DEGREE THREE COHOMOLOGICAL INVARIANTS OF SEMISIMPLE GROUPS

A. MERKURJEV

Abstract. We study the degree 3 cohomological invariants with coefficients in ℚ/ℤ(2) of a semisimple group over an arbitrary field. A list of all invariants of adjoint groups of inner type is given.

1. Introduction

1a. Cohomological invariants. Let G be a linear algebraic group over a field F (of arbitrary characteristic). The notion of an invariant of G was defined in [8] as follows. Consider functor

H^{1}(−, G) : Fields_{F} ⟶ Sets,

where Fields_{F} is the category of field extensions of F, taking a field K to the set H^{1}(K, G) of isomorphism classes of G-torsors over Spec K. Let

H : Fields_{F} ⟶ Abelian Groups

be another functor. An H-invariant of G is then a morphism of functors

I : H^{1}(−, G) ⟶ H.

We denote the group of H-invariants of G by Inv(G, H).

An invariant I ∈ Inv(G, H) is called normalized if I(X) = 0 for the trivial G-torsor X. The normalized invariants form a subgroup Inv(G, H)_{norm} of Inv(G, H) and there is a natural isomorphism

Inv(G, H) ≃ H(F) ⊕ Inv(G, H)_{norm}.

Of particular interest to us is the functor H which takes a field K/F to the Galois cohomology group H^{n}(K, ℚ/ℤ(j)), where the coefficients ℚ/ℤ(j), j ≥ 0, are defined as the direct sum of the colimit over n of the Galois modules μ_{m}^{⊗j}, where μ_{m} is the Galois module of m^{th} roots of unity, and a p-component in the case p = char(F) > 0 defined via logarithmic de Rham-Witt differentials (see [13, I.5.7], [14]).

We write Inv^{n}(G, ℚ/ℤ(j)) for the group of cohomological invariants of G of degree n with coefficients in ℚ/ℤ(j).

Date: March, 2013.

Key words and phrases. Semisimple groups, cohomological invariants, torsors, classifying space 2010 Mathematical Subject Classifications: 12G05, 11E72, 14F20, 14F42 .

The work has been supported by the NSF grant DMS #1160206.

If G is connected, then Inv^{1}(G, ℚ/ℤ(j))_{norm} = 0 (see [15, Proposition 31.15]). The degree 2 cohomological invariants with coefficients in ℚ/ℤ(1) (equivalently, the invariants with values in the Brauer group Br) of a smooth connected group were determined in [1]:

Inv^{2}(G, Br)_{norm} = Inv^{2}(G, ℚ/ℤ(1))_{norm} ≃ Pic(G).

In particular, for a semisimple group G we have

Inv^{2}(G, ℚ/ℤ(1))_{norm} ≃ Ĉ(F),

where Ĉ(F) is the group of characters defined over F of the kernel C of the universal cover G̃ → G by [21, Prop. 6.10].

The group of degree 3 invariants Inv^{3}(G, ℚ/ℤ(2))_{norm} was determined by Rost in the case when G is simply connected quasi-simple. This group is finite cyclic with a canonical generator called the Rost invariant (see [8, Part II]).

In the present paper, based on the results in [18], we extend Rost’s result to all semisimple groups.

Theorem. Let G be a semisimple group over a field F. Then there is an exact sequence

0 ⟶ CH^{2}(BG)_{tors} ⟶ H^{1}(F, Ĉ(1)) ⟶^{σ}

Inv^{3}(G, ℚ/ℤ(2))_{norm} ⟶ Q(G)/ Dec(G) ⟶ H^{2}(F, Ĉ(1)).

Here BG is the classifying space of G and Q(G)/ Dec(G) is the group defined in Section 3c in terms of the combinatorial data associated with G (the root system, weight and root lattices).

If G is simply connected, the character group Ĉ is trivial and we obtain Rost’s theorem mentioned above.

The main result has clearer form for adjoint groups G of inner type. In this case every character of C is defined over F, i.e., Ĉ = Ĉ(F) We show that the group Inv^{3}(G, ℚ/ℤ(2))_{dec} := Im(σ) of decomposable invariants (given by a cup-product with the degree 2 invariants), is canonically isomorphic to Ĉ ⊗ F^{×}. The factor group Inv^{3}(G, ℚ/ℤ(2))_{ind} of Inv^{3}(G, ℚ/ℤ(2))_{norm} by the decomposable invariants is nontrivial if and only if G has a simple component of type C_{n} or D_{n} (when n is divisible by 4), E_{6} or E_{7}. If G is simple, the group of indecomposable invariants is cyclic with a canonical generator restricting to a multiple of the Rost invariant.

We will use the following notation in the paper.

F is the base field,

F_{sep} a separable closure of F,

Γ_{F} = Gal(F_{sep}/F).

For a complex A of étale sheaves on a variety X, we write H^{∗}(X, A) for the étale (hyper-)cohomology group of X with values in A.

Acknowledgements: The authors would like to thank Vladimir Chernousov, Skip Garibaldi, Jean-Pierre Tignol and Burt Totaro for helpful discussions.

2. Preliminaries

2a. Cohomology of BG. Let G be a connected algebraic group over a field F and let V be a generically free representation of G such that there is an open G-invariant subscheme U ⊂ V and a G-torsor U → U/G such that U(F) ≠ ∅ (see [26, Remark 1.4]).

Let H be a (contravariant) functor from the category of smooth varieties over F to the category of abelian groups. Very often the value H(U/G) is independent (up to canonical isomorphism) of the choice of the representation V provided the codimension of V \ U in V is sufficiently large. This is the case, for example, if H = CH^{i}, the Chow group functor of cycles of codimension i (see [26] or [5]). We write H(BG) for H(U/G) and view U/G as an “approximation” for the “classifying space” BG of G.

We have the two maps p_{i}^{∗} : H(U/G) → H((U × U)/G), i = 1, 2, induced by the projections p_{i} : (U × U)/G → U/G. An element h ∈ H(U/G) is called balanced if p_{1}^{∗}(h) = p_{2}^{∗}(h). We write H(U/G)_{bal} for the subgroup of all balanced elements in H(U/G).

Write H^{n}(ℚ/ℤ(j)) for the Zariski sheaf on a smooth scheme X associated to the presheaf S ↦ H^{n}(S, ℚ/ℤ(j)).

Let u ∈ H_{Zar}^{0}(U/G, H^{n}(ℚ/ℤ(j)))_{bal}. Define an invariant I_{u} ∈ Inv^{n}(G, ℚ/ℤ(j)) as follows (see [1]). Let X be a G-torsor over a field extension K/F. Choose a point x ∈ (U/G)(K) such that X is isomorphic to the pull-back via x of the versal G-torsor U → U/G and set I_{u}(X) = x^{∗}(u), where

x^{∗} : H_{Zar}^{0}(U/G, H^{n}(ℚ/ℤ(j))) ⟶ H_{Zar}^{0}(Spec K, H^{n}(ℚ/ℤ(j))) = H^{n}(K, ℚ/ℤ(j))

is the pull-back homomorphism given by x : Spec(K) → U/G. The fact that the element u is balanced ensures that x^{∗}(u) does not depend on the choice of the point x (see [1, Lemma 3.2]).

Write \overline{H}_{Zar}^{0}(U/G, H^{n}(ℚ/ℤ(j))) for the factor group of H_{Zar}^{0}(U/G, H^{n}(ℚ/ℤ(j))) by the natural image of H^{n}(F, ℚ/ℤ(j)).

Proposition 2.1. ([1, Corollary 3.4]) The assignment u ↦ I_{u} yields an isomorphism

\overline{H}_{Zar}^{0}(U/G, H^{n}(ℚ/ℤ(j)))_{bal} ⟶^{∼} Inv^{n}(G, ℚ/ℤ(j))_{norm}.

2b. The map α_{G}. Let G be a semisimple group over F and let C be the kernel of the universal cover G̃ → G. For a character χ ∈ Ĉ(F) over F consider the push-out diagram

[figure: 1; C; χ; G̃; G; 1; 1; _{𝔾_{m}}; G′; G; 1.]

We define a map

α_{G} : H^{1}(F, G) ⟶ Hom(Ĉ(F), Br(F))

by α_{G}(ξ)(χ) = δ(ξ), where δ : H^{1}(F, G) → H^{2}(F, 𝔾_{m}) = Br(F) is the connecting map for the bottom row of the diagram.

Example 2.2. Let G = PGL_{n}. Then Ĉ = ℤ/nℤ and the map α_{G} takes the class [A] ∈ H^{1}(F, PGL_{n}) of a central simple algebra A of degree n to the homomorphism i + nℤ ↦ i[A] ∈ Br(F).

Let C′ be the center of G. Recall that there is the Tits homomorphism (see [15, Theorem 27.7])

β_{G} : Ĉ′(F) ⟶ Br(F).

A central simple algebra over F representing the class β_{G} for some χ ∈ Ĉ′(F) is called a Tits algebra of G over F.

