DEGREE THREE INVARIANTS FOR SEMISIMPLE GROUPS OF TYPES B, C, AND D

SANGHOON BAEK

Abstract. We determine the group of reductive cohomological degree 3 invariants of all split semisimple groups of types B, C, and D. We also present a complete description of the cohomological invariants. As an application, we show that the group of degree 3 unramified cohomology of the classifying space BG is trivial for all split semisimple groups G of types B, C, and D.

1. Introduction

A degree d cohomological invariant of an algebraic group G defined over a field F is a natural transformation of functors

G -torsors → H^{d}

on the category of field extensions over F, where the functor G -torsors takes a field K/F to the set G -torsors(K) of isomorphism classes of G-torsors over K and the functor H^{d} takes K to the Galois cohomology H^{d}(K) = H^{d}(K, ℚ/ℤ(d − 1)). All degree d invariants of G form a group Inv^{d}(G). This notion was introduced by Serre, and since then it has been intensively studied by Merkurjev and Rost for d = 3 [10, 19].

In this paper, we study degree 3 cohomological invariants of split semisimple groups of Dynkin types B, C, and D. Thus from now on we shall focus on degree 3 invariants. Let G be a split reductive group over a field F. An invariant in Inv^{3}(G) is called normalized if it vanishes on trivial G-torsors. Such invariants form a subgroup Inv^{3}(G)_{norm} of Inv^{3}(G), thus Inv^{3}(G) = Inv^{3}(G)_{norm} ⊕ H^{3}(F). A normalized invariant in Inv^{3}(G)_{norm} is called decomposable if it is given by a cup product of a degree 2 invariant with a constant invariant of degree 1. The subgroup of decomposable invariants of degree 3 is denoted by Inv^{3}(G)_{dec}. The quotient group Inv^{3}(G)_{norm}/ Inv^{3}(G)_{dec} is called the group of indecomposable invariants and is denoted by Inv^{3}(G)_{ind}. This group has been completely determined for all split simple groups in [10], [19], [4] and for some semisimple groups in [17], [1], [2], and [15].

Let G be a split semisimple group over F. A strict reductive envelope of G is a split reductive group G_{red} over F such that the derived subgroup of G_{red} is G and the center of G_{red} is a torus. Then, by [18, §10] the restriction map

Inv^{3}(G_{red})_{ind} → Inv^{3}(G)_{ind}

is injective and its image is independent of the choice of a strict reductive envelope G_{red}. This image is called the subgroup of reductive indecomposable invariants of G and is denoted by Inv^{3}(G)_{red}. Recently, this subgroup has been completely computed for all split simple groups in [13] and for all split semisimple groups of type A in [17].

In the present paper, we determine the group of reductive indecomposable invariants of all split semisimple groups of types B, C, and D, which completes the cohomological invariants of classical groups. In particular, if each component of the corresponding root system of type B (respectively, type C) has rank at least 2 (respectively, even rank), then the group of indecomposable invariants is also determined as follows (see Theorem 5.1, Theorem 5.5, Theorem 5.6, and Corollary 5.2):

Theorem 1.1. Let G be an arbitrary split semisimple group of one of the following types: B, C, and D, i.e., G = (∏_{i=1}^{m} Spin_{2n_{i}+1})/μ (n_{i} ≥ 1), (∏_{i=1}^{m} Sp_{2n_{i}})/μ (n_{i} ≥ 1), and (∏_{i=1}^{m} Spin_{2n_{i}})/μ (n_{i} ≥ 3) respectively for some central subgroup μ and m ≥ 1. Let R be the subgroup of Z whose quotient is the character group μ^{∗}, where

Z := ⨁_{i=1}^{m} Z_{i}, Z_{i} = \begin{cases}(ℤ/2ℤ)e_{i} & if G is of type B or C, \\ (ℤ/4ℤ)e_{i} & if G is of type D, n_{i} odd, \\ (ℤ/2ℤ)e_{i,1} ⨁(ℤ/2ℤ)e_{i,2} & if G is of type D, n_{i} even, \end{cases}

denotes the character group of the center of the corresponding simply connected semisimple group.

(1) Assume that G is of type B. Let l = dim R. Then,

Inv^{3}(G)_{red} = (ℤ/2ℤ)^{l−l_{1}−l_{2}},

where l_{1} = dim⟨e_{i} ∈ R | n_{i} ≤ 2⟩, l_{2} = dim⟨e_{i} + e_{j} ∈ R | e_{i}, e_{j} ∉ R, n_{i} = n_{j} = 1⟩. In particular, if n_{i} ≥ 2 for all 1 ≤ i ≤ m, then

Inv^{3}(G)_{ind} = Inv^{3}(G)_{red} = (ℤ/2ℤ)^{l−l_{1}}.

(2) Assume that G is of type C. Let s denote the number of ranks n_{i} divisible by 4 and l = dim (R ∩ (⨁_{4∤n_{i}} Z_{i})). Then,

Inv^{3}(G)_{red} = (ℤ/2ℤ)^{s+l−l_{1}−l_{2}},

where l_{1} = dim⟨e_{i} ∈ R⟩ and l_{2} = dim⟨e_{i} + e_{j} ∈ R | e_{i}, e_{j} ∉ R, n_{i} ≡ n_{j} ≡ 1 mod 2⟩. In particular, if n_{i} ≡ 0 mod 2 for all i, then

Inv^{3}(G)_{ind} = Inv^{3}(G)_{red} = (ℤ/2ℤ)^{s+l−l_{1}}.

(3) Assume that G is of type D. Let

R̄ = \{(r̄_{1}, …, r̄_{m}) ∈ ⨁_{i=1}^{m}(ℤ/2ℤ)ē_{i} | ∑_{i=1}^{m} r_{i} ∈ R\}, where r_{i} = \begin{cases}2r̄_{i}e_{i} & if n_{i} odd, \\ r̄_{i}e_{i,1} + r̄_{i}e_{i,2} & if n_{i} even, \end{cases}

R_{1,i} = R ∩ Z_{i} for odd n_{i}, and R′_{1,i} = R ∩ Z_{i} for even n_{i}. Set

R′ = R̄ ∩ ( ⨁_{4∤n_{i},R′_{1,i},R_{1,i}≠Z_{i}}(ℤ/2ℤ)ē_{i}) with l = dim R′, I_{1} = {i | Z_{i} = R_{1,i} or R′_{1,i}, n_{i} ≠ 3},

I_{2} = {i | R′_{1,i} = 0, 4|n_{i}} ∪ {i | R′_{1,i} = (ℤ/2ℤ)e_{i,1} or (ℤ/2ℤ)e_{i,2}, n_{i} ≥ 6, 4|n_{i}} with s_{i} = |I_{i}|.

Then, we have

Inv^{3}(G)_{red} = (ℤ/2ℤ)^{s_{1}+s_{2}+l−l_{1}−l_{2}}, where

l_{1} = |{i | 4 ∤ n_{i}, R_{1,i} = 2Z_{i} or R′_{1,i} = (ℤ/2ℤ)(e_{i,1} + e_{i,2})}|, l_{2} = dim⟨ē_{i} + ē_{j} | R_{1,i} = R_{1,j} = 0, 2e_{i} + 2e_{j} ∈ R⟩.

For each type of B, C, and D, our main theorem can be restated as follows (see Propositions 6.3, 6.7, 6.13): Assume that F is an algebraically closed field. For type B, let G_{red} = (∏_{i=1}^{m} Γ_{2n_{i}+1})/μ, where Γ_{2n_{i}+1} is the split even Clifford group [12, §23] and let

R → Inv^{3}(G_{red})_{norm}

be the homomorphism given by r ↦ e_{3}(ϕ[r]), where ϕ[r] is the quadratic form defined in Remark 6.2 and e_{3} denotes the Arason invariant. Then, this morphism is surjective and its kernel is the subspace

⟨e_{i}, e_{j} + e_{k} ∈ R | e_{j}, e_{k} ∉ R, n_{i} ≤ 2, n_{j} = n_{k} = 1⟩.

For type C, let G_{red} = (∏_{i=1}^{m} GSp_{2n_{i}})/μ, where GSp_{2n_{i}} is the group of symplectic similitudes [12, §12] and let

⨁_{4 |n_{i}}(ℤ/2ℤ)e_{i} ⨁ (R ∩ (⨁_{4∤n_{i}}(ℤ/2ℤ)e_{i})) → Inv^{3}(G_{red})_{norm}

be the homomorphism given by e_{i} ↦ Δ_{i} for i such that 4|n_{i} and r ↦ e_{3}(ϕ[r]) for r ∈ R ∩ (⨁_{4∤n_{i}}(ℤ/2ℤ)e_{i}), where ϕ[r] is the quadratic form defined in (59) and Δ_{i} is the invariant in (60) induced by the Garibaldi-Parimala-Tignol invariant [11]. Then, this morphism is surjective and its kernel is given by

⟨e_{i}, e_{j} + e_{k} ∈ R | e_{j}, e_{k} ∉ R, n_{j} ≡ n_{k} ≡ 1 mod 2⟩.

For type D, let G_{red} = (∏_{i=1}^{m} Ω_{2n_{i}})/μ, where Ω_{2n_{i}} is the extended Clifford group [12, §13] and let

⨁_{i∈I_{1}∪I_{2}}(ℤ/2ℤ)ē_{i} ⨁ R′ → Inv^{3}(G_{red})_{norm}

be the homomorphism given by ē_{i} ↦ e_{3,i} for i ∈ I_{1}, ē_{i} ↦ Δ′_{i} for i ∈ I_{2}, and r ↦ e_{3}(ϕ[r]) for r ∈ R′, where e_{3,i} denotes the invariant in (73) induced by the Arason invariant, Δ′_{i} denotes the invariant in (74) given by the invariant of PGO_{2n_{i}}^{+} (see [19, Theorem 4.7]), and ϕ[r] is the quadratic form defined in (59). Then, the morphism is surjective, and its kernel is given by

⟨ē_{i}, ē_{j} + ē_{k} ∈ R′ | ē_{j}, ē_{k} ∉ R′, n_{j} ≡ n_{k} ≡ 1 mod 2⟩.

Therefore, our main result (Theorem 1.1) tells us that for all split semisimple groups of types B, C, D there are essentially two types of degree three reductive invariants given by the Arason invariant e_{3} and the Garibaldi-Parimala-Tignol invariant Δ_{i} (and its analogue Δ′_{i}) and no other invariants exist.

An invariant α ∈ Inv^{3}(G) is said to be unramified if for any field extension K/F and any element η ∈ G -torsors(K), its value α(η) is contained in H_{nr}^{3}(K), where H_{nr}^{3}(K) denotes the subgroup in H^{3}(K) of all unramified elements defined by

H_{nr}^{3}(K) = ⋂_{v} Ker ( ∂_{v} : H^{3}(K) → H^{2}(F (v)) )

for all discrete valuations v on K/F and their residue homomorphisms ∂_{v}. The subgroup of all unramified invariant in Inv^{3}(G) will be denoted by Inv_{nr}^{3}(G). By a theorem of Rost, we have an isomorphism

(1)  Inv_{nr}^{3}(G) ≃ H_{nr}^{3}(F (BG)),

where BG is the classifying space of G (see [18], [25]).

