his student Heuser [12, 1976] studied partial generic splitting of central simple algebras, using the function fields of generalized Severi-Brauer varieties of prescribed level. It turned out that the splitting behavior of central simple algebras is much more uniform than that of quadratic forms (cf. 4.8 ii) and 5.8 below). As we were told, Knebusch, puzzled by this phenomenon, suggested already then to study partial generic splitting of linear algebraic groups.

Recently, also Blanchet [4, 1991] and Schofield/v. d. Bergh [26, 1991], [27, 1991] studied the partial generic splitting of central simple algebras by means of generalized Severi-Brauer varieties. As we point out in §4 these generalized Severi-Brauer varieties are the quotients of G by arbitrary maximal proper parabolic subgroups of G = SL_{r+1}(D) (here Θ consists of Δ minus one element).

Similarly, the layers of a generic splitting tower of a quadratic form q are, in our theory, achieved by the function fields of the quotients of G = SO(q) modulo its various maximal proper parabolic subgroups. This is discussed in §5.

Another uniformizing approach which establishes and generalizes the results of Amitsur and Knebusch above and which uses the techniques and terminologies of Jordan algebras and Jordan pairs has been given independently by Petersson [21, 1984] and by Jacobson [14, 1985] (for a survey, see [15, 1989]). Especially [14, §7, p. 591] gives results on generic splitting of involutorial simple associative algebras which can be transformed into special cases of our Theorem 6.1.

The authors want to express their gratitude to Manfred Knebusch who enthusiastically encouraged them to investigate partial generic splitting of algebraic groups.

We now briefly describe the contents of the various sections of this paper.

In §1 we collect some facts about varieties, splitting fields, reductive linear algebraic groups and their anisotropic kernels.

In §2 the generic splitting of algebraic tori is discussed.

In §3 we set up the framework of reductive groups and the rational theory of parabolic subgroups in order to prove the main results: Theorem 3.6 describes how to obtain a generic splitting field from a generic quasi-splitting field, Theorem 3.10 describes the fundamental properties of the varieties V_{Θ} (becoming rational exactly over specializations of their function fields). 3.11 – 3.17 prove the existence and describe the properties of generic Θ-splitting fields, 3.18 – 3.19 relate the general results to the respective anisotropic kernel. We conclude this chapter with the discussion of Witt’s first example (3.20) of generic splitting, namely the generic splitting of quaternion algebras.

In §4 we describe the generic Θ-splitting of groups of type^{1}A_{n}. We show how several essential results on generic splitting fields obtained by Amitsur, Roquette, Heuser, Blanchet and Schofield/v. d. Bergh can be deduced from our theory by taking proper maximal subsets Θ of Δ.

In §§5 and 6 we assume char (k) ≠ 2. In §5 we discuss the partial and total generic splitting of quadratic forms including its relations to the work of Knebusch. In §6 we describe the generic Θ-splitting of the classical groups of types ^{2}A_{n}, B_{n}, C_{n}, ^{1}D_{n} and ^{2}D_{n}.

In §7 we give, for all characteristics, generic splitting and quasi-splitting fields of arbitrary almost simple groups including the groups of exceptional types.

1. Some basic definitions and lemmas Let k be field and k̄ an algebraic closure of k.

In this paper, a k-variety V is an absolutely reduced quasi-projective scheme over k. Any algebraic k-group is supposed to have a k-variety structure in this sense. For any field extension k′ of k, we denote the set of k′-rational points of V by V (k′) = Hom_{k−scheme}(Spec k′, V) and we write V_{k′} := V ×_{k} k′ for the k′-scheme obtained from V by base extension with k′.

1.1 Lemma. If V is an absolutely irreducible k-variety, then k is algebraically closed in the function field k(V) and k(V) is separably generated over k.

For the proof see [20, Chap. II, §4, Prop. 4, p. 142].

Recall that a finitely generated field extension k′ of k is said to be regular if k̄ and k′ are linearly disjoint or, equivalently, if k is algebraically closed in k′, and k′ is separably generated over k, cf. [36, Chap. I.7, Th. 5, p. 18]. If V is an absolutely irreducible k-variety, then, by 1.1, for any field extension k′ of k, the free composite k′k(V) is uniquely determined up to k-isomorphism [13, Chap. IV, Cor. 1, p. 203, Th. 26, p. 209] and is isomorphic to k′(V_{k′}) [36, Chap. I.7, Th. 5, p. 18].

1.2 Definition. A field extension k′ of k is a k-specialization of an extension L of k if there is a k-place L → k′ ∪ {∞}.

1.3 Lemma. Let V be an absolutely irreducible projective k-variety. Let L, k′ be field extensions of k such that k′ is a k-specialization of L. Then V (L) ≠ ∅ implies V (k′) ≠ ∅.

Proof. There is a homogeneous ideal I in the polynomial ring k[X_{0}, …, X_{n}] for a suitable n together with a bijection of sets

V (K) →^{∼} {(x_{0} : …: x_{n}) ∈ ℙ^{n}(K) | f(x_{0}, …, x_{n}) = 0, ∀f ∈ I}

for every field extension K of k.

Let φ: L → k′ ∪ {∞} be the k-place describing k′ as a k-specialization of L, and let O_{φ} denote the valuation ring of φ.

Let x = (x_{0} : …: x_{n}) ∈ V (L). We choose j ∈ {0, …, n} such that the principal ideal x_{j}O_{φ} is maximal among the ideals x_{i}O_{φ}. This is possible because O_{φ} is a valuation ring, cf. [9, Chap. VI, §1, No. 2, Th. 1d].

Clearly, we have x_{j} ≠ 0, whence x = (x′_{0} : …: x′_{n}) with x′_{i} := x_{i}/x_{j} ∈ O_{φ} for all i = 0, …, n and x′_{j} = 1, and it follows that (φ(x′_{0}) : …: φ(x′_{n})) ∈ V (k′).

Let G be a connected affine algebraic k-group. This implies that Gk̄ is k̄-connected [11 I, Exp. VI_{2}, Prop. 2.1.1, p. 296].

1.4 Theorem. G has a maximal k-torus.

For the proof see [11 II, Exp. XIV, Th. 1.1, p. 296] or [6, Th. 18.2, p. 218].

We assume for the rest of this paragraph that G is reductive.

1.5 Definition. Let K be a field extension of k.

i) K is a splitting field of G if G_{K} has a maximal K-torus which splits over K. ii) K is a quasi-splitting field of G if G_{K} has a Borel subgroup defined over K. iii) A splitting field (resp. quasi-splitting field) K of G is said to be generic if every splitting field (resp.

quasi-splitting field) of G is a k-specialization of K.

For the notion of a split connected reductive affine algebraic group compare [6, 18.6, 18.7, p. 220ff] and [7, 8.1, 8.2, p. 481ff].

1.6 Remark. Obviously two generic splitting fields of G are k-equivalent to each other in the sense that they are k-specializations of each other. We shall show in 3.9 iii) that every k-specialization of a splitting field of G is also a splitting field of G. So we have the following result: If K and K′ are k-equivalent field extensions of k and if one of them is a generic splitting field of G, so is the other. In particular, K is a generic splitting field of G if this is true for some purely transcendental extension K({x_{i}}_{i∈I}) of K.

1.7 Remark. i) A field extension K of k is a splitting field of G if and only if rank (G_{K}) = rank (Gk̄) holds. We

write rank(G) for the k-rank of the k-group G, that is, the dimension of a maximal k-split k-torus of G (cf. [8, 4.21, p. 93]).

ii) If k is finite then G is quasi-split (cf. [6, 16.6, p. 211]), and the semi-simple groups over k are completely classified (cf. [30], [31]). Therefore we will always assume that the base field k is infinite except in §§1 and 2.

iii) It is known that G has a splitting field which is finite and separable over k. This follows from 1.4 and the fact that any k-torus has such a splitting field [6, 8.11, p. 117]. However, if K is a generic quasi-splitting field of G, then K does not split any nontrivial anisotropic k-torus of G (cf. Cor. 3.12 below).

1.8 Definition.

i) G is isotropic if it contains a non-trivial k-split k-torus and is anisotropic if rank (G) = 0. ii) If S is a maximal k-split k-torus of G and Z(S) its centralizer in G, then the derived group DZ(S)

is called a semisimple anisotropic kernel of G. If Z_{an} is the maximal anisotropic k-subtorus of the center of Z(S), then DZ(S) · Z_{an} is called a reductive anisotropic kernel of G.

Our notion of an (an-)isotropic group seems to be standard now, as it is used in [5, 6.4, p. 13], [32, 2.2, p. 39], [8, 4.23, p. 93] and [6, 20.1, p. 224]. It differs, however, from the definition in [29, 6.5, p. 476], where G is said to be anisotropic if its split k-subtori all are central.

1.9 Proposition.

i) The semisimple anisotropic kernels of G are precisely the subgroups occurring as derived groups of Levi k-subgroups of minimal parabolic k-subgroups of G. Any two such are conjugate under G(k).

ii) The anisotropic kernels of G are anisotropic k-groups. iii) G is quasi-split if and only if its semisimple anisotropic kernel is trivial.

Proof. i) If S is a maximal k-split k-subtorus of G, then Z(S) is a Levi k-subgroup of a minimal parabolic k-subgroup P of G by [8, 4.15, 4.16, p. 91]. Conversely, if P is a minimal parabolic k-subgroup, then the Levi k-subgroups of P are the centralizers of maximal k-split k-tori of G (contained in the radical of P), [8, 4.16, p. 91]. This proves the first assertion of i). The second follows from the fact that all maximal k-split k-tori are conjugate over k by [8, 4.21, p. 93] or [6, 20.9, p. 228].

ii) It suffices to prove the statement for semisimple anisotropic kernels. Let S be a maximal k-split k-torus of G. Being reductive, Z(S) is an almost direct product of its maximal semisimple subgroup DZ(S) and the identity component of its center which contains S (cf. [8, 2.2, p. 64]). The maximality of S now asserts that DZ(S) does not contain any nontrivial k-split k-torus.

iii) By definition, G is quasi-split if and only if it contains a Borel k-subgroup. Hence the statement follows from i).

1.10 Corollary (cf. [8, 4.17, p. 92]). G contains a non-central k-split k-torus if and only if it contains a proper parabolic k-subgroup.

2. Generic splitting of algebraic tori Let k be a field and let T be an algebraic k-torus (i.e., there is a field extension K of k such that T_{K} ≅ 𝔾_{m} ×_{K} ⋯ ×_{K} 𝔾_{m}, where the multiplicative K-group 𝔾_{m} is defined by 𝔾_{m}(K) = K^{∗}).

2.1 Lemma. Let L be a field extension of k which splits T. Then the subfield k̃ of L of elements which are separable algebraic over k also splits T.

Proof. Let ϱ: T → GL(W) be a faithful k-rational representation on a finite dimensional linear k-space W. By assumption there is an L-basis of W ⊗_{k} L such that every t ∈ T (L) is described by a diagonal matrix with respect to this basis [6, 8.2 Prop. (d), p. 112]. Hence, for every t ∈ T (k̃), the minimal polynomial m_{t}(X) ∈ k̃[X] decomposes into pairwise distinct linear factors

m_{t}(X) = ∏_{i}(X − α_{i}^{(t)}), with α_{i}^{(t)} ∈ L.

It follows that the α_{i}^{(t)} are separable over k̃, hence over k. Therefore every t ∈ T (k̃) is diagonizable over k̃. Since T (k̃) is commutative, there is a k̃-basis of W ⊗_{k} k̃ which diagonalizes T (k̃).

2.2 Proposition. Let L be a splitting field of T. Then any k-specialization k′ of L is a splitting field of T.

Proof. Let φ: L → k′ ∪{∞} be the place describing k′ as a k-specialization of L. By 2.1, the subfield k̃ of separable algebraic elements of L over k splits T. By [38, Cor. 1, p. 13], the restriction of φ to k̃ is injective, hence k′ splits T.

2.3 Theorem. An algebraic k-torus T has an algebraic generic splitting field, say F, which is unique up to k-isomorphism and is a finite Galois extension of k. Every splitting field of T contains a subfield isomorphic to F.

Proof. Let k̄ denote an algebraic closure of k, and define

F := ⋂{ L | k ⊆ L ⊆ k̄, L splitting field of T }.

Then F is a finite separable field extension of k because T has a finite separable splitting field (cf. [6, 8.11, p. 117], [7, Cor. 8.3, p. 482]).

Let K be a splitting field of T. By 2.1, the subfield k̃ ⊆ K of elements separable over k splits T. Then k̃ contains a subfield isomorphic to F by the definition of F.

We have to show that F splits T. Then it will follow from the above that F is a generic splitting field of T. Let A = k[T] be the affine coordinate ring of T. Let k_{1} be any extension of k. Then the set X (T_{k_{1}}) of characters of T defined over k_{1} are the k_{1}-group homomorphisms T_{k_{1}} → 𝔾_{m}. These are in one-to-one correspondence with the set of k_{1}-algebra homomorphisms k_{1}[X, X^{−1}] → A⊗_{k} k_{1} with an indeterminate X or equivalently, using restrictions, to the k-algebra homomorphisms k[X, X^{−1}] → A ⊗_{k} k_{1}. Let now k_{1}, k_{2} be extensions of k both contained in a field k_{3}. Then any character defined over k_{3} which, by restriction, gives a character defined over both k_{1} and k_{2}, will also give a character over k_{1} ∩ k_{2}, as its associated k-homomorphisms k[X, X^{−1}] → A ⊗_{k} k_{i} will map X into both A ⊗_{k} k_{i} for i = 1, 2 and hence into A ⊗_{k} (k_{1} ∩ k_{2}).

Now since any extension L ⊆ k̄ of k is a splitting field of T if and only if X (T_{L}) = X (Tk̄) [6, 8.2, Cor., p. 112], it follows that the intersection of two splitting fields of T which are contained in k̄ is also a splitting field of T. Hence F splits T. Since the same then is true for all the conjugates of F it follows from the definition of F that it is a Galois extension of k.

2.4 Example. Let char(k) ≠ 2 and a ∈ k^{∗}. Define a k-torus by

T (k) := \{( \begin{matrix}α & β \\ aβ & α \end{matrix} ) ∈ M_{2}(k) | α^{2} − aβ^{2} = 1\} .