In the following proposition we relate the maps α_{G} and βG̃.

Proposition 2.3. Let G be a semisimple group, X a G-torsor over F and χ ∈ Ĉ′(F), where C′ is the center of the universal cover G̃ of G. Let ^{X}G := Aut_{G}(X) be the twist of G by X and ^{X}G̃ the universal cover of ^{X}G. Then

α_{G}(X)(χ|_{C}) = β_{_{X}G̃}(χ) − βG̃(χ),

where C ⊂ C′ is the kernel of G̃ → G.

Proof. By [15, §31], there exist a unique (up to isomorphism) G-torsor Y such that the twist ^{Y}G = Aut_{G}(Y) is quasi-split and α_{G}(Y)(χ|_{C}) = −βG̃(χ). If ^{X}Y is the twist of Y by X, then Aut_{^{X}G}(^{X}Y) ≃ Aut_{G}(Y) is quasi-split. Hence α_{^{X}G}(^{X}Y)(χ|_{C}) = −β_{_{X}G̃}(χ). It follows from [15, Proposition 28.12] that α_{^{X}G}(^{X}Y) + α_{G}(X) = α_{G}(Y). □

2c. Admissible maps. Let G be a split simply connected group over F and Π a set of simple roots of G.

Proposition 2.4. (cf. [9, Proposition 5.5]) Let G be a split simply connected group over F, C the center of G. Let Π′ be a subset of Π and let G′ be the subgroup of G generated by the root subgroups of all roots in Π′. Then G′ is a simply connected group and C ⊂ G′ if and only if every fundamental weight w_{α} for α ∈ Π \ Π′ is contained in the root lattice Λ_{r} of G.

Proof. The group G′ is simply connected by [22, 5.4b]. The images of the co-roots α^{∗} : 𝔾_{m} → T for α ∈ Π′ generate the maximal torus T′ = G′ ∩ T of G′. Therefore, the character group Ω of the torus T/T′ coincides with

{λ ∈ T̂ such that ⟨λ, α^{∗}⟩ = 0 for all α ∈ Π′}

and hence Ω is generated by the fundamental weights w_{β} for all β ∈ Π \ Π′. We have T̂′ = Λ_{w}/Ω and Ĉ = Λ_{w}/Λ_{r}. Therefore, C ⊂ G′ ∩ T = T′ if and only if Ω ⊂ Λ_{r}. □

A homomorphism a : Ĉ(F) → Br(F) is called admissible if ind a(χ) divides ord(χ) for every χ ∈ Ĉ(F).

Example 2.5. Suppose G is the product of split adjoint groups of type A. By Example 2.2, every admissible map belongs to the image of α_{G}.

Proposition 2.6. Let G be a split adjoint group over F. Then every admissible map in Hom(Ĉ(F), Br(F)) belongs to the image of α_{G}.

Proof. Let Π′ be the subset of Π of all roots α such that w_{α} ∈ Λ_{r} and let G′ be the subgroup of G̃ generated by the root subgroups for all roots in Π′. Then by Proposition 2.4, G′ is a simply connected group such that C ⊂ G′. Let C′ be the center of G′ and set C′′ := C′/C. By Lemma 2.7 below, the top row in the commutative diagram

[figure: H^{1}(F, G′/C); α_{G′/C}; H^{1}(F, G′/C′); α_{G′/C′}; Hom(Ĉ′′(F), Br(F)); Hom(Ĉ(F), Br(F)); Hom(Ĉ′(F), Br(F)); Hom(Ĉ′′(F), Br(F))]

is exact.

Let a ∈ Hom(Ĉ(F), Br(F)) be an admissible map. Then the image a′ of a in Hom(Ĉ′(F), Br(F)) is also admissible. Inspection shows that every component of the Dynkin diagram of G′ is of type A. (A root α belongs to Π′ if and only if the i^{th} row of the inverse C^{−1} of the Cartan matrix is integer, see Section 4b.) By Example 2.5, a′ belongs to the image of α_{G′/C′}. A diagram chase shows that a belongs to the image of α_{G′/C}. The map α_{G′/C} is the composition of H^{1}(F, G′/C) → H^{1}(F, G) and α_{G}, hence a belongs to the image of α_{G}. □

Lemma 2.7. Let G_{1} → G_{2} be a central isogeny of split semisimple groups with the kernel C_{1}. Then the sequence

H^{1}(F, G_{1}) ⟶ H^{1}(F, G_{2}) ⟶ Hom(Ĉ_{1}(F), Br(F))

with the second map the composition of α_{G_{2}} and the restriction map on C_{1}, is exact.

Proof. The group C_{1} is diagonalizable as G_{1} is split. Let T be a split torus containing C_{1} as a subgroup. The push-out diagram

[figure: 1; C_{1}; χ; G_{1}; G_{2}; 1; 1; T; G_{3}; G_{2}; 1]

yields a commutative diagram

[figure: H^{1}(F, G_{1}); H^{1}(F, G_{2}); Hom(Ĉ_{1}(F), Br(F)); χ^{∗}; H^{1}(F, G_{3}); H^{1}(F, G_{2}); Hom(T̂(F), Br(F)).]

The bottom row is exact as Hom(T̂(F), Br(F)) = H^{2}(F, T). The left vertical arrow is surjective since H^{1}(F, Coker(χ)) = 1 by Hilbert’s Theorem 90. The result follows by diagram chase. □

2d. The morphism β_{f}. Let G be a semisimple group, C the kernel of the universal cover G̃ → G and f : X → Spec F a G-torsor. Write ℤ_{f}(1) for the cone of the natural morphism ℤ_{F} (1) → Rf_{∗}ℤ_{X}(1) of complexes of étale sheaves over Spec F, where ℤ(1) = 𝔾_{m}[−1]. The composition (see [18, §4])

β_{f} : Ĉ ≃ τ_{≤2}ℤ_{f}(1)[2] ⟶ ℤ_{f}(1)[2] ⟶ ℤ_{F} (1)[3]

yields a homomorphism

β_{f}^{∗} : Ĉ(F) ⟶ H^{3}(F, ℤ_{F} (1)) = Br(F).

In the following proposition we relate the maps β_{f}^{∗} and α_{G}.

Proposition 2.8. For a G-torsor f : X → Spec F, we have β_{f}^{∗} = α_{G}(X).

Proof. By [18, Example 6.12], the map β_{f}^{∗} coincides with the connecting homomorphism for the exact sequence

(2.1)  1 ⟶ F_{sep}^{×} ⟶ F_{sep}(X)^{×} ⟶ Div(X_{sep}) ⟶ Ĉ_{sep} ⟶ 0,

where Div is the divisor group (recall that Ĉ_{sep} = Pic(X_{sep})).

Consider first the case G = PGL_{n} and X = Isom(B, M_{n}) is the variety of isomorphisms between a central simple algebra B of degree n and the matrix algebra M_{n} over F. We have C = μ_{n} and Ĉ = ℤ/nℤ. The exact sequence (2.1) for the Severi-Brauer variety S of B in place of X gives the connecting homomorphism ℤ → Br(F) that takes 1 to the class [B] by [12, Theorem 5.4.10]. A natural map between the two exact sequences induced by the natural morphism X → S and Example 2.2 yield

(2.2)  β_{f}^{∗}(¯1) = [B] = α_{PGL_{n}}(X)(¯1).

Suppose now that G = PGL_{1}(A) for a central simple algebra A of degree n. Consider the PGL_{n}-torsor Y = Isom(A, M_{n}). Then G is the twist of PGL_{n} by Y. The G-torsor Z = Isom(B, A) is the twist of X by Y. It follows from [15, Proposition 28.12] that

(2.3)  α_{G}(Z)(¯1) = α_{PGL_{n}}(X)(¯1) − α_{PGL_{n}}(Y)(¯1) = [B] − [A].

The group homomorphism PGL_{1}(B) × PGL_{1}(A^{op}) → PGL_{1}(B ⊗ A^{op}) takes the torsor Z × Isom(A^{op}, A^{op}) to V := Isom(B ⊗ A^{op}, A ⊗ A^{op}). Let g and h be the structure morphisms for Z and V, respectively. It follows from (2.2) applied to β_{h}^{∗} and (2.3) that

(2.4)  β_{g}^{∗}(¯1) = β_{h}^{∗}(¯1) = [B] − [A] = α_{G}(Z)(¯1).

Now consider the general case. By [25, Théorème 3.3], for every χ ∈ Ĉ(F), there is a central simple algebra A (of degree n) over F and a commutative diagram

\xymatrix{1 \ar[r] & C \ar[r] \ar[d]_{χ} & G̃ \ar[r] \ar[d] & G \ar[r] \ar[d] & 1 \\ 1 \ar[r] & _{μ_{n}} \ar[r] & SL_{1}(A) \ar[r] & PGL_{1}(A) \ar[r] & 1.}

A G-torsor f : X → Spec F yields a PGL_{1}(A)-torsor, say k : W → Spec F. We have by (2.4),

β_{f}^{∗}(χ) = β_{k}^{∗}(¯1) = α_{PGL_{1}(A)}(W)(¯1) = α_{G}(X)(χ).  □
3. The group Inv^{3}(G, ℚ/ℤ(2))

In this section we determine the group Inv^{3}(G, ℚ/ℤ(2)) of degree 3 cohomological invariants of a semisimple group G.