A generalized version of Noether’s problem asks whether the classifying space BG of an algebraic group G is stably rational or retract rational (see [6], [16]). A way of detecting non-retract rationality is to use unramified cohomology as the following statement: the classifying space BG is not retract rational if there exists a nonconstant unramified invariant of degree d for some d [16]. In fact, Saltman gave the first counter example over an algebraically closed field to the original Noether’s question by providing certain finite groups which have a non-constant unramified invariant of degree 2 [21]. However, the generalized Noether’s problem is still open for a connected algebraic group over an algebraically closed field.

In [5], Bogomolov showed that connected groups have no nontrivial degree 2 unramified invariants, i.e., Inv_{nr}^{2}(G) = 0 for a connected group G. In [22] and [23], Saltman showed that the group Inv_{nr}^{3}(PGL_{n}) is trivial. Recently, Merkurjev has shown that the group Inv_{nr}^{3}(G) is trivial if G is a split simple group [18] or a split semisimple group of type A [14] over an algebraically field F of characteristic 0.

Using the main theorem above we determine the group of unramified invariants of a split semisimple groups of types B, C, and D (see Theorems 6.5, 6.10, 6.15).

Theorem 1.2. Let G = (∏_{i=1}^{m} Spin_{2n_{i}+1})/μ (n_{i} ≥ 1) or (∏_{i=1}^{m} Sp_{2n_{i}})/μ (n_{i} ≥ 1) or (∏_{i=1}^{m} Spin_{2n_{i}})/μ (n_{i} ≥ 3) defined over an algebraically closed field F of characteristic 0, m ≥ 1, where μ is an arbitrary central subgroup. Then, there are no nontrivial unramified degree 3 invariants for G, i.e., Inv_{nr}^{3}(G) = H_{nr}^{3}(F (BG)) = 0.

This paper is organized as follows. In Section 2 we recall some basic definitions and facts used in the rest of the paper. Sections 3-5 are devoted to the computation of the group of degree 3 invariants of a split semisimple group G of types B, C, and D. In the last section, we present a description of the degree 3 invariants of G and a proof of the second main result. Acknowledgements. I am grateful to Alexander Merkurjev for careful reading and numerous suggestions. I am also grateful to Jean-Pierre Tignol for helpful discussion. This work has been supported by National Research Foundation of Korea (NRF) funded by the Ministry of Science, ICT and Future Planning (2016R1C1B2010037).

2. Cohomological invariants of degree 3

In this section we recall some basic notions concerning degree 3 invariants following [10, 19]. We shall frequently use these in the following sections.

2.1. Invariant quadratic forms. Let G̃ be a split semisimple simply connected group of Dynkin type D, i.e., G̃ = G_{1} × ⋯ × G_{m} for some integer m ≥ 1, where each G_{i} is a split simple simply connected group of type D. Consider the natural action of the Weyl group W = W_{1} × ⋯ W_{m} of G̃ on the weight lattice Λ = Λ_{1} ⊕ ⋯ ⊕ Λ_{m}, where W_{i} (resp. Λ_{i}) is the Weyl group (resp. the weight lattice) of G_{i}. Then, the group of W -invariant quadratic forms S^{2}(Λ)^{W} on Λ, denoted by Q(G̃), is a sum of cyclic groups

Q(G̃) = ℤq_{1} ⊕ ⋯ ⊕ ℤq_{m},

where q_{i} is the normalized Killing form of G_{i} for 1 ≤ i ≤ m.

Consider an arbitrary split semisimple group G of Dynkin type D, i.e., G = G̃/μ, where μ is a central subgroup. Let T be a split maximal torus of G and let T^{∗} be the group of characters of T. Then, the subgroup Q(G) of W -invariant quadratic forms on T^{∗} is given by

(2)  Q(G) = S^{2}(T^{∗}) ∩ Q(G̃).

2.2. Degree 3 invariants. Consider the Chern class map c_{2} : ℤ[T^{∗}] → S^{2}(T^{∗}) defined by c_{2}(∑_{i} e^{λ_{i}}) = ∑_{i<j} λ_{i}λ_{j} [19, §3c], where ℤ[T^{∗}] is the group ring of the maximal torus T in Section 2.1 and λ_{i} ∈ T^{∗}. Since (T^{∗})^{W} = 0, the restriction of c_{2} induces a group homomorphism

(3)  c_{2} : ℤ[T^{∗}]^{W} → Q(G)

We shall write Dec(G) for the image of c_{2} in (3). For λ ∈ T^{∗}, we denote by ρ(λ) = ∑_{χ∈W (λ)} e^{χ}, where W (λ) is the W -orbit of λ. Then, the subgroup Dec(G) is generated by c_{2}(ρ(λ)) = −\frac{1}{2} ∑_{χ∈W (λ)} χ^{2}. By [19, Theorem 3.9], the indecomposable invariants of G is determined by the following exact sequence

0 → Inv^{3}(G)_{dec} → Inv^{3}(G)_{norm} → Q(G)/ Dec(G) → 0.

In particular, if F is algebraically closed, then we have Inv^{3}(G)_{norm} = Q(G)/ Dec(G).

3. The group Q(G) for semisimple groups G of types B, C, D

In the present section, we shall compute the group Q(G) for types B, C, and D.

3.1. Type B. Let G = (∏_{i=1}^{m} Spin_{2n_{i}+1})/μ be an (arbitrary) split semisimple group of type B, m, n_{i} ≥ 1, where μ ≃ (μ_{2})^{k} is a central subgroup for some k ≥ 0. Let T be the split maximal torus of G (i.e., T = (𝔾_{m}^{∑ n_{i}})/μ) and let

(4)  R = {r = (r_{1}, …, r_{m}) ∈ ⨁_{i=1}^{m}(ℤ/2ℤ)e_{i} | f_{p}(r) = 0, 1 ≤ p ≤ k}

be the subgroup of ⨁_{i=1}^{m}(ℤ/2ℤ)e_{i} whose quotient is the character group μ^{∗} for some linear polynomials f_{p} ∈ ℤ/2ℤ[t_{1}, …, t_{m}]. We shall simply write (ℤ/2ℤ)^{m} for ⨁_{i=1}^{m}(ℤ/2ℤ)e_{i}. Consider the following commutative diagram of exact sequences

(5)  [figure: 0; R; (ℤ/2ℤ)^{m}; μ^{∗}; 0; 0; T^{∗}; ∏_{i=1}^{m} ℤ^{n_{i}}; _{μ_{∗}}; 0]

where T^{∗} is the corresponding character group and the middle map ∏_{i=1}^{m} ℤ^{n_{i}} → (ℤ/2ℤ)^{m} is given by

(6)  ∑ a_{i,j}w_{i,j} ↦ (ā_{1,n_{1}}, …, ā_{m,n_{m}})

for 1 ≤ i ≤ m and 1 ≤ j ≤ n_{i}, where w_{i,j} denote the fundamental weights for the ith component of the root system of G. For the rest of this subsection, we simply write a_{i} and w_{i} for a_{i,n_{i}} and w_{i,n_{i}}, respectively. Then, it follows from (5) that

T^{∗} = {∑ a_{i,j}w_{i,j} | f_{p}(a_{1}, …, a_{m}) ≡ 0 mod 2}.

Let I = {1, …, m} and let I_{1} = {i ∈ I | f_{p}(e_{i}) = 0, 1 ≤ p ≤ k}, where {e_{1}, …, e_{m}} denotes the standard basis of ℤ^{m}. We write the relations f_{p}(a_{1}, …, a_{m}) ≡ 0 mod 2 as

(7)  (a_{i_{1}}, …, a_{i_{k}})^{T} = B · (a_{j_{1}}, …, a_{j_{l}})^{T} + (2c_{1}, …, 2c_{k})^{T}

for some distinct i_{1}, …, i_{k}, j_{1}, …, j_{l} such that {i_{1}, …, i_{k}, j_{1}, …, j_{l}} = I\I_{1} and some k × l binary matrix B = (b_{ij}) (i.e., b_{ij} = 0 or 1) with c_{p} ∈ ℤ. Then, we have

∑ a_{i,j}w_{i,j} = ∑_{1≤i≤m,1≤j≤n_{i}−1} a_{i,j}w_{i,j} + ∑_{i∈I_{1}} a_{i}w_{i} + ∑_{p=1}^{k} 2c_{p}w_{i_{p}} + ∑_{s=1}^{l} a_{j_{s}}(w_{j_{s}} + g_{s})

where g_{s} = (w_{i_{1}}, …, w_{i_{k}}) · B_{s} and B_{s} is the s-th column of B, thus we obtain the following ℤ-basis of T^{∗}:

(8)  {w_{i,j}}_{1≤i≤m,1≤j≤n_{i}−1} ∪ {w_{i}}_{i∈I_{1}} ∪ {2w_{i_{p}}}_{1≤p≤k} ∪ {w_{j_{s}} + g_{s}}_{1≤s≤l}.

Let v_{p} = 2w_{i_{p}} and h_{p}(t_{1}, …, t_{l}) = b_{p1}t_{1} + ⋯ + b_{pl}t_{l} ∈ ℤ/2ℤ[t_{1}, …, t_{l}] for 1 ≤ p ≤ k. Since the group Q(G̃) is generated by the normalized Killing forms

q_{i} = \begin{cases}2w_{i}^{2} − 2w_{i,n_{i}−1}w_{i} − ∑_{j=1}^{n_{i}−2} w_{i,j}w_{i,j+1} + ∑_{j=1}^{n_{i}−1} w_{i,j}^{2} & if n_{i} ≥ 1, \\ w_{i}^{2} & if n_{i} = 1 \end{cases}

for all 1 ≤ i ≤ m, any element of Q(G) is of the form q = ∑_{i=1}^{m} d_{i}q_{i} for some d_{i} ∈ ℤ. Therefore, with respect to the basis (8) we have

q = q′ + \frac{1}{4} ∑_{p=1}^{k} v_{p}^{2}[δ_{i_{p}}d_{i_{p}} + h_{p}(δ_{j_{1}}d_{j_{1}}, …, δ_{j_{l}}d_{j_{l}})] + \frac{1}{2} ∑_{1≤i<j≤k} v_{i}v_{j}h_{i}(δ_{j_{1}}d_{j_{1}}b_{j_{1}}, …, δ_{j_{l}}d_{j_{l}}b_{j_{l}})

for some quadratic form q′ with integer coefficients, where

δ_{i} = \begin{cases}2 & if n_{i} ≥ 2 with i ∈ I\I_{1}, \\ 1 & if n_{i} = 1 with i ∈ I\I_{1}. \end{cases}

Hence, by (2) we obtain q = ∑_{i=1}^{m} d_{i}q_{i} ∈ Q(G) if and only if

(9)  δ_{i_{p}}d_{i_{p}} + h_{p}(δ_{j_{1}}d_{j_{1}}, …, δ_{j_{l}}d_{j_{l}}) ≡ 0 mod 4

and

(10)  h_{p}(δ_{j_{1}}d_{j_{1}}b_{j_{1}}, …, δ_{j_{l}}d_{j_{l}}b_{j_{l}}) ≡ 0 mod 2

for all 1 ≤ p ≤ k. In particular, since two systems of equations {f_{p}(t_{1}, …, t_{m})} and {t_{i_{p}} + h_{p}(t_{j_{1}}, …, t_{j_{l}})} are equivalent we replace the condition (9) by

(11)  f_{p}(δ_{1}d_{1}, …, δ_{m}d_{m}) ≡ 0 mod 4,

where we set δ_{i} = 2 for i ∈ I_{1}.