Then it is easily checked that T splits over some field extension K of k if and only if a ∈ (K^{∗})^{2} and that k(\sqrt{a}) is a splitting field of T. In fact, it is just the generic splitting field F of T described in 2.3.

Let G be a connected affine algebraic k-group.

2.5 Corollary. Suppose G is reductive and quasi-split, and T is a maximal k-torus of G contained in a Borel k-subgroup B of G. Then the generic splitting field F of T is a generic splitting field of G, and every splitting field of G contains a field isomorphic to F.

Proof. Clearly F is a splitting field of G, since it splits one of its maximal tori. Let k′ be a splitting field of G. Then G_{k′} contains a maximal k′-torus which splits. This is contained in some Borel k′-subgroup of G_{k′} and since this is conjugate in G_{k′} to B_{k′}, the torus T_{k′} splits. Hence k′ is a splitting field of T and therefore contains F, by 2.3. This clearly implies that F is a generic splitting field of G.

3. Parabolic subgroups In this section k is an infinite field, k_{s} is the separable closure in an algebraic closure k̄ of k, and G is a connected reductive affine algebraic k-group.

Let K be a splitting field of G (for example K = k̄). Choose a maximal K-torus T of G_{K} which splits over K (cf. Definition 1.5). We denote by X (T) the character group Hom (T, 𝔾_{m}).

Let Φ_{K} = Φ(G_{K}, T) ⊆ X (T) be the set of roots of G_{K} with respect to T. For every α ∈ Φ_{K} there is a connected unipotent subgroup U_{α} of G_{K} such that T U_{α} = U_{α}T. Also, there is a K-isomorphism x_{α}: 𝔾_{a} → U_{α}, where the additive K-group 𝔾_{a} is defined by 𝔾_{a}(K) = K^{+}), such that

t x_{α}(u) t^{−1} = x_{α}(t^{α} u) (∀u ∈ K, t ∈ T (K))

(cf. [8, 2.3, p. 64] or [6, 18.6, p. 221]).

We choose an ordering of Φ_{K}, denote the set of positive roots by Φ_{K}^{+}, and let Δ_{K} ⊂ Φ_{K}^{+} be the basis (or the set of simple roots) of Φ_{K} with respect to that ordering.

For every subset Θ ⊆ Δ_{K} we have the so-called standard parabolic subgroup P_{Θ} of G_{K} (with respect to T) defined by

P_{Θ} := ⟨T, U_{α} | α ∈ Δ_{K} or − α ∈ Θ⟩.

It is known that the standard parabolic subgroups are in one-to-one correspondence with the conjugacy classes of parabolic subgroups of G_{K} [8, 4.6, p. 87]. Obviously we have P_{Δ_{K}} = G_{K}, and B := P_{∅} is the standard Borel subgroup of G_{K}. More generally, there is the following description of P_{Θ}, cf. [8, 4.2, p. 85f] or [6, Prop. 14.18, p. 197].

3.1 Remark. We denote by H^{∘} the connected component of the identity element in an algebraic group H. Let T_{Θ} = (⋂_{α∈Θ} Ker (α))^{∘}, let Z(T_{Θ}) be its centralizer in G_{K} and U_{Θ} = ⟨U_{α} | α ∈ u_{Θ}⟩ where u_{Θ} is the set of all α ∈ Φ_{K}^{+} which are not linear combinations of elements of Θ. Then P_{Θ} = Z(T_{Θ})U_{Θ} is the Levi decomposition of P_{Θ} with reductive part Z(T_{Θ}) and unipotent radical R_{u}(P_{Θ}) = U_{Θ}. If U_{Θ}^{−} = ⟨U_{α} | α ∈ u_{Θ}^{−}⟩ where u_{Θ}^{−} is the set of all α ∈ Φ_{K} \ Φ_{K}^{+} which are not linear combinations of elements of Θ, then P_{Θ}^{−} = Z(T_{Θ})U_{Θ}^{−} is, analogously, the Levi decomposition of the parabolic subgroup of G_{K} containig T which is opposite to P_{Θ}.

3.2 Lemma. Let P be a parabolic k-subgroup of G. Then the unipotent radical R_{u}(P) is, as a kvariety, isomorphic to an affine k-space, and G/P is a rational variety over k. If P_{K} is conjugate to P_{Θ} for Θ ⊆ Δ_{K}, then the dimension of G/P equals that of R_{u}(P) which is given by the number of elements of u_{Θ}.

Proof. Let P^{−} be a parabolic k-subgroup of G which is opposite to P. From [6, 14.21 (iii), p. 198f] we deduce that the product map in G induces a k̄-rational map R_{u}(P^{−})k̄ ×k̄ Pk̄ onto a k̄-open subvariety of Gk̄. By [6, 21.11, p. 233f and 21.20, p. 240] we find that R_{u}(P) and R_{u}(P^{−}) are affine k-spaces, and that G/P is a rational k-variety. It follows that dim G/P = dim R_{u}(P^{−}) which equals the cardinality of u_{Θ}^{−} and hence of u_{Θ}.

Let Δ denote the set of vertices of the Dynkin diagram of Gk̄ and let ι = ι_{K}: Δ → Δ_{K} denote the natural one-to-one correspondence.

3.3 Definition. Let k′ be a field extension of k contained in K and Θ ⊆ Δ. A parabolic subgroup P of G_{k′} is said to be of type Θ if P_{K} is conjugate to P_{ι(Θ)} in G_{K}.

3.4 Remark.

i) The type of a parabolic subgroup is independent of the choice of the splitting field K. To see this, let K_{1} be another splitting field of G which contains k′. Then any free composite K̃ of K, K_{1} over k′ is a splitting field of G as well. Hence, if T_{1} is a maximal split K_{1}-torus of G_{K_{1}}, then T_{K} and (T_{1})_{K} are conjugate over K̃ by [8, Th. 4.21, p. 93]. This conjugation induces an isomorphism i: X (T_{K}) → X ((T_{1})_{K_{1}}). Hence we obtain an ordered root system Φ(G_{K_{1}}, T_{1}) as the image of Φ_{K} under i, with basis i(Δ_{K}) as a set of simple roots of G_{K_{1}} with respect to T_{1}, and we have ι_{K_{1}} = i∘ι_{K}.

ii) For i = 1, 2, let k_{i} be two field extensions of k, and let P_{i} be parabolic subgroups of G_{k_{i}}. Then P_{1}, P_{2} are of the same type if and only if they are conjugate over some free composite of k_{1}, k_{2}. This follows from i) by using splitting field extensions K_{i} ⊇ k_{i} of G and from [8, Th. 4.13 c), p. 90].

iii) Because of i), we will henceforth identify Δ_{K} with Δ. By ii), there is a one-to-one correspondence of the subsets Θ ⊆ Δ and the conjugacy classes of parabolic subgroups of G_{K} for any splitting field K.

iv) Following [32, 2.3, p. 39] we define the ∗-action of the Galois group Γ = Gal (k_{s}/k) on Δ = Δ_{k_{s}} as follows. As G splits over k_{s}, parabolic subgroups of every type are defined over k_{s} and hence Γ operates on the set of their conjugacy classes. Via iii) we get an induced action on Δ, if we restrict this operation to the conjugacy classes of maximal parabolic subgroups of G_{k_{s}} which are in obvious one-to-one correspondence with the elements of Δ: The element corresponding to P_{Θ} is the unique one in Δ \ Θ. This gives the wanted ∗-action. The permutation of Δ corresponding to γ ∈ Γ will be denoted by γ^{∗}. The group G is of inner type if the ∗-action is trivial on Δ and of outer type otherwise.

v) Let S denote a maximal split k-subtorus of G contained in a maximal k-torus T of G. By i) we may assume that Φ = Φ(G_{k_{s}}, T_{k_{s}}). The set of roots of Δ which vanish on S is usually denoted by Δ_{0}, and the set of nontrivial restrictions of elements of Δ to S is _{k}Δ. Hence we have the restriction map res_{k} : Δ → _{k}Δ∪{0} with res _{k}^{−1}(_{k}Δ) = Δ\Δ_{0}. The set Δ_{0} is the set of simple roots of DZ(S). By 1.9 iii) G is quasi-split if and only if DZ(S) is trivial, which obviously is equivalent to Δ_{0} = ∅. On the other hand, the pre-images of single elements of _{k}Δ under res_{k} are precisely the equivalence classes of elements of Δ \ Δ_{0} under the ∗-action of Γ [32, 2.5.1, p. 40]. Hence we can conclude: If G is of inner type, then the map res_{k}, restricted to Δ \ Δ_{0}, is injective. Moreover, if G is of inner type and quasi-split, res_{k} : Δ → _{k}Δ is a bijection, hence the derived group D(G) of G is split.

3.5 Lemma. There is a finite Galois extension k_{inn} of k which is unique up to k-isomorphism with the following properties:

i) The group G_{k_{inn}} is of inner type. ii) Every field extension k′ of k such that G_{k′} is of inner type contains a subfield isomorphic to k_{inn}.

Proof. Clearly the subgroup Γ′ = { γ ∈ Γ | γ^{∗} = id } is normal of finite index in Γ = Gal (k_{s}/k). Hence its fixed field k_{inn} is a finite Galois extension of k such that G_{k_{inn}} is of inner type. Let k′_{s} be a separable closure of k′ containing k_{s}. If G_{k′} is of inner type, then the ∗-action of the Galois group Gal (k′_{s}/k′) on Δ is trivial. Hence (cf. 3.4 iii), iv)) the ∗-action of Gal (k′k_{s}/k′) ≅ Gal (k_{s}/(k′ ∩ k_{s})) is trivial as well, which implies k_{inn} ⊂ k′ ∩ k_{s}.

3.6 Theorem. Let k_{alg} be the composite in k_{s} of k_{inn} and the generic splitting field of the maximal central torus of G (cf. 2.3).

i) The free composite of k_{alg} and a generic quasi-splitting field of G is a generic splitting field of G. ii) Any splitting field of G contains a subfield k-isomorphic to k_{alg}.

Proof. Let F be a field obtained from an generic quasi-splitting field L of G as in i). Since G_{k_{alg}} is, by 3.5, of inner type and since it has a split maximal central torus, it follows from 3.4 v) that F is a splitting field of G_{k_{alg}} and hence of G. Let k′ be a splitting field of G. Then there is a k- place φ : L → k′ ∪ {∞}. Since k_{alg} is algebraic over k and since k_{alg} is contained in k′ by 3.5 ii) and 2.3 we have a trivial k_{alg}-place k_{alg} → k′ ∪ {∞} [38, Chap. VI.4, p. 13]. Thus φ can be extended to a k_{alg}-place F = L · k_{alg} → k′ ∪ {∞}. This implies that F is generic and also proves ii).

3.7 Lemma.

(1) If Θ ⊆ Δ is ∗-invariant, then there is a unique projective k-variety V_{Θ} with the following property: For any field extension k′ of k and any parabolic subgroup P′ of G_{k′} of type Θ one has V_{Θ} ×_{k} k′ ≅ G_{k′}/P′.

(2) For arbitrary Θ ⊆ Δ the following conditions i) – iii) are equivalent.

i) There is a parabolic subgroup of G of type Θ.

ii) Θ is ∗-invariant and V_{Θ}(k) ≠ ∅.

iii) Θ is ∗-invariant and Θ ⊇ Δ_{0} = {α ∈ Δ | res_{k}(α) = 0} (cf. 3.4 v)).

Proof. (1) Let V denote the k_{s}-variety given by the conjugacy class of parabolic subgroups of G_{k_{s}} of type Θ. By [8, 6.2 (3), p. 104], Θ is ∗-invariant if and only if V (k_{s}) is Γ-stable. By [7, 8.4, p. 482], the Γ-stability of V (k_{s}) implies that V is defined over k. For any field extension k′ of k, let k̄′ denote an algebraic closure of k′. Then, by [7, 7.2 (b), (i), p. 474], the set M = V (k̄′) is a homogeneous (G, k)-set represented by a k-variety V_{Θ} . Hence M is a homogeneous (G_{k′}, k′)-set represented by the k′-variety V_{Θ} ×_{k} k′ (cf. [7, 7.3, p. 475]). Let now P′ be a parabolic subgroup of G_{k′} of type Θ. Then V_{Θ} ×_{k} k′ ≅ G_{k′}/P′, since parabolic subgroups are self-normalizing. The uniqueness of V_{Θ} now follows from [7, 7.5 (i), p. 475].

(2) If i) holds then clearly Θ is ∗-invariant. Hence to prove the equivalence of i), ii), iii) we may assume the ∗-invariance of Θ. For any field extension k′ of k, the set V_{Θ}(k′) is the set of parabolic subgroups in G_{k′} of type Θ. This follows from [7, Prop. 7.6, p. 476] applied to the homogeneous (G, k)- set M above. Therefore i) is equivalent to ii). The equivalence of ii) and iii) follows from [8, 6.3 (1), p. 105] and [8, 6.8, p. 107].

3.8 Corollary. G is quasi-split if and only if G contains a parabolic k-subgroup of type Θ for every ∗-invariant subset Θ of Δ, and G is split if and only if it contains parabolic subgroups of every type and its maximal central torus splits.

Proof. We recall from 3.4 v) that G is quasi-split if and only if Δ_{0} = ∅. Hence the equivalence of i) and iii) in 3.7 says that every parabolic subgroup of ∗-invariant type occurs in the quasi-split case. The converse is trivial, as ∅ is ∗-invariant and the type of a Borel subgroup.

If G is split, then its maximal central torus splits and the ∗-action is trivial. It follows by the above that G has parabolic subgroups of every type. Conversely, if this is true, then certainly the ∗-action is trivial, G is quasi-split and therefore split by 3.4 v) if its maximal central torus splits.

Remark. Let Θ ⊆ Δ be ∗-invariant. For a parabolic subgroup P of G_{K} of type Θ the quotient G_{K}/P is a projective irreducible K-variety which defines, by 3.7, a k-variety V_{Θ} such that V_{Θ} ×_{k} K ≅ G_{K}/P. We will say that G_{K}/P is defined over k in spite of the fact that P is not necessarily defined over k. Note that V_{Θ} does not depend on the choice of the splitting field K.