Recall first a construction of degree two cohomological invariants of G with coefficients in ℚ/ℤ(1), or, equivalently, the invariants with values in the Brauer group. Every character χ ∈ Ĉ(F) yields an invariant I_{χ} of G of degree 2 with coefficients in ℚ/ℤ(1) defined by

I_{χ}(X) = α_{G}(X)(χ_{K}) ∈ Br(K).

By [1, Theorem 2.4], the assignment χ ↦ I_{χ} yields an isomorphism

Ĉ(F) ⟶^{∼} Inv^{2}(G, ℚ/ℤ(1))_{norm}.

3a. Representation ring. (See [25].) Write R(G) for the representation ring of G, i.e., R(G) is the Grothendieck group of the category of finite dimensional representations of G. As an abelian group R(G) is free with basis the isomorphism classes of irreducible representations.

Consider the weight lattice Λ of G (the character group of a maximal split torus over F_{sep}) as a Γ_{F} -lattice with respect to the ∗-action (see [24]). Let Γ′ be the (finite) factor group of Γ_{F} acting faithfully on Λ. Write Δ for the semidirect product of the Weyl group W of G and Γ′ with respect to the natural action of Γ′ on W. The group Δ acts naturally on Λ.

Assigning to a representation of G the formal sum of its weights, we get an injective homomorphism

ch : R(G) ⟶ ℤ[Λ]^{Δ}.

For any λ ∈ Λ write A_{λ} for the corresponding Tits algebra (over the field of definition of λ) and Δ(λ) for the sum ∑ e^{λ′} in ℤ[Λ]^{Δ}, where λ′ runs over the Δ-orbit of λ (we employ the exponential notation for ℤ[Λ]). By [8, Part II, Theorem 10.11], the image of R(G) in ℤ[Λ]^{Δ} is generated by ind(A_{λ}) · Δ(λ) over all λ ∈ Λ.

In particular, if G is quasi-split, all Tits algebras are trivial and hence ch is an isomorphism.

Example 3.1. Consider the variety X of maximal tori in G and the closed subscheme T ⊂ G × X of all pairs (g, T) with g ∈ T. The generic fiber of the projection T → X is a maximal torus in G_{F (X)}, it is called the generic maximal torus T_{gen} of G. By [27, Theorem 1], if G is split, the decomposition group of T_{gen} coincides with the Weyl group W. It follows that if G is quasi-split, then Δ is the decomposition group of T_{gen}. Moreover, ch is an isomorphism, hence the restriction homomorphism R(G) → R(T_{gen}) = ℤ[Λ]^{Δ} is an isomorphism for a quasi-split G.

3b. Root systems and invariant quadratic forms. Let {α_{1}, α_{2} … α_{n}} be a set of simple roots of an irreducible root system in a vector space V, {w_{1}, w_{2}, …, w_{n}} the corresponding fundamental weights generating the weight lattice Λ_{w} and W the Weyl group.

Consider the n-columns α := (α_{1}, α_{2}, …, α_{n})^{t} and w := (w_{1}, w_{2}, …, w_{n})^{t}. Then α = Cw, where C = (c_{ij}) is the Cartan matrix (see [2, Chapitre VI]). There is a (unique) W -invariant bilinear form on the dual space V^{∗} such that the length of a short co-root is equal to 1. Let D := diag(d_{1}, d_{2}, …, d_{n}) be the diagonal matrix with d_{i} the length of the i^{th} co-root. Then DC is a symmetric even integer matrix (i.e., the diagonal terms are even).

Note that if A is a symmetric n×n matrix over ℚ, then \frac{1}{2}w^{t}Aw is contained in Sym^{2}(Λ_{w}) if and only if the matrix A is even integer.

Consider the integer quadratic form

q := \frac{1}{2}w^{t}DCw ∈ Sym^{2}(Λ_{w})

on Λ_{r}^{∗}, where Λ_{r} is the root lattice. Recall that the Weyl group W acts naturally on Λ_{w}.

Lemma 3.2. The quadratic form q is W -invariant.

Proof. Let s_{i} be the reflection with respect to α_{i}. It suffices to prove that s_{i}(q) = q. We have s_{i}(w) = w − α_{i}e_{i}. Hence

s_{i}(q) = \frac{1}{2}(w − α_{i}e_{i})^{t}DC(w − α_{i}e_{i})
= q − α_{i}e_{i}^{t}D(Cw − \frac{1}{2}α_{i}Ce_{i})
= q − α_{i}d_{i}(e_{i}^{t}α − \frac{1}{2}α_{i}e_{i}^{t}Ce_{i})
= q − α_{i}d_{i}(α_{i} − \frac{1}{2}α_{i}c_{ii}) = q

as c_{ii} = 2. □

If α_{i}^{∗} is a short co-root, then q(α_{i}^{∗}) = d_{i} = 1 since ⟨w_{j}, α_{i}^{∗}⟩ = δ_{ji}. It follows that q is a (canonical) generator of the cyclic group Sym^{2}(Λ_{w})^{W}.

Example 3.3. For the root system of type A_{n−1}, n ≥ 2, we have Λ_{w} = ℤ^{n}/ℤe, where e = e_{1} + e_{2} + ⋯ + e_{n}. The root lattice Λ_{r} is generated by the simple roots ē_{1} −ē_{2}, ē_{2} −ē_{3}, …, ē_{n−1} −ē_{n}. The Weyl group W is the symmetric group S_{n} acting naturally on Λ_{w}. The generator of Sym^{2}(Λ_{w})^{W} is the form

q = − ∑_{i<j} x̄_{i}x̄_{j} = \frac{1}{2} ∑_{i=1}^{n} x̄_{i}^{2}.

The group Sym^{2}(Λ_{r})^{W} = Sym^{2}(Λ_{r}) ∩ Sym^{2}(Λ_{w})^{W} is also cyclic with the canonical generator a positive multiple of q.

Proposition 3.4. Let m be the smallest positive integer such that the matrix mDC^{−1} is even integer. Then mq is a generator of Sym^{2}(Λ_{r})^{W}.

Proof. Rewrite q in the form q = \frac{1}{2}(C^{−1}α)^{t}DC(C^{−1}α) = \frac{1}{2}α^{t}DC^{−1}α. The multiple mq is contained in Sym^{2}(Λ_{r}) if and only if the matrix mDC^{−1} is even integer. □

3c. The groups Dec(G) ⊂ Q(G). Let A be a lattice. Consider the abstract total Chern class homomorphism

c_{•} : ℤ[A] ⟶ Sym^{•}(A)[[t]]^{×}

defined by c_{•}(e^{a}) = 1 + at. We define the abstract Chern class maps

c_{i} : ℤ[A] ⟶ Sym^{i}(A), i ≥ 0,

by c_{•}(x) = ∑_{i≥0} c_{i}(x)t^{i}. Clearly, c_{0}(x) = 1,

c_{1}( ∑_{i} e^{a_{i}}) = ∑_{i} a_{i}, c_{2}( ∑_{i} e^{a_{i}}) = ∑_{i<j} a_{i}a_{j},

c_{1} is a homomorphism and

c_{2}(x + y) = c_{2}(x) + c_{1}(x)c_{1}(y) + c_{2}(y)

for all x, y ∈ ℤ[A].

If a group W acts on A, then all the c_{i} are W -equivariant.

Suppose that A^{W} = 0. Then c_{1} is zero on ℤ[A]^{W} and c_{2} yields a group homomorphism

(3.1)  c_{2} : ℤ[A]^{W} → Sym^{2}(A)^{W} .

We write Dec(A) for the image of this homomorphism. The group Dec(A) is generated by the decomposable elements ∑_{i<j} a_{i}a_{j}, where {a_{1}, a_{2}, ….a_{n}} is a W -invariant subset of A. We also have

(3.2)  c_{2}(xy) = rank(x)c_{2}(y) + rank(y)c_{2}(x)

for all x, y ∈ ℤ[A]^{W}, where rank : ℤ[A] → ℤ is the map e^{a} ↦ 1. If S ⊂ A is a finite W -invariant subset, then since ∑_{x∈S} x ∈ A^{W} = 0, we have

(3.3)  c_{2}( ∑_{a∈S} e^{a}) = −\frac{1}{2} ∑_{a∈S} a^{2}.

Let G be a semisimple group over F. Recall that the weight lattice Λ is a Δmodule (see Section (3a)). Note that Λ^{W} = 0, so we have the homomorphism of Γ_{F} -modules (3.1) with A = Λ.