Equivalently, we can compute Q(G) with respect to a basis of R as follows. Let

(12)  R_{1} = ⟨e_{i} | e_{i} ∈ R⟩ and R_{2} = ⟨e_{i} + e_{j} | e_{i} + e_{j} ∈ R, e_{i}, e_{j} ∉ R_{1}⟩

be the subspaces of R. We first choose {w_{i}}_{i∈I_{1}} as a part of basis of T^{∗}. Then, for the remaining part of a basis of T^{∗} we write a given basis of R as

(13)  (e_{j_{1}}, …, e_{j_{l}})^{T} = C(e_{i_{1}}, …, e_{i_{k}})^{T}

for some i_{1}, …, i_{k}, j_{1}, …, j_{l} with {i_{1}, …, i_{k}, j_{1}, …, j_{l}} = I\I_{1} and some l × k binary matrix C such that all basis elements of the form e_{i} + e_{j} in R_{2} is a part of (13). Then, we have the same ℤ-basis of T^{∗} as in (8) by replacing g_{s} in (8) with g_{s} = C_{s} · (w_{i_{1}}, …, w_{i_{k}}), where C_{s} is the s-th row of C. The rest of the computation is the same as in the previous one.

In particular, if either R = R_{1} ⊕ R_{2} or n_{i} ≥ 2 for all 1 ≤ i ≤ m, then the condition (10) becomes trivial, thus

Proposition 3.1. Let G = (∏_{i=1}^{m} Spin_{2n_{i}+1})/μ, m, n_{i} ≥ 1, where μ ≃ (μ_{2})^{k} is a central subgroup for some k ≥ 0. Let R = {r ∈ (ℤ/2ℤ)^{m} | f_{p}(r) = 0, 1 ≤ p ≤ k} be the subgroup of (μ_{2}^{m})^{∗} whose quotient is the character group μ^{∗} for some linear polynomials f_{j} ∈ ℤ/2ℤ[t_{1}, …, t_{m}]. Assume that either n_{i} ≥ 2 for all i or R = R_{1}⊕R_{2}, where R_{1} and R_{2} are the subgroups of R defined in (12). Then, we have

Q(G) = {∑_{i=1}^{m} d_{i}q_{i} | f_{p}(δ_{1}d_{1}, …, δ_{m}d_{m}) ≡ 0 mod 4}.

3.2. Type C. Let G = (∏_{i=1}^{m} Sp_{2n_{i}})/μ be a split semisimple group of type C, where m, n_{i} ≥ 1 and μ ≃ (μ_{2})^{k} is a central subgroup for some k ≥ 0. Let T be the split maximal torus of G and let R be the subgroup of (ℤ/2ℤ)^{m} as in (4). Then, we have the same commutative diagram (5), replacing the middle vertical map (6) by

∑ a_{i,j}e_{i,j} ↦ (∑_{j=1}^{n_{1}} ā_{1,j}, …, ∑_{j=1}^{n_{m}} ā_{m,j}),

where e_{i,j} denote the standard basis for the ith component of ∏_{i=1}^{m} ℤ^{n_{i}}. Then, by (5) we have

(14)  T^{∗} = {∑ a_{i,j}e_{i,j} | f_{p}(∑_{j=1}^{n_{1}} a_{1,j}, …, ∑_{j=1}^{n_{m}} a_{m,j}) ≡ 0 mod 2}.

We simply write e_{i} for e_{i,1}. Let e′_{i,j} = e_{i,j} − e_{i} for all 1 ≤ i ≤ m and 2 ≤ j ≤ n_{i} and let a_{i} = ∑_{j=1}^{n_{i}} a_{i,j}. Then, we apply the same argument as in type B so that we have the following ℤ-basis of T^{∗}

(15)  {e′_{i,j}}_{1≤i≤m, 2≤j≤n_{i}} ∪ {e_{i}}_{i∈I_{1}} ∪ {2e_{i_{p}}}_{1≤p≤k} ∪ {e_{j_{s}} + g_{s}}_{1≤s≤l},

where B is the binary matrix as in (7) and g_{s} = (e_{i_{1}}, …, e_{i_{k}}) · B_{s}.

Let v_{p} = 2e_{i_{p}} and let h_{p} be the polynomial defined as in type B. Since the normalized Killing forms are given by

q_{i} = e_{i,1}^{2} + ⋯ + e_{i,n_{i}}^{2},

for any q ∈ Q(G) there exist d_{i} ∈ ℤ such that q = ∑_{i=1}^{m} d_{i}q_{i}, thus with respect to the basis (15) we have

q = q′ + \frac{1}{4} ∑_{p=1}^{k} v_{p}^{2}[n_{i_{p}}d_{i_{p}} +h_{p}(n_{j_{1}}d_{j_{1}}, …, n_{j_{l}}d_{j_{l}})]+ \frac{1}{2} ∑_{1≤i<j≤k} v_{i}v_{j}h_{i}(n_{j_{1}}d_{j_{1}}b_{j_{1}}, …, n_{j_{l}}d_{j_{l}}b_{j_{l}})

for some quadratic form q′ with integer coefficients. Therefore, by the same argument as in type B we have q = ∑_{i=1}^{m} d_{i}q_{i} ∈ Q(G) if and only if (16)

h_{p}(n_{j_{1}}d_{j_{1}}b_{j_{1}}, …, n_{j_{l}}d_{j_{l}}b_{j_{l}}) ≡ 0 mod 2 and f_{p}(δ_{1}n_{1}d_{1}, …, δ_{m}n_{m}d_{m}) ≡ 0 mod 4

for all 1 ≤ p ≤ k, where

δ_{i} = \begin{cases}1 & if i ∈ I\I_{1}, \\ \frac{2}{n_{i}} & if i ∈ I_{1}. \end{cases}

Similar to the case of type B, if R = R_{1} ⊕ R_{2} or n_{i} is even for all 1 ≤ i ≤ m, then the first condition in (16) becomes obvious, thus

Proposition 3.2. Let G = (∏_{i=1}^{m} Sp_{2n_{i}})/μ, m, n_{i} ≥ 1, where μ ≃ (μ_{2})^{k} is a central subgroup for some k ≥ 0. Let R, R_{1}, and R_{2} be the groups as in (4) and (12). Assume that either n_{i} is even for all i or R = R_{1} ⊕ R_{2}. Then, we have

Q(G) = {∑_{i=1}^{m} d_{i}q_{i} | f_{p}(δ_{1}n_{1}d_{1}, …, δ_{m}n_{m}d_{m}) ≡ 0 mod 4}.

3.3. Type D. Let G = (∏_{i=1}^{m} Spin_{2n_{i}})/μ be a split semisimple group of type D, m ≥ 1, n_{i} ≥ 3, where μ ≃ (μ_{2})^{k_{1}} ×(μ_{4})^{k_{2}} is a subgroup of the center Z(∏_{i=1}^{m} Spin_{2n_{i}}) for some k_{1}, k_{2} ≥ 0. We shall denote the character group Z(∏_{i=1}^{m} Spin_{2n_{i}})^{∗} by

(17)  Z := ⨁_{i=1}^{m} Z_{i}, where Z_{i} = \begin{cases}(ℤ/4ℤ)e_{i} & if n_{i} odd, \\ (ℤ/2ℤ)e_{i,1} ⨁(ℤ/2ℤ)e_{i,2} & if n_{i} even. \end{cases}

Let T be the split maximal torus of G and let

R = {r ∈ Z | f_{p}(r) = 0, 1 ≤ p ≤ k}

be the subgroup of Z such that μ^{∗} ≃ Z/R for some linear polynomials f_{1}, …, f_{k} ∈ ℤ/4ℤ[T_{1}, …, T_{m}] with k = k_{1} + k_{2}, where T_{i} denotes a 2-tuple (t_{i1}, t_{i2}) of variables (resp. a variable t_{i}) if n_{i} is even (resp. odd) and the coefficients of t_{i1} and t_{i2} in f_{p} are either 0 or 2. Then, we have the same diagram (5), replacing the middle vertical map (6) by ∏_{i=1}^{m} ℤ^{n_{i}} → Z,

(18)  ∑_{j=1}^{n_{i}} a_{i,j}w_{i,j} ↦ A_{i} := \begin{cases}(\overline{a_{i,n_{i}−1} − a_{i,n_{i}} + 2S_{i}})e_{i} & if n_{i} odd, \\ (\overline{a_{i,n_{i}−1} + S_{i}})e_{i1} + (\overline{a_{i,n_{i}} + S_{i}})e_{i2} & if n_{i} even, \end{cases}

where S_{i} = ∑_{j=1}^{[(n_{i}−1)/2]} a_{i,2j−1} and w_{i,j} denote the fundamental weights for the ith component of the root system of G. Therefore, by (5) we have

(19)  T^{∗} = {∑ a_{i,j}w_{i,j} | f_{p}(∑_{i=1}^{m} A_{i}) = 0, 1 ≤ p ≤ k}.

Let I′_{1} = {i ∈ I | f_{p}(e_{i}) = 0 or f_{p}(e_{i,1}) = f_{p}(e_{i,2}) = 0 for all 1 ≤ p ≤ k} and I′ = I\I′_{1}. In view of the argument in the case of type B we may assume that each relation f_{p}(∑_{i=1}^{m} A_{i}) = 0 can be written as

δ_{p}a_{p} = b_{p} + 4c_{p}, where b_{p} = \begin{cases}δ_{p}a_{p} + f_{p}(∑_{i=1}^{m} A_{i}) & if a_{p} = a_{i,n_{i}} with odd n_{i}, \\ δ_{p}a_{p} − f_{p}(∑_{i=1}^{m} A_{i}) & otherwise, \end{cases}

for some distinct a_{p} ∈ {a_{i,n_{i}−1}, a_{i,n_{i}} | i ∈ I′} with δ_{p} ∈ {1, 2} and c_{p} ∈ ℤ such that the terms a_{1}, …, a_{k} do not appear in b_{1}, …, b_{k} and each coefficient of a_{i,l} in b_{p} is divisible by δ_{p}.

Let W_{1} = {w_{i,2j−1} | i ∈ I′, 1 ≤ j ≤ [(n_{i} − 1)/2]} ∪ {w_{i,n_{i}−1}, w_{i,n_{i}} | i ∈ I′}. We simply write w_{p} ∈ W_{1} for w_{i,n_{i}−1} (resp. w_{i,n_{i}}) if a_{p} = a_{i,n_{i}−1} (resp. a_{p} = a_{i,n_{i}}). Set

g_{i,l} = s_{1}(i, l)w_{1} + ⋯ + s_{k}(i, l)w_{k} and W′ = W_{1}\{w_{1}, …, w_{k}},

where s_{p}(i, l) denotes the coefficient of a_{i,l} in b_{p}/δ_{p}. Then, we obtain the following ℤ-basis of T^{∗}:

(20)  \{w_{i,j}\}_{i∈I_{1},∀j} ∪ \{w_{i,2j}\}_{i∈I′,1≤j≤[ \frac{n_{i}−2}{2} ]} ∪ \{\frac{4}{δ_{p}} w_{p}\}_{1≤p≤k} ∪ \{w_{i,l} + g_{i,l}\}_{w_{i,l}∈W′}.