3.9 Corollary. Let k′ and L be two field extensions of k such that k′ is a k-specialization of L. If P is a parabolic subgroup of G_{L}, then there is a parabolic subgroup of G_{k′} of the same type as P. Moreover we have the following:

i) rank (G_{k′}) ≥ rank (G_{L}). ii) Anisotropic reductive k-groups remain anisotropic under purely transcendental extensions of k. iii) Every k-specialization of a splitting field of G is a splitting field of G.

Proof. By assumption, we have a k-place φ : L → k′ ∪ {∞}, and V (L) ≠ ∅ with V being the quotient L-variety G_{L}/P. Clearly the type Θ of P is ∗-invariant with respect to the action of Gal (L_{s}/L). Since Gal (Lk_{inn}/L) ≅ Gal (k_{inn}/(L ∩ k_{inn})), we find that Θ is ∗-invariant with respect to Gal (k_{s}/k_{1}), where k_{1} := L ∩ k_{inn}. Therefore V is defined over k_{1} by 3.7. The field k_{1} is finite separable over k and consequently it is mapped isomorphically by φ onto a subfield of k′. Identifying this subfield with k_{1} we obtain that φ is a k_{1}-place, hence k′ is a k_{1}-specialization of L. By 1.3 we obtain that V (k′) ≠ ∅, hence G_{k′} contains a parabolic subgroup of type Θ.

We now prove i). Since G is an almost direct product of its maximal semisimple subgroup D(G) and a torus [8, 2.2, p. 63], it suffices to prove that assertion for tori and for semisimple groups. The statement for tori follows from 2.2 by induction on the dimension. If G is semisimple and P a minimal parabolic subgroup of G_{L} we find by the above that G_{k′} contains a parabolic subgroup of the type of P and hence a k′-split k′-torus whose rank equals rank (G_{L}) (cf. [6, 20.6, p. 225]) which proves the inequality.

ii) and iii) are immediate consequences of i).

3.10 Theorem. Let Θ ⊆ Δ be ∗-invariant and let V := V_{Θ} denote the corresponding k-variety as described in 3.7. Then the function field k(V) is a regular extension of k, and for every field extension k′ of k the following statements are equivalent:

i) V (k′) ≠ ∅. ii) The free composite of k′ and k(V) is a purely transcendental extension of k′. ii′) There is a k-linear embedding k(V) ↪ k′(X_{1}, …, X_{m}) of k(V) into a finitely generated purely

transcendental extension of k′. iii) k′ is a k-specialization of k(V).

Proof. It follows from 1.1 that k(V) is regular.

“i) ⇒ ii)”: If V (k′) ≠ ∅, then there is a parabolic subgroup Q of G_{k′} of type Θ by 3.7. By 3.2 the k′-variety V_{k′} ≅ G_{k′}/Q is rational over k′ which implies ii).

Clearly ii) implies ii′).

“ii′) ⇒ iii)”: There is a k′-place k′(X_{1}, …, X_{m}) → k′ ∪ {∞}, whose restriction to k(V) gives a k-place k(V) → k′ ∪ {∞}, hence iii).

“iii) ⇒ i)”: k(V) is the residue field at the generic point of V and hence V (k(V)) ≠ ∅ (cf. [20, Chap. II, §6, p. 161]). The assertion i) now follows from 1.3.

3.11 Theorem. The function field F := k(V_{∅}) is a generic quasi-splitting field of G. If G is semisimple of inner type, then F is a generic splitting field of G.

Proof. Clearly ∅ ⊆ Δ is ∗-invariant, and ∅ is the type of a Borel subgroup B of G_{K}. Hence V_{∅} is a k-variety such that G_{K}/B ≅ V_{∅} ×_{k} K by 3.7. Since V_{∅}(F) ≠ ∅, the field F is a quasi-splitting field of G. If k′ is a quasi-splitting field of G, then V_{∅}(k′) ≠ ∅ (by 1.5 ii)), hence k′ is a k-specialization of F by 3.10. Consequently F is a generic quasi-splitting field of G. By 3.4 v), every quasi-splitting field of an inner type semisimple group is a splitting field of that group.

3.12 Corollary. Let L be a generic quasi-splitting field of G. Then k is algebraically closed in L, and L does not split any nontrivial anisotropic k-torus of G.

Proof. Let F be the generic quasi-splitting field of G as defined in 3.11. If L is a generic quasi-splitting of G, then there is a k-place L → F ∪ {∞} by Definition 1.5 iii). Since any algebraic extension of k in L possesses only trivial k-places (cf. [38, Chap. VI, §4, p. 13]) and since k is algebraically closed in F by 1.1, it follows that k is algebraically closed in L. If L splits some k-torus T of G, then, by 2.1, the k-algebraic elements of L already form a splitting field of T.

We now generalize the notion of a quasi-splitting field.

3.13 Definition. Let F be a field extension of k and let Θ ⊆ Δ.

i) F is a Θ-splitting field of G if G_{F} contains a parabolic subgroup of type Θ. ii) A Θ-splitting field F of G is said to be generic, if every Θ-splitting field of G is a k-specialization

of F.

3.14 Remark. It follows from 3.9, that every k-specialization of a Θ-splitting field of G is a Θ-splitting field of G.

3.15 Examples.

i) A (generic) quasi-splitting field is a (generic) ∅-splitting field. ii) The field k is a generic Δ-splitting field of G. iii) For any ∗-invariant subset Θ ⊆ Δ the function field k(V_{Θ}) with V_{Θ} as in 3.7 is a generic Θ-splitting

field of G as follows from 3.10.

3.16 Theorem. Let Θ ⊆ Δ be any subset. Then there is a finite separable field extension k_{Θ} of k, contained in the field k_{inn} of 3.5, with the following properties:

i) Every Θ-splitting field of G contains a subfield isomorphic to k_{Θ}. ii) Θ is invariant with respect to the ∗-action of the Galois group Gal (k_{s}/k_{Θ}). iii) If V_{Θ} denotes the k_{Θ}-variety defined in 3.7, then the function field F_{Θ} := k_{Θ}(V_{Θ}) is a generic

Θ-splitting field of G. iv) The field F_{Θ} is regular over k if and only if k = k_{Θ}, hence if and only if Θ is invariant with respect

to the ∗-action of Gal (k_{s}/k).

Proof. Let Γ_{Θ} = {γ ∈ Gal (k_{s}/k) | γ^{∗}(Θ) = Θ} and let k_{Θ} be its fixed field. Since k_{inn} is the fixed field of Γ′ = {γ ∈ Gal (k_{s}/k) | γ^{∗} = id} and Γ′ ⊆ Γ_{Θ} it follows that k_{Θ} ⊆ k_{inn}.

i) Let k′ be a Θ-splitting field of G. Let k′_{s} be a separable closure of k′ containing k_{s}. By assumption Gal (k′_{s}/k′) leaves Θ ∗-invariant. Hence Gal (k′k_{s}/k′) ≅ Gal (k_{s}/(k′ ∩ k_{s})) leaves Θ ∗-invariant, which implies k_{Θ} ⊆ k′ ∩ k_{s}.

ii) This follows from the construction of k_{Θ}.

iii) Since V_{Θ}(F_{Θ}) ≠ ∅, Lemma 3.7 implies that F_{Θ} is a Θ-splitting field of G_{k_{Θ}} and hence of G. Let k′ be a Θ-splitting field of G. By i) we may assume that k′ is a field extension of k_{Θ}. Thus 3.10, “i) ⇒ iii)”, implies that k′ is a k_{Θ}-specialization of F_{Θ}, hence also a k-specialization of F_{Θ}. This proves iii).

iv) By 1.1, the field F_{Θ} is regular over k_{Θ}. Since k_{Θ} is algebraic over k, the first statement follows. If k = k_{Θ}, then Θ is invariant with respect to the ∗-action of Gal (k_{s}/k) by ii). If the latter is true, then, by 3.7, the variety V_{Θ} is defined over k, and the function field k(V_{Θ}) is a Θ-splitting field of G. By i) it contains k_{Θ}, which implies k = k_{Θ}, since k(V_{Θ}) is regular over k.

The following corollary illustrates the functorial behavior of the map Θ ↦ F_{Θ}.

3.17 Corollary. Let Θ′ ⊆ Θ be a pair of subsets of Δ. Then the following is true.

i) If k′ is a Θ′-splitting field of G, then k_{Θ}k′ is a k_{Θ}-specialization of F_{Θ}. ii) If F_{Θ} is a Θ′-splitting field of G and if k_{Θ} ⊆ k_{Θ′}, then F_{Θ} is a generic Θ′-splitting field of G. iii) If Θ is ∗-invariant and if F_{Θ} is a quasi-splitting field of G, then it is a generic quasi-splitting field

of G. iv) If F_{Θ} is a splitting field of G, then it is a generic splitting field of G.

Remark. The assumption that Θ is ∗-invariant in 3.17 iii) is necessary. See the example after 5.4.

Proof of 3.17. i) After replacing k by k_{Θ} we may assume that Θ is ∗-invariant, hence k_{Θ} = k. By 3.7, “i) ⇒ iii)”, we have Δ_{0} := {α ∈ Δ | res_{k′}(α) = 0} ⊆ Θ′. Since Θ′ ⊆ Θ, it follows that k′ is a Θ-splitting field of G by 3.7, “iii) ⇒ i)”. Thus k′ is a k-specialization of F_{Θ} by 3.15 iii).

ii) Let k′ be a Θ′-splitting field of G. Then k_{Θ} ⊆ k_{Θ′} ⊆ k′ by assumption and 3.16 i). So i) implies that k′ is a k-specialization of F_{Θ}.

iii) This follows from ii) and 3.16 iv) for Θ′ = ∅.

iv) Let k′ be a splitting field of G. By 3.6 ii), k′ and especially F_{Θ} both contain a copy of the Galois extension k_{alg} of k. Replacing k by k_{alg} we may assume that G is of inner type by 3.5. Hence we may apply iii) to find that k′ is a k-specialization of F_{Θ} which yields iv).

It seems to be natural to expect that Θ-splitting of any group G can be achieved by the corresponding Θ_{an}-splitting of the anisotropic kernel G_{an} of G, where Θ_{an} is the appropriate set of vertices of the Dynkin diagram of (G_{an})k̄. The precise meaning of this statement is given in 3.18, 3.19. It essentially is reflected by the fact that the generic Θ-splitting field of G is a purely transcendental extension of the corresponding Θ_{an}-splitting field of G_{an}, hence these two fields are obviously equivalent in the sense that they are k-specializations of each other. The reason for this is that G_{an} is given – up to a torus part, cf. 1.9 – by a Levi-subgroup of a minimal parabolic subgroup Q of G and that G/Q is a rational k-variety. This explains and generalizes an observation made by Roquette [23, Th. 4, p. 413], which will be discussed in 4.10 below.

3.18 Theorem. Let Q be a parabolic k-subgroup of G of type Δ′ ⊆ Δ and let L be a Levi k-subgroup of Q. Consider Δ′ as a root basis of L_{k_{s}}. Let Θ be a ∗-invariant subset of Δ. Let V (resp. V′) denote the projective k-varieties associated to Θ (with respect to G) (resp. to Θ′ (with respect to L)) according to 3.7. Then k(V) is isomorphic to a purely transcendental extension of k(V′).

Proof. In L we choose a maximal k-torus T of G containing a maximal k-split k-torus S of G, such that Z(S) ⊆ L (cf. [6, 20.6, p. 225]). In G_{k_{s}} we choose a Borel subgroup B such that T_{k_{s}} ⊂ B ⊆ Q_{k_{s}} and a parabolic subgroup P ⊆ G_{k_{s}} of type Θ with B ⊆ P. We identify Δ with the basis of a root system Φ(G_{k_{s}}, T_{k_{s}}) such that the parabolic subgroups of G_{k_{s}} containing B are those which are in standard position. Hence especially Q_{k_{s}} and P are standard parabolic subgroups of G_{k_{s}}. By [6, 21.13, p. 235], the subgroup B_{L} := B ∩L_{k_{s}} is a Borel subgroup of L_{k_{s}} which we will use to define the standard position of parabolic subgroups of L_{k_{s}}.

Since Q is defined over k, its type Δ′ and hence also Θ′ is ∗-invariant. P ∩L_{k_{s}} is a parabolic subgroup of L_{k_{s}} by [6, 21.13, p. 235]. It is obviously in standard position. From [8, 5.20, p. 102] it follows that its type is Θ′. We have V_{k_{s}} := V ×_{k} k_{s} ≅ G_{k_{s}}/P and V′_{k_{s}} := V′ ×_{k} k_{s} ≅ L_{k_{s}}/(P ∩ L_{k_{s}}) ≅ Q_{k_{s}}/(P ∩ Q_{k_{s}}). The k-embedding L → G induces a k_{s}-embedding ι : V′_{k_{s}} → V_{k_{s}} by g(P ∩ L_{k_{s}}) ↦ gP for g ∈ L(k_{s}). By construction, P = P_{Θ} in G_{k_{s}} and P ∩ L_{k_{s}} = P_{Θ′} in L_{k_{s}}.

We show that ι is Gal (k_{s}/k)-equivariant: For σ ∈ Gal (k_{s}/k) there is a unique w_{σ} in the Weyl group of Φ(G_{k_{s}}, T_{k_{s}}) such that w_{σ}σ(Δ) = Δ. For any root α ∈ Δ we then have w_{σ}σ(α) = σ^{∗}(α) (cf. [32, 2.3, p. 39]). Let n_{σ} be a representative of w_{σ} in the normalizer of T (k_{s}) in G(k_{s}). The orders defined on X (T_{k_{s}}) by Δ and by σ(Δ) induce the same order on X (S_{k_{s}}). Therefore it follows from [8, 6.6, p. 107] that n_{σ} ∈ Z(S)(k_{s}) ⊂ L(k_{s}). Thus ^{σ}P_{Θ} = n_{σ}P_{σ^{∗}(Θ)}n_{σ}^{−1} and ^{σ}P_{Θ′} = n_{σ}P_{σ^{∗}(Θ′)}n_{σ}^{−1}. Since the conjugacy class gP g^{−1} identifies with the coset gP in V and similarly for V′ we obtain by the ∗-invariance of Θ and Θ′ that ι(^{σ}(gP_{Θ′})) = ι(^{σ}g n_{σ}P_{Θ′}) = ^{σ}g n_{σ}P_{Θ} = ^{σ}g^{σ}P_{Θ} = ^{σ}(gP_{Θ}) = ^{σ}(ι(gP_{Θ′})) for any g ∈ L(k_{s}) which proves the Gal (k_{s}/k)-equivariance of ι.