Set

Q(G) := Sym^{2}(Λ)^{Δ} = (Sym^{2}(Λ)^{W} )^{Γ_{F}} .

and write Dec(G) for the image of the composition

(3.4)  τ : R(G) ⟶^{ch} ℤ[Λ]^{Δ} ⟶^{c_{2}} Sym^{2}(Λ)^{Δ} = Q(G).

Example 3.5. The map τ : R(SL_{n}) → Q(SL_{n}) takes the class of the tautological representation to the quadratic form ∑_{i<j} x̄_{i}x̄_{j} which is the negative of the canonical generator of Q(SL_{n}) (see Example 3.3).

It follows from Example 3.5 that if G is a quasi-simple group, then for a representation ρ of G, we have τ(ρ) = −N(ρ)q, where N(ρ) is the Dynkin index of ρ (see [7]). Hence the image of Dec(G) under τ is equal to n_{G}ℤq, where n_{G} is the gcd of the Dynkin indexes of all the representations of G. The numbers n_{G} for split adjoint groups G of types B_{n}, C_{n} and E_{7} were computed in [7] (see also Section 4b).

A loop in G is a group homomorphism 𝔾_{m} → G_{sep} over F_{sep} (see [15, §31]). By [8, Part II, §7]), the group Q(G) has an intrinsic description as the group of all Γ_{F} -invariant quadratic integral-valued functions on the set of all loops in G. It follows that a homomorphism G → G′ of semisimple groups yields a group homomorphism Q(G′) → Q(G). The functoriality of the Chern class shows that this homomorphism takes Dec(G′) into Dec(G).

3d. The key diagram. Let V be a generically free representation of G such that there is an open G-invariant subscheme U ⊂ V and a G-torsor U → U/G such that U(F) ≠ ∅ (see Section 2a). We assume in addition that V \ U is of codimension at least 3.

By [14, Th. 1.1], there is an exact sequence

0 ⟶ CH^{2}(U^{n}/G) ⟶ \overline{H}^{4}(U^{n}/G, ℤ(2)) ⟶ \overline{H}_{Zar}^{0}(U^{n}/G, H^{3}(ℚ/ℤ(2))) ⟶ 0

for every n. We can view this as an exact sequence of cosimplicial groups. The group CH^{2}(U^{n}/G) is independent of n, so it represents a constant cosimplicial groups CH^{2}(BG). Therefore, we have an exact sequence

0 ⟶ CH^{2}(BG) ⟶ \overline{H}^{4}(U/G, ℤ(2))_{bal} ⟶ \overline{H}_{Zar}^{0}(U/G, H^{3}(ℚ/ℤ(2)))_{bal} ⟶ 0.

The right group in the sequence is canonically isomorphic to Inv^{3}(G, ℚ/ℤ(2))_{norm} by Proposition 2.1, and hence is independent of V. Therefore, the middle term is also independent of V and we write \overline{H}^{4}(BG, ℤ(2)) for \overline{H}^{4}(U/G, ℤ(2))_{bal}. Therefore, we have the exact row in the following diagram with the exact column given by [18, Theorem 5.3]:

\xymatrix{ &  & 0 \ar[d] \\  &  & H^{1}(F, Ĉ(1)) \ar[d] \ar@{-->}[dr]^{σ} \\ 0 \ar[r] & CH^{2}(BG) \ar[r] \ar[dr]_{γ} & \overline{H}^{4}(BG, ℤ(2)) \ar[r] \ar[d] & Inv^{3}(G, ℚ/ℤ(2))_{norm} \ar[r] & 0 \\  &  & Q(G) \ar[d]^{θ_{G}^{∗}} \\  &  & H^{2}(F, Ĉ(1)),}

where Ĉ(1) is the derived tensor product Ĉ ⊗^{L} ℤ_{Y} (1) in the derived category of et́ale sheaves on F. Explicitly (see [18, Section 4c]),

Ĉ(1) = Tor_{1}^{ℤ}(Ĉ_{sep}, F_{sep}^{×}) ⊕ (Ĉ_{sep} ⊗ F_{sep}^{×})[−1].

Example 3.6. The group SL_{n} is special simply connected, hence Ĉ = 0 and Inv^{3}(SL_{n}, ℚ/ℤ(2))_{norm} = 0. It follows that we have isomorphisms of infinite cyclic groups

γ : CH^{2}(B SL_{n}) →^{∼} \overline{H}^{4}(B SL_{n}, ℤ(2)) →^{∼} Q(SL_{n}).

The group CH^{2}(B SL_{n}) is generated by c_{2} of the tautological representation by [20, §2].

3e. The map σ. The map σ is defined as follows (see [18, §5]). Let f : X → Spec K be a G-torsor over a field extension K/F, so we have a morphism β_{f} : Ĉ → ℤ_{K}(1)[3] as in Section 2d, and therefore, the composition

Ĉ(1) = Ĉ ⊗^{L} ℤ_{F} (1) ⟶^{β_{f} ⊗^{L}Id} (ℤ_{K}(1) ⊗^{L} ℤ_{F} (1))[3] ⟶ ℤ_{K}(2)[3],

which induces a homomorphism H^{1}(F, Ĉ(1)) → H^{4}(K, ℤ(2)) = H^{3}(K, ℚ/ℤ(2)). Then the value of the invariant σ(α) for an element α ∈ H^{1}(F, Ĉ(1)) is equal to the image of α under this homomorphism.

Let χ ∈ Ĉ(F) and a ∈ F^{×}. By [18, Remark 5.2], we have χ ∪ (a) ∈ H^{1}(F, Ĉ(1)) and therefore, σ(χ∪(a)) is the invariant taking a G-torsor X over K to α_{G}(X)(χ_{K}) ∪ (a) ∈ H^{3}(K, ℚ/ℤ(2)). Here the cup-product is taken with respect to the pairing

Br(K) ⊗ K^{×} = H^{2}(K, ℚ/ℤ(1)) ⊗ H^{1}(K, ℤ(1)) ⟶ H^{3}(K, ℚ/ℤ(2)).

3f. The map γ. We will determine the map γ in the key diagram.

Lemma 3.7. The maps γ and \overline{H}^{4}(BG, ℤ(2)) → Q(G) are functorial in G. Proof. In [18] the map γ is given by the composition

CH^{2}(BG) ⟶ H^{4}(BG, ℤ(2)) →^{∼} H^{3}(BG, ℤ_{f}(2)) →^{∼}

H^{3}(BG, τ_{≤3}ℤ_{f}(2)) ⟶ H_{Zar}^{1}(BG, K_{2})^{Γ_{F}} → D(G),

where ℤ_{f}(2) is the cone of ℤ_{BG}(2) → Rf_{∗}ℤ_{EG}(2) for the versal G-torsor f : EG → BG and the group D(G) containing Q(G) is defined in [18]. The first four homomorphisms are functorial in G, and the last one is functorial as was shown in [8, Page 116] in the case G is simply connected. The proof also goes through for an arbitrary semisimple G. □ Lemma 3.8. The composition of the second Chern class map

R(G) ⟶ K_{0}(BG) ⟶^{c_{2}} CH^{2}(BG)

with the diagonal morphism γ in the diagram coincides with the map τ in (3.4) up to sign. The image of γ coincides with Dec(G). Proof. As Q(G) injects when the base field gets extended, for the proof of the first statement we may assume that F is separably closed. Let ρ : G → SL_{n} be a representation. Write x_{1}, x_{2}, …, x_{n} for the characters of ρ in the weight lattice Λ. Consider the diagram

[figure: R(SL_{n}); c_{2}; τ; Q(SL_{n}); CH^{2}(B SL_{n}); γ; R(G); c_{2}; τ; Q(G); CH^{2}(BG); γ]

with the vertical homomorphisms induced by ρ. The vertical faces of the diagram are commutative by Lemma 3.7 and the functoriality of c_{2} and the character map ch. By Example 3.5, the top map τ takes the class of the tautological representation ι of SL_{n} to the a generator of Q(SL_{n}). By Example 3.6, γ in the top of the diagram is an isomorphism taking the canonical generator of CH^{2}(B SL_{n}) to a generator of Q(SL_{n}). It follows that τ(ι) and γ(c_{2}(ι)) in the top face of the diagram are equal up to sign. The class of ρ in R(G) is the image of τ under the left vertical homomorphism. It follows that τ(ρ) and γ(c_{2}(ρ)) in the bottom face of the diagram are also equal to sign.

The second statement follows from the first and the surjectivity of the second Chern class map R(G) → CH^{2}(BG) (see [6, Appendix C] and [26, Corollary 3.2]). □

3g. Main theorem. The following theorem describes the group of degree 3 cohomological invariants with coefficients in ℚ/ℤ(2) of an arbitrary semisimple group.

Theorem 3.9. Let G be a semisimple group over a field F. Then there is an exact sequence

0 ⟶ CH^{2}(BG)_{tors} ⟶ H^{1}(F, Ĉ(1)) ⟶^{σ}

Inv^{3}(G, ℚ/ℤ(2))_{norm} ⟶ Q(G)/ Dec(G) ⟶^{θ_{G}^{∗}} H^{2}(F, Ĉ(1)).

