TITS INDICES OVER SEMILOCAL RINGS

V. PETROV AND A. STAVROVA

Abstract. We present a simplified version of Tits’ proof of the classification of semisimple algebraic groups, which remains valid over semilocal rings. We also provide explicit conditions on anisotropic groups to appear as anisotropic kernels of semisimple groups of a given index.

1. Introduction

In his famous paper [Ti66] Tits has shown that any semisimple group G over a field is determined by its anisotropic kernel and a combinatorial structure called the Tits index of G. Some arguments were sketched or omitted there, and in full details appeared in [Ti90]. The goal of the present paper is to show that this result remains true over arbitrary connected semilocal rings. We do not use the case of fields, but rather provide a shortened and simplified version of Tits’ arguments.

Our argument relies on the fact that, given a good enough representation of an algebraic group over a splitting covering, one can twist it to a representation into some Azumaya algebra, called a Tits algebra (cf. [Ti71]), over the base ring. We show that this in fact holds over arbitrary schemes (Theorem 1). Then in Theorem 2 we give a necessary and sufficient criterion for a semisimple group scheme H to be the derived subgroup of a Levi subgroup of a parabolic subgroup of a given type. After that, the list of all possible indices is easily obtained by induction, and the existence of a group with a given index shows to be equivalent to the existence of an anisotropic group subject to some explicitly stated restrictions (§ 5, Theorem 3).

2. Semisimple group schemes

In this section we reproduce some definitions and results of [SGA].

Let S be a scheme (not necessarily separated). A group scheme G over S is called reductive if it is affine and smooth over S, and its geometric fibers G\overline{k(s)} are connected reductive groups in the usual sense for all s ∈ S (Exp. XIX Déf. 4.7^{1}). When S is reduced, the smoothness can be replaced by the condition that G is finitely presented over S and the dimension of a fiber is locally constant (see Exp. VI_{B}, Cor. 4.4). The type of G at s ∈ S is the root datum of G\overline{k(s)}. The type is locally constant (Exp. XXII Prop. 2.8). To simplify the exposition, in the sequel we consider reductive group schemes of constant type only. Thus the type of a reductive group scheme G is a root datum R = (Φ, Λ, Φ^{∗}, Λ^{∗}), where Φ is a root system, called the root system of G, Λ is a ℤ-lattice containing Φ, called the lattice of weights of G, and Φ^{∗} and Λ^{∗} are the dual objects (Exp. XXI Déf. 1.1.1). We usually include in the type the Dynkin diagram D of Φ, whose vertices can be identified with a system of simple roots of Φ.

Over any scheme S there exists a unique split group scheme G_{0} of a given type R, which actually comes from a group scheme over Spec ℤ known as the Chevalley – Demazure group scheme (Exp. XXV Thm. 1.1). Quasi-split group schemes over S of the same type as G_{0} are parametrized by H^{1}(S, Aut (R)) (Exp. XXIV Thm. 3.11). All cohomology groups we consider are with respect to the fpqc topology (but note that H^{1}(S, H) = H_{ét}^{1}(S, H) when H is smooth).

1All references starting with Exp. YZ refer to [SGA].

Every semisimple group scheme G is an inner twisted form of a uniquely determined quasi-split group G_{qs}, given by a cocycle ξ ∈ Z^{1}(S, G_{qs}^{ad}), where G_{qs}^{ad} is the adjoint group acting on G_{qs} by inner automorphisms; cocycles in the same class in H^{1}(S, G_{qs}^{ad}) produce isomorphic group schemes (Exp. XXIV 3.12.1).

A Dynkin diagram D is nothing but a finite set of vertices together with a subset E ⊆ D × D of edges and a length function D → {1, 2, 3} (in other words, a colored graph). The scheme-theoretic counterpart of this notion is called a Dynkin scheme (Exp. XXIV § 3). So a Dynkin scheme over S is a twisted finite scheme D over S together with a subscheme E ⊆ D ×_{S} D and a map D → {1, 2, 3}_{S}. Isomorphisms, base extensions and constant Dynkin schemes are defined in an obvious way. We denote by D_{S} the constant Dynkin scheme over S corresponding to a Dynkin diagram D. By Aut (D) we always mean the scheme of automorphisms of D over S as a Dynkin scheme; it is a twisted constant group scheme over S.

With each semisimple group scheme G one associates the Dynkin scheme Dyn(G), in such a way that Dyn(G) is isomorphic to Dyn(G_{qs}); if G_{qs} is given by a cocycle ξ ∈ Z^{1}(S, Aut (R)) then Dyn(G_{qs}) is a twisted form of D_{S} corresponding to the image of ξ in Z^{1}(S, Aut (D)) induced by the canonical map Aut (R) → Aut (D) (Exp. XXIV 3.7). When G_{qs} is simply connected or adjoint, the latter map is an isomorphism.

Let T/S be a Galois covering that splits Dyn(G), i.e. Dyn(G)_{T} ≃ D_{T}. For example, one can take as T the torsor corresponding to the cocycle in Z^{1}(S, Aut (D)). Every element σ ∈ Aut (T/S) acts on Dyn(G)_{T} and therefore defines some φ_{σ} ∈ Aut (D)(T) such that the diagram

\xymatrix{D_{T} \ar[d] \ar[r]^{φ_{σ}} & D_{T} \ar[d] \\ T \ar[r]^{σ} & T}

commutes. By Galois descent this action (which will be called the ∗-action) completely determines Dyn(G). If S is connected, the ∗-action can be considered as an action of Aut (T/S) on the Dynkin diagram D, and extends by linearity to the ∗-action on Λ.

A subgroup scheme P of G is called parabolic if it is smooth and P\overline{k(s)} is a parabolic subgroup of G\overline{k(s)} in the usual sense for every s ∈ S (Exp. XXVI Déf. 1.1). To a parabolic subgroup P one can attach the type 𝔱(P) of P which is a clopen subscheme of Dyn(G) (Exp. XXVI 3.2). Note that the clopen subschemes of Dyn(G) are in one-to-one correspondence with the ∗-invariant clopen subschemes of D_{T}, where T/S is as above.

3. Representations

By a representation of a group scheme G over S we mean a homomorphism of algebraic monoids ρ: G → A, where A is an Azumaya algebra (more formally, a sheaf of Azumaya algebras) of constant rank over S.

We will say that a representation ρ : G → A is absolutely irreducible if it satisfies the following property: for any extension S′/S and any element a ∈ A(S′) there exists a faithfully flat extension S′′/S′ such that a considered as an element of A(S′′) belongs to the S′′-span of G(S′′). It is easy to see that this property is stable under a base extension and is local in the fpqc topology.

Lemma 1. Let ρ: G → A be a representation of a semisimple group scheme G over a scheme S. Then ρ is absolutely irreducible if and only if for any extension Spec k → S, where k is separably closed field, ρ_{k} : G_{k} → A_{k} is irreducible in the usual sense.

Proof. If ρ_{k} is reducible for some separably closed field k, then all elements of G(k) are presented by block-triangular matrices in A_{k} ≃ M_{n}(k) after an appropriate choice of basis. Clearly, the representation is not absolutely irreducible in this case.

Conversely, assume that ρ_{k} is irreducible for every separably closed field k. Since the property of being absolutely irreducible is local, we may pass to an étale neighbourhood, and therefore assume that the base scheme is a spectrum of a strictly Henselian ring R with the maximal ideal M. Note that A is isomorphic to the matrix ring M_{n}(R), and it suffices to show that the R-span of G(R) coincides with M_{n}(R). But k = R/M is separably closed, so the representation of G in M_{n}(k) is irreducible, hence the k-span of G(k) coincides with M_{n}(k). Now the result follows from the Nakayama lemma and the fact that the map G(R) → G(k) is surjective. □

Let G_{0} be a split semisimple group scheme over a scheme S, and let G_{0} → End (V) be a representation of G_{0} on a projective module (more formally, a locally free sheaf of modules) V of constant rank over S. Fix a maximal split torus T_{0} of G_{0} and let Λ and Λ_{r} be its lattices of weights and roots respectively. Then V decomposes into a direct sum ⨁_{λ∈Λ} V_{λ} such that for any t ∈ T_{0}(S′), where S′ is a scheme over S, and any v ∈ V_{λ}(S′) one has ρ(t)v = λ(t)v (Exp. I Prop. 4.7.3). A character λ with V_{λ} ≠ 0 is called a weight of V.

Lemma 2. For any g ∈ G_{0}(S′) one has ρ(g)(V_{λ}) ≤ ⨁_{μ−λ∈Λ_{r}} V_{μ}.

Proof. Changing the base we may assume S′ = S. Let g be an element of G_{0}(S). We need to show that pr_{μ}(ρ(g)(V_{λ})) = 0 when μ − λ ∉ Λ_{r}, pr_{μ} stands for the projection onto V_{μ}. Since V_{λ} is finitely generated, we may assume that S = Spec k is the spectrum of an algebraically closed field.

Let v be an element of V_{λ}. Set v_{ν} = pr_{ν}(v); for every c ∈ Cent(G_{0})(S) we have

ρ(g)ρ(c)v = ρ(g)(λ(c)v) = ∑_{ν} λ(c)v_{ν}

and

ρ(c)ρ(g)v = ∑_{ν} ν(c)v_{ν}.

But gc = cg, therefore (μ(c) − λ(c))v_{μ} = 0 for any c ∈ Cent(G_{0})(S). If v_{μ} ≠ 0 then the restrictions of μ and λ to Cent(G_{0}) coincide, i.e. μ − λ belongs to Λ_{r} (see Exp. XXII Rem. 4.1.8). □

Now let ρ: G_{qs} → End (V) be a representation of a quasi-split group G_{qs} on a projective module V of constant rank over S. As above, fix a maximal torus T of G_{qs}, and let Λ and Λ_{r} be the lattices of weights and roots of G_{qs}. Let ∐ S_{τ} → S be an fpqc covering by connected schemes, such that each (G_{qs})_{S_{τ}} is split. Over each S_{τ} the weights of V ×_{S} S_{τ} with respect to T_{S_{τ}} form a ∗-invariant subset of Λ. From now on, we will consider only representations of constant type in the sense that this subset doesn’t depend on S_{τ} (the condition automatically holds when S is connected); its elements are called the weights of V. We say that a weight λ ∈ Λ is ∗-invariant, if it is ∗-invariant over each S_{τ}. This is the case if and only if λ descends to a character of T over the base scheme. We denote by Λ^{∗} and Λ_{r}^{∗} the sublattices of ∗-invariant elements in Λ and Λ_{r} respectively. By [Ti71, 3.1] we have (Λ / Λ_{r})^{∗} = Λ^{∗} / Λ_{r}^{∗}.

A representation ρ: G_{qs} → End (V) will be called center preserving if ρ(Cent(G_{qs})) ⊆ Cent(End (V)). In this case ρ induces a homomorphism ρ^{ad}: G_{qs}^{ad} → PGL(V).

Lemma 3. (1) V is center preserving if and only if every two weights of V differ by an

element of Λ_{r}.

(2) The dual V^{∗} to a center preserving representation V is center preserving.

(3) Tensor product V_{1} ⊗ V_{2} of center preserving representations V_{1} and V_{2} is center preserving.

(4) The class of a weight λ of a center preserving representation V in Λ / Λ_{r} is ∗- invariant.

(5) If λ be a weight of a representation V of a quasi-split group such that the class of λ in Λ / Λ_{r} is ∗-invariant, then V decomposes as a direct sum of two represetations V = W ⊕ W′, such that λ is a weight of W and W is center preserving.

Proof. 1. Since the condition ρ(Cent(G_{qs})) ⊆ Cent(End (V)) is local with respect to fpqc topology, we can assume that G_{qs} = G_{0} is split. Then V is center preserving if and only if the restrictions of every two weights λ and μ to Cent(G_{0}) coincide. This exactly means that λ − μ belongs to Λ_{r} (Exp. XXII Rem. 4.1.8).

2,3,4. Follows from 1.