Let v_{p} = \frac{4}{δ_{p}} w_{p} and v_{i,l} = w_{i,l} + g_{i,l}. Assume that for each p, w_{p} is a fundamental weight for the i_{p}-th component of the root system of G. As the normalized Killing forms are given by

q_{i} = ( ∑_{j=1}^{n_{i}} w_{i,j}^{2}) − (w_{i,n_{i}−2}w_{i,n_{i}} + ∑_{j=1}^{n_{i}−2} w_{i,j}w_{i,j+1}),

for any q ∈ Q(G) there exist d_{i} ∈ ℤ such that q = ∑_{i=1}^{m} d_{i}q_{i}. Hence, with respect to the basis (20) we obtain

q =q′ + \frac{1}{16} ∑_{p=1}^{k} v_{p}^{2}δ_{p}^{2}[d_{i_{p}} + ∑_{w_{i,l}∈W′} d_{i}s_{p}(i, l)^{2}] + \frac{1}{8} ∑_{1≤p<u≤k} v_{p}v_{u}δ_{p}δ_{u}[ ∑_{w_{i,l}∈W′} d_{i}s_{p}(i, l)s_{u}(i, l)]

− \frac{1}{2} ∑_{p=1}^{k} v_{p}δ_{p}[ ∑_{w_{i,l}∈W′} v_{il}d_{i}s_{p}(i, l)]

for some quadratic form q′ with integer coefficients. Hence, q = ∑_{i=1}^{m} d_{i}q_{i} ∈ Q(G) if and only if

δ_{p}^{2}[d_{i_{p}} + ∑_{w_{i,l}∈W′} d_{i}s_{p}(i, l)^{2}] ≡ 0 mod 16, ∑_{w_{i,l}∈W′} d_{i}δ_{p}δ_{u}s_{p}(i, l)s_{u}(i, l) ≡ 0 mod 8, (21)

and d_{i}δ_{p}s_{p}(i, l) ≡ 0 mod 2 (22)

for all 1 ≤ p ≤ k, 1 ≤ p < u ≤ k, and all (i, l) such that w_{i,l} ∈ W′.

Let c_{i,1}(p), c_{i,2}(p), c_{i}(p) denote the coefficients of t_{i,1}, t_{i,2}, t_{i} in f_{p}, respectively. Note that c_{i,1}(p) and c_{i,2}(p) are either 0 or 2. Since

δ_{p}^{2} + ∑_{l} δ_{p}^{2}s_{p}(i_{p}, l)^{2} = ∑_{l} δ_{p}^{2}s_{p}(i, l)^{2} = \begin{cases}8 & if c_{i}(p) = 2 or c_{i,1}(p) + c_{i,2}(p) = 4, \\ 2n_{i} & if c_{i}(p) = ±1 or c_{i,1}(p) + c_{i,2}(p) = 2 \end{cases}

for all p and i ≠ i_{p}, where the sums range over all l such that w_{i,l} ∈ W′, the first equation in (21) is equivalent to the following equation

(23)  f_{p}(T_{1}, …, T_{m}) ≡ 0 mod 8, where t_{i} = \begin{cases}±n_{i}d_{i} & if c_{i}(p) = ±1, \\ 2d_{i} & if c_{i}(p) = 2, \end{cases}

t_{i,1} = \begin{cases}\frac{n_{i}d_{i}}{2} & if c_{i,1}(p) = 2, c_{i,2}(p) = 0, \\ d_{i} & if c_{i,1}(p) + c_{i,2}(p) = 4, \end{cases} and t_{i,2} = \begin{cases}\frac{n_{i}d_{i}}{2} & if c_{i,1}(p) = 0, c_{i,2}(p) = 2, \\ d_{i} & if c_{i,1}(p) + c_{i,2}(p) = 4 \end{cases}

for all i ∈ I′ and we set t_{i} = 4d_{i}, t_{i1} = t_{i2} = 2d_{i} for all i ∈ I′_{1}. Since we have

∑_{l} s_{p}(i, l)s_{u}(i, l) ≡ \begin{cases}±2n_{i} mod 8 & if c_{i}(p)c_{i}(u) ≡ ±1 mod 4, \\ 4  mod 8 & if c_{i}(p)c_{i}(u) ≡ 2 mod 4, \\ 0  mod 8 & otherwise \end{cases}

for all 1 ≤ p < u ≤ k such that δ_{p} = δ_{u} = 1, where the sum ranges over all l such that w_{i,l} ∈ W′, the second equation in (21) is equivalent to

(24)  ∑_{{i∈I′|c_{i}(p)c_{i}(u)≡±1 mod 4}} 2d_{i} + ∑_{{i∈I′|c_{i}(p)c_{i}(u)≡2 mod 4}} 4d_{i} ≡ 0 mod 8

if δ_{p} = δ_{u} = 1 and

4 ∑_{i∈I′′} d_{i} ≡ 0 mod 8

for some subset I′′ of I′ otherwise.

4. The subgroup Dec(G) for semisimple groups G of types B, C, D

In this section we will compute the subgroup Dec(G) of decomposable elements of G for types B, C, and D. In this section we shall denote by T and T^{∗} the maximal split torus of G and its character group, respectively and we denote by Λ and Λ_{r} the weight lattice and the root lattice of G, respectively. The Weyl group of G will be denoted by W.

4.1. Type B. Consider a split semisimple group G = (∏_{i=1}^{m} Spin_{2n_{i}+1})/μ of type B, where m, n_{i} ≥ 1 and μ ≃ (μ_{2})^{k} is a central subgroup for some k ≥ 0. Let I = {1, …, m}.

We first consider the case where G is simply connected (i.e. G = G̃), equivalently k = 0. Since

(25)  Dec(G_{1} × G_{2}) = Dec(G_{1}) × Dec(G_{2})

for any two semisimple groups G_{1} and G_{2}, it suffices to compute Dec(Spin_{2n+1}). Observe that Dec(Spin_{3}) = ℤq as c_{2}(ρ(w_{1})) = −q and Dec(Spin_{5}) = ℤq as c_{2}(ρ(w_{2})) = −q. Similarly, c_{2}(ρ(w_{1})) = −2q ∈ Dec(Spin_{2n+1}) for any n ≥ 2. As the Weyl group of Spin_{2n+1} contains a normal subgroup (ℤ/2ℤ)^{n} generated by sign switching, we see that 2 | c_{2}(ρ(λ)) for any λ ∈ Λ (c.f. [10, Part II, §13]), thus Dec(Spin_{2n+1}) = 2ℤq. Therefore,

(26)  Dec(G̃) = δ′_{1}ℤq_{1} ⊕ ⋯ ⊕ δ′_{m}ℤq_{m}, where δ′_{i} = \begin{cases}2 & if n_{i} ≥ 3, \\ 1 & if n_{i} = 1, 2. \end{cases}

Now we assume that G is adjoint (i.e. G = Ḡ), equivalently, k = m. Then, Dec(O_{3}^{+}) = 4ℤq as c_{2}(ρ(2w_{1})) = −4q. Similarly, by the same argument as in the simply connected case, we see that Dec(O_{2n+1}^{+}) = 2ℤq for n ≥ 2 (see [19, Theorem 4.5]). Hence,

(27)  Dec(Ḡ) = δ′′_{1}ℤq_{1} ⊕ ⋯ ⊕ δ′′_{m}ℤq_{m}, where δ′′_{i} = \begin{cases}2 & if n_{i} ≥ 2, \\ 4 & if n_{i} = 1. \end{cases}

In general, we show that the subgroup Dec(G) is determined by certain subgroups of R introduced in Section 3.

Proposition 4.1. Let G = (∏_{i=1}^{m} Spin_{2n_{i}+1})/μ, m, n_{i} ≥ 1, where μ is a central subgroup. Let R be the subgroup of (μ_{2}^{m})^{∗} = (ℤ/2ℤ)^{m} such that μ^{∗} = (μ_{2}^{m})^{∗}/R. Let R′_{1} = ⟨e_{i} ∈ R | n_{i} ≤ 2⟩ and R′_{2} = ⟨e_{i} + e_{j} ∈ R | e_{i}, e_{j} ∉ R, n_{i} = n_{j} = 1⟩ be two subspaces of R with dim R′_{1} = l_{1} and dim R′_{2} = l_{2}. Then,

(28)  Dec(G) = ( ⨁_{e_{i}∈R′_{1}} ℤq_{i}) ⊕ ( ⨁_{n_{i}≥2, e_{i}∉R′_{1}} 2ℤq_{i}) ⊕ (⨁_{r=1}^{l_{2}} 2ℤq′_{r}) ⊕ (⨁_{s=1}^{l_{3}} 4ℤq′′_{s}),

where l_{3} = m − l_{1} − l_{2} − |{i | n_{i} ≥ 2, e_{i} ∉ R′_{1}}| and q′_{r} (resp. q′′_{s}) is of the form q_{i} + q_{j} (resp. q_{i}) for some i, j such that ⟨q′_{r}, q′′_{s} | 1 ≤ r ≤ l_{2}, 1 ≤ s ≤ l_{3}⟩ = ⟨q_{i} | n_{i} = 1, e_{i} ∉ R′_{1}⟩ over ℤ.

Proof. It follows from (26) and (27) that we have

(29)  δ′′_{1}ℤq_{1} ⊕ ⋯ ⊕ δ′′_{m}ℤq_{m} ⊆ Dec(G) ⊆ δ′_{1}ℤq_{1} ⊕ ⋯ ⊕ δ′_{m}ℤq_{m}.

By a simple computation, we obtain

(30)  − c_{2}(ρ(χ)) = \begin{cases}a_{i}^{2}q_{i} & if χ = a_{i}w_{i,1}, n_{i} = 1, \\ 2(a_{i}^{2}q_{i} + a_{j}^{2}q_{j}) & if χ = a_{i}w_{i,1} + a_{j}w_{j,1}, n_{i} = n_{j} = 1 \end{cases}

for any nonzero integers a_{i}, a_{j} and

(31)  − c_{2}(ρ(χ)) = (2a_{i,1}^{2} + a_{i,2}^{2} + 2a_{i,1}a_{i,2})q_{i} if χ = a_{i,1}w_{i,1} + a_{i,2}w_{i,2}, n_{i} = 2

for any integers a_{i,1}, a_{i,2}. Let us denote the right hand side of equation (28) by D. We write D = ⨁ D_{u}, where D_{u} denotes u-th direct summand of D for 1 ≤ u ≤ 4. First, we show that D ⊆ Dec(G). If e_{i} ∈ R′_{1}, then by (8) we have w_{i,1}, w_{i,2} ∈ T^{∗}, thus by (30) and (31) D_{1} ⊆ Dec(G). Similarly, if e_{i} + e_{j} ∈ R, then by (8), w_{i,1} + w_{j,1} ∈ T^{∗}, thus by (30) D_{3} ⊆ Dec(G). Finally, it follows from (29) that D_{2} ⊕ D_{4} ⊆ Dec(G).