Proof. Follows from the key diagram above and Lemma 3.8 as Q(G) is torsion free and H^{1}(F, Ĉ(1)) is torsion. □

Remark 3.10. The map θ_{G}^{∗} is trivial if G is split or adjoint of inner type (see [18, Proposition 4.1 and Remark 5.5]).

The exact sequence in Theorem 3.9 is functorial in G. More precisely, let G → G′ be a homomorphism of semisimple groups extending to a homomorphism C → C′ of the kernels of the universal covers. By Lemma 3.7, the diagram

\xymatrix{H^{1}(F, Ĉ′(1)) \ar[r]^{σ′} \ar[d] & Inv^{3}(G′, ℚ/ℤ(2))_{norm} \ar[r] \ar[d] & Q(G′)/ Dec(G′) \ar[d] \\ H^{1}(F, Ĉ(1)) \ar[r]^{σ} & Inv^{3}(G, ℚ/ℤ(2))_{norm} \ar[r] & Q(G)/ Dec(G)}

is commutative.

Write Inv^{3}(G, ℚ/ℤ(2))_{dec} for the image of σ. We call these invariants decomposable. Thus, we have an exact sequence

0 ⟶ CH^{2}(BG)_{tors} ⟶ H^{1}(F, Ĉ(1)) ⟶^{σ} Inv^{3}(G, ℚ/ℤ(2))_{dec} ⟶ 0.

We don’t know if the group CH^{2}(BG)_{tors} is trivial, but it is always finite.

Proposition 3.11. The group CH^{2}(BG) is finitely generated. In particular, CH^{2}(BG)_{tors} is finite.

Proof. By [25, Théorème 3.3] and Section 3a, we have

ℤ[Λ_{r}]^{Δ} ⊂ R(G) ⊂ ℤ[Λ_{w}].

The Noetherian ring ℤ[Λ_{r}] is finite over ℤ[Λ_{r}]^{Δ}, hence ℤ[Λ_{r}]^{Δ} is Noetherian. The ℤ[Λ_{r}]^{Δ}-algebra ℤ[Λ_{w}] is finite, hence so is R(G). It follows that the ring R(G) is Noetherian. Let I be the kernel of the rank map R(G) → ℤ. Since I is finitely generated, the factor group R(G)/I^{2} is finitely generated. By (3.2), the second Chern class factors through a surjective homomorphism R(G)/I^{2} → CH^{2}(BG), whence the result. □

We will show in Section 4a that the group CH^{2}(BG)_{tors} is trivial if G is adjoint of inner type.

The factor group

Inv^{3}(G, ℚ/ℤ(2))_{ind} := Inv^{3}(G, ℚ/ℤ(2))/ Inv^{3}(G, ℚ/ℤ(2))_{dec}

is called the group of indecomposable invariants. Thus, we have an exact sequence

0 ⟶ Inv^{3}(G, ℚ/ℤ(2))_{ind} ⟶ Q(G)/ Dec(G) ⟶^{θ_{G}^{∗}} H^{2}(F, Ĉ(1)).

If G is simply connected quasi-simple, all decomposable invariants are trivial, and the group Inv^{3}(G, ℚ/ℤ(2)) = Inv^{3}(G, ℚ/ℤ(2))_{ind} ≃ Q(G)/ Dec(G) is cyclic generated by the Rost invariant R_{G}. The order of the Rost number n_{G} of R_{G} is determined in [8, Part II].

4. Groups of inner type

Let G be a semisimple group over F. A group G′ is called an inner form of G if there is a G-torsor X over F such that G′ is the twist of G by X, or equivalently, G′ ≃ Aut_{G}(X). The choice of the torsor X yields a canonical bijection φ : H^{1}(K, G′) →^{∼} H^{1}(K, G) for every field extension K/F (see [15, Proposition 8.8]). Therefore, we have an isomorphism Inv^{n}(G, ℚ/ℤ(j)) →^{∼} Inv^{n}(G′, ℚ/ℤ(j)). Note that this isomorphism does not preserve normalized invariants as φ does not preserve trivial torsors. Precisely, φ takes the class of a trivial torsor to the class of X. We modify the isomorphism to get an isomorphism

(4.1)  Inv^{n}(G, ℚ/ℤ(j))_{norm} →^{∼} Inv^{n}(G′, ℚ/ℤ(j))_{norm},

taking an invariant I of G to an invariant I′ of G′ satisfying

I′(X′) = I(φ(X′)) − I(X).

4a. Decomposable invariants. Let G be a semisimple group of inner type. Then Ĉ is a diagonalizable finite group.

Lemma 4.1. There is a natural isomorphism H^{1}(F, Ĉ(1)) ≃ Ĉ ⊗ F^{×}.

Proof. Write Ĉ ≃ R/S, where R and S are lattices. In the exact sequence

H^{1}(F, S(1)) ⟶ H^{1}(F, R(1)) ⟶ H^{1}(F, Ĉ(1)) ⟶ H^{2}(F, S(1))

the first two terms are S ⊗ F^{×} and R ⊗ F^{×}, respectively, and the last term is equal to S ⊗H^{2}(F, ℤ(1)) = 0 by Hilbert’s Theorem 90. The result follows. □

Recall that under the isomorphism in Lemma 4.1, the map σ in Theorem 3.9 is defined as follows. For every χ ∈ Ĉ and a ∈ F^{×}, the invariant σ(χ∪(a)) takes a G-torsor X over a field extension K/F to α_{G}(X)(χ_{K})∪(a) ∈ H^{3}(K, ℚ/ℤ(2)) (see Section 3e).

Theorem 4.2. Let G be a semisimple adjoint group of inner type over a field F. Then the homomorphism

σ : Ĉ ⊗ F^{×} ⟶ Inv^{3}(G, ℚ/ℤ(2))_{dec}

is an isomorphism. Equivalently, the group CH^{2}(BG) is torsion-free.

Proof. As G is an inner form of a split group, by (4.1), we may assume that G is split. The group Ĉ is a direct sum of cyclic subgroups generated by χ_{1}, …, χ_{m}, respectively. Let a_{1}, …, a_{m} ∈ F^{×} be such that the element u := ∑ χ_{i} ⊗a_{i} belongs to the kernel of σ. It suffices to show that a_{i} ∈ (F^{×})^{s_{i}}, where s_{i} := ord(χ_{i}) for all i.

Fix an integer i. For a field extension K/F and any ρ ∈ H^{1}(K, ℚ/ℤ) of order s_{i}, consider the admissible map f : Ĉ → Br(K(t)) for the field K(t) of rational functions over K, defined by

f(χ_{j}) = \{ \begin{matrix}ρ ∪ (t), & in Br(K(t)) if j = i; \\ 0, & otherwise. \end{matrix}

By Proposition 2.6, there is a G-torsor X over K(t) satisfying α_{G}(X)(χ_{j}) = f(χ_{j}) for all j. As u ∈ Ker(σ), we have

0 = σ(u)(X) = ∑_{j} α_{G}(X)(χ_{j}) ∪ (a_{j}) = ρ ∪ (t) ∪ (a_{i})

in H^{3}(K(t), ℚ/ℤ(2)). Taking residue at t (see [8, Part II, Appendix A]),

H_{nr}^{3}(K(t), ℚ/ℤ(2)) ⟶ H^{2}(K, ℚ/ℤ(1)) = Br(K),

we get ρ ∪ (a_{i}) = 0 in Br(K). By Lemma 4.3 below, we have a ∈ (F^{×})^{s_{i}}. □

Lemma 4.3. Let a ∈ F^{×} and s > 0 be such that for every field extension K/F and every ρ ∈ H^{1}(K, ℚ/ℤ) of order s one has ρ∪(a) = 0 in H^{2}(K, ℚ/ℤ(1)) = Br(K). Then a ∈ (F^{×})^{s}.

Proof. Let H = ℤ/sℤ. Choose an H-torsor X → Y with smooth Y, Pic(X) = 0 and F [X]^{×} = F^{×}. (For example, take an approximation of EH → BH.) By [3] or [17], there is an exact sequence

Pic(X)^{H} ⟶ H^{2}(H, F [X]^{×}) ⟶ Br(Y),

which yields an injective map F^{×}/F^{×s} → Br(F (Y)) as H^{2}(H, F [X]^{×}) = H^{2}(H, F^{×}) = F^{×}/F^{×s} and Br(Y) injects into Br(F (Y)) by [19, Corollary 2.6]. This map takes a to ρ ∪ (a), where ρ ∈ H^{1}(F (Y), ℚ/ℤ) corresponds to the cyclic extension F (X)/F (Y). As ρ ∪ (a) = 0 by assumption, we have a ∈ (F^{×})^{s}. □

4b. Indecomposable invariants. In this section we compute the groups of indecomposable invariants of adjoint groups of inner type.