Hence ι is defined over k [6, AG.14.3, p. 31], i. e., it is obtained by base extension from a k-embedding ι : V′ → V. Let Q^{−} and P^{−} denote parabolic subgroups of G (resp. G_{k_{s}}) which are opposite to Q and P.

By [6, 14.21, p. 198], the product maps

R_{u}(P^{−}) ×_{k_{s}} P → G_{k_{s}}, (R_{u}(P^{−}) ∩ L_{k_{s}}) ×_{k_{s}} (P ∩ L_{k_{s}}) → L_{k_{s}}

induce k_{s}-isomorphisms of their pre-images onto open dense subvarieties of G_{k_{s}} and L_{k_{s}}. Hence we obtain morphisms of k_{s}-varieties

R_{u}(P^{−}) → V_{k_{s}}, R_{u}(P^{−}) ∩ L_{k_{s}} → V′_{k_{s}}

which are k_{s}-isomorphisms of their pre-images onto open dense subvarieties of V_{k_{s}} and V′_{k_{s}}. Also the product map

(R_{u}(P^{−}) ∩ R_{u}(Q_{k_{s}}^{−})) ×_{k_{s}} (R_{u}(P^{−}) ∩ L_{k_{s}}) → R_{u}(P^{−})

is an isomorphism of k_{s}-varieties which can be seen as follows: By 3.1, we have

\begin{aligned}R_{u}(P^{−}) & = ⟨ U_{α} | α ∈ u_{Θ}^{−} ⟩, \\ R_{u}(P^{−}) ∩ L_{k_{s}} & = ⟨ U_{α} | α ∈ u_{Θ}^{−} ∩ ⟨Δ′⟩ ⟩, \\ R_{u}(P^{−}) ∩ R_{u}(Q_{k_{s}}^{−}) & = ⟨ U_{α} | α ∈ u_{Θ}^{−} \ ⟨Δ′⟩ ⟩, \end{aligned}

where u_{Θ}^{−} is the set of all negative roots which are not linear combinations of elements of Θ, and where ⟨Δ′⟩ is the set of roots which are linear combinations of elements of Δ′. It follows from [6, 21.9 (ii), p. 232] that the three groups above are the direct span of their respectively generating groups U_{α} since each of their underlying sets of roots α is closed in the sense that it contains the sum of each two of its elements if this sum itself is a root. Therefore, as varieties, each of the above three groups is k_{s}-isomorphic to an affine space (cf. [6, 21.20 (i), p. 240]) and the product map induces a k_{s}-isomorphism.

Hence we obtain the following commutative diagram of k_{s}-morphisms, each of which is an isomorphism onto an open dense subvariety.

(R_{u}(P^{−}) ∩ R_{u}(Q_{k_{s}}^{−})) ×_{k_{s}} (R_{u}(P^{−}) ∩ L_{k_{s}}) ⟶ R_{u}(P^{−})
↓  ↓
(R_{u}(P^{−}) ∩ R_{u}(Q_{k_{s}}^{−})) ×_{k_{s}} V′_{k_{s}}  ⟶  V_{k_{s}}

Here the latter horizontal map is just given by (g, h(P ∩ L_{k_{s}})) ↦ ghP.

Let ψ = u_{Θ}^{−} \ ⟨Δ′⟩, so that R_{u}(P^{−}) ∩ R_{u}(Q_{k_{s}}^{−}) = ⟨U_{α} | α ∈ ψ⟩ as above. As Θ and Δ′ are both ∗-invariant, R_{u}(P^{−}) ∩ R_{u}(Q_{k_{s}}^{−}) is Gal (k_{s}/k)-invariant and hence defined over k (cf. [6, AG.14.4, p. 32]). Therefore we have a k-subvariety U of R_{u}(Q^{−}) such that U ×_{k} k_{s} = R_{u}(P^{−}) ∩ R_{u}(Q_{k_{s}}^{−}). The image of ψ under res_{k} (cf. 3.4 v)) is a closed set of roots of G over k. Hence we conclude using [6, 21.20 (i), p. 240] that U is isomorphic, as a k-variety, to an affine k-space. Thus we obtain a morphism of k-varieties U ×_{k} V′ → V which is an isomorphism onto an open and dense k-subvariety. Therefore k(V) ≅ k(U) ⊗_{k} k(V′), and since U is isomorphic to an affine k-space, the theorem is proved.

Let Q be a minimal parabolic k-subgroup of G, with Levi subgroup L and type Δ_{0} ⊆ Δ (cf. 3.4 v)). Then, by 1.9, the derived group G_{an} of L is a semisimple anisotropic kernel of G. Let Θ ⊆ Δ be ∗-invariant. As above, Θ_{an} := Θ ∩ Δ_{0} is ∗-invariant and can be considered as a set of roots of L_{k_{s}} and of (G_{an})_{k_{s}}. If P is a parabolic subgroup of G_{k_{s}} of type Θ then P_{L} := P ∩ L_{k_{s}} and P_{an} := P ∩ (G_{an})_{k_{s}} are parabolic subgroups of type Θ_{an} of L_{k_{s}} (resp. (G_{an})_{k_{s}}). Consequently, the associated quotient varieties G_{k_{s}}/P, L_{k_{s}}/P_{L}, (G_{an})_{k_{s}}/P_{an} are defined over k by 3.7. We denote the respective k-structures by V_{Θ}, V_{L,Θ_{an}}, V_{Θ_{an}}. Since L is the product of its maximal central torus and G_{an}, the natural k-embedding G_{an} → L defines a k_{s}-isomorphism (G_{an})_{k_{s}}/P_{an} → L_{k_{s}}/P_{L} which is Gal (k_{s}/k)-equivariant. Therefore it induces a k-isomorphism of the k-varieties V_{Θ_{an}} → V_{L,Θ_{an}} (cf. [6, AG.14.3, p. 31]). Hence k(V_{Θ_{an}}) is naturally isomorphic to k(V_{L,Θ_{an}}). By 3.18 we find that k(V_{Θ}) is purely transcendental over k(V_{L,Θ_{an}}). Hence we conclude:

3.19 Corollary. For any ∗-invariant Θ ⊆ Δ, the generic Θ-splitting field k(V_{Θ}) of G is a purely transcendental extension of the corresponding induced generic Θ_{an}-splitting field k(V_{Θ_{an}}) of the semisimple anisotropic kernel G_{an} of G.

3.20 Example (Witt [37]). We first consider the case char(k) ≠ 2 which has been investigated by Witt and which is the origin of the theory of generic splitting.

Let a, b ∈ k^{∗} be such that D = (a, b) is a quaternion algebra over k, that is, an Azumaya algebra over k of k-dimension 4. One can choose a k-basis {1, u, v, uv} of D such that the multiplication in D is given by u^{2} = a, v^{2} = b, vu = −uv. Let G = SL_{1}(D) be the kernel of the reduced norm N_{red} of D over k restricted to the group GL_{1}(D) of invertible elements. G is an anisotropic k-form of (SL_{2})_{k} if D is non-split. The Dynkin diagram of Gk̄ consists of a single vertex only, hence the only conjugacy class of proper parabolic subgroups is given by the class of Borel subgroups, which can be represented, over k̄, by the k̄-group B of upper triangular matrices of determinant 1.

Now G operates k-morphically and k-linearly on the affine k-space D by conjugation. This operation gives an operation on the projective k-space ℙ(D)_{k} ≅ ℙ_{k}^{3}. Let V denote the k̄-subvariety of nilpotent lines of ℙ(D)k̄. It is easily checked that G(k̄) operates transitively on V (k̄) and that B is the stabilizer subgroup of the nilpotent line of V (k̄) represented by the matrix

\binom{0 1}{0 0} ∈ M_{2}(k̄) ≅ D ⊗_{k} k̄.

Hence V ≅ Gk̄/B. By 3.7 we know that V is defined over k, but we here will give an elementary argument for this fact which will give us the equation with coefficients in k defining the complete curve V. If x = ξ + η_{1}u + η_{2}v + η_{3}uv ∈ D, then the reduced norm N_{red} and the reduced trace S_{red} are given by the formulae N_{red}(x) = ξ^{2} − η_{1}^{2}a − η_{2}^{2}b + η_{3}^{2}ab and S_{red}(x) = 2ξ. The variety of nilpotent elements of Dk̄ is defined by the equations N_{red}(x) = S_{red}(x) = 0. These are equivalent to ξ = 0 and η_{1}^{2}a + η_{2}^{2}b = η_{3}^{2}ab.

Hence a defining equation for k(V) is given by X^{2}a + Y^{2}b = ab. This is the function field associated to D = (a, b) by Witt [37, p. 464].

The field k(V) is isomorphic to F := k(Z)(y) with an indeterminate Z and y = \sqrt{aZ^{2} + b} (with X = b/y, Y = aZ/y, and y = b/X, Z = bY/aX). If D splits over k, then clearly F is a splitting field of G. So let D be non-split. Since D_{k(Z)} ≅ (−a/b, aZ^{2} + b), we may assume that F is a maximal commutative subfield of D_{k(Z)}. We then obtain a maximal anisotropic k(Z)-torus T in G_{k(Z)} defined as the kernel of Norm_{F/k(Z)} (restricted to the invertible elements) (cf. 2.4). The torus T is not defined over k. It splits over F and hence F is a splitting field of G.

Every splitting field of G is a splitting field of D and vice versa. If now L is such a splitting field, then, over L, the element b is a norm from the L-algebra L[X]/(X^{2} − a) (cf. [19, Th. 15.7, p. 149]), hence there are elements ξ_{1}, ξ_{2} ∈ L such that ξ_{1}^{2} = aξ_{2}^{2} + b. We then have a k-place φ: F → L ∪ {∞} with φ(Z) = ξ_{2} and φ(y) = ξ_{1}. Hence F is a generic splitting field of G.

We now assume char(k) to be arbitrary. Then for a, b ∈ k with b ≠ 0 we obtain a quaternion k-algebra D with k-basis {1, u, v, uv} and multiplication defined by u^{2} = u + a, v^{2} = b, vu = (1 − u)v which is a full 2×2-matrix ring over k if and only if the equation b = ξ^{2} +ξη −aη^{2} has a solution ξ, η ∈ k [1, Th. 26, p. 146], or equivalently, if b is a norm from the separable extension k[X]/(X^{2} − X − a).

It is easily checked that the reduced norm and trace of D for x = ξ + η_{1}u + η_{2}v + η_{3}uv ∈ D is given by the formulae N_{red}(x) = ξ^{2} + ξη_{1} − η_{1}^{2}a − (η_{2}^{2} + η_{2}η_{3} − η_{3}^{2}a)b and S_{red}(x) = 2ξ + η_{1}. As above, we get the variety V ≅ Gk̄/B of nilpotent lines of Dk̄ by the equations N_{red}(x) = S_{red}(x) = 0.

If char(k) = 2, these equations are equivalent to the k-equation ξ^{2}b + η_{2}^{2} + η_{2}η_{3} + η_{3}^{2}a = 0. By [28, XIV, §5, Example, p. 221] this is the homogeneous equation defining the Severi-Brauer variety associated to D.

4. Generic splitting of Azumaya algebras over fields

Let A be an Azumaya algebra over an infinite field k, that is, A is a finite dimensional central simple k-algebra, and, by Wedderburn’s theorem, there is a unique integer r ≥ 0 and a central division algebra D over k which is unique up to k-isomorphism such that A ≅ M_{r+1}(D). Let d = ind (A) denote the index of A (that is, dim_{k}D = d^{2}), and let n be defined by

n + 1 = d(r + 1).

Then the semisimple k-group G := SL_{r+1}(D) has the k-rank r and the absolute rank n (cf. [6, 23.2, p. 254f]).

Let K be a splitting field of G. We then have G_{K} ≅ SL_{n+1,K}, and a maximal K-split torus T of G_{K} is given by the set of diagonal matrices

t = diag (t_{1}, t_{2}, …, t_{n+1}) := ( \begin{matrix}t_{1} & 0 & … & 0 \\ 0 & t_{2} & … & 0 \\ ⋮ & ⋮ & ... & ⋮ \\ 0 & 0 & … & t_{n+1} \end{matrix} ) with det(t) = 1 .

A basis Δ of the root system Φ(G_{K}, T) is given by the K-rational characters α_{i}(t) := t_{i} t_{i+1}^{−1}, for i = 1, …, n, and its Dynkin diagram is given by

{∘ \atop α_{1}} {∘ \atop α_{2}} ⋯ {∘ \atop α_{n}}.

Since G is of inner type ^{1}A_{n} we have a k-variety V_{Θ} for any subset Θ ⊆ Δ by 3.7. The function field F_{Θ} = k(V_{Θ}) has the properties described in 3.10 and is a generic Θ-splitting field of G by 3.16.

4.1 Theorem. Let {α_{i_{1}}, …, α_{i_{l}}} = Δ \ Θ for Θ ⊆ Δ. Then, for every field extension k′ of k, we have V_{Θ}(k′) ≠ ∅ if and only if ind (A ⊗_{k} k′) divides gcd (d, i_{1}, …, i_{l}).

Proof. It suffices to show the equivalence for the case k′ = k. Using the description of the relative Dynkin diagram of ^{1}A_{n} as given in [32, Table II, p. 55] we see that Δ \ Δ_{0} = {α_{d}, α_{2d}, …, α_{rd}} with Δ_{0} as in 3.4 v). It follows that d | gcd (d, i_{1}, …, i_{l}) if and only if Δ \ Θ ⊆ Δ \ Δ_{0}. However, the latter condition is equivalent to V_{Θ}(k) ≠ ∅ by 3.7, “iii) ⇔ ii)”.