On the other hand, a character λ in the weight lattice Λ = ⨁_{i=1}^{m} Λ_{i} of G can be written as

(32)  λ = λ_{i_{1}} + ⋯ + λ_{i_{t}} = ∑_{j∈J} λ_{i_{j}} + ∑_{j∈K} λ_{i_{j}}

for some nonzero characters λ_{i_{j}} ∈ Λ_{i_{j}} and some subsets J = {1 ≤ j ≤ t | n_{i_{j}} = 1} and K = {1 ≤ j ≤ t | n_{i_{j}} ≥ 2} of I. We show that c_{2}(ρ(λ)) ∈ D for all λ ∈ T^{∗}. First, assume that t = 1, i.e., λ = a_{i,1}w_{i,1} + ⋯ + a_{i,n_{i}}w_{i,n_{i}} for some i and a_{i,1}, …, a_{i,n_{i}} ∈ ℤ. If a_{i,n_{i}} is even, then λ ∈ (Λ_{i})_{r}, thus by (27) we have c_{2}(ρ(λ)) ∈ D_{2} ⊕ D_{4}. Otherwise, as λ ∈ T^{∗} is equivalent to e_{i} ∈ R, by (26) we get c_{2}(ρ(λ)) ∈ D_{1} ⊕ D_{2}.

Now we assume that t = 2 and n_{i_{1}} = n_{i_{2}} = 1, i.e., λ = a_{i}w_{i,1} + a_{j}w_{j,1} for some i, j and a_{i}, a_{j} ∈ ℤ\{0} with n_{i} = n_{j} = 1. If both a_{i} and a_{j} are even, then λ ∈ (Λ_{i})_{r}⊕(Λ_{j})_{r}, so c_{2}(ρ(λ)) ∈ D_{3} ⊕ D_{4}. If a_{i} is even and a_{j} is odd, then as λ ∈ T^{∗} if and only if e_{j} ∈ R′_{1}, we get c_{2}(ρ(λ)) ∈ D_{1} ⊕ D_{3} ⊕ D_{4}. Similarly, if both a_{i} and a_{j} are odd, then by (30) we have c_{2}(ρ(λ)) ∈ D_{3}.

Finally, assume that either t ≥ 3 or t = 2 with n_{i_{1}}n_{i_{2}} ≠ 1. Then, by the action of the normal subgroups (ℤ/2ℤ)^{n_{i}} of the Weyl group generated by sign switching, we see that the coefficient at each e_{i_{j},l} in the expansion of c_{2}(ρ(λ)) is divisible by 4 and 2 for j ∈ J and j ∈ K, respectively, i.e.,

(33)  c_{2}(ρ(λ)) = 4(∑_{j∈J} a_{j}q_{j}) + 2(∑_{j∈K} b_{j}q_{j})

for some a_{i}, b_{i} ∈ ℤ. Hence, c_{2}(ρ(λ)) ∈ D, i.e., Dec(G) ⊆ D. □

4.2. Type C. Let G = (∏_{i=1}^{m} Sp_{2n_{i}})/μ be a split semisimple group of type C, m, n_{i} ≥ 1, where μ is a central subgroup. As c_{2}(ρ(e_{1})) = −q, we have Dec(Sp_{2n}) = ℤq. Similarly, as c_{2}(ρ(2e_{1})) = −4q and c_{2}(ρ(e_{1} + e_{2})) = −2(n − 1)q, we have \frac{4}{gcd(2,n)}q ∈ Dec(PGSp_{2n}). Moreover, since the Weyl group of Sp_{2n} contains a normal subgroup (ℤ/2ℤ)^{n} generated by sign switching, we see that \frac{4}{gcd(2,n)} | c_{2}(ρ(λ)) for any λ ∈ Λ_{r} (c.f. [10, Part II, §14]), thus Dec(PGSp_{2n}) = \frac{4}{gcd(2,n)}ℤq (see [19, §4b]). Therefore, by (25) we have

(34) δ′′_{1}ℤq_{1} ⊕⋯⊕δ′′_{m}ℤq_{m} ⊆ Dec(G) ⊆ ℤq_{1} ⊕⋯⊕ℤq_{m}, where δ′′_{i} = \begin{cases}4 & if n_{i} odd, \\ 2 & if n_{i} even. \end{cases}

Similar to the case of type B, we determine the subgroup Dec(G) for type C.

Proposition 4.2. Let G = (∏_{i=1}^{m} Sp_{2n_{i}})/μ, m, n_{i} ≥ 1, where μ ≃ (μ_{2})^{k} is a central subgroup. Let R be the subgroup of (μ_{2}^{m})^{∗} = (ℤ/2ℤ)^{m} such that μ^{∗} = (μ_{2}^{m})^{∗}/R. Let R′′_{2} = ⟨e_{i} + e_{j} ∈ R | e_{i}, e_{j} ∉ R, n_{i} ≡ n_{j} ≡ 1 mod 2⟩ be a subspace of R with dim R′′_{2} = l_{2}. Then,

(35)  Dec(G) = (⨁_{e_{i}∈R} ℤq_{i}) ⊕ (  ⨁_{n_{i}≡0 mod 2, e_{i}∉R} 2ℤq_{i}) ⊕ (⨁_{r=1}^{l_{2}} 2ℤq′_{r}) ⊕ (⨁_{s=1}^{l_{3}} 4ℤq′′_{s}),

where l_{3} = |{i | n_{i} ≡ 1 mod 2, e_{i} ∉ R}| − l_{2} and q′_{r} (resp. q′′_{s}) is of the form q_{i} + q_{j} (resp. q_{i}) for some i, j such that ⟨q′_{r}, q′′_{s} | 1 ≤ r ≤ l_{2}, 1 ≤ s ≤ l_{3}⟩ = ⟨q_{i} | n_{i} ≡ 1 mod 2, e_{i} ∉ R⟩ over ℤ.

Proof. Let T be the split maximal torus of G. Then, by (14) we have

(36)  T^{∗} = {∑ a_{i,j}e′_{i,j} + ∑ a_{i}e_{i,1} | f_{p}(a_{1}, …, a_{m}) ≡ 0 mod 2},

where e′_{i,j} = e_{i,j} − e_{i,1} for all 1 ≤ i ≤ m and 2 ≤ j ≤ n_{i}. First note that we have

(37)  − c_{2}(ρ(χ)) = \begin{cases}a_{i}^{2}q_{i} & if χ ∈ W (a_{i}e_{i,1}), \\ 2(n_{j}a_{i}^{2}q_{i} + n_{i}a_{j}^{2}q_{j}) & if χ ∈ W (a_{i}e_{i,1} + a_{j}e_{j,1}) \end{cases}

for any nonzero integers a_{i} and a_{j}. We shall denote by D the right hand side of equation (35) and write D = ⨁ D_{u}, where D_{u} denotes u-th direct summand of D for 1 ≤ u ≤ 4. If e_{i} ∈ R, then by (36) we get e_{i,1} ∈ T^{∗}, thus by (37) D_{1} ⊆ Dec(G). Similarly, by (34) we have D_{2} ⊕ D_{4} ⊆ Dec(G). Let e_{i} + e_{j} ∈ R′′_{2}. Then, by (36) we have e_{i,1} + e_{j,1} ∈ T^{∗}. As both n_{i} and n_{j} are odd, by (37) we get 2q_{i} + 2q_{j} ∈ Dec(G), i.e., D_{2} ⊆ Dec(G). Therefore, we get D ⊆ Dec(G).

Conversely, we shall now show that c_{2}(ρ(λ)) ∈ D for all λ ∈ T^{∗}. Let λ be a character written as in (32) for some subsets

(38) J = {1 ≤ j ≤ t | n_{i_{j}} ≡ 1 mod 2} and K = {1 ≤ j ≤ t | n_{i_{j}} ≡ 0 mod 2}

of I. For each λ_{i} = a_{i,1}e_{i,1} + ⋯ + a_{i,n_{i}}e_{i,n_{i}} ∈ Λ_{i} we shall denote by |λ_{i}| the number of nonzero coefficients in λ_{i}. We first assume that t = 1, i.e., λ = a_{i,1}w_{i,1} + ⋯ + a_{i,n_{i}}e_{i,n_{i}} for some i and a_{i,1}, …, a_{i,n_{i}} ∈ ℤ. Let a_{i} = a_{i,1} + ⋯ + a_{i,n_{i}}. By the same argument as in the proof of Proposition 4.1, we have c_{2}(ρ(λ)) ∈ D_{2} ⊕ D_{4} (resp. c_{2}(ρ(λ)) ∈ D_{1}) if a_{i} is even (resp. odd). Now we assume that t = 2 with |λ_{i_{1}}| + |λ_{i_{2}}| = 2, i.e., λ = a_{i}e_{i,1} + a_{j}e_{j,1} for some i, j and a_{i}, a_{j} ∈ ℤ\{0}. Then, by the same argument as in the proof of Proposition 4.1 we see from (37) that c_{2}(ρ(λ)) ∈ D.

Assume that either t ≥ 3 or t = 2 with |λ_{i_{1}}| + |λ_{i_{2}}| ≥ 3. Then, as before it follows from the action of the normal subgroups (ℤ/2ℤ)^{n_{i}} of W that

c_{2}(ρ(λ)) = 4(∑_{i∈J} a_{i}q_{i}) + 2(∑_{i∈K} b_{i}q_{i})

for some a_{i}, b_{i} ∈ ℤ. Therefore, we get c_{2}(ρ(λ)) ∈ D, thus Dec(G) ⊆ D. □

4.3. Type D. Let G = (∏_{i=1}^{m} Spin_{2n_{i}})/μ be a split semisimple group of type D, where m ≥ 1, n_{i} ≥ 3 and μ is a central subgroup. Consider the case when G is simple (i.e., m = 1 and n_{1} = n). First of all, as

(39) c_{2}(ρ(ω_{1})) = −2q, c_{2}(ρ(2ω_{1})) = −8q, c_{2}(ρ(ω_{2})) = \begin{cases}−4(n − 1)q & if n ≥ 4, \\ −q & if n = 3, \end{cases}

we have 2ℤq ⊆ Dec(Spin_{2n}) for n ≥ 4, Dec(Spin_{6}) = ℤq, \frac{8}{gcd(2,n)}ℤq ⊆ Dec(PGO_{2n}^{+}). On the other hand, as the Weyl group of Spin_{2n} contains a normal subgroup (ℤ/2ℤ)^{n−1} generated by sign switching of even number of coordinates, we see that 2 | c_{2}(ρ(λ)) for any λ ∈ Λ with n ≥ 4 and \frac{8}{gcd(2,n)} | c_{2}(ρ(λ′)) for all λ′ ∈ Λ_{r} with n ≥ 3 (c.f. [10, Part II, §15]), thus Dec(Spin_{2n}) = 2ℤq for any n ≥ 4 and Dec(PGO_{2n}^{+}) = \frac{8}{gcd(2,n)}ℤq for any n ≥ 3 (see [19, §4b]). Hence, by (25) we obtain

(40)  δ′′_{1}ℤq_{1} ⊕ ⋯ ⊕ δ′′_{m}ℤq_{m} ⊆ Dec(G) ⊆ δ′_{1}ℤq_{1} ⊕ ⋯ ⊕ δ′_{m}ℤq_{m}, where

δ′′_{i} = \begin{cases}8 & if n_{i} odd, \\ 4 & if n_{i} even, \end{cases} and δ′_{i} = \begin{cases}2 & if n_{i} ≥ 4, \\ 1 & if n_{i} = 3. \end{cases}

For the remaining simple groups O_{2n}^{+} and HSpin_{2n} (n even), we also have 2ℤq ⊆ Dec(O_{2n}^{+}) and 4ℤq ⊆ Dec(HSpin_{2n}) by (39). Moreover, if n = 4, then we have

(41)  c_{2}(ρ(ω_{3})) = c_{2}(ρ(ω_{4})) = −2q,

thus 2ℤq ⊆ Dec(HSpin_{8}). Then, by the action of the Weyl group as above we obtain Dec(O_{2n}^{+}) = 2ℤq for all n ≥ 3, Dec(HSpin_{2n}) = 4ℤq for even n ≥ 6, and Dec(HSpin_{8}) = 2ℤq ([4, Theorem 5.1]). In general, we determine the subgroup Dec(G) for type D.