Type A_{n−1}

In the split case we have G = PGL_{n}, the projective general linear group, n ≥ 2, Λ_{w} = ℤ^{n}/ℤe, where e = e_{1} + e_{2} + ⋯ + e_{n}. The root lattice is generated by the simple roots ē_{1} − ē_{2}, ē_{2} − ē_{3}, …, ē_{n−1} − ē_{n}, Ĉ = Λ_{w}/Λ_{r} ≃ ℤ/nℤ. The generator of Sym^{2}(Λ_{w})^{W} is the form

q = − ∑_{i<j} x̄_{i}x̄_{j} = \frac{1}{2} ∑ x̄_{i}^{2}.

The matrix D (see Section 3b) is the identity matrix I_{n}. The inverses of Cartan matrices here and below are taken from [4, Appendix F]:

C^{−1} = \frac{1}{n} ( \begin{matrix}n − 1 & n − 2 & n − 3 & ⋮ & 2 & 1 \\ n − 2 & 2(n − 2) & 2(n − 3) & ⋮ & 4 & 2 \\ n − 3 & 2(n − 3) & 3(n − 3) & ⋮ & 6 & 3 \\ ⋯ & ⋯ & ⋯ & & ⋯ & ⋯ \\ 2 & 4 & 6 & ⋮ & 2(n − 2) & n − 2 \\ 1 & 2 & 3 & ⋮ & n − 2 & n − 1 \end{matrix} )

By Proposition 3.4,

Q(G) = Sym^{2}(Λ_{r})^{W} = \{ \begin{matrix}2nℤq, & if n is even; \\ nℤq, & if n is odd. \end{matrix}

If a := ∑_{i,j=1}^{n} e^{x̄_{i}−x̄_{j}} ∈ ℤ[Λ_{r}]^{W}, we have by (3.3),

c_{2}(a) = \frac{1}{2} ∑(x̄_{i} − x̄_{j})^{2} = n ∑ x̄_{i}^{2} = 2nq ∈ Dec(G).

It follows that Dec(G) = Q(G) if n is even.

Suppose that n is odd. If b = ∑_{i=1}^{n} e^{nx̄_{i}} ∈ ℤ[Λ_{r}]^{W}, we have by (3.3),

c_{2}(b) = \frac{1}{2} ∑(nx̄_{i})^{2} = n^{2}q ∈ Dec(G).

As n is odd, gcd(2n, n^{2}) = n, hence nq ∈ Dec(G) and again Dec(G) = Q(G). Thus, Inv^{3}(G, ℚ/ℤ(2))_{ind} = Q(G)/ Dec(G) = 0. A G-torsor is given by a central simple algebra A of degree n (here and

below see [15]). The twist of G by A is the group PGL_{1}(A). The Tits classes of algebras for this group are the multiples of [A] in Br(F). In view of Proposition 2.3 and 4.1, we have

Theorem 4.4. Let G = PGL_{1}(A) for a central simple algebra A over F. Then

Inv^{3}(G, ℚ/ℤ(2))_{norm} ≃ F^{×}/F^{×n}.

An element x ∈ F^{×} corresponds to the invariant taking a central simple algebra A′ of degree n to the cup-product ([A′] − [A]) ∪ (x).

Type B_{n}

In the split case we have G = O_{2n+1}^{+}, the special orthogonal group, n ≥ 2, Λ_{w} = ℤ^{n} + ℤe, where e = \frac{1}{2}(e_{1} + e_{2} + ⋯ + e_{n}), Λ_{r} = ℤ^{n} and Ĉ ≃ ℤ/2ℤ. The generator of Sym^{2}(Λ_{w})^{W} is the form q = \frac{1}{2} ∑_{i} x_{i}^{2} and D = diag(1, 1, … 1, 2),

C^{−1} = ( \begin{matrix}1 & 1 & 1 & ⋮ & 1 & 1 & 1 \\ 1 & 2 & 2 & ⋮ & 2 & 2 & 2 \\ 1 & 2 & 3 & ⋮ & 3 & 3 & 3 \\ ⋯ & ⋯ & ⋯ & & ⋯ & ⋯ & ⋯ \\ 1 & 2 & 3 & ⋮ & n − 2 & n − 2 & n − 2 \\ 1 & 2 & 3 & ⋮ & n − 2 & n − 1 & n − 1 \\ 1/2 & 1 & 3/2 & ⋮ & (n − 2)/2 & (n − 1)/2 & n/2 \end{matrix} )

By Proposition 3.4, Q(G) = Sym^{2}(Λ_{r})^{W} = 2ℤq.

If a := ∑_{i=1}^{n}(e^{x_{i}} + e^{−x_{i}}) ∈ ℤ[Λ_{r}]^{W}, we have

c_{2}(a) = \frac{1}{2} ∑(x_{i}^{2} + (−x_{i})^{2}) = 2q ∈ Dec(G).

It follows that Dec(G) = Q(G), so Inv^{3}(G, ℚ/ℤ(2))_{ind} = Q(G)/ Dec(G) = 0.

A G-torsor is given by the similarity class of a nondegenerate quadratic form p of dimension 2n + 1. The twist of G by p is the special orthogonal group O^{+}(p) of the form p. The only nontrivial Tits class of algebras for this group is the class of the even Clifford algebra C_{0}(p) of p. In view of Proposition 2.3 and 4.1, we have

Theorem 4.5. Let G = O^{+}(p) for a nondegenerate quadratic form p of dimension 2n + 1. Then

Inv^{3}(G, ℚ/ℤ(2))_{norm} ≃ F^{×}/F^{×2}.

An element x ∈ F^{×} corresponds to the invariant taking the similarity class of a nondegenerate quadratic form p′ of dimension 2n + 1 to the cup-product ([C_{0}(p′)] − [C_{0}(p)]) ∪ (x).

Type C_{n}

In the split case we have G = PGSp_{2n}, the projective symplectic group, n ≥ 3, Λ_{w} = ℤ^{n}, Λ_{r} consists of all ∑ a_{i}e_{i} with ∑ a_{i} even, Ĉ ≃ ℤ/2ℤ. The generator of Sym^{2}(Λ_{w})^{W} is q = ∑_{i} x_{i}^{2}. D = diag(2, 2, … 2, 1) and

C^{−1} = ( \begin{matrix}1 & 1 & 1 & ⋮ & 1 & 1 & 1/2 \\ 1 & 2 & 2 & ⋮ & 2 & 2 & 1 \\ 1 & 2 & 3 & ⋮ & 3 & 3 & 3/2 \\ ⋯ & ⋯ & ⋯ & & ⋯ & ⋯ & ⋯ \\ 1 & 2 & 3 & ⋮ & n − 2 & n − 2 & (n − 2)/2 \\ 1 & 2 & 3 & ⋮ & n − 2 & n − 1 & (n − 1)/2 \\ 1 & 2 & 3 & ⋮ & n − 2 & n − 1 & n/2 \end{matrix} )

By Proposition 3.4,

Q(G) = Sym^{2}(Λ_{r})^{W} = \{ \begin{matrix}ℤq, & if n ≡ 0 modulo 4; \\ 2ℤq, & if n ≡ 2 modulo 4; \\ 4ℤq, & if n is odd. \end{matrix}

If a := ∑_{i}(e^{2x_{i}} + e^{−2x_{i}}) ∈ ℤ[Λ_{r}]^{W}, we have

c_{2}(a) = ∑(2x_{i})^{2} = 4q ∈ Dec(G).

It follows that Dec(G) = Q(G) if n is odd.

Suppose that n is even. If b := ∑_{i≠j}(e^{x_{i}+x_{j}} + e^{x_{i}−x_{j}}) ∈ ℤ[Λ_{r}]^{W}, we have

c_{2}(b) = \frac{1}{2} ∑_{i≠j}[(x_{i} − x_{j})^{2} + (x_{i} + x_{j})^{2}] = 2(n − 1)q ∈ Dec(G).

As n is even, gcd(4, 2(n − 1)) = 2, we have 2q ∈ Dec(G). On the other hand, by [8, Part II, Lemma 14.2], Dec(G) ⊂ 2qℤ, therefore, Dec(G) = 2qℤ.

It follows that

Inv^{3}(G, ℚ/ℤ(2))_{ind} = Q(G)/ Dec(G) = \{ \begin{matrix}(ℤ/2ℤ)q, & if n ≡ 0 modulo 4; \\ 0, & otherwise. \end{matrix}

A G-torsor is given by a pair (A, σ), where A is a central simple algebra of degree 2n and σ is a symplectic involution on A. The twist of G by (A, σ) is the projective symplectic group PGSp(A, σ). The only nontrivial Tits class of algebras for this group is the class of the algebra A. In view of Proposition 2.3 and 4.1, we have

Theorem 4.6. Let G = PGSp(A, σ) for a a central simple algebra of degree 2n with symplectic involution σ. Then

Inv^{3}(G, ℚ/ℤ(2))_{dec} ≃ F^{×}/F^{×2}.