5. Consider a covering ∐ S_{τ} → S by connected schemes that splits G_{qs}. Lemma 2 implies that the submodules W = ⨁_{μ∈λ+Λ_{r}} V_{μ} and W′ = ⨁_{μ∉λ+Λ_{r}} V_{μ} are (G_{qs})_{S_{τ}} -invariant. Since λ+Λ_{r} is ∗-invariant, these modules are also ∗-invariant. By the faithfully flat descent we see that W and W′ are defined over the base scheme S and G_{qs}-invariant, and V is the direct sum of W and W′. The summand W is center preserving by 1. □

Lemma 4. Let G_{qs} be a quasi-split group over a scheme S. Then any weight λ ∈ Λ^{∗} is congruent modulo Λ_{r}^{∗} to a weight of some absolutely irreducible representation G_{qs} → End (V).

Proof. It is known (see [B, Ch. VI, Exerc. 5 du §2]) that any weight is equivalent modulo Λ_{r} to a minuscule weight; on the other hand by [Ti71, 3.1] we have (Λ / Λ_{r})^{∗} = Λ^{∗} / Λ_{r}^{∗}. So we may assume that λ is a ∗-invariant minuscule weight.

Consider first the split group G_{0} over ℤ. Recall briefly the construction of the Weyl module V (λ) (see [Jan] for details). We start from the finite dimensional irreducible (G_{0})_{ℂ}- module with the highest weight λ; we fix a vector v_{+} of the weight λ (which is unique up to a scalar). Denote by 𝔘 the universal enveloping algebra of the Lie algebra of (G_{0})_{ℂ}, by 𝔘^{+} and 𝔘^{−} its subalgebras generated by the positive (respectively, negative) root subspaces, and by 𝔘_{ℤ}, 𝔘_{ℤ}^{+}, 𝔘_{ℤ}^{−} their ℤ-forms used in the Chevalley construction. Then V (λ) is defined as 𝔘_{ℤ}^{−} v_{+}. If λ is minuscule, V (λ)_{k} is irreducible for any field k, so V (λ) is an absolutely irreducible representation of G_{0}.

Let Γ be a group of outer automorphisms of G_{0} preserving some fixed épinglage of G_{0} and λ. Then any element γ ∈ Γ induces an automorphism of 𝔘_{ℤ} which preserves 𝔘_{ℤ}^{+} and 𝔘_{ℤ}^{−}. Since γ preserves λ, the representations dρ: 𝔘 → End (V (λ)_{ℂ}) and dρ∘γ: 𝔘 → End (V (λ)_{ℂ}) are equivalent, therefore there exists φ ∈ GL(V (λ)_{ℂ}) such that γ(g)φ(v) = φ(gv) for every v ∈ V (λ)_{ℂ} and g ∈ 𝔘; moreover, φ is unique up to a scalar. It is easy to see that φ preserves the line generated by v_{+}, and we can normalize φ in such a way that φ(v_{+}) = v_{+}. Now

φ(𝔘_{ℤ}^{−} v_{+}) ≤ γ(𝔘_{ℤ}^{−})φ(v_{+}) = 𝔘_{ℤ}^{−} v_{+},

so φ induces an automorphism of V (λ) compatible with γ and preserving v_{+}. Since 𝔘_{ℤ} is the universal enveloping algebra of the Lie algebra of G_{0}, φ is an equivalence of the representations ρ: G_{0} → End (V (λ)) and ρ ∘ γ: G_{0} → End (V (λ)). Moreover, since φ is uniquely determined by γ, we obtain a homomorphism ψ: Γ → GL(V (λ)).

Now G_{qs} is constructed by a cocycle ξ ∈ Z^{1}(S, Γ), and the cocycle ψ_{∗}(ξ) allows us to construct a projective module V together with a representation G_{qs} → End (V), as claimed. □

Lemma 5. If G_{qs} → A is an absolutely irreducible representation of a quasi-split group G_{qs} over a scheme S, then A ≃ End (V) for some projective module V over S, and the representation is center preserving.

Proof. Consider an affine covering ∐ S_{τ} → S which splits G. Passing to a finer covering, we can also assume that for each τ A_{S_{τ}} is isomorphic to M_{n}(S_{τ}), a matrix algebra over S_{τ}. The decomposition of S_{τ}^{n} into a direct sum of the 1-dimensional highest weight subspace and its complement determines an idempotent e_{τ} in M_{n}(S_{τ}). These idempotents can be glued together to an idempotent e ∈ A defined over the base scheme S. Now the natural map A → End (Ae) is a homomorphism between Azumaya algebras of the same rank, so it is an isomorphism by [K, Ch. III, Cor. 5.1.18]. Finally, Lemma 1, together with the Schur lemma, implies that an absolutely irreducible representation of a quasi-split group into End (V) is center preserving. □

4. Tits algebras

Theorem 1. Let G be a semisimple group scheme of constant type over S, and let G_{qs} be the corresponding quasi-split group with the lattice of weights Λ.

(1) Suppose that G is given by a cocycle ξ ∈ Z^{1}(S, G_{qs}^{ad}). For every center preserving representation ρ: G_{qs} → End (V) there exists a representation ρ_{ξ}: G → A_{ρ} into an Azumaya algebra A_{ρ} over S which becomes equal to ρ over a covering of S that quasi-splits G.

(2) For every absolutely irreducible representation ρ′: G → A and every cocycle ξ ∈ Z^{1}(S, G_{qs}^{ad}) that gives G, there exists an absolutely irreducible representation ρ: G_{qs} → End (V) such that ρ′ = ρ_{ξ}.

(3) In the setting of 1, the class [A_{ρ}] in the Brauer group Br(S) depends only on G and on the class of any weight λ of ρ in Λ / Λ_{r}, and not on the particular choices of ρ and ξ. Its image in H^{2}(S, G_{m}) coincides with λ_{∗}(δ([ξ])), where λ_{∗}: H^{2}(S, Cent(G_{qs})) → H^{2}(S, G_{m}) corresponds to the restriction of λ to Cent(G_{qs}), and δ is the connecting homomorphism in the long exact sequence associated to the sequence

\xymatrix{1 \ar[r] & Cent(G_{qs}) \ar[r] & G_{qs} \ar[r] & G_{qs}^{ad} \ar[r] & 1.}

Proof. 1. The cocycle ξ is presented by elements g_{στ} ∈ G_{qs}^{ad}(S_{σ} ×_{S} S_{τ}) for some covering ∐ S_{τ} → S that quasi-splits G. Now the diagram (we write S_{στ} instead of S_{σ} ×_{S} S_{τ} for brevity)

\xymatrix{(G_{qs})_{S_{στ}} \ar[d]_{g_{στ}} \ar[r]^{id_{Sσ} ×ρ_{Sτ}} & End (V)_{S_{στ}} \ar[d]^{ρ_{Sστ}^{ad} (g_{στ})} \\ (G_{qs})_{S_{στ}} \ar[r]^{ρ_{Sσ} ×id_{Sτ}} & End (V)_{S_{στ}} .}

commutes, and the faithfully flat descent gives the homomorphism ρ_{ξ}: G → A_{ρ}, where A_{ρ} is a twisted form of End (V), that is an Azumaya algebra.

2. Let η be some cocycle in Z^{1}(S, G^{ad}) corresponding to G_{qs}; η is presented by elements h_{στ} ∈ G^{ad}(S_{σ} ×_{S} S_{τ}). Using the same construction as in the first part (with interchanged G and G_{qs}) we obtain an absolutely irreducible representation ρ: G_{qs} → B, where B ≃ End (V) for some V by Lemma 5. We will show that, for a suitable η, the representation ρ_{ξ} of G coincides with ρ′.

Denote by φ_{σ} the fixed isomorphisms from (G_{qs})_{S_{σ}} to G_{S_{σ}}, and by ψ_{σ} the isomorphisms between A_{S_{σ}} and End (V)_{S_{σ}} constructed by the descent. We have the following commutative diagram:

[figure: (G_{qs})_{S_{στ}}; id_{Sσ} ×φ_{τ}; G_{S_{στ}}; id_{Sσ} ×ρ′_{Sτ}; h_{στ}; A_{S_{στ}}; id_{Sσ} ×ψ_{τ}; ρ′_{Sστ}^{ad} (h_{στ}); End (V)_{S_{στ}}; (G_{qs})_{S_{στ}}; φ_{σ}×id_{Sτ}; G_{S_{στ}}; ρ′_{Sστ}; A_{S_{στ}}; ψ_{σ}×id_{Sτ}; End (V)_{S_{στ}} ,]

and we want to show that the rightmost square in the diagram

[figure: G_{S_{στ}}; id_{Sσ} ×φ_{τ}^{−1}; (G_{qs})_{S_{στ}}; id_{Sσ} ×ρ_{Sτ}; g_{στ}; End (V)_{S_{στ}}; id_{Sσ} ×ψ_{τ}^{−1}; ρ_{Sστ}^{ad} (g_{στ}); A_{S_{στ}}; G_{S_{στ}}; φ_{σ}^{−1}×id_{Sτ}; (G_{qs})_{S_{στ}}; ρ_{Sστ}; End (V)_{S_{στ}}; ψ_{σ}^{−1}×id_{Sτ}; A_{S_{στ}}]

is commutative. For a suitable choice of η, the first two squares are commutative, and the big rectangle is commutative as well, since the top and the bottom maps coincide with ρ′. Now the claim follows from the fact that ρ′ is absolutely irreducible.

3. The cohomological class in H^{1}(S, PGL(V)) corresponding to A_{ρ} is nothing but ρ_{∗}^{ad}([ξ]). Now the last assertion of the Theorem follows from the commutativity of the diagram

\xymatrix{H^{1}(S, G_{qs}^{ad}) \ar[d]_{ρ_{∗}^{ad}} \ar[r]^{δ} & H^{2}(S, Cent(G_{qs})) \ar[d]^{λ_{∗}} \\ H^{1}(S, PGL(V)) \ar[r] & H^{2}(S, G_{m}),}

which comes from the diagram

\xymatrix{1 \ar[r] & Cent(G_{qs}) \ar[d]_{λ} \ar[r] & G_{qs} \ar[d]_{ρ} \ar[r] & G_{qs}^{ad} \ar[d]_{ρ^{ad}} \ar[r] & 1 \\ 1 \ar[r] & G_{m} \ar[r] & GL(V) \ar[r] & PGL(V) \ar[r] & 1.}

So once ξ is fixed, the class of A_{ρ} depends only on the class of λ in Λ/ Λ_{r}. By Lemma 3 the class of λ is ∗-invariant, and hence λ is congruent modulo Λ_{r} to a ∗-invariant weight μ ∈ Λ. Now Lemma 4 and part 2 imply that λ_{∗}(δ([ξ])) = μ_{∗}(δ([ξ])) doesn’t depend on the particular choice of ξ, since any representation ρ : G_{qs} → End (V) one choses has the same set of weights. □

The Azumaya algebra A_{ρ} will be called the Tits algebra of a center preserving representation ρ of G_{qs}. We denote by β_{G} the corresponding homomorphism from Λ^{∗} / Λ_{r}^{∗} ≃ (Λ / Λ_{r})^{∗} to Br(S),

β_{G}: Λ^{∗} / Λ_{r}^{∗} → Br(S)
λ ↦ [A_{ρ}].

To see that β_{G} is indeed a homomorphism one can use either the tensor product of representations or the fact that Br(S) is a subgroup in H^{2}(S, G_{m}).

Dyn(G) is the disjoint union of its minimal clopen subschemes which will be called the orbits for brevity; they indeed correspond to the orbits of the ∗-action on the set of simple roots.

Assume that G is simply connected. Let T be a fixed torus of G_{qs}. Over a splitting covering we have two canonical homomorphisms

α, ω: Dyn(G) → Hom (T, G_{m})

which associate to each vertex i of the Dynkin diagram the simple root α_{i} (resp. the fundamental weight ω_{i}); the faitfully flat descent shows that these homomorphisms are defined over the base scheme S as well.

Let O be an orbit in Dyn(G); taking composition with the inclusion map O → Dyn(G) we obtain a root α_{O}: T_{O} → G_{m} (resp., a weight ω_{O}: T_{O} → G_{m}) of G_{O} which will be called the canonical root (resp., the canonical weight) corresponding to O (cf. Exp. XXIV 3.8). It is easy to see that α_{O} and ω_{O} are ∗-invariant weights of G_{O}.

Note that we have homomorphisms^{2}

ᾱ_{O}, ω̄_{O}: T → R_{O/S}(G_{m})

which are the compositions of R_{O/S}(α_{O}) and R_{O/S}(ω_{O}) with the canonical homomorphism T → R_{O/S}(T_{O}).