Proposition 4.3. Let G = (∏_{i=1}^{m} Spin_{2n_{i}})/μ, m ≥ 1, n_{i} ≥ 3, where μ is a central subgroup. Let R be the subgroup of (17) such that μ^{∗} = Z/R, R_{1,i} = R ∩ Z_{i} for odd n_{i}, and R′_{1,i} = R ∩ Z_{i} for even n_{i}. Set

I′_{1} = {i | R_{1,i} ≠ 0, n_{i} ≠ 3} ∪ {i | R_{1,i} = 2Z_{i}, n_{i} = 3} ∪ {i | R′_{1,i} ≠ 0, n_{i} = 4} ∪ {i | e_{i,1} + e_{i,2} ∈ R′_{1,i}, n_{i} ≥ 6}, I′_{2} = {i | R′_{1,i} = 0} ∪ {i | e_{i,1} + e_{i,2} ∉ R′_{1,i} ≠ 0, n_{i} ≥ 6}.

Then, we have (42)

Dec(G) = ( ⨁_{R_{1,i}=Z_{i}, n_{i}=3} ℤq_{i}) ⊕ ( ⨁_{i∈I′_{1}} 2ℤq_{i}) ⊕ ( ⨁_{i∈I′_{2}} 4ℤq_{i}) ⊕ ( ⨁_{r=1}^{l_{2}} 4ℤq′_{r}) ⊕ ( ⨁_{s=1}^{l_{3}} 8ℤq′′_{s}),

where l_{2} = dim_{ℤ/2ℤ}⟨e_{i} + e_{j} | 2e_{i} + 2e_{j} ∈ R, R_{1,i} = R_{1,j} = 0⟩, l_{3} = |{i | R_{1,i} = 0}| − l_{2}, and q′_{r} (resp. q′′_{s}) is of the form q_{i} + q_{j} (resp. q_{i}) for some i, j such that ⟨q_{i} | R_{1,i} = 0⟩ = ⟨q′_{r}, q′′_{s} | 1 ≤ r ≤ l_{2}, 1 ≤ s ≤ l_{3}⟩ over ℤ.

Proof. Let μ ≃ (μ_{2})^{k_{1}} × (μ_{4})^{k_{2}} be a central subgroup for some k_{1}, k_{2} ≥ 0 with k = k_{1} + k_{2}. We denote by D the right hand side of equation (42) and we write D = ⨁ D_{u}, where D_{u} denotes u-th direct summand of D for 1 ≤ u ≤ 5. If e_{i} ∈ R with n_{i} = 3, then by (39) D_{1} ⊆ Dec(G). If 2e_{i} ∈ R or e_{i,1} + e_{i,2} ∈ R, then by (19) we have w_{i,1} ∈ T^{∗}, thus by (39) 2q_{i} ∈ Dec(G). Similarly, if e_{i1} ∈ R′_{1,i} (resp. e_{i,2} ∈ R′_{1,i}) with n_{i} = 4, then by (19) we have w_{i,3} ∈ T^{∗} (resp. w_{i,4} ∈ T^{∗}), thus by (41) 2q_{i} ∈ Dec(G). Therefore, D_{2} ⊆ Dec(G). By a simple calculation, we have

(43)  − c_{2}(ρ(χ)) = 4(n_{j}a_{i}^{2}q_{i} + n_{i}a_{j}^{2}q_{j}) if χ ∈ W (a_{i}w_{i,1} + a_{j}w_{j,1})

for any nonzero integers a_{i} and a_{j}. If 2e_{i} + 2e_{j} ∈ R for some i ≠ j, then again by (19) we obtain w_{i,1} + w_{j,1} ∈ T^{∗}. As both n_{i} and n_{j} are odd, by (43) D_{4} ⊆ Dec(G). Finally, it follows by (40) that D_{3} ⊕ D_{5} ⊆ Dec(G), thus D ⊆ Dec(G).

Now we prove that c_{2}(ρ(λ)) ∈ D for all λ ∈ T^{∗}. Let λ be a character written as in (32) for some subsets J and K in (38). Assume that t = 1, i.e., λ = a_{i,1}w_{i,1} + ⋯ + a_{i,n_{i}}w_{i,n_{i}}. Applying the same argument as in the proof of Proposition 4.2 we obtain

c_{2}(ρ(λ)) ∈ \begin{cases}D_{4} ⊕ D_{5} & if A_{i} = 0 with odd n_{i}, \\ D_{2} & if A_{i} ≠ 0 with odd n_{i} ≥ 5; or A_{i} = 2e_{i} with n_{i} = 3, \\ D_{1} & if A_{i} = ±e_{i} with n_{i} = 3, \end{cases}

where A_{i} denotes the image of λ in Z as defined in (18) and

c_{2}(ρ(λ)) ∈ \begin{cases}D_{3} & if A_{i} = 0 with even n_{i}; or A_{i} ≠ e_{i,1} + e_{i,2} with even n_{i} ≥ 6, \\ D_{2} & if A_{i} ≠ 0 with n_{i} = 4; or A_{i} = e_{i,1} + e_{i,2} with even n_{i} ≥ 6. \end{cases}

We assume that t = 2 with λ_{i_{1}} with |λ_{i_{1}}| + |λ_{i_{2}}| = 2. Then, by the same argument as in the proof of Proposition 4.1 together with (43) we get c_{2}(ρ(λ)) ∈ D. Finally, Assume that either t ≥ 3 or t = 2 with |λ_{i_{1}}| + |λ_{i_{2}}| ≥ 3. Then, by the action of the normal subgroups (ℤ/2ℤ)^{n_{i}−1} of the Weyl group of G we obtain

c_{2}(ρ(λ)) = 8(∑_{i∈J} a_{i}q_{i}) + 4(∑_{i∈K} b_{i}q_{i})

for some a_{i}, b_{i} ∈ ℤ, thus, c_{2}(ρ(λ)) ∈ Dec(G). Hence, Dec(G) ⊆ D. □

5. Degree 3 invariants for semisimple groups G of types B, C, D

We now determine the group of reductive indecomposable invariants of split semisimple groups of types B, C, and D by using the results of Section 3, Propositions 4.1, 4.2, and 4.3.

5.1. Type B.

Theorem 5.1. Let G = (∏_{i=1}^{m} Spin_{2n_{i}+1})/μ, m, n_{i} ≥ 1, where μ ≃ (μ_{2})^{k} is a central subgroup. Let R be the subgroup of (μ_{2}^{m})^{∗} = (ℤ/2ℤ)^{m} whose quotient is the character group μ^{∗}. Then,

Inv^{3}(G)_{red} = (ℤ/2ℤ)^{m−k−l_{1}−l_{2}}, where

l_{1} = dim⟨e_{i} ∈ R | n_{i} ≤ 2⟩ and l_{2} = dim⟨e_{i} + e_{j} ∈ R |, e_{i}, e_{j} ∉ R, n_{i} = n_{j} = 1⟩.

Proof. Let R = {r = (r_{1}, …, r_{m}) ∈ (ℤ/2ℤ)^{m} | f_{p}(r) = 0, 1 ≤ p ≤ k} be the subgroup of (ℤ/2ℤ)^{m} whose quotient is the character group μ^{∗} for some linear polynomials f_{p} ∈ ℤ/2ℤ[t_{1}, …, t_{m}]. Let α_{i,j} denote the simple roots of the ith component of the root system of G and let θ_{i,j} be the square of the length of the coroot of α_{i,j}. Then, we have

θ_{i,j} = \begin{cases}2 & if j = n_{i} ≥ 2, \\ 1 & if n_{i} ≥ 2, 1 ≤ j ≤ n_{i} − 1; or j = n_{i} = 1. \end{cases}

By [13, Proposition 7.1], an indecomposable invariant of G corresponding to q = ∑_{i=1}^{m} d_{i}q_{i} ∈ Q(G) is reductive indecomposable if and only if the order |w̄_{i,j}| in Λ/T^{∗} divides θ_{i,j}d_{i} for all i and j.

Since |w̄_{i,1}| = 1 with n_{i} = 1 is equivalent to e_{i} ∈ R and

|w̄_{i,j}| ≤ \begin{cases}2 & if j = n_{i} ≥ 2, \\ 1 & if n_{i} ≥ 2, 1 ≤ j ≤ n_{i} − 1, \end{cases}

we see that the equation (10) becomes trivial and we may assume that the term δ_{i}d_{i}(= d_{i}) appears in the equation (11) is divisible by 2. Therefore, any reductive indecomposable invariant of G corresponding to q = ∑_{i=1}^{m} d_{i}q_{i} ∈ Q(G) satisfies

f_{p}(\frac{δ_{1}d_{1}}{2} , …, \frac{δ_{m}d_{m}}{2} ) ≡ 0 mod 2, where δ_{i} = \begin{cases}2 & if n_{i} ≥ 2 or e_{i} ∈ R, \\ 1 & if n_{i} = 1 and e_{i} ∉ R. \end{cases}

for all p, thus we have

(44)  Inv^{3}(G)_{red} = \frac{\{∑_{i=1}^{m} d_{i}q_{i} | f_{p}(\frac{δ_{1}d_{1}}{2} , …, \frac{δ_{m}d_{m}}{2} ) ≡ 0 mod 2\}}{Dec(G)}.

Let R′ = R ∩ (⨁_{e_{i}∉R}(ℤ/2ℤ)e_{i}). Then, the group in the numerator of (44) is generated by

\{q_{i} | e_{i} ∈ R\} ∪ \{∑_{i=1}^{m} (\frac{2r_{i}}{δ_{i}} )q_{i} |r = (r_{1}, …, r_{m}) ∈ R′\} ∪ \{(\frac{4}{δ_{i}} )q_{i} | e_{i} ∉ R\}.

Hence, the statement for the group of indecomposable reductive invariants follows by Proposition 4.1. □

In particular, under the assumption that the ranks of all components of the root system of G are at least 2 we have the following result.

Corollary 5.2. Let G = (∏_{i=1}^{m} Spin_{2n_{i}+1})/μ, m, n_{i} ≥ 1, where μ ≃ (μ_{2})^{k} is a central subgroup. Let R be the subgroup of (μ_{2}^{m})^{∗} = (ℤ/2ℤ)^{m} whose quotient is the character group μ^{∗}. Assume that n_{i} ≥ 2 for all 1 ≤ i ≤ m. Then,

Inv^{3}(G)_{ind} = Inv^{3}(G)_{red} = (ℤ/2ℤ)^{m−k−l},

where l = dim⟨e_{i} ∈ R | n_{i} = 2⟩.

Proof. By Theorem 5.1, it suffices to show that Inv^{3}(G)_{ind} ⊆ Inv^{3}(G)_{red}. Since n_{i} ≥ 2 for all 1 ≤ i ≤ m, the inclusion follows directly from the proof of Theorem 5.1. □

Remark 5.3. One can directly compute Inv^{3}(G)_{ind} using Propositions 3.1 and 4.1.