An element x ∈ F^{×} corresponds to the invariant taking a pair (A′, σ′) to the cup-product ([A′] − [A]) ∪ (x).

If n is not divisible by 4, we have Inv^{3}(G, ℚ/ℤ(2))_{norm} = Inv^{3}(G, ℚ/ℤ(2))_{dec}. If n is divisible by 4, the group Inv^{3}(G, ℚ/ℤ(2))_{ind} is cyclic of order 2.

In the case n is divisible by 4 and char(F) ≠ 2 an invariant I of order 2 generating Inv^{3}(G, ℚ/ℤ(2))_{ind} was constructed in [11, §4]. Thus, in this case we have

Inv^{3}(G, ℚ/ℤ(2))_{norm} = Inv^{3}(G, ℚ/ℤ(2))_{dec} ⊕ (ℤ/2ℤ)I ≃ F^{×}/F^{×2} ⊕ (ℤ/2ℤ).

Type D_{n}

In the split case we have G = PGO_{2n}^{+}, the projective orthogonal group, n ≥ 4, Λ_{w} = ℤ^{n} + ℤe, where e = \frac{1}{2}(e_{1} + e_{2} + ⋯ + e_{n}), Λ_{r} consists of all ∑ a_{i}e_{i} with ∑ a_{i} even, C̃ is isomorphic to ℤ/2ℤ ⊕ ℤ/2ℤ if n is even and to ℤ/4ℤ if n is odd. The generator of Sym^{2}(Λ_{w})^{W} is the form q = \frac{1}{2} ∑_{i} x_{i}^{2} and D = I_{n},

C^{−1} = ( \begin{matrix}1 & 1 & 1 & ⋮ & 1 & 1/2 & 1/2 \\ 1 & 2 & 2 & ⋮ & 2 & 1 & 1 \\ 1 & 2 & 3 & ⋮ & 3 & 3/2 & 3/2 \\ ⋯ & ⋯ & ⋯ & & ⋯ & ⋯ & ⋯ \\ 1 & 2 & 3 & ⋮ & n − 2 & (n − 2)/2 & (n − 2)/2 \\ 1/2 & 1 & 3/2 & ⋮ & (n − 2)/2 & n/4 & (n − 2)/4 \\ 1/2 & 1 & 3/2 & ⋮ & (n − 2)/2 & (n − 2)/4 & n/4 \end{matrix} )

By Proposition 3.4,

Q(G) = Sym^{2}(Λ_{r})^{W} = \{ \begin{matrix}2ℤq, & if n ≡ 0 modulo 4; \\ 4ℤq, & if n ≡ 2 modulo 4; \\ 8ℤq, & if n is odd. \end{matrix}

If a := ∑_{i}(e^{2x_{i}} + e^{−2x_{i}}) ∈ ℤ[Λ_{r}]^{W}, we have

c_{2}(a) = ∑(2x_{i})^{2} = 8q ∈ Dec(G).

It follows that Dec(G) = Q(G) if n is odd.

Suppose that n is even. If b := ∑_{i≠j}(e^{x_{i}+x_{j}} + e^{x_{i}−x_{j}}) ∈ ℤ[Λ_{r}]^{W}, we have

c_{2}(b) = \frac{1}{2} ∑_{i≠j} [(x_{i} − x_{j})^{2} + (x_{i} + x_{j})^{2}] = 4(n − 1)q ∈ Dec(G).

As n is even, gcd(8, 4(n − 1)) = 4, we have 4q ∈ Dec(G). On the other hand, by [8, Part II, Lemma 15.2], Dec(G) ⊂ 4ℤq, therefore, Dec(G) = 4ℤq.

It follows that

Inv^{3}(G, ℚ/ℤ(2))_{ind} = Q(G)/ Dec(G) = \{ \begin{matrix}(2ℤ/4ℤ)q, & if n ≡ 0 modulo 4; \\ 0, & otherwise. \end{matrix}

A G-torsor is given by a quadruple (A, σ, f, e), where A is a central simple algebra of degree 2n, (σ, f) is a quadratic pair on A of trivial discriminant and e an idempotent in the center of the Clifford algebra C(A, σ, f). The twist of G by (A, σ, f, e) is the projective orthogonal group PGO^{+}(A, σ, f). The nontrivial Tits classes of algebras for this group are the class of the algebra A and the classes of the two components C^{±}(A, σ, f) of the Clifford algebra. In view of Proposition 2.3 and 4.1, we have

Theorem 4.7. Let G = PGO^{+}(A, σ, f) for a a central simple algebra of degree 2n with quadratic pair (σ, f) of trivial discriminant. Then

Inv^{3}(G, ℚ/ℤ(2))_{dec} ≃ \{ \begin{matrix}(F^{×}/F^{×2}) ⊕ (F^{×}/F^{×2}), & if n is even; \\ F^{×}/F^{×4}, & if n is odd. \end{matrix}

If n is even and x^{+}, x^{−} ∈ F^{×}, then the corresponding invariant takes a quadruple (A′, σ′, f′, e′) to

([C^{+}(A′, σ′, f′)]−[C^{+}(A, σ, f)])∪(x^{+})+([C^{−}(A′, σ′, f′)]−[C^{−}(A, σ, f)])∪(x^{−}).

If n is even and x ∈ F^{×}, then the corresponding invariant takes a quadruple (A′, σ′, f′, e′) to ([C^{+}(A′, σ′, f′)] − [C^{+}(A, σ, f)]) ∪ (x).

If n is not divisible by 4, we have Inv^{3}(G, ℚ/ℤ(2))_{norm} = Inv^{3}(G, ℚ/ℤ(2))_{dec}. If n is divisible by 4, the group Inv^{3}(G, ℚ/ℤ(2))_{ind} is cyclic of order 2.

In the case n is divisible by 4 and char(F) ≠ 2 we sketch below a construction of a nontrivial indecomposable invariant I of order 2 for a split adjoint group G = PGO_{2n}^{+}. A G-torsor X over F is given by a triple (A, σ, e), where A is a central simple algebra over F with an orthogonal involution σ of trivial discriminant and e is a nontrivial idempotent of the center of the Clifford algebra of (A, σ) (see [15, §29F]). We need to determine the value of I(X) in H^{3}(F, ℚ/ℤ(2)).

We have G = Aut(A, σ, e) = PGO^{+}(A, σ). The exact sequence

1 ⟶ μ_{2} ⟶ O^{+}(A, σ) ⟶ PGO^{+}(A, σ) ⟶ 1,

where O^{+}(A, σ) is the special orthogonal group, yields an exact sequence

H^{1}(F, O^{+}(A, σ)) ⟶^{φ} H^{1}(F, PGO^{+}(A, σ)) ⟶^{δ} Br(F).

The reduction method used in [11] for the construction of an indecomposable degree 3 invariant for a symplectic involution works as well in the orthogonal case. It reduces the general situation to the case ind(A) ≤ 4. In this case the algebra A is isomorphic to M_{2}(B) for a central simple algebra B as 2n is divisible by 8 and hence it admits a hyperbolic involution σ′. By [15, Proposition 8.31], one of the two components of the Clifford algebra C(A, σ′) is split. Let e′ be the corresponding idempotent in the center of C(A, σ′). (If both components split, then A is split by [15, Theorem 9.12], and we let e′ be any of the two idempotents.)

The element δ(A, σ′, e′) is trivial, hence (A, σ′, e′) = φ(v) for some v ∈ H^{1}(F, O^{+}(A, σ)). The set H^{1}(F, O^{+}(A, σ)) is described in the [15, §29.27] as the set of equivalence classes of pairs (a, x) ∈ A×F such that a is σ-symmetric invertible element and x^{2} = Nrd(a). Thus, v = (a, x) for such a pair (a, x) and we set I(X) = [A] ∪ (x).

Type E_{6}

We have Ĉ ≃ ℤ/3ℤ and D = I_{6},

C^{−1} = \frac{1}{3} ( \begin{matrix}4 & 5 & 6 & 4 & 2 & 3 \\ 5 & 10 & 12 & 8 & 4 & 6 \\ 6 & 12 & 18 & 12 & 6 & 9 \\ 4 & 8 & 12 & 10 & 5 & 6 \\ 2 & 4 & 6 & 5 & 4 & 3 \\ 3 & 6 & 9 & 6 & 3 & 6 \end{matrix} )

By Proposition 3.4, Q(G) = Sym^{2}(Λ_{r})^{W} = 3ℤq.

Write δ_{i} ∈ ℤ[Λ_{w}]^{W} for the sum of elements in the W -orbit of e^{w_{i}}. We have c_{2}(δ_{1}) = 6q, c_{2}(δ_{2}) = 24q, c_{2}(δ_{3}) = 150q by [16, §2] and rank(δ_{1}) = [W (E_{6}) : W (D_{5})] = 27, rank(δ_{3}) = [W (E_{6}) : W (A_{1} + A_{4})] = 216. Note that δ_{2} and δ_{1}w_{3} belong to ℤ[Λ_{r}]^{W}. By (3.2),

c_{2}(δ_{1}δ_{3}) = rank(δ_{1})c_{2}(δ_{3}) + rank(δ_{3})c_{2}(δ_{1}) = 27 · 150q + 216 · 6q = 5346q.