2For the Weil restriction we use a more common notation R_{S′/S} instead of ∏_{S′/S} of [SGA].

Proposition 1. Let G be simply connected, L be a Levi subgroup of a parabolic subgroup P in G and L′ be the corresponding Levi subgroup of the corresponding parabolic subgroup P′ in G^{ad}. Then we have the isomorphisms

∏_{O: O⊂𝔱(P)} ω̄_{O}: Cent(L) ≃ ∏_{O: O⊂𝔱(P)} R_{O/S}(G_{m});

∏_{O: O⊂𝔱(P)} ᾱ_{O}: Cent(L′) ≃ ∏_{O: O⊂𝔱(P)} R_{O/S}(G_{m})

(cf. Exp. XXIV Prop. 3.13).

Proof. Indeed, composing these maps with the isomorphism

∏_{O: O⊂𝔱(P)} R_{O/S}(G_{m}) → R_{𝔱(P)/S}(G_{m})

we obtain the maps Cent(L) → R_{𝔱(P)/S}(G_{m}) and Cent(L′) → R_{𝔱(P)/S}(G_{m}). It is easy to see that they are isomorphisms over a splitting covering of G. □

Proposition 2. In the settings of Theorem 1 assume that G is simply connected and Pic(Dyn(G)) = 0. Then [ξ] comes from an element in H^{1}(S, G_{qs}) if and only if β_{G_{O}}(ω_{O}) = 0 for each orbit O.

Proof. If [ξ] belongs to the image of H^{1}(S, G_{qs}) → H^{1}(S, G_{qs}^{ad}) then δ([ξ]_{O}) = 0 and therefore β_{G_{O}} = 0 for each O. Conversely, assume that β_{G_{O}}(ω_{O}) = 0 for each O. Proposition 1 applied to the Borel subgroup implies that T_{qs} ≃ ∏_{O} R_{O/S}(G_{m}) and T_{ad} ≃ ∏_{O} R_{O/S}(G_{m}). Now the Shapiro lemma (cf. Exp. XXIV Prop. 8.2) implies that the image of δ([ξ]) in H^{2}(S, T_{qs}) is trivial, while H^{1}(S, T_{qs}^{ad}) = Pic(Dyn(G)) = 0. Now the claim follows from the exact sequence

\xymatrix{H^{1}(S, T_{qs}^{ad}) \ar[r] & H^{2}(S, Cent(G_{qs})) \ar[r] & H^{2}(S, T_{qs}),}

which comes from the sequence

\xymatrix{1 \ar[r] & Cent(G_{qs}) \ar[r] & T_{qs} \ar[r] & T_{qs}^{ad} \ar[r] & 1.}
□

Theorem 2. (1) Let G be a semisimple group scheme of constant type over S, P be its parabolic subgroup admitting a Levi subgroup L, H be the derived subgroup of L. Denote by G_{qs} and H_{qs} the corresponding quasi-split groups and by Λ the lattice of weights of G_{qs}. For every λ ∈ Λ^{∗} denote by λ′ the restriction of λ to the maximal torus of H_{qs}. Then β_{G}(λ) = β_{H}(λ′). In particular, for any α ∈ Λ_{r}^{∗} one has β_{H}(α′) = 0.

(2) Let G_{qs} be a quasi-split simply connected group, P_{qs} be a standard parabolic subgroup of G_{qs}, L_{qs} be its standard Levi part, H_{qs} be the derived subgroup of L_{qs}. Assume that H is an inner form of H_{qs}, satisfying the conditions β_{H_{O}}(α′_{O}) = 0 for all O ⊂ 𝔱(P_{qs}). Then there exist an inner form G of G_{qs} and its parabolic subgroup P admitting a Levi subgroup L, such that over a quasi-splitting covering the pair L ≤ G becomes isomorphic to L_{qs} ≤ G_{qs}, and the derived subgroup of L is isomorphic to H.

(3) In the setting of 2, assume that Pic(Dyn(S)) = 0. Then such a G is unique up to an isomorphism.

Proof. 1. Let ξ be a cocycle in Z^{1}(S, G_{qs}^{ad}) corresponding to G, given by elements g_{στ} ∈ G_{qs}^{ad}(S_{σ} ×_{S} S_{τ}) for some covering ∐ S_{τ} → S that quasi-splits G. Over each S_{τ} one can (possibly, passing to a finer covering) conjugate P_{S_{τ}} and L_{S_{τ}} by some element of G_{qs}^{ad} to P_{qs} and L_{qs}, where P_{qs} is a standard parabolic subgroup of G_{qs} and L_{qs} is its standard Levi subgroup. Adjusting ξ by the coboundary given by these elements, we can assume that all g_{στ} ’s belong to L′_{qs}, where L′_{qs} is the image of L_{qs} in G_{qs}^{ad}, by Exp. XXVI Prop. 1.15 and Cor. 1.8 (cf. Exp. XXVI 3.21)

Let ρ: G_{qs} → GL(V) be a center preserving representation with a weight λ. Consider its restriction to H_{qs} and denote by U the center preserving direct summand corresponding to the weight λ′ and by U′ its complement invariant under H_{qs} (see Lemma 3, part 5). Denote by T_{qs} the standard maximal torus of L_{qs} and by T′_{qs} its intersection with H_{qs}. Note that U and U′ being sums of weight subspaces of T′_{qs} are stable under T_{qs} and, therefore, are invariant under the action of L_{qs}. Therefore, the map H^{1}(S, L′_{qs}) → H^{1}(S, PGL(V) factors through H^{1}(S, GL(U)×GL(U′)/ G_{m}), where G_{m} is embedded into GL(U)×GL(U′) diagonally.

Now the claim is obtained by comparing the diagrams

[figure: H^{1}(S, (GL(U) × GL(U′))/ G_{m}); H^{2}(S, G_{m}); H^{1}(S, PGL(V)); H^{2}(S, G_{m})]

and

[figure: H^{1}(S, (GL(U) × GL(U′))/ G_{m}); H^{2}(S, G_{m}); H^{1}(S, PGL(U)); H^{2}(S, G_{m}),]

which come from the sequences

[figure: 1; G_{m}; GL(U) × GL(U′); (GL(U) × GL(U′))/ G_{m}; 1; 1; G_{m}; GL(V); PGL(V); 1.]

and

[figure: 1; G_{m}; GL(U) × GL(U′); (GL(U) × GL(U′))/ G_{m}; 1; 1; G_{m}; GL(U); PGL(U); 1.]

2. Let ζ be a cocycle in Z^{1}(S, H_{qs}^{ad}) = Z^{1}(S, L_{qs}^{ad}) corresponding to H. Denote by L′_{qs} and H′_{qs} the images of L_{qs} and H_{qs} in G_{qs}^{ad}. Let us compute the image δ([ζ]) ∈ H^{2}(S, Cent(L′_{qs})). Using the assumption, Theorem 1, and the commutative diagram

[figure: H^{1}(S, H_{qs}^{ad}); δ; H^{2}(S, Cent(H′_{qs})); H^{1}(S, L_{qs}^{ad}); δ; H^{2}(S, Cent(L′_{qs})),]

we see that α_{O}(δ([ζ_{O}])) = 0 for any O ⊂ 𝔱(P_{qs}). Now Proposition 1 and the Shapiro lemma show that δ([ζ]) = 0. It means that [ζ] comes from some [ξ] ∈ H^{1}(S, L′_{qs}), and the image of [ξ] in H^{1}(S, G_{qs}^{ad}) defines a desired group G.

3. Let G be such a group; denote by ξ a corresponding cocycle in Z^{1}(S, G_{qs}^{ad}). As we have seen earlier, [ξ] comes from an element of H^{1}(S, L′_{qs}), say [ζ]. We have to show that [ζ] (and a fortioti [ξ]) is completely determined by its image in H^{1}(S, L_{qs}^{ad}), or, in other words, that the canonical map π_{∗}: H^{1}(S, L′_{qs}) → H^{1}(S, L_{qs}^{ad}) is injective. Since Cent(L′_{qs}) is central in L′_{qs}, H^{1}(S, Cent(L′_{qs})) acts on H^{1}(S, L′_{qs}), and the orbits of the action coincide with the fibers of π_{∗}. But H^{1}(S, Cent(L′_{qs})) by Proposition 1 and the Shapiro lemma injects to Pic(Dyn(G)), which is trivial by the assumption. □

5. Tits indices

From now on we assume that S = Spec R, where R is a connected semilocal ring. Recall that in this case all minimal parabolic subgroups P_{min} of G are conjugate under G(S) and hence have the same type 𝔱_{min} = 𝔱(P_{min}), which is a clopen subscheme of Dyn(G). If T/S is a Galois covering that splits Dyn(G), (𝔱_{min})_{T} is a clopen ∗-invariant subscheme of D_{T}. By Exp. XXVI Lemme 3.8 P ↦ 𝔱(P) is a bijection between parabolic subgroups P of G containing P_{min} and clopen subschemes 𝔱 of Dyn(G) containing 𝔱_{min}.

Since S is affine, for any parabolic subgroup P of G there exists a Levi subgroup L (Exp. XXVI Cor. 2.3) of P, and a unique parabolic subgroup P^{−} which is opposite to P with respect to L, i.e. satisfies P^{−} ∩ P = L (Exp. XXVI Th. 4.3.2). The type 𝔱(P^{−}) is the image s_{G}(𝔱(P)) of 𝔱(P) under some automorphism s_{G} of Dyn(G), called the opposition involution (Exp. XXIV Prop. 3.16.6 and Exp. XXVI 4.3.1; cf. [Ti66] 1.5.1). The corresponding automorphism s_{G} ∈ Aut (D) is induced by the automorphism α ↦ −w_{0}(α) of the root system Φ of G_{0}, where w_{0} is the unique element of the maximal length in the Weyl group of Φ. In fact s_{G} acts nontrivially only on components of types A_{n}, n ≥ 2, D_{2n+1}, n ≥ 1, or E_{6}, in which cases it acts as the unique nontrivial automorphism of this component.

Clearly, we have 𝔱_{min} = s_{G}(𝔱_{min}), since if P = P_{min} is a minimal parabolic subgroup, then P^{−} is also minimal, and we must have both 𝔱_{min} ⊆ s_{G}(𝔱_{min}) and s_{G}(𝔱_{min}) ⊆ 𝔱_{min}. It follows that J = s_{G}(J).

We now start the classification of semisimple algebraic groups over S = Spec R. The problem allows two immediate reductions. First, every semisimple group G is completely determined by its root datum and the corresponding simply connected group G^{sc}, so we can assume that G is simply connected.

Second, if the Dynkin diagram D of G is not connected (as a graph), we can present D as the disjoint union of its isotypic components D_{t} (it means that we collect isomorphic components together), and then we have a canonical decomposition G ≃ ∏ G_{t}, where G_{t} is a group over S with the Dynkin diagram D_{t} (Exp. XXIV Prop. 5.5). Further, if D_{t} is the disjoint union of n_{t} copies of a connected graph D_{0, t}, there exists a canonical étale extension S_{t}/S of degree n_{t} and a group G_{0, t} over S_{t} such that G_{t} ≃ R_{S_{t}/S}(G_{0, t}) (Exp. XXIV Prop. 5.9). So we can assume that D is connected, that is, G is a simple algebraic group.

The assumption that S = Spec R is connected allows us to identify D_{T} with D, and a clopen ∗-invariant subscheme of D_{T} with a ∗-invariant subset of D. Let J ⊆ D be the complement of the subset corresponding to 𝔱_{min}. Then the Tits index of G is the pair (D, J), together with a ∗-action on D, represented by a subgroup Γ of Aut (D). Usually we indicate Γ by writing its order as the upper left index attached to D (for example, ^{2}E_{6}, ^{6}D_{4} and so on). The group is of inner type, if Γ = {1}. The group G is quasi-split, if J = D, and split, if it is quasi-split and the ∗-action is trivial. When J = ∅ we say that G is anisotropic.