4.2 Corollary.

i) Let L and k′ be two field extensions of k such that k′ is a k-specialization of L. Then ind (A ⊗_{k} k′) divides ind (A ⊗_{k} L).

ii) For d = ind (A) and i_{1}, …, i_{l} as in 4.1 we have ind (A ⊗_{k} F_{Θ}) = gcd (d, i_{1}, …, i_{l}). iii) Let i divide d. Then there is a parabolic subgroup P of G_{k_{s}} such that V = G_{k_{s}}/P is defined over

k and that ind (A ⊗_{k} k(V)) = i. Every field extension k′ of k with ind (A ⊗_{k} k′) dividing i is a k-specialization of k(V). A possible choice is P = P_{Δ_{i}} for Δ_{i} = Δ \ {α_{i}}.

Proof. i) For i = 1, …, n let V_{i} := V_{Δ_{i}} with Δ_{i} = Δ \ {α_{i}}, and let F_{i} := k(V_{i}). It follows from 4.1 that, for any extension k_{1} of k, the set of all j ∈ {1, …, n} with V_{j}(k_{1}) ≠ ∅ consists precisely of the multiples of ind (A ⊗_{k} k_{1}). Applying this to the fields L, k′ yields i), since V_{j}(L) ≠ ∅ implies V_{j}(k′) ≠ ∅ by 1.3.

ii) We have V_{Θ}(F_{Θ}) ≠ ∅ which implies ind (A ⊗_{k} F_{Θ}) | g := gcd (d, i_{1}, …, i_{l}) by 4.1. Let p be a prime dividing g and p^{s} the highest power of p which divides g. It suffices to show that p^{s} divides ind (A ⊗_{k} F_{Θ}). There is finite separable field extension k′ of k such that p^{s} = ind (A ⊗_{k} k′) (cf. [22, 14.4, Lemma b, p. 260]). Thus 4.1 and 3.10 yield that k′ is a k-specialization of F_{Θ}. Now i) implies that p^{s} divides ind (A ⊗_{k} F_{Θ}).

iii) For P = P_{Δ_{i}} the first statement follows immediately from ii). The second statement follows from 4.1 and 3.10.

The generic splitting field k(V_{∅}) of G (cf. 3.11) is of transcendence degree n(n + 1)/2 (apply 4.4 below with Θ = ∅). If n > 1, then there are generic splitting fields of G of smaller transcendence degree, as follows from 4.3 and 4.4.

4.3 Corollary. If the greatest common divisor of i_{1}, …, i_{l} and d is 1 then the function field F_{Θ} := k(V_{Θ}) is a generic splitting field of G = SL_{r+1}(D) .

Proof. By 4.2 ii) we find that F_{Θ} is a splitting field of A and hence of G. Cor. 3.17 iv) now implies that F_{Θ} is a generic splitting field of G.

4.4 Proposition. For Θ ⊂ Δ, let V_{Θ} be the corresponding k-variety as defined in 3.7. If {α_{i_{1}}, …, α_{i_{l}}} = Δ \ Θ with i_{1} < ⋯ < i_{l} and if i_{0} := 0, then

dim V_{Θ} = ∑_{j=1}^{l}(i_{j} − i_{j−1})(n + 1 − i_{j}).

Moreover, V_{Θ} ×_{k} K is isomorphic, as a K-variety, to the projective variety Flag_{Θ}(K^{n+1}) of flags of subspaces of the (n + 1)-dimensional affine K-space {0} = U_{0} ⊂ U_{1} ⊂ ⋯ ⊂ U_{l} with dim_{K}U_{j} = i_{j} for j = 0, …, l.

Proof. Let {e_{1}, …, e_{n+1}} denote the standard basis of K^{n+1}. The group G_{K} ≅ (SL_{n+1})_{K} operates K-morphically and transitively on Flag_{Θ}(K^{n+1}), and the stabilizer subgroup P of the flag

{0} = U_{0} ⊂ U_{1} ⊂ ⋯ ⊂ U_{l}, U_{j} = Ke_{1} ⊕ ⋯ ⊕ Ke_{i_{j}} ⊆ K^{n+1}, for j = 1, …, l

is defined by the matrices in SL_{n+1}(K) of shape (A_{jj′})_{j,j′=1,...,l+1}. Here A_{jj′} is an (i_{j}−i_{j−1})×(i_{j′}−i_{j′−1})- matrix for j, j′ = 1, …, l + 1, where we define i_{l+1} := n + 1, and A_{jj′} = 0 for j > j′. Since P contains the Borel subgroup B of G_{K} defined by the upper triangular matrices, it is a parabolic subgroup of G_{K}. The dimension of its unipotent radical R_{u}(P) is equal to the sum of the number of entries of all matrices A_{jj′} for j < j′. By 3.2, this yields the right hand side of the formula for the dimension of V_{Θ}.

By 3.1, the reductive part of P_{Θ} is the centralizer Z(T_{Θ}) in G_{K} of the K-torus

T_{Θ} := ( ⋂_{ν=1,ν≠i_{1},...,i_{l}}^{n} Ker (α_{ν}))^{∘} .

Hence T_{Θ}(K) consists of diagonal matrices

t = diag (\underbrace{t_{i_{1}}, …, t_{i_{1}}}_{i_{1} times}, \underbrace{t_{i_{2}}, …, t_{i_{2}}}_{i_{2}−i_{1} times}, …, \underbrace{t_{i_{l+1}}, …, t_{i_{l+1}}}_{i_{l+1}−i_{l} times})

with det(t) = 1 and i_{l+1} = n + 1. Now it is easily checked that the Levi subgroup of P is the centralizer Z(T_{Θ}) of T_{Θ} in G_{K}. Therefore we obtain

P = Z(T_{Θ})R_{u}(P) = Z(T_{Θ}) · B = Z(T_{Θ})R_{u}(P_{Θ}) = P_{Θ}

by 3.1. Hence P = P_{Θ}, which proves 4.4, since G_{K}/P ≅ Flag_{Θ}(K^{n+1}).

We now restrict our attention to proper maximal subsets of Δ. Set

Θ := Δ_{i} := Δ \ {α_{i}}, V_{i} = V_{Δ_{i}}, F_{i} := k(V_{i})

for i ∈ {1, …, n}. The following corollary is a direct consequence of 4.4.

4.5 Corollary. For i = 1, …, n we have dim V_{i} = i(n + 1 − i), and V_{i} is, as a K-variety, isomorphic to the Graßmann variety Grass_{i}(K^{n+1}).

4.6 Corollary. For i = 1, …, n the equality ind (A ⊗_{k} F_{i}) = gcd (d, i) holds, and for every field extension k′ of k we have V_{i}(k′) ≠ ∅ if and only if ind (A ⊗_{k} k′) divides i. In particular, F_{jd} is a purely transcendental extension of k for j = 1, …, r.

Proof. The equality holds by 4.2, the rest of 4.6 follows from 4.1 and 3.10.

As it was mentioned in [4, p. 103], the generalized Severi-Brauer varieties described there are precisely the k-forms of the Graßmann varieties from 4.5, except in the case 2 | (n+1) and i = (n+1)/2, where also an outer form of Grass_{i}(k^{n+1}) exists. This will naturally occur in the theory of the generic splitting of special unitary groups of type ^{2}A_{n} which will be discussed in §6 (cf. 6.5).

4.7 Corollary. For every field extension L of k, the following statements are equivalent:

i) L is a splitting field of G. ii) V_{1} ×_{k} L ≅ ℙ_{L}^{n}. iii) V_{1}(L) ≠ ∅. The statements i), ii), iii) remain equivalent if V_{1} is replaced by V_{n} in ii) and iii).

Proof. The statement i) implies ii) by 4.4. Obviously, ii) implies iii). If iii) holds, then d_{L} := ind (A ⊗_{k} L) | 1 (resp. n) by 4.6. Since d_{L} | (n + 1), it follows in both cases that L is a splitting field of A and hence of G.

4.8 Remark. i) Since the separable closure k_{s} is a splitting field of G (cf. 1.7 iii)) it follows from 4.7 that V_{i} ×_{k} k_{s} ≅ ℙ_{k_{s}}^{n} for i = 1, n, hence V_{1} and V_{n} are n-dimensional Severi-Brauer varieties over k [28, Chap. X, §6, p. 168]. More generally, all V_{i} for i = 1, …, n are isomorphic to “generalized Severi-Brauer varieties” introduced in 1976 by Heuser (for i | (n + 1)), [12, p.30, 46], and later (1991) for all i by Blanchet [4, p. 100, 102] and Schofield/v. d. Bergh [26]. The generalized Severi-Brauer varieties are the varieties W_{i} of rank i left ideals of A in [12] and [26] (and right ideals in [4]). Using the isomorphism A⊗_{k} K ≅ M_{n+1}(K), one verifies similarly as in the proof of 4.4 that G_{K} operates transitively on W_{i} and that P_{Δ_{i}} stabilizes a rank i left ideal of M_{n+1}(K) under left multiplication. Hence W_{i} ≅ G_{K}/P_{Δ_{i}} = V_{i} for i = 1, …, n. (Note that P_{Δ_{i}} are precisely the proper maximal parabolic subgroups of G_{K}.) It follows that the fields F_{i} are the generic partial splitting fields of A, introduced by Heuser [12, Def. 7, p. 22 and p. 63], Blanchet [4, Def. 3 and Th. 2, p. 103] and Schofield/v. d. Bergh [26, Sec. 3]. The statement ind (A ⊗_{k} F_{i}) = gcd (d, i) and the equivalence of 4.6 was proved by Heuser [12, p. 73, p. 43], Blanchet [4, Th. 3, p. 104, Prop. 3, p. 103] and Schofield/v. d. Bergh [27, Th. 2.5]. Blanchet and Schofield/v. d. Bergh also proved the equivalence of “i)” and “ii)” of 3.10 for the special case V = V_{i}.

ii) The assertion ind (A ⊗_{k} F_{i}) = gcd (d, i) with F_{i} = k(V_{i}) in 4.6 shows a significant difference in the behavior of the generic splitting of Azumaya algebras and that of quadratic forms as discussed in the next paragraph (cf. 5.8).

4.9 Remark. Taking i = 1 (or i = n) we obtain from 4.6, 4.7 and 3.10 the results of Amitsur [2, 9.1, p. 26] (see also [3, Th. 2, p. 1]), which were later proved by Roquette [23, Th. 2, p. 413] with methods from Galois cohomology and which generalize the result of Witt on quaternion algebras (cf. Example 3.20).

Amitsur also showed that the automorphism group of F over k is isomorphic to A^{∗}/k^{∗}. This can be shown in the following way: The automorphism group of V_{1} is certainly a k-form of the group (PGL_{n+1})_{k} which is isomorphic to (GL_{n+1})_{k} modulo its center C((GL_{n+1})_{k}). Obviously it contains the group GL_{1}(A)/C(GL_{1}(A)) ≅ GL_{r+1}(D)/C(GL_{r+1}(D)) which is a k-form of the group above. For dimension reasons, this is already the full automorphism group of V_{1}. But its k-rational points are just given by A^{∗}/C(A^{∗}) ≅ A^{∗}/k^{∗}.

The corollary in [3, p. 3] characterizes splitting fields K of A by the condition that k(V_{1}) is contained in a purely transcendental extension of K. This condition is, by 3.10, equivalent to V_{1}(K) ≠ ∅, hence the assertion of the corollary follows from 4.6.

4.10 Remark. In [23, p. 424f], Roquette associates to every Galois-2-cocycle γ ∈ H^{2}(Gal (K/k), K^{∗}) (where K is a finite Galois extension of k), and every multiple m of its Schur index d a “Brauer field” F_{m}(γ) of transcendence degree m − 1 over k.

In our terminology, the cocycle γ defines a central k-division algebra D of index d, the multiple m of d is just n + 1 = (r + 1)d. These data define the semisimple group G = SL_{r+1}(D), and the Brauer field F_{m}(γ) defined by Roquette is precisely the function field k(V_{1}), where V_{1} is a Severi-Brauer variety satisfying G_{K}/P_{Δ_{1}} = V_{1} ×_{k} K. Clearly we hereby obtain an infinite series of generic splitting fields of D as r ranges over all non-negative integers. It can easily be deduced from 3.18 that, for m′ ≤ m, the field F_{m}(γ) is a purely transcendental extension of the field F_{m′}(γ), which is the content of [23, Th. 4, p. 413]. In particular, all the fields F_{m}(γ) are purely transcendental over the smallest one, F_{1}(γ), which is isomorphic to the generic splitting field of G_{an} as constructed in 3.19, since the semisimple anisotropic kernel of SL_{r+1}(D) is a direct product of r + 1 copies of SL_{1}(D).

5. Generic splitting of quadratic forms

Let k be an infinite field with char(k) ≠ 2 and let (M, q) be a regular quadratic k-space of dimension m, that is, M is an m-dimensional k-vector space and q is a quadratic form with nondegenerate associated bilinear form ( , ) such that q(x + y) = q(x) + q(y) + (x, y) holds for all x, y ∈ M. The discriminant d(M) of (M, q) is defined to be the square class (−1)^{[m/2]} det((u_{i}, u_{j})_{i,j=1,...,m})k^{∗}^{2} ∈ k^{∗}/k^{∗}^{2}. (Here {u_{1}, …, u_{m}} denotes a k-basis of M.) We have a Witt decomposition of M into mutually orthogonal subspaces

M = ( ⊥_{i=1}^{r} H_{i}) ⊥ M_{an}

where H_{i} is a hyperbolic plane for i = 1, …, r and (M_{an}, q_{an}) with q_{an} := q|_{Man} is a maximal anisotropic subspace of (M, q) which is unique up to k-isometry and is called an anisotropic kernel of the quadratic space (M, q). The integer r ≥ 0 is the Witt index of (M, q), and we have m = 2r + dim_{k} M_{an} and d(M) = d(M_{an}). It is convenient to choose a k-basis {e_{1}, …, e_{m}} of M as follows. For i = 1, …, r, let {e_{i}, e_{m−i+1}} ⊂ M be a basis of H_{i} such that q(e_{i}) = q(e_{m−i+1}) = 0, (e_{i}, e_{m−i+1}) = 1, and let {e_{i} | i = r + 1, …, m − r} be any basis of (M_{an}, q_{an}). A basis like this we will call a Witt basis of (M, q). We mention that {e_{1}, …, e_{r}} (as well as {e_{m−r+1}, …, e_{m}}) generate a maximal totally isotropic subspace of (M, q).