We present below another particular case of Theorem 5.1 (and Theorem 5.5), which follows by the exceptional isomorphism A_{1} = B_{1} = C_{1}. This result in turn determine the reductive invariants of semisimple groups of type A (see [17, Theorem 7.1]).

Corollary 5.4. Let G = (∏_{i=1}^{m} SL_{2})/μ, m ≥ 1, where μ ≃ (μ_{2})^{k} is a central subgroup. Let R be the subgroup of (μ_{2}^{m})^{∗} = (ℤ/2ℤ)^{m} whose quotient is the character group μ^{∗}. Then,

Inv^{3}(G)_{red} = (ℤ/2ℤ)^{m−k−l_{1}−l_{2}},

where l_{1} = dim⟨e_{i} ∈ R⟩ and l_{2} = dim⟨e_{i} + e_{j} ∈ R | e_{i}, e_{j} ∉ R⟩.

5.2. Type C.

Theorem 5.5. Let G = (∏_{i=1}^{m} Sp_{2n_{i}})/μ, m, n_{i} ≥ 1, where μ ≃ (μ_{2})^{k} is a central subgroup. Let R be the subgroup of (μ_{2}^{m})^{∗} = ⨁_{i=1}^{m}(ℤ/2ℤ)e_{i} whose quotient is the character group μ^{∗} and let s denote the number of ranks n_{i} which are divisible by 4. Then,

Inv^{3}(G)_{red} = (ℤ/2ℤ)^{s+l−l_{1}−l_{2}}, where

l_{1} = dim⟨e_{i} | e_{i} ∈ R⟩, l_{2} = dim⟨e_{i} + e_{j} | e_{i} + e_{j} ∈ R, e_{i}, e_{j} ∉ R, n_{i} ≡ n_{j} ≡ 1 mod 2⟩, and l = dim (R∩(⨁_{4∤n_{i}}(ℤ/2ℤ)e_{i})). In particular, if n_{i} ≡ 0 mod 2 for all 1 ≤ i ≤ m, then

Inv^{3}(G)_{ind} = Inv^{3}(G)_{red} = (ℤ/2ℤ)^{s+l−l_{1}}.

Proof. We apply arguments similar to the proof of type B. Let θ_{ij} be the square of the length of the coroot corresponding to the simple root of ith component of the root system of G. Then, we have

θ_{i,j} = \begin{cases}1 & if j = n_{i} ≥ 1, \\ 2 & otherwise. \end{cases}

Note that |w̄_{i,n_{i}}| = 2 if and only if n_{i} is odd and the element e_{i,n_{i}} has order 2 in Λ/T^{∗}. Moreover, by (14) the latter is equivalent to e_{i} ∉ R. Hence, by [13, Proposition 7.1] an indecomposable invariant of G corresponding to q = ∑_{i=1} d_{i}q_{i} ∈ Q(G) is reductive indecomposable if and only if 2|d_{i} for all odd n_{i} such that e_{i} ∉ R. Therefore, any reductive indecomposable invariant of G corresponding to q = ∑_{i=1} d_{i}q_{i} ∈ Q(G) obviously satisfies the first equation of (16) and the second equation of (16) divided by 2, i.e.,

f_{p}(\frac{δ_{1}n_{1}d_{1}}{2} , …, \frac{δ_{m}n_{m}d_{m}}{2} ) ≡ 0 mod 2, where δ_{i} = \begin{cases}1 & if e_{i} ∉ R, \\ \frac{2}{n_{i}} & if e_{i} ∈ R \end{cases}

for all 1 ≤ p ≤ k, thus,

(45)  Inv^{3}(G)_{red} = \{∑_{i=1}^{m} d_{i}q_{i} | f_{p}(\frac{δ_{1}n_{1}d_{1}}{2} , …, \frac{δ_{m}n_{m}d_{m}}{2} ) ≡ 0 mod 2\}/ Dec(G).

Let R′ = R ∩ (⨁_{4∤n_{i},e_{i}∉R}(ℤ/2ℤ)e_{i}), where R is the subgroup of ⨁_{i=1}^{m}(ℤ/2ℤ)e_{i} as in (4). Then, we easily see that the group in the numerator of (45) is generated by

\{q_{i} | e_{i} ∈ R or 4|n_{i}\} ∪ \{∑_{i=1}^{m} ϵ_{i}r_{i}q_{i} |r = (r_{i}) ∈ R′\} ∪ \{2ϵ_{i}q_{i} | e_{i} ∉ R\}, ϵ_{i} = \begin{cases}1 & if 2|n_{i}, \\ 2 & if 2 ∤ n_{i}. \end{cases}

Hence, the statement immediately follows from Proposition 4.2. If n_{i} is even for all i, then the same argument together with Proposition 3.2 shows the result for the group of indecomposable invariants. □

5.3. Type D.

Theorem 5.6. Let G = (∏_{i=1}^{m} Spin_{2n_{i}})/μ, m ≥ 1, n_{i} ≥ 3, where μ is a central subgroup. Let R be the subgroup of the character group Z defined in (17) such that μ^{∗} = Z/R, R_{1,i} = R ∩ Z_{i} for odd n_{i}, R′_{1,i} = R ∩ Z_{i} for even n_{i}, and let

R̄ = \{(r̄_{1}, …, r̄_{m}) ∈ ⨁_{i=1}^{m}(ℤ/2ℤ)ē_{i} | ∑_{i=1}^{m} r_{i} ∈ R\}, r_{i} := \begin{cases}2r̄_{i}e_{i} & if n_{i} odd, \\ r̄_{i}e_{i,1} + r̄_{i}e_{i,2} & if n_{i} even, \end{cases}

where Z := ⨁_{i=1}^{m} Z_{i} with Z_{i} = \begin{cases}(ℤ/4ℤ)e_{i} & if n_{i} odd, \\ (ℤ/2ℤ)e_{i,1} ⨁(ℤ/2ℤ)e_{i,2} & if n_{i} even. \end{cases}

denote the character group of the center of ∏_{i=1}^{m} Spin_{2n_{i}}. Set

R′ = R̄ ∩ ( ⨁_{4∤n_{i},R′_{1,i},R_{1,i}≠Z_{i}}(ℤ/2ℤ)ē_{i}) with l = dim R′, I_{1} = {i | Z_{i} = R_{1,i} or R′_{1,i}, n_{i} ≠ 3},

I_{2} = {i | R′_{1,i} = 0, 4|n_{i}} ∪ {i | R′_{1,i} =(ℤ/2ℤ)e_{i,1} or (ℤ/2ℤ)e_{i,2}, n_{i} ≥ 6, 4|n_{i}} with s_{i} = |I_{i}|.

Then, we have

Inv^{3}(G)_{red} = (ℤ/2ℤ)^{s_{1}+s_{2}+l−l_{1}−l_{2}}, where

l_{1} = |{i | 4 ∤ n_{i}, R_{1,i} = 2Z_{i} or R′_{1i} = (ℤ/2ℤ)(e_{i,1} + e_{i,2})}|, and l_{2} = dim⟨ē_{i} + ē_{j}|2e_{i} + 2e_{j} ∈ R, R_{1,i} = R_{1,j} = 0⟩.

Proof. Let Z denote the character group of the center of ∏_{i=1}^{m} Spin_{2n_{i}} as in (17). Let μ be a central subgroup such that μ ≃ (μ_{2})^{k_{1}} × (μ_{4})^{k_{2}} for some k_{1}, k_{2} ≥ 0 and let R = {r ∈ Z | f_{p}(r) = 0, 1 ≤ p ≤ k} be the subgroup of Z such that μ^{∗} ≃ Z/R for some linear polynomials f_{p} ∈ ℤ/4ℤ[T_{1}, …, T_{m}] with k = k_{1} + k_{2}. We shall use the description of Q(G) in Section 3.3.

Let θ_{i,j} denote the square of the length of the jth coroot of the ith component of the root system of G. Then, θ_{i,j} = 1 for all 1 ≤ i ≤ m and 1 ≤ j ≤ n_{i}. Note that the order of the fundamental weight w_{i,j} in Λ/T^{∗} is trivial for all j if and only if

Z_{i} = \begin{cases}R_{1,i} & if n_{i} odd, \\ R′_{1,i} & if n_{i} even. \end{cases}

Moreover, if c_{i}(p) = ±1 for some 1 ≤ p ≤ k, where c_{i}(p) denotes the coefficient of t_{i} in f_{p}, then R_{1i} = 0, thus by (19) 2w_{i,n_{i}} ∉ T^{∗}, i.e., |w̄_{i,n_{i}}| = 4. Hence, by [13, Proposition 7.1] any reductive indecomposable invariant of G corresponding to q = ∑_{i=1} d_{i}q_{i} ∈ Q(G) satisfies (22) and (24). Therefore, it follows by (23) that

(46)  Inv^{3}(G)_{red} = \frac{\{∑_{i=1}^{m} d_{i}q_{i} | f̄_{p}(ϵ_{1}d_{1}, ⋯, ϵ_{m}d_{m}) ≡ 0 mod 2\}}{Dec(G)}

where, f̄_{p} ∈ ℤ/2ℤ[t_{1}, …, t_{m}] denotes the image of f_{p} under the following map

ℤ/4ℤ[T] → ℤ/4ℤ[t_{1}, …, t_{m}] → ℤ/2ℤ[t_{1}, …, t_{m}] given by 2t_{i1}, 2t_{i2} ↦ t_{i}, t_{i} ↦ t_{i}

and ϵ_{i} = \begin{cases}1 & if Z_{i} = R_{1,i} or R′_{1,i}, \\ \frac{1}{2} & if c_{i}(p) = 2 or c_{i1}(p) + c_{i2}(p) = 4, \\ \frac{n_{i}}{4} & otherwise. \end{cases}

Let R̄ = {r̄ = (r̄_{1}, …, r̄_{m}) ∈ ⨁_{i=1}^{m}(ℤ/2ℤ)ē_{i} | f̄_{p}(r̄) ≡ 0 mod 2}, equivalently

R̄ = \{(r̄_{1}, …, r̄_{m}) ∈ (ℤ/2ℤ)^{m} | ∑_{i=1}^{m} r_{i} ∈ R\}, where r_{i} := \begin{cases}2r̄_{i}e_{i} & if n_{i} odd, \\ r̄_{i}e_{i,1} + r̄_{i}e_{i,2} & if n_{i} even \end{cases}

and let R′ = R̄ ∩ ( ⨁_{4∤n_{i},R′_{1,i},R_{1,i}≠Z_{i}}(ℤ/2ℤ)ē_{i}). Observe that f̄_{p}(ē_{i}) ≡ 0 mod 2 for all p with n_{i} odd if and only if either c_{i}(p) = 0 or 2 for all p (i.e., f_{p}(e_{i}) ≡ 0 or f_{p}(2e_{i}) ≡ 0 mod 4, respectively) and this, in turn, is equivalent to R_{1,i} = Z_{i} or 2Z_{i}. Similarly, f̄_{p}(ē_{i}) ≡ 0 mod 2 for all p with n_{i} even if and only if either c_{i1}(p) = c_{i2}(p) = 0 or c_{i1}(p) = c_{i2}(p) = 2 for all p (i.e., f_{p}(e_{i1}) ≡ f_{p}(e_{i2}) ≡ 0 or f_{p}(e_{i1} + e_{i2}) ≡ 0 mod 4, respectively) and this, in turn, is equivalent to R′_{1,i} = Z_{i} or (ℤ/2ℤ)(e_{i1} + e_{i2}). Therefore, we see that the group in the numerator of (46) is generated by