Our reasoning will be based on Theorem 2, which implies that a semisimple algebraic group is determined, up to an isomorphism, by the type 𝔱_{min} = J of a minimal parabolic subgroup, the anisotropic semisimple group (the anisotropic kernel of G) subject to certain conditions on Tits algebras, which is the derived subgroup of a Levi subgroup of a minimal parabolic, and by the quasi-split group G_{qs}. In its turn, G_{qs} is determined by the Dynkin diagram D and the ∗-action on it. Thus the classification consists in listing all possible Tits indices of simple algebraic groups, and, for any given index, the conditions on the corresponding anisotropic kernels. Whenever it does not require any extra technique, we describe the isomorphism classes of anisotropic kernels in terms of more intuitive algebraic structures, like Azumaya algebras over R or étale extensions R′/R of a given degree.

Our numbering of the vertices of Dynkin diagrams follows [B]. We represent Tits indices graphically by Dynkin diagrams D with the vertices in J being circled. We also use the Tits notation ^{m}X_{n,r}^{k} for the groups of specific indices (see [Ti66]). Unless explicitly stated otherwise, E denotes an Azumaya algebra over R.

We begin with simple groups of type A_{n}. The split simple simply connected group scheme of type A_{n} over R is SL_{n+1}(R); the corresponding adjoint group is PGL_{n+1}(R) = Aut (M_{n+1}(R)). So the simple simply connected groups of inner type A_{n} are of the form SL_{1}(A), where A is an Azumaya algebra over R of degree n + 1, uniquely determined up to an isomorphism. Obviously A is the Tits algebra of SL_{1}(A) corresponding to the natural representation of SL_{n+1}(R) in R^{n+1}; so [A] = β_{SL_{1}(A)}(ω_{1}).

Lemma 6. Assume that SL_{1}(E) and SL_{1}(E′) are anisotropic, and [E] = [E′] in Br(R). Then E ≃ E′.

Proof. Since projective modules over R are free, [E] = [E′] means that M_{n}(E) ≃ M_{m}(E′) for some n and m. Consider the simple group G = SL_{n}(E) ≃ SL_{m}(E′). Then SL_{1}(E)^{n} and SL_{1}(E′)^{m} are anisotropic groups which are the derived subgroups of Levi subgroups of some parabolic subgroups of G. Hence these parabolic subgroups are minimal, and SL_{1}(E)^{n} and SL_{1}(E′)^{m} are semisimple groups of the same type; in particular, m = n and the degrees of E and E′ are equal. This implies E ≃ E′ by [K, Ch. III Prop. 5.2.3 2)]. □

Theorem 3 (^{1}A_{n}). Every simple simply connected group G of inner type A_{n} over R has the form SL_{r+1}(E) for a uniquely determined r ≥ 0 and an Azumaya algebra E over R such that SL_{1}(E) is anisotropic. The Tits index of G is (^{1}A_{n}, J), where J = {d, 2d, …, rd}, d is the degree of E and n + 1 = (r + 1)d:

(^{1}A_{n, r}^{(d)})  [figure: •; •; •; •; •; •; •; •; •; •; •; d; 2d; rd]

Proof. The proof goes by induction on n. Let (^{1}A_{n}, J) be the Tits index of G and m be the least element of J (in the case J = ∅ G is anisotropic, and we can set d = n, r = 0). Denote the derived subgroup of the Levi part of the parabolic subgroup of type D \ {m} by G′. By Theorem 2, we have β_{G′}(α′_{m}) = 0, and G is uniquely determined by G′ satisfying this condition.

Assume first that 1 < m < n. Then G′ ≃ SL_{1}(E) × H, where E has degree m, SL_{1}(E) is anisotropic, and H is a simple simply connected group of inner type A_{n−d}. By the induction hypothesis, H ≃ SL_{r}(E′) for some r, where SL_{1}(E′) is anisotropic, and the Tits index of H is (^{1}A_{n−m}, J′) with J′ = {d, 2d, …, (r − 1)d}, where d = deg E′, n − m + 1 = rd. The Cartan matrix of A_{n} shows that α_{m} = 2ω_{m} − ω_{m−1} − ω_{m+1}, so

0 = β_{G′}(α′_{m}) = β_{SL_{1}(E)}(ω_{1}) − β_{H}(ω_{1}) = [E] − [E′].

By Lemma 6, E ≃ E′ and hence m = d. In this case G ≃ SL_{r+1}(E).

In the case m = 1 G′ is a simply simply connected group of inner type A_{n−1}. By the induction hypothesis, G′ ≃ SL_{r}(E′) for some r and E′. The Cartan matrix shows that α_{m} = 2ω_{1} − ω_{2}, and

0 = β_{G′}(α′_{m}) = −β_{G′}(ω_{1}) = −[E′],

therefore G′ splits and so does G. In this case G ≃ SL_{n+1}(R).

In the case m = n G′ = SL_{1}(E) is anisotropic. But the Cartan matrix shows that α_{m} = 2ω_{n} − ω_{n−1}, and

0 = β_{SL_{1}(E)}(α′_{m}) = β_{SL_{1}(E)}(ω_{1}) = [E],

a contradiction. □

The above result implies that for any Azumaya algebra A over R, the group G = SL_{1}(A) is isomorphic to SL_{r+1}(E), with E an Azumaya algebra such that SL_{1}(E) is anisotropic. In this case the degree of E is called the index of A and is denoted by ind A; obviously ind A divides deg A. The exponent exp A of A is the order of [A] in Br(R). We will need the following result:

Proposition 3. For any A exp A divides ind A, and they have the same prime factors.

Proof. The first part follows from the fact that [A] = [E] = β_{SL_{1}(E)}(ω_{1}), and (deg E)ω_{1} belongs to Λ_{r}. The second part follows from [Gab, Ch. II, Thm. 1]. □

Let R′/R be an étale extension of degree n. We can interpret the corestriction homomorphism cores_{R′/R}: Br(R′) → Br(R) as follows. If A is an Azumaya algebra over R′ of degree d, R_{R′/R}(SL_{1}(A)) is a group of type nA_{d−1} over R, with the ∗-action permuting the copies of A_{d−1}. Now cores_{R′/R}([A]) = β_{R_{R′/R}(SL_{1}(A))}(ω), where ω is the sum of the fundamental weights ω_{1} corresponding to each copy of A_{d−1} (cf. [Ti71, § 5.3]).

Theorem 4 (^{2}A_{n}). Every simple simply connected group G of type ^{2}A_{n} over R has the Tits index (^{2}A_{n}, J), where J = {d, 2d, …, rd, n + 1 − rd, …, n + 1 − 2d, n + 1 − d} for some r ≥ 0, d > 0 such that d | n + 1, 2rd ≤ n + 1:

(^{2}A_{n, r}^{(d)})

[figure: •; •; •; •; •; •; •; •; •; •; •; •; •; •; •; •; d; rd; n + 1 − rd; n + 1 − d]

Isomorphism classes of groups of a given index bijectively correspond to isomorphism classes of:

• simple simply connected anisotropic groups H of type^{2}A_{n−2rd} over R with β_{H_{O}}(ω_{1}) = [E], ind E = d, O is the orbit corresponding to {1, n − 2rd}, when n − 2rd ≥ 2;

• pairs consisting of an Azumaya algebras A over R and a connected quadratic étale extension R′/R such that ind A = deg A = 2 and ind A_{R′} = d, when n − 2rd = 1;

• Azumaya algebras E over a connected quadratic étale extension R′/R with ind E = deg E = d, cores_{R′/R}([E]) = 0, when n − 2rd ≤ 0.

Proof. Let (^{2}A_{n}, J) be the Tits index of G. Note that if j belongs to J then so does n + 1 − j. Let m be the greatest element of J which is not greater than \frac{n+1}{2} (in the case J = ∅ G is anisotropic, and we can set r = 0). Denote the orbit corresponding to m by O, and the derived subgroup of the Levi part of the parabolic subgroup of type D \ O by G′. By Theorem 2 we have β_{G′_{O}}(α′_{O}) = 0, and G is uniquely determined by G′ satisfying this condition.

Assume first that 1 < m < \frac{n−1}{2}. Then O as a scheme is a connected quadratic étale extension of R, G′ ≃ H_{1} × H, where H is a simple simply connected anisotropic group of type ^{2}A_{n−2m}, H_{1} is a group of outer type A_{m−1} + A_{m−1} with the ∗-action permuting the copies of A_{m−1}. Conseqeuently, H_{1} ≃ R_{O/R}(SL_{r}(E)) for some Azumaya algebra E over O, ind E = deg E = d, dr = m. The Cartan matrix shows that α_{m} = 2ω_{m} − ω_{m−1} − ω_{m+1}, so

0 = β_{G′_{O}}(α′_{O}) = β_{SL_{r}(E)}(ω_{1}) − β_{H_{O}}(ω_{1}) = [E] − β_{H_{O}}(ω_{1}).

Note that H_{O} is of inner type A_{n−2m}, and therefore d must divide n − 2m + 1 = n − 2rd + 1. We have J = {d, …, rd, n + 1 − rd, …, n + 1 − d}.

In the case m = \frac{n−1}{2}, O as a scheme is a connected quadratic étale extension R′/R, G′ ≃ R_{R′/R}(SL_{r}(E)) × SL_{1}(A), E is an Azumaya algebra over R′, ind E = deg E = d, dr = m, A is an Azumaya algebra over R, ind A = deg A = 2. The Cartan matrix shows that α_{m} = 2ω_{m} − ω_{m−1} − ω_{m+1}, so

0 = β_{G′_{O}}(α′_{O}) = β_{SL_{r}(E)}(ω_{1}) − β_{SL_{1}(A)_{O}}(ω_{1}) = [E] − [A_{R′}].

Since ind A = 2, d = 1 or 2 in this case, so d | n + 1. We have J = {d, …, rd, n + 1 − rd, …, n + 1 − d} again.

In the case m = \frac{n}{2}, O as a scheme is a connected quadratic étale extension R′/R, G′ ≃ R_{R′/R}(SL_{r}(E)), E is an Azumaya algebra over R′, ind E = deg E = d, dr = m. The Cartan matrix shows that α_{m} = 2ω_{m} − ω_{m−1} − ω_{m+1}, so

0 = β_{G′_{O}}(α′_{O}) = β_{SL_{r}(E)}(ω_{1}) = [E].

Hence E = R′ and d = 1. G is quasi-split in this case.

In the case m = \frac{n+1}{2}, O as a scheme is isomorphic to R, G′ ≃ R_{R′/R}(SL_{r}(E)) for some Azumaya algebra E over a connected quadratic étale extension R′/R with ind E = deg E = d, rd = m. The Cartan matrix shows that α_{m} = 2ω_{m} − ω_{m−1} − ω_{m+1}, so

0 = β_{G′_{O}}(α′_{O}) = cores_{R′/R}(β_{SL_{r}(E)}(ω_{1})) = cores_{R′/R}([E]).

We have J = {d, …, (r − 1)d, rd = (n + 1) − rd, …, n + 1 − d}.

In the case m = 1, O as a scheme is a connected quadratic étale extension of R, G′ = H is a simple simply connected anisotropic group of type ^{2}A_{n−2}. The Cartan matrix show that α_{m} = 2ω_{1} − ω_{2}, so

0 = β_{G′_{O}}(α′_{O}) = β_{H_{O}}(ω_{1}).

We have J = {1, n} in this case. □

Theorem 3 (B_{n}). Every simple simply connected group of type B_{n} over R, n ≥ 2, has the Tits index (B_{n}, J), where J = {1, 2, …, r} for some r ≥ 0:

(B_{n, r})  [figure: •; •; •; •; •; 1; r]

Isomorphism classes of groups of a given index bijectively correspond to isomorphism classes of:

• simple simply connected anisotropic groups of type B_{n−r} over R, when n − r ≥ 2;

• Azumaya algebras A over R with ind A = deg A = 2, when n − r = 1. If n = r then G is split.

Proof. Let (B_{n}, J) be the Tits index of G and m be the greatest element of J (in the case J = ∅ G is anisotropic, and we can set r = 0). Denote the derived subgroup of the Levi part of the parabolic subgroup of type D \ {m} by G′. By Theorem 2 we have β_{G′}(α′_{m}) = 0, and G is uniquely determined by G′ satisfying this condition.

Assume first that 1 < m < n − 1. Then G′ ≃ SL_{r}(E) × H, where ind E = deg E = d, dr = m, H is a simple simply connected anisotropic group of type B_{n−m}. The Cartan matrix shows that α_{m} = 2ω_{m} − ω_{m−1} − ω_{m+1}, so

0 = β_{G′}(α′_{m}) = β_{SL_{r}(E)}(ω_{1}) = [E],

that is E = R, d = 1, J = {1, …, r}.