As gcd(24, 5346) = 6, we have 6q ∈ Dec(G). On the other hand, c_{2}(δ_{i}) ∈ 6ℤq for all i by [16, §2], hence Dec(G) = 6ℤq. Thus,

Inv^{3}(G, ℚ/ℤ(2))_{ind} = Q(G)/ Dec(G) = (3ℤ/6ℤ)q.

Note that the exponents of the groups Inv^{3}(G)_{dec} and Inv^{3}(G)_{ind} are relatively prime.

Theorem 4.8. Let G be an adjoint group of type E_{6} of inner type. Then

Inv^{3}(G, ℚ/ℤ(2))_{norm} ≃ (F^{×}/F^{×3}) ⊕ (ℤ/2ℤ).

It follows from the computation that the pull-back of the generator of Inv^{3}(G)_{ind} to Inv^{3}(G̃)_{norm} is 3 times the Rost invariant RG̃. This was observed in [10, Proposition 7.2] in the case char(F) ≠ 2.

Type E_{7}

We have Ĉ ≃ ℤ/2ℤ and D = I_{7},

C^{−1} = \frac{1}{2} ( \begin{matrix}4 & 6 & 8 & 6 & 4 & 2 & 4 \\ 6 & 12 & 16 & 12 & 8 & 4 & 8 \\ 8 & 16 & 24 & 18 & 12 & 6 & 12 \\ 6 & 12 & 18 & 15 & 10 & 5 & 9 \\ 4 & 8 & 12 & 10 & 8 & 4 & 6 \\ 2 & 4 & 6 & 5 & 4 & 3 & 3 \\ 4 & 8 & 12 & 9 & 6 & 3 & 7 \end{matrix} )

By Proposition 3.4, Q(G) = Sym^{2}(Λ_{r})^{W} = 4ℤq.

We have c_{2}(δ_{1}) = 36q and c_{2}(δ_{7}) = 12q by [16, §2] and rank(δ_{7}) = [W (E_{7}) : W (E_{6})] = 56. Note that δ_{1} and δ_{7}^{2} belong to ℤ[Λ_{r}]^{W}.

By (3.2),

c_{2}(δ_{7}^{2}) = 2 rank(δ_{7})c_{2}(δ_{7}) = 2 · 56 · 12q = 1344.

As gcd(36, 1344) = 12, we have 12q ∈ Dec(G). On the other hand, c_{2}(δ_{i}) ∈ 12ℤq for all i by [16, §2], hence Dec(G) = 12ℤq. Thus,

Inv^{3}(G, ℚ/ℤ(2))_{ind} = Q(G)/ Dec(G) = (4ℤ/12ℤ)q.

Theorem 4.9. Let G be an adjoint group of type E_{7} of inner type. Then

Inv^{3}(G, ℚ/ℤ(2))_{norm} ≃ (F^{×}/F^{×2}) ⊕ (ℤ/3ℤ).

It follows from the computation that the pull-back of the generator of Inv^{3}(G)_{ind} to Inv^{3}(G̃)_{norm} is 4 times the Rost invariant RG̃. This was observed in [10, Proposition 7.2] in the case char(F) ≠ 3.

Every inner semisimple group of the types G_{2}, F_{4} and E_{8} is simply connected. Then the group Inv^{3}(G, ℚ/ℤ(2))_{norm} is of order 2, 6 and 60, respectively (see [8, Part II]).

Recall that the groups Inv^{3}(G)_{ind} are all the same for all twisted forms of G. This is not the case for Inv^{3}(G̃)_{ind} = Inv^{3}(G̃). Write G̃_{gen} for a “generic” twisted form of G̃ (see [10, §6]). For such groups the Rost number nG̃_{gen} is the largest possible. Their values can be found in [8, Part II].

Theorem 4.10. Let G be an adjoint semisimple group of inner type, G̃ → G a universal cover. Then the map

Inv^{3}(G)_{ind} ≃ Inv^{3}(G_{gen})_{ind} ⟶ Inv^{3}(G̃_{gen})_{ind} = Inv^{3}(G̃_{gen}) = (ℤ/nG̃_{gen}ℤ)RG̃_{gen}

is injective. In the case G is simple, the group Inv^{3}(G)_{ind} is nonzero only in the following cases:

C_{n}, n is divisible by 4: Inv^{3}(G)_{ind} = (ℤ/2ℤ)RG̃,

D_{n}, n is divisible by 4: Inv^{3}(G)_{ind} = (2ℤ/4ℤ)RG̃,

E_{6}: Inv^{3}(G)_{ind} = (3ℤ/6ℤ)RG̃,

E_{7}: Inv^{3}(G)_{ind} = (4ℤ/12ℤ)RG̃.

5. Restriction to the generic maximal torus

Let G be a semisimple group over F and T_{gen} the generic maximal torus of G defined over F (X), where X is the variety of maximal tori in G (see Example 3.1). We can restrict invariants of G to invariant of T_{gen} via the composition

Inv^{n}(G, ℚ/ℤ(j)) ⟶ Inv^{n}(G_{F (X)}, ℚ/ℤ(j)) ⟶^{Res} Inv^{n}(T_{gen}, ℚ/ℤ(j)).

The degree 3 invariants of algebraic tori have been studied in [1].

Suppose that G is quasi-split. Then the character group of T_{gen} is isomorphic to the weight lattice Λ with the Δ-action (see Example 3.1). The exact sequence 0 → Λ → Λ_{w} → Ĉ → 0, Example 3.1, Theorem 3.9 and [1, Theorem 4.3] yield a diagram

[figure: H^{1}(F, Ĉ(1)); Inv^{3}(G, ℚ/ℤ(2))_{norm}; ℤ[Λ]^{Δ}/ Dec(Λ); H^{2}(F (X), T̂_{gen}(1)); Inv^{3}(T_{gen}, ℚ/ℤ(2))_{norm}; ℤ[Λ]^{Δ}/ Dec(Λ).]

Theorem 5.1. Let G be a quasi-split group over a perfect field F, T_{gen} the generic maximal torus. Then the homomorphism

Inv^{n}(G, ℚ/ℤ(j)) ⟶ Inv^{n}(T_{gen}, ℚ/ℤ(j))

is injective, i.e., every invariant of G is determined by its restriction on the generic maximal torus.

Proof. Consider the morphism T → X as in Example 3.1. Let V be a generically free representation of G such that there is an open G-invariant subscheme U ⊂ V and a G-torsor U → U/G. The group scheme T over X acts naturally on U × X. Consider the factor scheme (U × X)/T . In fact, we can view this as a variety as follows. Let T_{0} be a quasi-split maximal torus in G. The Weyl group W of T_{0} acts on (U/T_{0}) × (G/T_{0}) by w(T_{0}u, gT_{0}) = (T_{0}wu, gw^{−1}T_{0}). Then (U × X)/T can be viewed as a factor variety ((U/T_{0}) × (G/T_{0}))/W. Note that the function field of (U × X)/T is isomorphic to the function field of U_{F (X)}/T_{gen} over F (X).

We claim that the natural morphism

f : (U × X)/T ⟶ U/G

is surjective on K-points for any field extension K/F. A K-point of U/G is a G-orbit O ⊂ U defined over K. As F is perfect, by [23, Theorem 11.1], there is a maximal torus T ⊂ G and a T -orbit O′ ⊂ O defined over K. Then the pair (O′, T) determines a point of ((U × X)/T )(K) over O. The claim is proved.

It follows from the claim that the generic fiber of f has a rational point (over F (U/G)). Therefore, the natural homomorphism

(5.1)  H^{n}(F (U/G), ℚ/ℤ(j)) ⟶ H^{n}(F (X)(U_{F (X)}/T_{gen}), ℚ/ℤ(j))

is injective.

Let I ∈ Inv^{n}(G, ℚ/ℤ(j)) be an invariant with trivial restriction on T_{gen}. Let p_{gen} be the generic fiber of p : U → U/G and let q_{gen} be the generic fiber of q : U_{F (X)} → U_{F (X)}/T_{gen}. Then the pull-back of p_{gen} with respect to the field extension F (X)(U_{F (X)}/T_{gen})/F (U/G) is isomorphic to the pull-back of q_{gen} under the change of group homomorphism T_{gen} → G. It follows that

0 = Res(I)(q_{gen}) = I(p_{gen})_{F (X)(U_{F (X)}/T_{gen})}.

As (5.1) is injective, we have I(p_{gen}) = 0 in H^{n}(F (U/G), ℚ/ℤ(j)) and hence I = 0 by [8, Part II, Theorem 3.3] or [1, Theorem 2.2]. □

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Department of Mathematics, University of California, Los Angeles, CA 90095-1555, USA

  E-mail address: merkurev@math.ucla.edu