Let G := SO(q) be the special orthogonal group of (M, q). If m = 2, then G is a k-torus and its generic splitting field is described in 2.3 and 2.5. Hence we now assume m ≥ 3. This implies that G is semisimple. The following proposition is obtained from [6, 23.4, p. 256f] and Definition 1.8.

5.1 Proposition. Let (M, q) be a regular quadratic k-space of Witt index r. Then r is the rank of G = SO(q), and a maximal k-split k-torus S of G is given, with respect to a Witt basis {e_{i}} of (M, q), by the diagonal matrices

s = diag (s_{1}, …, s_{r}, 1, …, 1, s_{r}^{−1}, …, s_{1}^{−1}) ∈ GL(M), s_{1}, …, s_{r} ∈ k^{∗}.

A reductive anisotropic kernel of G is given by G_{an} = SO(q_{an}), where (M_{an}, q_{an}) is the anisotropic kernel of (M, q). More precisely, we have Z(S) = S ×_{k} G_{an}, and G_{an} is semisimple if and only if dim_{k} M_{an} ≥ 3, and is an anisotropic k-torus of rank 1 if and only if dim_{k} M_{an} = 2, in which case G is quasi-split but not split. G is split over k if and only if dim_{k} M_{an} ≤ 1.

Let K be any splitting field of G. Then the rank of G_{K} is n := [m/2]. We modify the Witt basis given above over k into one over K by setting e′_{i} = e_{i} for i ∉ {r + 1, …, m − r} and by replacing the basis e_{i} of M_{an} for i = r + 1, …, m − r by a Witt basis e′_{i} of (M_{an} ⊗_{k} K, q_{an} ⊗_{k} K) such that q(e′_{i}) = q(e′_{m−i+1}) = 0, (e′_{i}, e′_{m−i+1}) = 1 for i = r + 1, …, n. Let T be the K-torus of G_{K} which is given with respect to the new basis by the diagonal matrices

t = diag (t_{1}, …, t_{n}, ˆ1, t_{n}^{−1}, …, t_{1}^{−1}) ∈ GL(M ⊗_{k} K), t_{1}, …, t_{n} ∈ K^{∗}.

(Here the symbol ˆ1 means that the component 1 occurs (resp. does not occur) in the middle according as m being odd (resp. even).) Then, by the above, T is a maximal torus of G_{K} which splits completely and contains S_{K} .

A basis Δ of the root system Φ = Φ(G_{K}, T) is given by the K-rational characters α_{i}(t) := t_{i} t_{i+1}^{−1}, for i = 1, …, n − 1, and, in addition,

α_{n}(t) = \{ \begin{matrix}t_{n}, & if m is odd, i. e., G is of type B_{n}; \\ t_{n−1}t_{n}, & if m is even, i. e., G is of type D_{n}. \end{matrix}

The Dynkin diagram of G_{K} is, respectively, given by

{∘ \atop α_{1}}  {∘ \atop α_{2}} ⋯ {∘ \atop α_{n−1}}> {∘ \atop α_{n}}  in case B_{n}

and

{∘ \atop α_{1}} {∘ \atop α_{2}} ⋯{∘ \atop α_{n−2}}[figure: ∘ α_{n−1}; ∘ α_{n}] in case D_{n}.

If m is odd or d(M) = 1, then G is of inner type, while G is of outer type ^{2}D_{n} for m even and d(M) ≠ 1. [32, 2.3, p. 39 and Table II, p. 56f].

For i = 1, …, n, define standard parabolic subgroups of G_{K} by

P_{i} = P_{Δ_{i}}, where Δ_{i} := \{ \begin{matrix}Δ \ \{α_{i}\} & if G is of inner type or i ≤ n − 2; \\ Δ \ \{α_{n−1}, α_{n}\} & if G is of outer type and i = n − 1. \end{matrix}

(Intentionally, we leave P_{n} undefined in the outer type case.) Then P_{i} is, for every i, a proper parabolic subgroup of G_{K} such that G_{K}/P_{i} ≅ V_{i} ×_{k} K for some k-variety V_{i} and such that P_{i} is maximal with this property. This follows by 3.7, since in the outer type case, the subset {α_{n−1}, α_{n}} ⊂ Δ is the only equivalence class under the ∗-action which contains more than one element [32, Table II, p. 57]. Also we have P_{i} ⊃ B = P_{∅} = ⟨T, U_{α} | α ∈ Δ⟩, and B is the stabilizer of the complete isotropic flag of K-spaces given by U_{i} = Ke′_{1} ⊕ ⋯ ⊕ Ke′_{i} for i = 1, …, n.

5.2 Lemma. Let i ∈ {1, …, n} if G is of inner type, and i ∈ {1, …, n − 1} otherwise.

i) Let the Witt index of (M,q) be at least i. Then P_{i} is a k-parabolic subgroup of G. Moreover, the k-variety I_{i} of i-dimensional totally isotropic k-subspaces of M is equal to V_{i} except when G is of inner type D_{n} and i ≥ n − 1, in which case M splits totally and I_{n} equals V_{n−1} ∪ V_{n}.

ii) Conversely, if there is a parabolic k-subgroup of type Δ_{i}, then there is a totally isotropic subspace U of M of dimension greater than or equal to i, hence the Witt index of (M, q) is at least i.

Proof. i) By Witt’s cancellation theorem on quadratic forms, the operation of G on I_{i} is transitive unless we are in the exceptional case where I_{n} has two orbits.

For the Witt basis {e_{1}, …, e_{m}}, the k-subspace U = ke_{1} ⊕ ⋯ ⊕ ke_{i} of M is totally isotropic and U ⊗_{k} K = U_{i}, hence the stabilizer subgroup P_{U} of U is a k-subgroup of G such that P_{U,K} ⊃ B. Therefore it is a parabolic k-subgroup of G.

The group P_{U}(k) consists of matrices (A_{jj′})_{j,j′=1,2,3} where A_{11}, A_{33} are i × i-matrices, while A_{22} is a (m − 2i) × (m − 2i)-matrix and A_{jj′} = 0 for j > j′. From the definition of the Witt basis, we conclude the identities A_{33} = IA_{11}^{−t}I and Q = A_{22}^{t}QA_{22}. Here I denotes the i × i-matrix with 1’s in the antidiagonal and zero elsewhere, while Q is the matrix describing the bilinear form on the subspace M_{0} generated by e_{i+1}, …, e_{m−i}. This implies that A_{33} is uniquely determined by A_{11} ∈ GL_{i}(k) with det(A_{33}) = det(A_{11})^{−1} and that A_{22} ∈ G_{0}(k), where G_{0} := SO(M_{0}, q|_{M_{0}}). Hence P_{U} has a Levi k-subgroup isomorphic to GL_{i} × G_{0} given by the matrices

( \begin{matrix}A_{11} & 0 & 0 \\ 0 & A_{22} & 0 \\ 0 & 0 & IA_{11}^{−t}I \end{matrix} ) .

The torus T_{Δ_{i}} = (⋂_{α∈Δ_{i}} Ker (α))^{∘} is a k-torus and given by the diagonal matrices

diag (t_{1}, …, t_{i}, 1, …, 1, t_{i}^{−1}, …, t_{1}^{−1})

with t_{1} = ⋯ = t_{i} except when G is of type D_{n} and i ≥ n − 1, in which case T_{Δ_{i}} is given by the matrices

diag (t_{1}, …, t_{n−1}, ∗, ∗, t_{n−1}^{−1}, …, t_{1}^{−1})

with t_{1} = ⋯ = t_{n−1}, where ∗, ∗ denotes entries which show up as diagonal elements only over a splitting field of G and which, otherwise, have to be replaced by 2 × 2 matrices which represent an anisotropic torus.

Now it is easily checked that, in any case, the Levi subgroup of P_{U} is the centralizer of T_{Δ_{i}} in G. Hence P_{U,K} = (Z(T_{Δ_{i}})R_{u}(P_{U}))_{K} = Z(T_{Δ_{i}})_{K} · B = Z(T_{Δ_{i}})_{K} · R_{u}(P_{i}) = P_{i} by 3.1. This proves i).

ii) Applying the arguments above to the pair K, K instead of k, K we find that the existence of a parabolic k-subgroup of type Δ_{i} implies the existence of a totally isotropic k-subspace U of dimension i in M.

By 3.15 iii) the function field F_{i} = k(V_{i}) is a generic Δ_{i}-splitting field of G.

5.3 Theorem. Let i ∈ {1, …, n} if G is of inner type, and i ∈ {1, …, n − 1} otherwise. The field F_{i} = k(V_{i}) is a generic field for splitting off at least i hyperbolic planes from the underlying quadratic space. Namely, (M ⊗_{k} F_{i}, q ⊗_{k} F_{i}) has the Witt index ≥ i, and for every field extension L of k the quadratic space (M ⊗_{k} L, q ⊗_{k} L) has the Witt index ≥ i if and only if L is a k-specialization of F_{i}.

Proof. Since V_{i}(F_{i}) ≠ ∅ it follows from 5.2 ii) that (M ⊗_{k} F_{i}, q ⊗_{k} F_{i}) has the Witt index ≥ i. By 5.2 and 3.10 we have: (M ⊗_{k} L, q ⊗_{k} L) has the Witt index ≥ i if and only if V_{i}(L) ≠ ∅, and the latter holds if and only if L is a k-specialization of F_{i}.

If G is of outer type, there is no regular extension of k which splits (M,q) totally (that is, gives the maximal Witt index); this follows from 3.5.

5.4 Corollary. If G is of inner type, then k(V_{∅}) and F_{n} are generic splitting fields of G. If G is of inner type and m is even, then also F_{n−1} is a generic splitting field of G. If G is of outer type, then k(V_{∅}) and F_{n−1} are generic quasi-splitting fields of G, and the fields k(\sqrt{d(M)}) · F_{n−1} and k(\sqrt{d(M)}) · k(V_{∅}) are generic splitting fields of G.

Proof. It follows from 3.11 that k(V_{∅}) is a generic quasi-splitting field of G and that this is a generic splitting field if G is of inner type. In that case also F_{n} is a generic splitting field which is obvious from 5.1, 5.3, and 3.17 iv). If G is of inner type and m is even, then the discriminant of (M, q) is 1. Hence if L is a field extension of k such that the Witt index of (M ⊗_{k} L, q ⊗_{k} L) is ≥ n − 1, then M ⊗_{k} L ≅ H ⊥ M′ with a hyperbolic L-space H and a regular L-space M′ of dimension 2 and discriminant 1, which therefore is a hyperbolic plane. Hence G_{L} splits. The last statement follows from 3.6 in combination with 3.17, since for n ≥ 3 the field k_{alg} of 3.6 coincides with k(\sqrt{d(M)}).

Example. To clarify the situation in the case of a non-∗-invariant Θ we take m even and Θ = \{α_{1}, …, α_{n−1}\}. This is, in the outer type case, not ∗-invariant, and F_{Θ} ⊇ k_{Θ} = k(\sqrt{d(M)}) (cf. 3.16). Clearly G_{k_{Θ}} is of inner type, F_{Θ} is a splitting field of G_{k_{Θ}} by 5.4 and hence also of G. Of course F_{Θ} is then a fortiori a quasi-splitting field of G. However, it is not a generic quasi-splitting field of G by 3.12, since k is not algebraically closed in F_{Θ}.

5.5 Corollary. The transcendence degree of F_{i} is given by the following formulae:

trdeg F_{i} = \{ \begin{matrix}i(4n − 3i + 1)/2 & if m is odd, 1 ≤ i ≤ n; \\ i(4n − 3i − 1)/2 & if m is even, 1 ≤ i ≤ n − 2; \\ n(n − 1)/2 & if m is even, G is of inner type, n − 1 ≤ i ≤ n; \\ (n + 2)(n − 1)/2 & if m is even, G is of outer type, i = n − 1; \end{matrix}

trdeg k(V_{∅}) = \{ \begin{matrix}n^{2} & if m is odd; \\ n(n − 1) & if m is even. \end{matrix}

There is an increasing sequence of (non-canonical) k-linear embeddings

F_{1} ↪ ⋯ ↪ F_{i} ↪ F_{i+1} ↪ ⋯ ↪ F_{n′} ↪ k(V_{∅})

with n′ = n or n − 1 according as G is inner or not.

Proof. By 3.2, the dimension of V_{i} is the cardinality of u_{Δ_{i}}. This can be computed by using the explicit descriptions of the root systems of types B_{n} and D_{n} as given in [10, p. 252 and p. 256] or [33, p. 30, p. 35]. (Note: The description of positive roots of D_{n} in [10, p. 256] is erroneous. For a correct description cf. [33, p. 35].)

By 5.3, F_{i+1} splits off at least i + 1 hyperbolic planes of (M, q). Hence V_{i}(F_{i+1}) ≠ ∅ by 5.2. Thus, by 3.10, there is a k-linear embedding of F_{i} into a purely transcendental extension of F_{i+1}. By the above we have trdeg F_{i+1} ≥ trdeg F_{i}, hence it follows from [24, Lemma 1, p. 209] that there is a k-linear embedding F_{i} ↪ F_{i+1}. A similar argument gives the k-linear embedding F_{n′} ↪ k(V_{∅}). Alternatively, we here can use the natural map induced by the inclusion B ↪ P_{n′} for a proper choice of a Borel group B.

5.6 Corollary. We have F_{1} = k(V_{1}) ≅ k(q)_{0}, where k(q)_{0} is a generic zero field as defined by Knebusch [16, 3.2, p. 69, and p. 71].

Proof. V_{1} is, by 5.2, the variety of the isotropic lines in M, which can be defined by the equation q(x) = 0 for x ∈ M.

5.7 Corollary. Let i = 1, …, n′ where n′ = [m/2] or [m/2] − 1 according as G is of inner or outer type. Assume L is an arbitrary field extension of k. Then (M ⊗_{k} L, q ⊗_{k} L) is of index ≥ i if and only if the free composite F_{i} · L is purely transcendental over L. In particular, (M, q) is of index ≥ i if and only if F_{i} is a purely transcendental extension of k.

This follows from 5.3 and 3.10. Corollary 5.7 was obtained by Knebusch for i = 1 [16. 3.8 and 3.10, p. 72].