In the case m = n − 1 G′ ≃ SL_{r}(E) × SL_{1}(A), where ind E = deg E = d, dr = n − 1, ind A = deg A = 2. The Cartan matrix shows that α_{m} = 2ω_{n−1} − ω_{n−2} − 2ω_{n}, so

0 = β_{G′}(α′_{m}) = β_{SL_{r}(E)}(ω_{1}) = [E],

that is E = R, d = 1, J = {1, …, n − 1}.

In the case m = n G′ ≃ SL_{r}(E), where ind E = deg E = d, dr = n. The Cartan matrix shows that α_{m} = 2ω_{n} − ω_{n−1}, so

0 = β_{G′}(α′_{m}) = β_{SL_{r}(E)}(ω_{1}) = [E],

that is E = R, d = 1, J = {1, …, n}. G is split in this case.

In the case m = 1 G′ = H is a simple simply connected anisotropic group of type B_{n−1}. The Cartan matrix shows that α_{m} = 2ω_{1} − ω_{2}, and the condition β_{G′}(α′_{m}) = 0 is vacuous. We have J = {1} in this case. □

The split simple simply connected group scheme of type C_{n} over R is Sp_{2n}(R).

Proposition 4. Assume that G is a simple simply connected group of type C_{n} over R, β_{G}(ω_{1}) = [E], ind E = d. Then d = 2^{k} for some k ≥ 0 and d | 2n. If d = 1 then G is split.

Proof. We have 2[E] = 0, since 2ω_{1} belongs to Λ_{r}. Now Proposition 3 implies that d = 2^{k}.

The vector representation ρ: Sp_{2n}(R) → End (R^{2n}) is center preserving and has a weight ω_{1}; so [A_{ρ}] = [E]. But A_{ρ} has degree 2n, so d | 2n.

If d = 1 then by Proposition 2 G corresponds to an element of H^{1}(R, Sp_{2n}), and the latter is trivial by [K, Ch. I, Cor. 4.1.2]. □

Theorem 3 (C_{n}). Every simple simply connected group G of type C_{n} over R, n ≥ 2, has the Tits index (C_{n}, J), where J = {d, 2d, …, rd} for some r ≥ 0, d > 0 such that d = 2^{k} | 2n, rd ≤ n, and r = n when d = 1:

(C_{n, r}^{(d)})  [figure: •; •; •; •; •; •; •; •; •; d; rd]

Isomorphism classes of groups of a given index bijectively correspond to isomorphism classes of:

• simple simply connected anisotropic groups H of type C_{n−rd} over R with β_{H}(ω_{1}) = [E], ind E = 2, when n − rd ≥ 2;

• Azumaya algebras E over R with ind E = deg E = d and exp E ≤ 2, when n−rd ≤ 1.

Proof. Let (C_{n}, J) be the Tits index of G and m be the greatest element of J (in the case J = ∅ G is anisotropic, and we can set r = 0). Denote the derived subgroup of the Levi part of the parabolic subgroup of type D \ {m} by G′. By Theorem 2 we have β_{G′}(α′_{m}) = 0, and G is uniquely determined by G′ satisfying this condition.

Assume first that 1 < m < n − 1. Then G′ ≃ SL_{r}(E) × H, where ind E = deg E = d, dr = m, H is a simple imply connected anisotropic group of type C_{n−m}. Note that [E] = β_{G}(ω_{1}), hence d = 2^{k} | 2n and d > 1 by Proposition 4. The Cartan matrix shows that α_{m} = 2ω_{m} − ω_{m−1} − ω_{m+1}, so

0 = β_{G′}(α′_{m}) = β_{SL_{r}(E)}(ω_{1}) − β_{H}(ω_{1}) = [E] − β_{H}(ω_{1}).

We have J = {d, …, rd}.

In the case m = n − 1 G′ ≃ SL_{r}(E) × SL_{1}(A), where ind E = deg E = d, dr = n − 1, d = 2^{k} | 2n, d > 1, ind A = deg A = 2. The Cartan matrix shows that α_{m} = 2ω_{n−1} − ω_{n−2} − ω_{n}, so

0 = β_{G′}(α′_{m}) = β_{SL_{r}(E)}(ω_{1}) − β_{SL_{1}(A)}(ω_{1}) = [E] − [A].

Hence [E] = [A] and d = 2, so d | 2n. We have J = {d, …, rd} again.

In the case m = n G′ ≃ SL_{r}(E), where ind E = deg E = d, dr = n, d = 2^{k}. The Cartan matrix shows that α_{m} = 2ω_{n} − 2ω_{n−1}, so

0 = β_{G′}(α′_{m}) = 2β_{SL_{r}(E)}(ω_{1}) = 2[E],

that is exp E ≤ 2. We have J = {d, …, rd = n}.

In the case m = 1 G′ = H is a simple simply connected anisotropic group of type C_{n−1}. The Cartan matrix shows that α_{m} = 2ω_{1} − ω_{2}, so

0 = β_{G′}(α′_{m}) = −β_{H}(ω_{1}).

Now Proposition 4 implies that H is split, a contradiction. □

The split simple simply connected group scheme of type D_{n} over R is Spin_{2n}(R).

Proposition 5. Assume that G is a simple simply connected group of type ^{1}D_{n} or ^{2}D_{n} over R, n ≥ 4, β_{G}(ω_{1}) = [E], ind E = d. Then d = 2^{k} for some k ≥ 0 and d | 2n.

Proof. We have 2[E] = 0, since 2ω_{1} belongs to Λ_{r}. Now Proposition 3 implies that d = 2^{k}.

The vector representation ρ: Spin_{2n}(R) → End (R^{2n}) is center preserving and has a weight ω; so [A_{ρ}] = [E]. But A_{ρ} has degree 2n, so d | 2n. □

Theorem 3 (^{1}D_{n}). Every simple simply connected group G of inner type D_{n} over R, n ≥ 4, has the Tits index (^{1}D_{n}, J), where J = {d, 2d, …, rd} (possibly, after interchanging n − 1 and n) for some r ≥ 0, d > 0 such that d = 2^{k} | 2n, rd ≤ n, n ≠ rd + 1:

(^{1}D_{n, r}^{(d)})  [figure: •; •; •; •; •; •; •; •; •; d; rd; •]

Isomorphism classes of groups of a given index bijectively correspond to isomorphism classes of:

• simple simply connected anisotropic groups H of inner type D_{n−rd} over R with β_{H}(ω_{1}) = [E], ind E = 2, when n − rd ≥ 4;

• Azumaya algebras A over R with ind A = deg A = 4 and 2[A] = [E], ind E = d, when n − rd = 3;

• pairs of Azumaya algebras A_{1} and A_{2} over R with ind A_{1} = deg A_{1} = ind A_{2} = deg A_{2} = 2 and [A_{1}] + [A_{2}] = [E], ind E = d, when n − rd = 2;

• Azumaya algebras E over R with ind E = deg E = d and exp E ≤ 2, when n = rd.

Proof. Let (^{1}D_{n}, J) be the Tits index of G and m be the greatest element of J (in the case J = ∅ G is anisotropic, and we can set r = 0). Denote the derived subgroup of the Levi part of the parabolic subgroup of type D \ {m} by G′. By Theorem 2 we have β_{G′}(α′_{m}) = 0, and G is uniquely determined by G′ satisfying this condition.

Assume first that 1 < m < n − 3. Then G′ ≃ SL_{r}(E) × H, where ind E = deg E = d, dr = m, H is a simple simply connected anisotropic group of type D_{n−m}. Note that [E] = β_{G}(ω_{1}), hence d = 2^{k} | 2n by Proposition 5. The Cartan matrix shows that α_{m} = 2ω_{m} − ω_{m−1} − ω_{m+1}, so

0 = β_{G′}(α′_{m}) = β_{SL_{r}(E)}(ω_{1}) − β_{H}(ω_{1}) = [E] − β_{H}(ω_{1}).

We have J = {d, …, rd}.

In the case m = n − 3 G′ ≃ SL_{r}(E) × SL_{1}(A), where ind E = deg E = d, dr = n − 3, d = 2^{k} | 2n (so d = 1 or 2), ind A = deg A = 4. The Cartan matrix shows that α_{m} = 2ω_{n−3} − ω_{n−4} − ω_{n−2}, so

0 = β_{G′}(α′_{m}) = β_{SL_{r}(E)}(ω_{1}) − β_{SL_{1}(A)}(ω_{2}) = [E] − 2[A].

We have J = {d, …, rd = n − 3}.

In the case m = n − 2 G′ ≃ SL_{r}(E) × SL_{1}(A_{1}) × SL_{1}(A_{2}), where ind E = deg E = d, dr = n − 2, d = 2^{k} | 2n (so d = 1, 2, or 4), ind A_{1} = deg A_{1} = ind A_{2} = deg A_{2} = 2. The Cartan matrix shows that α_{m} = 2ω_{n−2} − ω_{n−3} − ω_{n−1} − ω_{n}, so

0 = β_{G′}(α′_{m}) = β_{SL_{r}(E)}(ω_{1}) − β_{SL_{1}(A_{1})}(ω_{1}) − β_{SL_{1}(A_{2})}(ω_{1}) = [E] − [A_{1}] − [A_{2}].

We have J = {d, …, rd = n − 2}.

In the case m = n − 1 we can interchange n − 1 and n and so assume that m = n.

In the case m = n G′ ≃ SL_{r}(E), where ind E = deg E = d, dr = n, d = 2^{k}. The Cartan matrix shows that α_{m} = 2ω_{n} − ω_{n−2}, so

0 = β_{G′}(α′_{m}) = β_{SL_{r}(E)}(ω_{2}) = 2[E],

hence exp E ≤ 2. We have J = {d, …, rd = n}.

In the case m = 1 G′ = H is a simple simply connected anisotropic group of type D_{n−1}. The Cartan matrix shows that α_{m} = 2ω_{1} − ω_{2}, so

0 = β_{G′}(α′_{m}) = −β_{H}(ω_{1}).

We have J = {1} in this case. □

Proposition 6. Assume that G is a simple simply connected group of inner type D_{5} over R with β_{G}(ω_{4}) = 0. Then G is of index ^{1}D_{5, 1}^{(1)} or ^{1}D_{5, 5}^{(1)}.

Proof. Indeed, by Theorem 2, part 2, we can construct a group G′ of type E_{6} over R such that G is the derived subgroup of the Levi subgroup of the parabolic subgroup corresponding to the subset {2, 3, 4, 5, 6}. Let ^{1}E_{6} be the Tits index of G′; then 1 belongs to J and therefore 6 = s_{G′}(1) belongs to J as well. Therefore, G has the index ^{1}D_{5, r}^{(1)} for some r ≥ 1. If r ≥ 2, consider the parabolic subgroup of type {2, 3, 4} in G; the derived subgroup of its Levi part is SL_{1}(A) for some Asumaya algebra A over R with deg A = 4. By Theorem 2, part 1, we have

0 = β_{G}(ω_{4}) = β_{SL_{1}(A)}(ω_{1}) = [A],

so G is split and has the index ^{1}D_{5, 5}^{(1)} in this case. □

Theorem 3 (^{2}D_{n}). Every simple simply connected group G of type ^{2}D_{n}, n ≥ 4, has the Tits index (^{2}D_{n}, J), where J = {d, 2d, …, rd}, except for J = {d, 2d, …, (r − 1)d, n − 1, n} in the case n − rd = 1, for some r ≥ 0, d > 0 such that d = 2^{k} | 2n, rd ≤ n − 1:

(^{2}D_{n, r}^{(d)})  [figure: •; •; •; •; •; •; •; •; •; d; rd; •]

Isomorphism classes of groups of a given index bijectively correspond to isomorphism classes of:

• simple simply connected anisotropic groups H of type ^{2}D_{n−rd} over R with β_{H}(ω_{1}) = [E], ind E = d, when n − rd ≥ 4;

• simple simply connected anisotropic groups H of type ^{2}A_{3} over R with β_{H}(ω_{2}) = [E], ind E = d, when n − rd = 3;

• Azumaya algebras A over a connected quadratic étale extension R′/R with ind A = deg A = 2 and cores_{R′/R}([A]) = [E], ind E = d, when n − rd = 2;

• pairs consisting of an Azumaya algebra E over R with ind E = deg E = d and a connected quadratic étale extension R′/R such that [E_{R′}] = 0, when n − rd = 1.

Proof. Let (^{2}D_{n}, J) be the Tits index of G and m be the greatest element of J (in the case J = ∅ G is anisotropic, and we can set r = 0). Denote the orbit corresponding to m by O, and the derived subgroup of the Levi part of the parabolic subgroup of type D \ O by G′. By Theorem 2 we have β_{G′_{O}}(α′_{O}) = 0, and G is uniquely determined by G′ satisfying this condition.

Assume first that 1 < m < n − 3. Then O as a scheme is isomorphic to R, G′ ≃ SL_{r}(E) × H, where ind E = deg E = d, dr = m, H is a simple simply connected anisotropic group of type ^{2}D_{n−m}. Note that [E] = β_{G}(ω_{1}), hence d = 2^{k} | 2n by Proposition 5. The Cartan matrix shows that α_{m} = 2ω_{m} − ω_{m−1} − ω_{m+1}, so

0 = β_{G′_{O}}(α′_{O}) = β_{SL_{r}(E)}(ω_{1}) − β_{H}(ω_{1}) = [E] − β_{H}(ω_{1}).

We have J = {d, …, rd}.

In the case m = n−3 G′ ≃ SL_{r}(E)×H, where ind E = deg E = d, dr = n−3, d = 2^{k} | 2n (so d = 1 or d = 2), H is a simple simply connected anisotropic group of type ^{2}A_{3}. The Cartan matrix shows that α_{m} = 2ω_{n−3} − ω_{n−4} − ω_{n−2}, so

0 = β_{G′_{O}}(α′_{O}) = β_{SL_{r}(E)}(ω_{1}) − β_{H}(ω_{2}) = [E] − β_{H}(ω_{2}).

We have J = {d, …, rd = n − 3}.

In the case m = n − 2 O as a scheme is isomorphic to R, G′ ≃ SL_{r}(E) × R_{R′/R}(SL_{1}(A)), where ind E = deg E = d, dr = n − 2, d = 2^{k} | 2n (so d = 1, 2, or 4), A is an Azumaya algebra over a connected quadratic étale extension R′/R, ind A = deg A = 2. The Cartan matrix shows that α_{m} = 2ω_{n−2} − ω_{n−3} − ω_{n−1} − ω_{n}, so

0 = β_{G′_{O}}(α′_{O}) = β_{SL_{r}(E)}(ω_{1}) − cores_{R′/R}(β_{SL_{1}(A)}(ω_{1})) = [E] − cores_{R′/R}([A]).

We have J = {d, …, rd = n − 2}.

In the case m = n − 1 n also belongs to J, a contradiction.

In the case m = n O as a scheme is a connected quadratic étale extension R′/R, G′ ≃ SL_{r}(E), where ind E = deg E = d, dr = n − 1, d = 2^{k} | 2n (so d = 1 or 2). The Cartan matrix shows that α_{m} = 2ω_{n} − ω_{n−2}, so

0 = β_{G′_{O}}(α′_{O}) = β_{SL_{r}(E)_{O}}(ω_{1}) = [E_{R′}].

We have J = {d, …, (r − 1)d, n − 1, n} in this case.

In the case m = 1 O as a scheme is isomorphic to R, G′ = H is a simple simply connected anisotropic group of type ^{2}D_{n−1}. The Cartan matrix shows that α_{m} = 2ω_{1} − ω_{2}, so

0 = β_{G′_{O}}(α′_{O}) = −β_{H}(ω_{1}).

We have J = {1} in this case. □

Theorem 3 (^{3}D_{4} and ^{6}D_{4}). Every simple simply connected group G of type ^{3}D_{4} or ^{6}D_{4} over R has one of the following Tits indices:

(^{3}D_{4, 0}^{28}, ^{6}D_{4, 0}^{28})  [figure: •; •; •; •; •; •; •; •]

(^{3}D_{4, 1}^{9}, ^{6}D_{4, 1}^{9})  [figure: •; •; •; •; •; •; •; •]

(^{3}D_{4, 2}^{2}, ^{6}D_{4, 2}^{2})  [figure: •; •; •; •; •; •; •; •]

Isomorphism classes of groups of a given index bijectively correspond to isomorphism classes of:

• Azumaya algebras A over a connected cubic cyclic (resp., noncyclic) étale extension R′/R with ind A = deg A = 2 and cores_{R′/R}([A]) = 0, in the case of ^{3}D_{4, 1}^{9} (resp., 6D_{4, 1}^{9});

• connected cubic cyclic (resp., noncyclic) étale extensions R′/R, in the case of ^{3}D_{4, 2}^{2} (resp., ^{6}D_{4, 2}^{2}).

Proof. Let (^{3}D_{4}, J) (or (^{6}D_{4}, J)) be the Tits index of G and m be the least element of J (in the case J = ∅ G is anisotropic, that is of index ^{3}D_{4, 0}^{28} or ^{6}D_{4, 0}^{28}). Denote the orbit corresponding to m by O, and the derived subgroup of the Levi part of the parabolic subgroup of type D \ O by G′. By Theorem 2 we have β_{G′_{O}}(α′_{O}) = 0, and G is uniquely determined by G′ satisfying this condition.

In the case m = 1 O as a scheme is a connected cubic étale extension R′/R, G′ ≃ SL_{1}(E), where ind E = deg E = 2. The Cartan matrix shows that α_{m} = 2ω_{1} − ω_{2}, so

0 = β_{G′_{O}}(α′_{O}) = −β_{SL_{1}(E)_{O}}(ω_{1}) = [E_{R′}].

But then 3[E] = cores_{R′/R}([E_{R′}]) = 0, and 2[E] = 0 by Proposition 3, hence E = R. In this case G is quasi-split, that is of index ^{3}D_{4, 4}^{2} or ^{6}D_{4, 4}^{2}.

In the case m = 2 O as a scheme is isomorphic to R, G′ ≃ R_{R′/R}(SL_{1}(A)), where A is an Azumaya algebra over a connected cubic étale extension R′/R, ind A = deg A = 2. The Cartan matrix shows that α_{m} = 2ω_{2} − ω_{1} − ω_{3} − ω_{4}, so

0 = β_{G′_{O}}(α′_{O}) = − cores_{R′/R}(β_{SL_{1}(A)}(ω_{1})) = − cores_{R′/R}([A]).

In this case G is of index ^{3}D_{4, 1}^{9} or ^{6}D_{4, 1}^{9}. □

Theorem 3 (^{1}E_{6}). Every simple simply connected group G of inner type E_{6} over R has one of the following Tits indices:

(^{1}E_{6, 0}^{78})  •  •  •  •  • •

(^{1}E_{6, 2}^{28})  [figure: •; •; •; •; •; •]

(^{1}E_{6, 2}^{16})  [figure: •; •; •; •; •; •]

(^{1}E_{6, 6}^{0})  [figure: •; •; •; •; •; •]

Isomorphism classes of groups of a given index bijectively correspond to isomorphism classes of:

• simple simply connected anisotropic groups H of type D_{4} over R with β_{H} = 0, in the case of ^{1}E_{6, 2}^{28};

• Azumaya algebras A over R with ind A = deg A = 3, in the case of ^{1}E_{6, 2}^{16}.

The only group of index ^{1}E_{6, 6}^{0} is split.

Proof. Let (^{1}E_{6}, J) be the Tits index of G and m be the least element of J (in the case J = ∅ G is anisotropic, that is of index ^{1}E_{6, 0}^{78}). Denote the derived subgroup of the Levi part of the parabolic subgroup of type D \ {m} by G′. By Theorem 2 we have β_{G′}(α′_{m}) = 0, and G is uniquely determined by G′ satisfying this condition. Since J is invariant under s_{G}, m ≠ 5, 6.

In the case m = 1 G′ is a simple simply connected group of inner type D_{5}. The Cartan matrix shows that α_{m} = 2ω_{1} − ω_{3}, so

0 = β_{G′}(α′_{m}) = β_{G′}(ω_{4}).

By Proposition 6 G′ is of index ^{1}D_{5, 1}^{(1)} or ^{1}D_{5, 5}^{(1)}. In the first case G has the index ^{1}E_{6, 2}^{28} and is uniquely determined by a simple simply connected group H of inner type D_{4} with β_{H}(ω_{1}) = 0 and β_{H}(ω_{3}) = β_{G}(ω_{4}) = 0, that is β_{H} = 0. In the second case G is split, that is of index ^{1}E_{6, 6}^{0}.

In the case m = 2 G′ ≃ SL_{r}(A), where A is an Azumaya algebra over R with ind A = deg A = d, d ≠ 1 dr = 6. The Cartan matrix shows that α_{m} = 2ω_{2} − ω_{4}, so

0 = β_{G′}(α′_{m}) = β_{SL_{r}(A)}(ω_{3}) = 3[A].

Therefore, exp A = 3, and by Proposition 3 d = 3. The index of G is ^{1}E_{6, 2}^{16} in this case.

In the case m = 3 G′ ≃ SL_{1}(A) × SL_{1}(B), where A and B are Azumaya algebras over R with deg A = ind A = 2 and deg B = 5. The Cartan matrix shows that α_{m} = 2ω_{3} − ω_{1} − ω_{4}, so

0 = β_{G′}(α′_{m}) = −β_{SL_{1}(A)}(ω_{1}) − β_{SL_{1}(B)}(ω_{2}) = −[A] − 2[B].

But by Proposition 3 5[B] = 0 and 2[A] = 0; hence [A] = 0, a contradiction.

In the case m = 4 G′ ≃ SL_{1}(A)×SL_{1}(B)×SL_{1}(C), where A, B, C are Azumaya algebras over R with deg A = deg B = 3, deg C = ind C = 2. The Cartan matrix shows that α_{m} = 2ω_{4} − ω_{2} − ω_{3} − ω_{5}, so

0 = β_{G′}(α′_{m}) = β_{SL_{1}(A)}(ω_{1}) − β_{SL_{1}(B)}(ω_{1}) − β_{SL_{1}(C)}(ω_{1}) = [A] − [B] − [C].

But by Proposition 3 3[A] = 3[B] = 0 and 2[C] = 0; hence [C] = 0, a contradiction. □

Theorem 3 (^{2}E_{6}). Every simple simply connected group G of type ^{2}E_{6} over R has one of the following Tits indices:

(^{2}E_{6, 0}^{78})  [figure: •; •; •; •; •; •]

(^{2}E_{6, 1}^{35})  [figure: •; •; •; •; •; •]

(^{2}E_{6, 1}^{29})  [figure: •; •; •; •; •; •]

(^{2}E_{6, 2}^{16′})  [figure: •; •; •; •; •; •]

(^{2}E_{6, 2}^{16′′})  [figure: •; •; •; •; •; •]

(^{2}E_{6, 4}^{2})  [figure: •; •; •; •; •; •]

Isomorphism classes of groups of a given index bijectively correspond to isomorphism classes of:

• simple simply connected anisotropic groups H of type ^{2}A_{5} over R with β_{H}(ω_{3}) = 0, in the case of ^{2}E_{6, 1}^{35};

• simple simply connected anisotropic groups H of type ^{2}D_{4} over R with β_{H_{O}}(ω_{3}) = 0, O is the orbit corresponding to {3, 4}, in the case of ^{2}E_{6, 1}^{29};

• simple simply connected anisotropic groups H of type ^{2}A_{3} over R with β_{H}(ω_{2}) = 0 and β_{H_{O}}(ω_{1}) = 0, O is the orbit corresponding to {1, 3}, in the case of ^{2}E_{6, 2}^{16′};

• Azumaya algebras A over a connected quadratic étale extension R′/R with ind A = deg A = 3 and cores_{R′/R}([A]) = 0, in the case ^{2}E_{6, 2}^{16′′};

• connected quadratic étale extensions R′/R, in the case of ^{2}E_{6, 4}^{2}.

Proof. Let (^{2}E_{6}, J) be the Tits index of G and m be the least element of J (in the case J = ∅ G is anisotropic, that is of index ^{2}E_{6, 0}^{78}). Denote the orbit corresponding to m by O, and the derived subgroup of the Levi part of the parabolic subgroup of type D \ O by G′. By Theorem 2 we have β_{G′_{O}}(α′_{O}) = 0, and G is uniquely determined by G′ satisfying this condition.