5.8 Remark. It is easily seen that a suitable subsequence {F_{i_{j}}} of the sequence {F_{i}} in 5.3 is a so-called generic splitting tower as originated by Knebusch [16, §5, p. 78]: Let n′ = n if G is of inner type and let n′ = n − 1 otherwise. We define i_{j} inductively. Let i_{0} = 0 and F_{i_{0}} := k. If i_{j} ≤ n′ is defined let i_{j+1} ∈ {1, …, n′} be the smallest number such that the Witt index of (M ⊗_{k} F_{i_{j+1}}, q ⊗_{k} F_{i_{j+1}}) is bigger than that of (M ⊗_{k} F_{i_{j}}, q ⊗_{k} F_{i_{j}}). In the inner case the sequence F_{i_{j}}, j ≥ 1, is a generic splitting tower. If G is of outer type and F := F_{i_{j′}} is the last element of this sequence, then the anisotropic kernel of (M ⊗_{k} F, q ⊗_{k} F) is a binary form, hence its special orthogonal group is an anisotropic F -torus which is generically split by the field F (\sqrt{d(M)}) (cf. 2.4). In this case we define F_{i_{j′+1}} := F (\sqrt{d(M)}) as the last element of the sequence.

Knebusch gives in [16, Example 5.7, p. 80] an example of an anisotropic form of arbitrary dimension together with a generic splitting tower {K_{i}} such that every layer splits off precisely one hyperbolic plane. Clearly, for such a form, the sequence {F_{i}} is also a generic splitting tower. We have K_{1} = F_{1}, however, for i > 1, the transcendence degree of K_{i} exceeds that of F_{i} by i(i − 1)/2 if m is odd or i ≤ n − 2, and if m is even and i = n − 1, by (n − 1)n/2 in the inner case and by (n − 1)(n − 2)/2 in the outer case.

On the other hand it is easy to see that there are forms for which the sequence {F_{i}} degenerates completely in the sense that F_{1} already is a generic splitting field of SO(φ). For example, any Pfister form φ has the property that it is hyperbolic already if it is isotropic [25, 4, Cor. 1.5, p. 144]. This implies that all the associated fields F_{i} are k-specializations of each other. Since the special orthogonal group SO(φ) is of inner type if the dimension of φ is ≥ 4 (the discriminant of a Pfister form of dimension ≥ 4 is 1), it follows from 3.17 iii) that the F_{i} are all generic splitting fields of SO(φ).

As has also been observed by Knebusch, a generic zero field of any orthogonal summand ψ of a Pfister form φ with dim ψ = (dim φ)/2 + 1 is a generic splitting field of SO(φ).

This seems to indicate that in general it might be difficult to find a generic splitting field with minimal transcendence degree for an arbitrary reductive group.

As a consequence of Theorem 5.3 we obtain a corollary which can be also derived from [16, Th. 3.3, p. 69] by using a generic splitting tower of Knebusch (cf. [25, 4. Cor. 6.10, p. 160]).

5.9 Corollary. Let L be an arbitrary field extension of k. If i > 0 is the Witt index of (M ⊗_{k} L, q ⊗_{k} L), then i is the Witt index of (M ⊗_{k} F_{i}, q ⊗_{k} F_{i}).

Proof. The field L is a k-specialization of F_{i} by 5.3. However, L is not a k-specialization of F_{i+1}, for otherwise the Witt index of (M ⊗_{k} L, q ⊗_{k} L) would be at least i + 1 by 5.3. Thus the result follows from 5.3.

6. Generic splitting of the classical groups In §§4 and 5 we studied the generic splitting of groups of type ^{1}A_{n}, B_{n}, and certain cases of type ^{1}D_{n} and ^{2}D_{n} (namely, those for which the underlying central k-division algebra is k itself).

In this section we investigate the generic Θ-splitting of G for arbitrary ∗-invariant subsets Θ of Δ and G of types ^{2}A_{n}, B_{n}, C_{n}, ^{1}D_{n}, ^{2}D_{n} in a uniform manner. This is possible because all these groups are special unitary groups of certain (skew-) Hermitian forms over some finite dimensional division algebras over k.

Let k be an infinite field of char(k) ≠ 2. Suppose E is a field extension of degree 1 or 2 over k and D is a central division E-algebra of degree d over E. Let σ : D → D, a ↦ a^{σ}, be an involution on D, so that σ is E-linear, of order ≤ 2 and (ab)^{σ} = b^{σ}a^{σ} for all a, b ∈ D. Assume that M is an m-dimensional right D-vector space and that h : M × M → D is a non-degenerate ϵ-σ-Hermitian form on M with ϵ = ±1. In particular, we have h(xa, yb) = a^{σ}h(x, y)b, h(y, x) = ϵh(x, y)^{σ} for x, y ∈ M, a, b ∈ D. The pair (M, h) is called an ϵ-σ-Hermitian space.

Let now G := SU(h) be the special unitary group of (M, h). Then the index r of (M, h) is the k-rank of G (cf. [6, 23.9, p. 266]).

If G is of type ^{2}A_{n}, then the involution on D is of second type, hence E is separable of degree 2 over k. In this case we let n ≥ 1. If G is of type B_{n}, then d = 1 and D = E = k and we may assume n ≥ 2. If G is of type C_{n} or D_{n}, we have E = k. We may then assume n ≥ 3.

Let K be a splitting field of G (for example, K is a separable closure of k). The group G_{K} is isomorphic to (SL_{n+1})_{K} (resp. (SO_{2n+1})_{K}, (Sp_{2n})_{K}, (SO_{2n})_{K}) if G is of type ^{2}A_{n} (resp. B_{n}, C_{n}, D_{n}).

Then the absolute rank n of G is the rank of G_{K} and is given by the formulae n + 1 = md in case 2A_{n} and n = [md/2] in the other cases.

In the case ^{2}A_{n} we take the maximal K-split K-torus T given by

t = diag (t_{1}, …, t_{n+1}) ∈ G(K), t_{1}, …, t_{n+1} ∈ K^{∗}

and the basis Δ from §4 for the root system Φ = Φ(G_{K}, T), which is given by α_{i}(t) := t_{i} t_{i+1}^{−1}, for i = 1, …, n.

In the cases B_{n}, C_{n}, D_{n} we proceed as follows. Similarly as in §5 we can use a Witt basis of the underlying bilinear K-space K^{md} to embed G_{K} into (SL_{md})_{K}. Then a maximal K-split K-subtorus T of G_{K} is defined by the set of diagonal matrices (cf. [6, 23.9, p. 266])

t = diag (t_{1}, …, t_{n}, ˆ1, t_{n}^{−1}, …, t_{1}^{−1}) ∈ G(K), t_{1}, …, t_{n} ∈ K^{∗}.

(Here the symbol ˆ1 means that the component 1 occurs (resp. does not occur) in the middle according as G is of type B_{n} or not.) A basis Δ = {α_{1}, …, α_{n}} of the root system Φ = Φ(G_{K}, T) is given by α_{i}(t) := t_{i} t_{i+1}^{−1}, for i = 1, …, n − 1, and, in addition,

α_{n}(t) = \{ \begin{matrix}t_{n}, & if G is of type B_{n}; \\ t_{n}^{2}, & if G is of type C_{n}; \\ t_{n−1}t_{n}, & if G is of type D_{n} \end{matrix}

(cf. [33, p. 30, 32, 35] and [10, p. 252, 254, 256]).

If deg (E/k) = 2, then for a field extension k′ of k, the k-algebra D ⊗_{k} k′ is an Azumaya E ⊗_{k} k′- algebra if E ⊗_{k} k′ is a field, or it is a direct sum of two copies of an Azumaya k′-algebra A′ if E ⊗_{k} k′ ≅ k′ ⊕ k′. In Theorem 6.1 below we use the following notation:

ind (D ⊗_{k} k′) := \{ \begin{matrix}ind_{E⊗_{k}k′}(D ⊗_{k} k′) & if E ⊗_{k} k′ is a field; \\ ind_{k′}(A′) & otherwise. \end{matrix}

We now prove a theorem corresponding to 4.1 for special unitary groups G. If G is of outer type, we have to replace the set Δ\Θ occurring in 4.1 by a suitable set of representatives in Δ\Θ of ∗-orbits. The function field F_{Θ} := k_{Θ}(V_{Θ}) is a generic Θ-splitting field of G by 3.16. It has the equivalent properties listed in 3.10 if Θ is ∗-invariant.

6.1 Theorem. Let G be of type ^{2}A_{n}, B_{n}, C_{n},^{1}D_{n} or ^{2}D_{n}. For each ∗-invariant subset Θ ⊂ Δ let {α_{i_{1}}, …, α_{i_{l}}} be the set of representatives of ∗-orbits of Δ \ Θ such that i_{ν} ∈ {1, …, n − 1} in case ^{2}D_{n} and i_{ν} ∈ {1, …, [(n + 1)/2]} in case ^{2}A_{n}, and let {α_{i_{1}}, …, α_{i_{l}}} = Δ \ Θ otherwise. If V_{Θ} is the k-variety associated to Θ according to 3.7, then for every field extension k′ of k we have V_{Θ}(k′) ≠ ∅ if and only if d′ := ind (D ⊗_{k} k′) divides gcd (d, i_{1}, …, i_{l}) and max (i_{1}, …, i_{l}) ≤ d′ · rank (G_{k′}).

Proof. If G_{k′} is of inner type ^{1}A_{n} the equivalence follows from 4.1, since the last condition in 4.1 implies the condition max (i_{1}, …, i_{l}) ≤ d′ · rank (G_{k′}). Therefore it suffices to show the equivalence for the case k′ = k. Namely, we have to show: The index d = ind (D) divides gcd (d, i_{1}, …, i_{l}) and max (i_{1}, …, i_{l}) ≤ dr if and only if V_{Θ}(k) ≠ ∅.

A maximal k-split torus S of G is given, with respect to a Witt basis {e_{i}} of (M, h), by the diagonal matrices s = diag (s_{1}, …, s_{r}, 1, …, 1, s_{r}^{−1}, …, s_{1}^{−1}) ∈ M_{m}(D) with s_{j} ∈ k^{∗} for j = 1, …, r (cf. [6, 23.9, p. 266]). Let k_{s} be a separable closure of k.

Using a Witt basis over k_{s} we may obtain an embedding G_{k_{s}} ↪ (SL_{md})_{k_{s}} such that a maximal k_{s}-torus T_{k_{s}} of G_{k_{s}} is, respectively, described by matrices diag (t_{1}, …, t_{n+1}) in case ^{2}A_{n} and diag (t_{1}, …, t_{n}, ˆ1, t_{n}^{−1}, …, t_{1}^{−1}) otherwise with t_{1}, …, t_{n+1} ∈ k_{s}^{∗}, and S_{k_{s}}, as a subtorus of T_{k_{s}}, is given by the following matrices

s = diag (\underbrace{s_{1}, …, s_{1}}_{d times}, …, \underbrace{s_{r}, …, s_{r}}_{d times}, \underbrace{1, …, 1,}_{(m−2r)d times} \underbrace{s_{r}^{−1}, …, s_{r}^{−1}}_{d times}, …, \underbrace{s_{1}^{−1}, …, s_{1}^{−1}}_{d times})

with s_{1}, …, s_{r} as above.

We first evaluate the cases B_{n}, C_{n}, D_{n}. If G is of type B_{n}, then d = 1 and we find, for i ∈ {1, …, n},

α_{i}(s) = \{ \begin{matrix}s_{r} & if i = r ≤ n \\ s_{i}s_{i+1}^{−1} & if 1 ≤ i < r \\ 1 & if i > r. \end{matrix}

If G is of type D_{n} and n = rd + 1, then d ≤ 2 and α_{n−1}(s) = α_{n}(s) = s_{r}. Hence G cannot be of inner type in this case and is necessarily of type ^{2}D_{n}.

We find in case d > 1

α_{i}(s) = \{ \begin{matrix}s_{r} & if i = rd < n \\ s_{r}^{2} & if i = rd = n \\ s_{j}s_{j+1}^{−1} & if i = jd, 1 ≤ j < r \\ 1 & if d |/ i or i > rd \end{matrix}

for i ∈ {1, …, n − 1} if G is of type ^{2}D_{n} and i ∈ {1, …, n} otherwise.

If d = 1 and G is of type D_{n} we have the same formula with the exception that α_{i}(s) = s_{n−1}s_{n}, if i = r = n.

If d = 1 and G is of type C_{n}, then r = n (cf. [35, §91, p. 31]), and we find that α_{i}|_{S} is not trivial for all i = 1, …, r = n.

In the case ^{2}A_{n} we obtain similarly, for i = 1, …, [(n + 1)/2],

α_{i}(s) = α_{n+1−i}(s) = \{ \begin{matrix}s_{r} & if i = rd < (n + 1)/2 \\ s_{r}^{2} & if i = rd = (n + 1)/2 \\ s_{j}s_{j+1}^{−1} & if i = jd, 1 ≤ j < r \\ 1 & if d |/ i or i > rd. \end{matrix}

Using the notation of 3.4 v) we now see that

Δ \ Δ_{0} = \{ \begin{matrix}\{α_{jd}, α_{n+1−jd} | j = 1, …, r\} & in case ^{2}A_{n} \\ \{α_{jd} | j = 1, …, r\} ∪ \{α_{n}\} & in case ^{2}D_{n}, if d ≤ 2 and n = rd + 1 \\ \{α_{jd} | j = 1, …, r\} & otherwise \end{matrix}

for d ≥ 1. It follows that d | gcd (d, i_{1}, …, i_{l}) and max (i_{1}, …, i_{l}) ≤ rd if and only if Δ \ Θ ⊆ Δ \ Δ_{0}. The latter condition is equivalent to V_{Θ}(k) ≠ ∅ by 3.7, “iii) ⇔ ii)”.

For any α_{i} ∈ Δ let Δ_{i} be the maximal ∗-invariant subset of Δ which does not contain α_{i}. It follows from [34, Table II, p. 55ff] or also from the above calculations that all those sets can be described as follows.

Δ_{i} = \{ \begin{matrix}Δ \ \{α_{i}, α_{n+1−i}\} & if G is of type ^{2}A_{n} and i ∈ \{1, …, [(n + 1)/2]\}; \\ Δ \ \{α_{i}\} & if G is of type B_{n}, C_{n} or ^{1}D_{n} or i < n − 1; \\ Δ \ \{α_{n−1}, α_{n}\} & if G is of type ^{2}D_{n} and i = n − 1. \end{matrix}

We emphasize that, in case ^{2}A_{n}, the set Δ_{i} is always of order n−2 except if 2 | (n+1) and i = (n+1)/2, in which case it is of order n − 1 since α_{i} = α_{n+1−i}.