In the case m = 1 O as a scheme is a connected quadratic étale extension R′/R, G′ is a simple simply connected group of type ^{2}D_{4}. The Cartan matrix shows that α_{m} = 2ω_{1} − ω_{3}, so

0 = β_{G′_{O}}(α′_{O}) = −β_{G′_{O}}(ω_{3}).

By Theorem 3, case ^{2}D_{n}, G′ has one of the following indices: ^{2}D_{4, 0}, ^{2}D_{4, 1}^{(1)}, ^{2}D_{4, 2}^{(1)}, ^{2}D_{4, 1}^{(2)}, 2D_{4, 4}^{(1)}.

In the case of ^{2}D_{4, 0} G has the index ^{2}E_{6, 1}^{29} and is uniquely determined by H = G′.

In the case of ^{2}D_{4, 1}^{(1)} G has the index ^{2}E_{6, 2}^{16′} and is uniquely determined by the a simple simply connected group H of type ^{2}A_{3} with β_{H}(ω_{2}) = 0 and β_{H_{O}}(ω_{1}) = β_{G′_{O}}(ω_{3}) = 0.

In the cases of ^{2}D_{4, 2}^{(1)} and ^{2}D_{4, 1}^{(2)} G′ is determined by an Azumaya algebra A over R′ with deg A = ind A = 2. But [A] = β_{G′_{O}}(ω_{3}) = 0, a contradiction.

In the case of ^{2}D_{4, 4}^{(1)} G is quasi-split, that is of index ^{2}E_{6, 4}^{2}.

In the case m = 2 O as a scheme is isomorphic to R, G′ is a simple simply connected group of type ^{2}A_{5}. The Cartan matrix shows that α_{m} = 2ω_{2} − ω_{4}, so

0 = β_{G′_{O}}(α′_{O}) = −β_{G′}(ω_{3}).

By Theorem 3, case ^{2}D_{n}, G′ has one of the following indices: ^{2}A_{5, 0}, ^{2}A_{5, 2}^{(1)}, ^{2}A_{5, 3}^{(1)}. In the case of ^{2}A_{5, 0} G has the index ^{2}E_{6, 1}^{35} and is uniquely determined by H = G′. In the case of ^{2}A_{5, 2}^{(1)} G′ is determined by an Azumaya algebra A over R with deg A =

ind A = 2. But [A] = β_{G′_{O}}(ω_{3}) = 0, a contradiction.

In the case of ^{2}A_{5, 3}^{(1)} G has the index ^{2}E_{6, 2}^{16′′} and is uniquely determined by an Azumaya algebra A over a connected quadratic étale extension R′/R with ind A = deg A = 3 and cores_{R′/R}([A]) = 0.

In the case m = 3 O as a scheme is a connected quadratic étale extension R′/R, G′ ≃ R_{R′/R}(SL_{1}(A)) × SL_{1}(E), where A is an Azumaya algebra over R′ with ind A = deg A = 2, E is an Azumaya algebra over R with ind E = deg E = 3. The Cartan matrix shows that α_{m} = 2ω_{3} − ω_{1} − ω_{4}, so

0 = β_{G′_{O}}(α′_{O}) = −β_{SL_{1}(A)}(ω_{1}) − β_{SL_{1}(E)_{O}}(ω_{1}) = −[A] − [D_{R′}].

But by Proposition 3 3[E] = 0 and 2[A] = 0, so [A] = 0, a contradiction.

In the case m = 4 O as a scheme is isomorphic to R, G′ ≃ R_{R′/R}(SL_{1}(A)) × SL_{1}(E), where A is an Azumaya algebra over R′ with ind A = deg A = 3, E is an Azumaya algebra over R with ind E = deg E = 2. The Cartan matrix shows that α_{m} = 2ω_{4} − ω_{2} − ω_{3} − ω_{5}, so

0 = β_{G′_{O}}(α′_{O}) = cores_{R′/R}(β_{SL_{1}(A)}(ω_{1})) − β_{SL_{1}(E)}(ω_{1}) = cores_{R′/R}([A]) − [E].

But by Proposition 3 3[A] = 0 and 2[E] = 0, so [E] = 0, a contradiction. □

Theorem 3 (E_{7}). Every simple simply connected group G of type E_{7} over R has one of the following Tits indices:

(E_{7, 0}^{133})  •  •  •  •  •  • •

(E_{7, 1}^{78})  [figure: •; •; •; •; •; •; •]

(E_{7, 1}^{66})  [figure: •; •; •; •; •; •; •]

(E_{7, 1}^{48})  [figure: •; •; •; •; •; •; •]

(E_{7, 2}^{31})  [figure: •; •; •; •; •; •; •]

(E_{7, 3}^{28})  [figure: •; •; •; •; •; •; •]

(E_{7, 4}^{9})  [figure: •; •; •; •; •; •; •]

(E_{7, 7}^{0})  [figure: •; •; •; •; •; •; •]

Isomorphism classes of groups of a given index bijectively correspond to isomorphism classes of:

• simple simply connected anisotropic groups H of type ^{1}E_{6} over R with β_{H} = 0, in the case of E_{7, 1}^{78};

• simple simply connected anisotropic groups H of type ^{1}D_{6} over R with β_{H}(ω_{5}) = 0, in the case of E_{7, 1}^{66};

• simple simply connected anisotropic groups H of type ^{1}D_{5} over R with β_{H}(ω_{4}) = [E], ind E = 2, in the case of E_{7, 1}^{48};

• simple simply connected anisotropic groups H of type ^{1}D_{4} over R with β_{H}(ω_{1}) = 0 and β_{H}(ω_{3}) = [E], ind E = 2, in the case of E_{7, 2}^{31}; • simple simply connected anisotropic groups H of type ^{1}D_{4} over R with β_{H} = 0, in

the case of E_{7, 3}^{28};

• Azumaya algebras A over R with ind A = deg A = 2, in the case of E_{7, 4}^{9}.

The only group of index E_{7, 7}^{0} is split.

Proof. Let (E_{7}, J) be the Tits index of G and m be the least element of J (in the case J = ∅ G is anisotropic, that is of index E_{7, 0}^{133}). Denote the derived subgroup of the Levi part of the parabolic subgroup of type D \ {m} by G′. By Theorem 2 we have β_{G′}(α′_{m}) = 0, and G is uniquely determined by G′ satisfying this condition.

In the case m = 1 G′ is a simple simply connected group of inner type D_{6}. The Cartan matrix shows that α_{m} = 2ω_{1} − ω_{3}, so

0 = β_{G′}(α′_{m}) = β_{G′}(ω_{5}).

By Theorem 3, case ^{1}D_{n}, together with Proposition 6, G′ has one of the following indices: 1D_{6, 0}, ^{1}D_{6, 2}^{(1)}, ^{1}D_{6, 1}^{(2)}, ^{1}D_{6, 2}^{(2)}, ^{1}D_{6, 3}^{(2)}, ^{1}D_{6, 1}^{(4)}, ^{1}D_{6, 6}^{(1)}.

In the case of ^{1}D_{6, 0} G has the index E_{7, 1}^{78} and is uniquely determined by H = G′.

In the case of ^{1}D_{6, 2}^{(1)} G has the index E_{7, 3}^{28} and is uniquely determined by a simple simply connected group H of inner type D_{4} with β_{H}(ω_{1}) = 0 and β_{H}(ω_{3}) = β_{G′}(ω_{5}) = 0, that is β_{H} = 0.

In the case of ^{1}D_{6, 1}^{(2)} G has the index E_{7, 1}^{48} and is uniquely determined by a simple simply connected group H of inner type D_{4} with β_{H}(ω_{1}) = [E], ind E = 2, and β_{H}(ω_{3}) = β_{G′}(ω_{5}) = 0. We may change enumeration of weights in such a way that β_{H}(ω_{3}) = [E] and β_{H}(ω_{1}) = 0.

In the cases of ^{1}D_{6, 2}^{(2)} and ^{1}D_{6, 1}^{(4)} G′ is determined by Azumaya algebras A_{1} and A_{2} over R with ind A_{1} = deg A_{1} = ind A_{2} = deg A_{2} = 2. But [A_{1}] = β_{G′}(ω_{5}) = 0, a contradiction.

In the case of ^{1}D_{6, 3}^{(2)} G′ is determined by an Azumaya algebra A over R with ind A = deg A = 2. There are two possibilities: J = {1, 3, 4, 6} and J = {1, 2, 4, 6}. In the first case G has the index E_{7, 4}^{9}. In the second case [A] = β_{G′}(ω_{5}) = 0, a contradiction.

In the case of ^{1}D_{6, 6}^{(1)} G is split, that is of index E_{7, 7}^{0}.

In the case m = 2 G′ ≃ SL_{1}(A), where A is an Azumaya algebra over R with deg A = 7. The Cartan matrix shows that α_{m} = 2ω_{2} − ω_{4}, so

0 = β_{G′}(α′_{m}) = −β_{SL_{1}(A)}(ω_{3}) = −3[A].

But by Proposition 3 7[A] = 0, hence [A] = 0, a contradiction.

In the case m = 3 G′ ≃ SL_{1}(A) × SL_{1}(B), where A and B are Azumaya algebras over R with deg A = ind A = 2 and deg B = 6. The Cartan matrix shows that α_{m} = 2ω_{3} − ω_{1} − ω_{4}, so

0 = β_{G′}(α′_{m}) = −β_{SL_{1}(A)}(ω_{1}) − β_{SL_{1}(B)}(ω_{2}) = −[A] − 2[B].

But by Proposition 3 6[B] = 0 and 2[A] = 0; hence [A] = 0, a contradiction.

In the case m = 4 G′ ≃ SL_{1}(A)×SL_{1}(B)×SL_{1}(C), where A, B, C are Azumaya algebras over R with deg A = ind A = 3, deg B = 4, deg C = ind C = 2. The Cartan matrix shows that α_{m} = 2ω_{4} − ω_{2} − ω_{3} − ω_{5}, so

0 = β_{G′}(α′_{m}) = β_{SL_{1}(A)}(ω_{1}) − β_{SL_{1}(B)}(ω_{1}) − β_{SL_{1}(C)}(ω_{1}) = [A] − [B] − [C].

But by Proposition 3 3[A] = 0, 4[B] = 0 and 2[C] = 0; hence [A] = 0, a contradiction.

In the case m = 5 G′ ≃ SL_{1}(A) × SL_{1}(B), where A and B are Azumaya algebras over R with deg A = ind A = 5 and deg B = 3. The Cartan matrix shows that α_{m} = 2ω_{3} − ω_{1} − ω_{4}, so

0 = β_{G′}(α′_{m}) = −β_{SL_{1}(A)}(ω_{3}) − β_{SL_{1}(B)}(ω_{1}) = −3[A] − [B].

But by Proposition 3 3[B] = 0 and 5[A] = 0; hence [A] = 0, a contradiction.

In the case m = 6 G′ ≃ H × SL_{1}(E), where H is a simple simply connected anisotropic group of inner type D_{5}, E is an Azumaya algebra over R with deg E = 2. The Cartan matrix shows that α_{m} = 2ω_{6} − ω_{5} − ω_{7}, so

0 = β_{G′}(α′_{m}) = −β_{H}(ω_{4}) − β_{SL_{1}(E)}(ω_{1}) = −β_{H}(ω_{4}) − [E].

If [E] = 0 then H is isotropic by Proposition 6, a contradiction. Therefore, ind E = 2. The index of G is E_{7, 1}^{48} in this case.

In the case m = 7 G′ = H is a simple simply connected anisotropic group of inner type E_{6}. The Cartan matrix shows that α_{m} = 2ω_{7} − ω_{6}, so

0 = β_{G′}(α′_{m}) = −β_{H}(ω_{6}),

that is β_{H} = 0. The index of G is E_{7, 1}^{78} in this case. □

Theorem 3 (E_{8}). Every simple simply connected group G of type E_{8} over R has one of the following Tits indices:

(E_{8, 0}^{248})  •  •  •  •  •  •  • •

(E_{8, 1}^{133})  [figure: •; •; •; •; •; •; •; •]

(E_{8, 1}^{91})  [figure: •; •; •; •; •; •; •; •]

(E_{8, 2}^{78})  [figure: •; •; •; •; •; •; •; •]

(E_{8, 2}^{66})  [figure: •; •; •; •; •; •; •; •]

(E_{8, 4}^{28})  [figure: •; •; •; •; •; •; •; •]

(E_{8, 8}^{0})  [figure: •; •; •; •; •; •; •; •]

Isomorphism classes of groups of a given index bijectively correspond to isomorphism classes of:

• simple simply connected anisotropic groups H of type E_{7} over R with β_{H} = 0, in the case of E_{8, 1}^{133};

• simple simply connected anisotropic groups H of type ^{1}D_{7} over R with β_{H} = 0, in the case of E_{8, 1}^{91}; • simple simply connected anisotropic groups H of type ^{1}E_{6} over R with β_{H} = 0, in

the case of E_{8, 2}^{78};

• simple simply connected anisotropic groups H of type ^{1}D_{6} over R with β_{H} = 0, in the case of E_{8, 2}^{66};

• simple simply connected anisotropic groups H of type ^{1}D_{4} over R with β_{H} = 0, in

the case of E_{8, 4}^{28}. The only group of index E_{8, 8}^{0} is split.

Proof. Let (E_{8}, J) be the Tits index of G and m be the least element of J (in the case J = ∅ G is anisotropic, that is of index E_{8, 0}^{248}). Denote the derived subgroup of the Levi part of the parabolic subgroup of type D \ {m} by G′. By Theorem 2 we have β_{G′}(α′_{m}) = 0, and G is uniquely determined by G′ satisfying this condition.

In the case m = 1 G′ is a simple simply connected group of inner type D_{7}. The Cartan matrix shows that α_{m} = 2ω_{1} − ω_{3}, so

0 = β_{G′}(α′_{m}) = β_{G′}(ω_{6}),

that is β_{G′} = 0. By Theorem 3, case ^{1}D_{n}, together with Proposition 6, G′ has one of the following indices: ^{1}D_{7, 0}, ^{1}D_{7, 1}^{(1)}, ^{1}D_{7, 3}^{(1)}, ^{1}D_{7, 7}^{(1)}.

In the case of ^{1}D_{7, 0} G has the index E_{8, 1}^{91} and is uniquely determined by H = G′.

In the case of ^{1}D_{7, 1}^{(1)} G has the index E_{8, 2}^{66} and is uniquely determined by a simple simply connected group H of inner type D_{6} with β_{H}(ω_{1}) = 0 and β_{H}(ω_{5}) = β_{G′}(ω_{6}) = 0, that is β_{H} = 0.

In the case of ^{1}D_{7, 3}^{(1)} G has the index E_{8, 4}^{28} and is uniquely determined by a simple simply connected group H of inner type D_{4} with β_{H}(ω_{1}) = 0 and β_{H}(ω_{3}) = β_{G′}(ω_{6}) = 0, that is β_{H} = 0.

In the case of ^{1}D_{7, 7}^{(1)} G is split, that is of index E_{8, 8}^{0}.

In the case m = 2 G′ ≃ SL_{1}(A), where A is an Azumaya algebra over R with deg A = 8. The Cartan matrix shows that α_{m} = 2ω_{2} − ω_{4}, so

0 = β_{G′}(α′_{m}) = −β_{SL_{1}(A)}(ω_{3}) = −3[A].

But by Proposition 3 8[A] = 0, hence [A] = 0, a contradiction.

In the case m = 3 G′ ≃ SL_{1}(A) × SL_{1}(B), where A and B are Azumaya algebras over R with deg A = ind A = 2 and deg B = 7. The Cartan matrix shows that α_{m} = 2ω_{3} − ω_{1} − ω_{4}, so

0 = β_{G′}(α′_{m}) = −β_{SL_{1}(A)}(ω_{1}) − β_{SL_{1}(B)}(ω_{2}) = −[A] − 2[B].

But by Proposition 3 7[B] = 0 and 2[A] = 0; hence [A] = 0, a contradiction.

In the case m = 4 G′ ≃ SL_{1}(A)×SL_{1}(B)×SL_{1}(C), where A, B, C are Azumaya algebras over R with deg A = ind A = 3, deg B = 5, deg C = ind C = 2. The Cartan matrix shows that α_{m} = 2ω_{4} − ω_{2} − ω_{3} − ω_{5}, so

0 = β_{G′}(α′_{m}) = β_{SL_{1}(A)}(ω_{1}) − β_{SL_{1}(B)}(ω_{1}) − β_{SL_{1}(C)}(ω_{1}) = [A] − [B] − [C].

But by Proposition 3 3[A] = 0, 5[B] = 0 and 2[C] = 0; hence 3[B] + [C] = 0 and [C] = 0, a contradiction.

In the case m = 5 G′ ≃ SL_{1}(A) × SL_{1}(B), where A and B are Azumaya algebras over R with deg A = ind A = 5 and deg B = 4. The Cartan matrix shows that α_{m} = 2ω_{3} − ω_{1} − ω_{4}, so

0 = β_{G′}(α′_{m}) = −β_{SL_{1}(A)}(ω_{3}) − β_{SL_{1}(B)}(ω_{1}) = −3[A] − [B].

But by Proposition 3 4[B] = 0 and 5[A] = 0; hence [A] = 0, a contradiction.

In the case m = 6 G′ ≃ H × SL_{1}(E), where H is a simple simply connected anisotropic group of inner type D_{5}, E is an Azumaya algebra over R with deg E = 3. The Cartan matrix shows that α_{m} = 2ω_{6} − ω_{5} − ω_{7}, so

0 = β_{G′}(α′_{m}) = −β_{H}(ω_{4}) − β_{SL_{1}(E)}(ω_{1}) = −β_{H}(ω_{4}) − [E].

But by Proposition 3 3[E] = 0, hence β_{H}(ω_{4}) = 0. Therefore, H is isotropic by Proposition 6, a contradiction.

In the case m = 7 G′ ≃ H × SL_{1}(E), where H is a simple simply connected anisotropic group of inner type E_{6}, E is an Azumaya algebra over R with deg A = 2. The Cartan matrix shows that α_{m} = 2ω_{7} − ω_{6} − ω_{8}, so

0 = β_{G′}(α′_{m}) = −β_{H}(ω_{6}) − β_{SL_{1}(E)}(ω_{1}) = −β_{H}(ω_{6}) − [E].

But by Proposition 3 2[E] = 0, hence β_{H} = 0 and [E] = 0. The index of G is E_{8, 2}^{78} in this case.

In the case m = 8 G′ = H is a simple simply connected anisotropic group of type E_{7}. The Cartan matrix shows that α_{m} = 2ω_{8} − ω_{7}, so

0 = β_{G′}(α′_{m}) = −β_{H}(ω_{7}),

that is β_{H} = 0. The index of G is E_{8, 1}^{133} in this case. □

Theorem 3 (F_{4}). Every simple simply connected group G of type F_{4} over R has one of the following Tits indices:

(F_{4, 0}^{52})  [figure: •; •; •; •]

(F_{4, 1}^{21})  [figure: •; •; •; •]

(F_{4, 4}^{0})  [figure: •; •; •; •]

Isomorphism classes of groups of index F_{4, 1}^{21} bijectively correspond to isomorphism classes of simple simply connected anisotropic groups H of type B_{3} over R with β_{H} = 0. The only group of index F_{4, 4}^{0} is split.

Proof. Let (F_{4}, J) be the Tits index of G and m be the least element of J (in the case J = ∅ G is anisotropic, that is of index F_{4, 0}^{52}). Denote the derived subgroup of the Levi part of the parabolic subgroup of type D \ {m} by G′. By Theorem 2 we have β_{G′}(α′_{m}) = 0, and G is uniquely determined by G′ satisfying this condition.

In the case m = 1 G′ is a simple simply connected group of type C_{3}. The Cartan matrix shows that α_{1} = 2ω_{1} − ω_{2}, so

0 = β_{G′}(α′_{m}) = −β_{G′}(ω_{1}).

By Proposition 4 G′ is split and so is G. The index of G is F_{4, 4}^{0} in this case.

In the case m = 2 G′ ≃ SL_{1}(A) × SL_{1}(B), where A and B are Azumaya algebras over R with deg A = ind A = 2 and deg B = 3. The Cartan matrix shows that α_{m} = 2ω_{2} −ω_{1} −2ω_{3}, so

0 = β_{G′}(α′_{m}) = −β_{SL_{1}(A)}(ω_{1}) − 2β_{SL_{1}(B)}(ω_{1}) = −[A] − 2[B].

But by Proposition 3 3[B] = 0 and 2[A] = 0; hence [A] = 0, a contradiction.

In the case m = 2 G′ ≃ SL_{1}(A) × SL_{1}(B), where A and B are Azumaya algebras over R with deg A = ind A = 3 and deg B = 2. The Cartan matrix shows that α_{m} = 2ω_{3} − ω_{2} − ω_{4}, so

0 = β_{G′}(α′_{m}) = β_{SL_{1}(A)}(ω_{1}) − β_{SL_{1}(B)}(ω_{1}) = [A] − [B].

But by Proposition 3 2[A] = 0 and 3[B] = 0; hence [A] = 0, a contradiction.

In the case m = 4 G′ = H is a simple simply connected group of type B_{3}. The Cartan matrix shows that α_{1} = 2ω_{4} − ω_{3}, so

0 = β_{G′}(α′_{m}) = −β_{H}(ω_{3}),

that is β_{H} = 0. The index of G is F_{4, 1}^{21} in this case. □

Theorem 3 (G_{2}). Every simple simply connected group G of type G_{2} over R has one of the following Tits indices:

(G_{2, 0}^{14})  [figure: •; •]

(G_{2, 2}^{0})  [figure: •; •]

The only group of index G_{2, 2}^{0} is split.

Proof. Let (G_{2}, J) be the Tits index of G and m be the least element of J (in the case J = ∅ G is anisotropic, that is of index G_{2, 0}^{14}). Denote the derived subgroup of the Levi part of the parabolic subgroup of type D \ {m} by G′. By Theorem 2 we have β_{G′}(α′_{m}) = 0.

In the case m = 1 G′ ≃ SL_{1}(A), where A is an Azumaya algebra over R with deg A = 2. The Cartan matrix shows that α_{m} = 2ω_{1} − ω_{2}, so

0 = β_{G′}(α′_{m}) = −β_{SL_{1}(A)}(ω_{1}) = −[A].

Therefore, G′ is split and so is G. The index of G is G_{2, 2}^{0} in this case.

In the case m = 1 G′ ≃ SL_{1}(A), where A is an Azumaya algebra over R with ind A = deg A = 2. The Cartan matrix shows that α_{m} = 2ω_{2} − 3ω_{1}, so

0 = β_{G′}(α′_{m}) = −3β_{SL_{1}(A)}(ω_{1}) = −3[A].

But by Proposition 3 2[A] = 0, hence [A] = 0, a contradiction. □

References

[B] N. Bourbaki, Groupes et algèbres de Lie. Chapitres 4–6, Hermann, Paris, 1968.

[SGA] M. Demazure, A. Grothendieck, Schémas en groupes, Lecture Notes in Mathematics, Vol. 151–153, Springer-Verlag, Berlin-Heidelberg-New York, 1970.

[Gab] O. Gabber, Some theorems on Azumaya algebras, Lecture Notes in Mathematics, Vol. 844, Springer, Berlin-New York, 1981, 129–209.

[Jan] J.C. Jantzen, Darstellungen halbeinfacher Gruppen und kontravariante Formen, J. Reine Angew. Math. 290 (1977), 117–141.

[K] M.-A. Knus, Quadratic and Hermitian forms over rings, Springer-Verlag, Berlin, 1991.

[Ti66] J. Tits, Classification of algebraic semisimple groups, Algebraic groups and discontinuous subgroups, Proc. Sympos. Pure Math. 9, Amer. Math. Soc., Providence RI, 1966, 33–62.

[Ti71] J. Tits. Représentations linéares irréductibles d’un groupe réductif sur un corps quelconque, J. Reine Angew. Math. 247 (1971), 196–220.

[Ti90] J. Tits. Strongly inner anisotropic forms of simple algebraic groups, J. of Algebra 131 (1990), 648– 677.

PIMS, University of Alberta, Edmonton, Canada

   E-mail address: victorapetrov@googlemail.com

St. Petersburg State University, St. Petersburg, Russia

   E-mail address: a stavrova@mail.ru