In all cases, let V_{i} := V_{Δ_{i}} be the k-variety associated to Δ_{i} according to 3.7, and let F_{i} = k(V_{i}) be its function field. Applying 6.1 for Θ = Δ_{i} we obtain the following corollary.

6.2 Corollary. Let i ∈ {1, …, [(n+1)/2]} if G is of type ^{2}A_{n}, let i ∈ {1, …, n} if G is of type B_{n}, C_{n} or 1D_{n}, and i ∈ {1, …, n − 1} if G is of type ^{2}D_{n}. Then for every field extension k′ of k we have V_{i}(k′) ≠ ∅ if and only if d′ := ind (D ⊗_{k} k′) divides i and 1 ≤ i/d′ ≤ rank (G_{k′}).

We now list generic (quasi-)splitting fields of G with low transcendence degrees. Most, but not all of them, are defined by maximal proper ∗-invariant subsets Θ of Δ. For G of type B_{n} this is discussed in 5.4.

6.3 Corollary. Let the notation be as in 6.2.

i) Let G be of type ^{2}A_{n} and let n′ = [(n + 1)/2]. If gcd (n′, d) = 1, then F_{n′} is a generic quasi-splitting field of G. More generally, let n′′ ∈ {1, …, n′} be some integer such that gcd (n′, n′′, d) = 1 and let Θ := Δ \ {α_{n′}, α_{n′′}, α_{n+1−n′}, α_{n+1−n′′}}. Then k(V_{Θ}) is also a generic quasi-splitting field of G. Moreover, the fields E · F_{n′} and E · k(V_{Θ}) are generic splitting fields of G, respectively.

ii) Let G be of type C_{n}. Then every (generic) splitting field of D is a (generic) splitting field of G. For

every odd i ∈ {1, …, n} the field F_{i} = k(V_{i}) is a generic splitting field of G. iii) Let G be of type ^{1}D_{n}. Then F_{n−1} is a generic splitting field of G. If n is odd or d = 1, then F_{n} is

also a generic splitting field of G. iv) Let G be of type ^{2}D_{n}. If n is even or if d = 1, then F_{n−1} is a generic quasi-splitting field of G.

If n is odd and Θ = Δ \ {α_{n−2}, α_{n−1}, α_{n}}, then k(V_{Θ}) is a generic quasi-splitting field. If d(M) denotes the discriminant of the Hermitian space (M, h), then a generic splitting field is given by k(\sqrt{d(M)}) · F_{n−1} if n is even or d = 1 and by k(\sqrt{d(M)}) · k(V_{Θ}) if n is odd.

Proof. We recall once and for all that, by 3.17 iv) (resp. iii) ), some field F_{Θ} is a generic splitting field (resp. quasi-splitting field) of G if it is a splitting field (resp. quasi-splitting field) of G.

i) Let k_{1} be either F_{n′} or k(V_{Θ}). Then it follows from 6.1 or 6.2 that rank (G_{k_{1}}) ≥ n′. Therefore G_{k_{1}} is a special unitary group (SU_{n+1})_{k_{1}} of maximal rank and hence quasi-split. Now it follows from 3.6 that E · k_{1} is a generic splitting field, since E is isomorphic to the field k_{alg}.

ii) If d = 1, then G splits [35, §91, p. 31]. Hence the first assertion follows. Since V_{i}(F_{i}) ≠ ∅ and d is a power of 2, it follows from 6.1 that ind (D ⊗_{k} F_{i}) = 1 for odd i. Thus G_{F_{i}} splits.

iii) Since V_{i}(F_{i}) ≠ ∅ we have d_{i} := ind (D ⊗_{k} F_{i}) | gcd (d, i) by 6.1. If n is even, then d_{n−1} = 1, and if n is odd, then d_{n} = 1, since d is a power of 2. Hence if n is odd we obtain n ≤ rank (G_{F_{n}}) from 6.1, therefore G_{F_{n}} splits.

We have (n − 1)/d_{n−1} ≤ rank (G_{F_{n−1}}) by 6.1. Therefore, if d_{n−1} = 1, then G_{F_{n−1}} is the orthogonal group of a quadratic form of dimension 2n with discriminant 1 and of Witt index ≥ n − 1. Hence the form is hyperbolic, which implies that G_{F_{n−1}} splits.

It remains to show that d_{n−1} = 1 for odd n. If n is odd, then d ≤ 2, because d | 2n and d is a power of 2. Hence d_{n−1} | 2. Assume d_{n−1} = 2. Then the rank r_{n−1} of G_{F_{n−1}} is at least (n − 1)/2 which implies n = r_{n−1}d_{n−1} + 1. This is impossible since G is of inner type ^{1}D_{n} (cf. [32. Table II, p. 56]). Hence necessarily d_{n−1} = 1.

iv) If n is even or d = 1, then, as above, d_{i} := ind (D ⊗_{k} F_{i}) = 1, hence G_{F_{n−1}} is the orthogonal group of a quadratic form of dimension 2n over F_{n−1} with Witt index n − 1 and discriminant d(M) ≠ 1, since F_{n−1} is regular over k. Hence G_{F_{n−1}} is quasi-split. If n is odd, then d ≤ 2 (cf. the proof of iii)). Obviously Θ is ∗-invariant. Hence we may apply 6.1 to find that d′ = ind (D ⊗_{k} F_{Θ}) = 1 and n − 1 ≤ rank (G_{F_{Θ}}). Therefore G_{F_{Θ}} is of rank n − 1 which means that it is quasi-split. Since G is semisimple, we have k(\sqrt{d(M)}) = k_{alg}, where k_{alg} is given by 3.6. The rest of the statement follows from 3.6.

6.4 Remark. For groups of outer type there are other non-regular generic splitting fields with possibly lower transcendence degrees: Let Θ ⊂ Δ be any not necessarily ∗-invariant subset. By 3.16 there is a generic Θ-splitting field F_{Θ} = k_{Θ}(V_{Θ}) of G. Applying the results of §4 (resp. §5) to G_{k_{Θ}} for special proper maximal subsets Θ we obtain for example:

i) In case ^{2}A_{n}: Let Θ = Δ \ {α_{1}}. Then k_{Θ} = E and V_{Θ} is an n-dimensional Severi-Brauer variety over E and the function field E(V_{Θ}) is a generic splitting field of G_{E_{Θ}}, and hence of G.

ii) In case ^{2}D_{n}: Let Θ = Δ \ \{α_{n−1}\}. Then k_{Θ} = k(\sqrt{d(M)}) and V_{Θ} is an n(n − 1)/2-dimensional variety (cf. 5.5) over k_{Θ} and the function field k_{Θ}(V_{Θ}) is a generic splitting field of G_{k_{Θ}} by 6.3 iii), and hence of G.

6.5 Remark. In case ^{2}A_{n} and 2 | (n + 1) we find an outer form of a generalized Severi-Brauer variety as discussed in §4: Let i := (n + 1)/2 and Δ_{i} = Δ \ {α_{i}}. Then Δ_{i} is ∗-invariant, hence the associated variety V_{Δ_{i}} is defined over k (cf. 3.7). This is the outer form of Grass_{i}(k^{n+1}) mentioned after 4.6.

7. Generic splitting of almost simple groups In this paragraph we will give generic splitting and quasi-splitting fields of the absolutely almost simple k-groups including the exceptional groups and groups over fields of characteristic 2 which have been excluded in §§5 and 6. We emphazise that the notions of quasi-splitting field and splitting field coincide in the case of semi-simple groups of inner type, as it follows from the last statement of 3.4 v). In the outer type case, one may obtain generic splitting fields out of generic quasi-splitting fields by applying 3.6. Therefore, in this case we only give quasi-splitting fields. But the same method can also be used to construct generic splitting fields directly.

Let G be any almost simple k-group. Let T be a maximal k-torus of G. Let Δ = {α_{1}, …, α_{n}} be the set of simple roots of Gk̄ with respect to Tk̄ and some ordering of the root system. We will give generic splitting or quasi-splitting fields of G in terms of its Dynkin diagram (over a splitting field) by describing the maximal subsets Θ of Δ such that F_{Θ} is a generic splitting or quasi-splitting field. We will use the abbreviations Δ_{i} = Δ \ {α_{i}} and F_{i} = F_{Δ_{i}}.

7.1 Lemma. Let Θ ⊂ Δ. Then res_{F_{Θ}}(α) ≠ 0 for all α ∈ Δ \ Θ.

Proof. As F_{Θ} contains k_{Θ}, we may assume that k = k_{Θ}. Hence we may assume that Θ is ∗-invariant. Then our claim will follow from 3.7,“i) ⇒ iii)”, applied to F_{Θ} instead of k.

In the following, we will indicate how to use 7.1 together with 3.17 iii) or iv) and the information encoded in the index of G as described in [32, §2, p. 38ff] just for the particular case ^{1}A_{n}, since the considerations in all the other cases are quite similar.

Dynkin diagram of type A_{n}:

{∘ \atop α_{1}}  {∘ \atop α_{2}} ⋯ {∘ \atop α_{n}}

Generic splitting fields in case ^{1}A_{n} are given by F_{i} for any i which is coprime to the index d of the underlying central k-division algebra D (cf. 4.3). In order to see this, let G denote an almost simple k-group of type ^{1}A_{n} and let k′ be a field extension of k.

We will verify that, for any i ∈ {1, …, n} coprime to d, the condition res_{k′}(α_{i}) ≠ 0 implies that G_{k′} is split. It follows from the description of the index of G_{k′} in the sense of [32, §2, p. 38ff] that res_{k′}(α_{i}) ≠ 0 if and only if i is a multiple of d′ := ind (D ⊗_{k} k′). As d′ divides both d and i, and since gcd (d, i) = 1 by assumption, we find d′ = 1, which implies that G_{k′} is split.

In particular, for k′ = F_{i}, it follows from 7.1 that res_{F_{i}}(α_{i}) ≠ 0. By the above, G_{F_{i}} splits, and 3.17 iii) or iv) proves that F_{i} is a generic splitting field of G.

For ^{2}A_{n}, we use the notation of [33, p.55] or of §4 and let ϱ = [(n + 1)/2]. Generic quasi-splitting fields in case ^{2}A_{n} are given by Δ \ {α_{ϱ}, α_{n+1−ϱ}} if gcd (ϱ, d) = 1 and by Δ \ {α_{ϱ}, α_{n+1−ϱ}, α_{ϱ−1}, α_{n+2−ϱ}} if gcd (ϱ, d) ≠ 1 (cf. 6.3 i)).

Dynkin diagram of type B_{n}:

{∘ \atop α_{1}} {∘ \atop α_{2}} ⋯ {∘ \atop α_{n−1}}> {∘ \atop α_{n}}

Generic splitting fields are given by Δ_{n} (cf. 5.4).

Dynkin diagram of type C_{n}:

{∘ \atop α_{1}} {∘ \atop α_{2}} ⋯ {∘ \atop α_{n−1}}< {∘ \atop α_{n}}

Generic splitting fields are given by Δ_{i} for any i which is coprime to the index of the underlying division algebra. As this is a power of two, i just has to be odd in this case (cf. 6.3 ii)).

Dynkin diagram of type D_{n}:

{∘ \atop α_{1}}  {∘ \atop α_{2}} ⋯{∘ \atop α_{n−2}}[figure: ∘ α_{n−1}; ∘ α_{n}]

Generic splitting fields for ^{1}D_{n} are given by Δ_{n−1} (cf. 6.3 iii)). For the outer case we again use the notation of [33, p.57] which is consistent with §§5 and 6.

Generic quasi-splitting fields for ^{2}D_{n} are given by Δ \ {α_{n−1}, α_{n}} if d = 1 or 2 | n and by Δ \ {α_{n−2}, α_{n−1}, α_{n}} if n is odd (which implies that d ≤ 2 since d is a power of 2 and divides 2n) and d = 2 (cf. 6.3 iv)).

Dynkin diagram of type ^{3}D_{4},^{6}D_{4}:

{∘ \atop α_{1}}  {∘ \atop α_{2}}[figure: ∘ α_{3}; ∘ α_{4}]

A generic quasi-splitting field is given by Θ = {α_{2}}. Dynkin diagram of type E_{6}:

{∘ \atop α_{1}}  {∘ \atop α_{2}}  α_{6} ; \begin{matrix}∘ \\ ∘ \end{matrix} ; α_{3}  {∘ \atop α_{4}}  {∘ \atop α_{5}}

Generic quasi-splitting fields are given in case ^{1}E_{6} by Δ_{2}, Δ_{4}, in case ^{2}E_{6} by Δ \ {α_{2}, α_{4}}. Dynkin diagram of type E_{7}:

{∘ \atop α_{1}}  {∘ \atop α_{2}}  {∘ \atop α_{3}}  α_{7} ; \begin{matrix}∘ \\ ∘ \end{matrix} ; α_{4}  {∘ \atop α_{5}}  {∘ \atop α_{6}}

Generic splitting fields are given by Δ_{3}, Δ_{5}, Δ_{7}. Dynkin diagram of type E_{8}:

{∘ \atop α_{1}}  {∘ \atop α_{2}}  {∘ \atop α_{3}}  {∘ \atop α_{4}}  α_{8} ; \begin{matrix}∘ \\ ∘ \end{matrix} ; α_{5}  {∘ \atop α_{6}}  {∘ \atop α_{7}}

Generic splitting fields are given by Δ_{4}, Δ_{5}, Δ_{6}, Δ_{8}. Dynkin diagram of type F_{4}:

{∘ \atop α_{1}}  {∘ \atop α_{2}} < {∘ \atop α_{3}}  {∘ \atop α_{4}}

Generic splitting fields are given by Δ_{2}, Δ_{3}, Δ_{4}. Dynkin diagram of type G_{2}:

{∘ \atop α_{1}} < {∘ \atop α_{2}}

Generic splitting fields are given by Δ_{1}, Δ_{2}.

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Fakultät für Mathematik Universität Bielefeld 4800 Bielefeld 1 Germany