{2q_{i} | R_{1,i} = 2Z_{i} or R′_{1,i} = (ℤ/2ℤ)(e_{i1} + e_{i2}) or e_{i,1} + e_{i,2} ∉ R′_{1,i}, 4|n_{i}} ∪ {8q_{i} | R_{1,i} = 0}

∪ {q_{i} | Z_{i} = R_{1,i} or R′_{1,i}} ∪ {4q_{i} | e_{i,1} + e_{i,2} ∉ R′_{1,i}, 4 ∤ n_{i}} ∪ {∑_{i=1}^{m} ϵ′_{i}r′_{i}q_{i}}

for all r′ = (r′_{1}, …, r′_{m}) ∈ R′\{ē_{i} | 1 ≤ i ≤ m}, where ϵ′_{i} = 2 (resp. 4) if n_{i} is even (resp. odd). Therefore, the statement immediately follows by Proposition 4.3. □

6. Unramified invariants for semisimple groups G of types B, C, D

In this section, we first describe torsors for the corresponding reductive groups in Lemmas 6.1, 6.6, and 6.11. Then, using this together with Theorems 5.1, 5.5, and 5.6, we present a complete description of the corresponding cohomological invariants in Propositions 6.3, 6.7, and 6.13. Finally, using such descriptions, we show that there are no nontrivial unramified degree 3 invariants for semisimple groups of types B, C, and D (see Theorems 6.5, 6.10, 6.15). In this section, we assume that the base field F is of characteristic 0. We shall write ⟨a_{1}, …, a_{n}⟩ = ⟨a_{1}⟩ ⊥ ⋯ ⊥ ⟨a_{n}⟩ for the diagonal quadratic form a_{1}x_{1}^{2} + ⋯ + a_{n}x_{n}^{2} and write ⟨⟨a_{1}, …, a_{n}⟩⟩ = ⟨1, −a_{1}⟩ ⊗ ⋯ ⊗ ⟨1, −a_{n}⟩ for the n-fold Pfister form.

6.1. Type B.

Lemma 6.1. Let G = (∏_{i=1}^{m} Spin_{2n_{i}+1})/μ, m, n_{i} ≥ 1, where μ is a central subgroup. Let R be the subgroup of (μ_{2}^{m})^{∗} = (ℤ/2ℤ)^{m} whose quotient is the character group μ^{∗}. Set G_{red} = (∏_{i=1}^{m} Γ_{2n_{i}+1})/μ, where Γ_{2n_{i}+1} is the split even Clifford group. Then, for any field extension K/F the first Galois cohomology set H^{1}(K, G_{red}) is bijective to the set of m-tuples of quadratic forms (ϕ_{1}, …, ϕ_{m}) with dim ϕ_{i} = 2n_{i} + 1, disc ϕ_{i} = 1 such that for all r = e_{i_{1}} + ⋯ + e_{i_{s}} ∈ R, i_{1} < ⋯ < i_{s},

(47)  I^{3}(K) ∋ \begin{cases}⊥_{p=1}^{s} (−1)^{p}ϕ_{i_{p}} & if s is even, \\ (⊥_{p=1}^{s} (−1)^{p}ϕ_{i_{p}}) ⊥ ⟨1⟩ & otherwise, \end{cases}

where disc ϕ_{i} denotes the discriminant of ϕ_{i} and I^{3}(K) denotes the cubic power of the fundamental ideal I(K) in the Witt ring of K.

Proof. Let G_{red} = (∏_{i=1}^{m} Γ_{2n_{i}+1})/μ, where Γ_{2n_{i}+1} denotes the split even Clifford group. Consider the natural exact sequence

1 → (𝔾_{m})^{m}/μ → G_{red} → ∏_{i=1}^{m} O_{2n_{i}+1}^{+} → 1.

Then, by Hilbert Theorem 90 and [24, Proposition 42], this sequence yields a bijection between the set H^{1}(F, G_{red}) and the kernel of the connecting map which factors as

H^{1}(F, ∏_{i=1}^{m} O_{2n_{i}+1}^{+}) → H^{2}(F, (μ_{2})^{m}) = Br_{2}(F)^{m} →^{τ} H^{2}(F, (μ_{2})^{m}/μ),

where the first map sends an m-tuple of quadratic forms (ϕ_{1}, …, ϕ_{m}) with dim ϕ_{i} = 2n_{i} + 1, disc(ϕ_{i}) = 1 to the m-tuple (C_{0}(ϕ_{1}), …, C_{0}(ϕ_{m})) of even Clifford algebras C_{0}(ϕ_{i}) associated to ϕ_{i} and the map τ is induced by the natural surjection (μ_{2})^{m} → (μ_{2})^{m}/μ. Since (C_{0}(ϕ_{1}), …, C_{0}(ϕ_{m})) ∈ Ker(τ) if and only if it is contained in the kernel of the composition

(48)  H^{2}(F, (μ_{2})^{m}) →^{τ} H^{2}(F, (μ_{2})^{m}/μ) →^{r_{∗}} H^{2}(F, 𝔾_{m})

for all r ∈ R = ((μ_{2})^{m}/μ)^{∗}, we have

(49) H^{1}(F, G_{red}) ≃ {(ϕ_{1}, …, ϕ_{m}) | dim ϕ_{i} = 2n_{i} + 1, disc ϕ_{i} = 1, ∑_{i=1}^{m} r_{i}C_{0}(ϕ_{i}) = 0}

for all r = (r_{i}) ∈ R.

Write an element r ∈ R as r = e_{i_{1}} + ⋯ + e_{i_{s}} for some i_{1} < ⋯ < i_{s}, so that the condition ∑_{i=1}^{m} r_{i}C_{0}(ϕ_{i}) = 0 in (49) is equal to ∑_{p=1}^{s} C_{0}(ϕ_{i_{p}}) = 0. Assume that s is even. Since disc(−ϕ_{i_{p}} ⊥ ϕ_{i_{p+1}}) = 1 for any 1 ≤ p ≤ s/2,

C_{0}(ψ) = C_{0}(−ψ), and C_{0}(ϕ) + C_{0}(ϕ′) = C(ϕ ⊥ ϕ′)

for any quadratic form ψ and any odd-dimensional quadratic forms ϕ and ϕ′, where C(ϕ ⊥ ϕ′) is the corresponding Clifford algebra, we have

0 = ∑_{p=1}^{s} C_{0}(ϕ_{i_{p}}) = C(−ϕ_{i_{1}} ⊥ ϕ_{i_{2}} ⊥ ⋯ ⊥ −ϕ_{i_{s−1}} ⊥ ϕ_{i_{s}}),

which is equivalent to (−ϕ_{i_{1}} ⊥ ϕ_{i_{2}}) ⊥ ⋯ ⊥ (−ϕ_{i_{s−1}} ⊥ ϕ_{i_{s}}) ∈ I^{3}(F) by [9, Theorem 14.3]. Now we assume that s is odd. Since C_{0}(ϕ ⊥ ⟨1⟩) = C_{0}(ϕ) for any odddimensional quadratic form ϕ and disc(−ϕ_{i_{s}} ⊥ ⟨1⟩) = 1, the same argument shows that (−ϕ_{i_{1}} ⊥ ϕ_{i_{2}}) ⊥ ⋯ ⊥ (−ϕ_{i_{s−2}} ⊥ ϕ_{i_{s−1}}) ⊥ (−ϕ_{i_{s}} ⊥ ⟨1⟩) ∈ I^{3}(F). □

Remark 6.2. If we assume that −1 ∈ (F^{×})^{2}, then the condition (47) in Lemma 6.1 can be simplified without sign changes as follows:

H^{1}(K, G_{red}) ≃ {ϕ := (ϕ_{1}, …, ϕ_{m}) | dim ϕ_{i} = 2n_{i} + 1, disc ϕ_{i} = 1, ϕ[r] ∈ I^{3}(K)}

for all r = (r_{i}) ∈ R, where

ϕ[r] := \begin{cases}⊥_{i=1}^{m} r_{i}ϕ_{i} & if ∑_{i=1}^{m} r_{i} ≡ 0 mod 2, \\ (⊥_{i=1}^{m} r_{i}ϕ_{i}) ⊥ ⟨1⟩ & otherwise. \end{cases}

Proposition 6.3. Let G = (∏_{i=1}^{m} Spin_{2n_{i}+1})/μ defined over an algebraically closed field F, where m, n_{i} ≥ 1, μ is a central subgroup. Let R be the subgroup of (μ_{2}^{m})^{∗} whose quotient is the character group μ^{∗}. Set G_{red} = (∏_{i=1}^{m} Γ_{2n_{i}+1})/μ, where Γ_{2n_{i}+1} is the split even Clifford group. Then, every normalized invariant in Inv^{3}(G_{red}) is of the form e_{3}(ϕ[r]) for some r ∈ R, where ϕ[r] is the quadratic form defined in Remark 6.2 and e_{3} : I^{3}(K) → H^{3}(K) denotes the Arason invariant over a field extension K/F. Moreover, we have

(50)  Inv^{3}(G_{red})_{norm} ≃ \frac{R}{⟨e_{i}, e_{j} + e_{k} ∈ R | e_{j}, e_{k} ∉ R, n_{i} ≤ 2, n_{j} = n_{k} = 1⟩}.

Proof. Observe that Inv^{3}(G_{red})_{norm} = Inv^{3}(G_{red})_{ind} as F is algebraically closed. Since ϕ[r] ∈ I^{3}(K) for any r ∈ R, the Arason invariant gives a normalized invariant of G_{red} of order dividing 2 that sends an m-tuple ϕ ∈ H^{1}(K, G_{red}) to e_{3}(ϕ[r]) ∈ H^{3}(K).

Let r ∈ R′_{1}+R′_{2}, where R′_{1} and R′_{2} denote the subgroups of R defined in Proposition 4.1. Then, as every 4 and 6-dimensional quadratic forms in I^{3}(K) are hyperbolic, the invariant e_{3}(ϕ[r]) vanishes.

Now we show that the invariant e_{3}(ϕ[r]) is nontrivial for any r ∈ R\(R′_{1} + R′_{2}). Let G′_{red} = (Γ_{3})^{m}/μ. If R is a subgroup such that every element r in R has at least 3 nonzero components, then by [14, Lemma 4.3] and the exceptional isomorphism A_{1} = B_{1}, any invariant of G′_{red} is nontrivial. Hence, it follows from the map

Inv^{3}(G_{red}) → Inv^{3}(G′_{red})

induced by the standard embedding Γ_{3} → Γ_{2n_{i}+1} that every invariant e_{3}(ϕ[r]) is nontrivial. Otherwise, by the proof of Lemma 6.4 each invariant e_{3}(ϕ[r]) is nontrivial, thus the statements follow from Theorem 5.1. □

Recall from Section 3 the following subgroups of R.

R_{1} = ⟨e_{i} ∈ R⟩ and R_{2} = ⟨e_{i} + e_{j} ∈ R | e_{i}, e_{j} ∉ R_{1}⟩.

We shall need the following key lemma.

Lemma 6.4. Let G = (∏_{i=1}^{m} Spin_{2n_{i}+1})/μ defined over an algebraically closed field F, where m, n_{i} ≥ 1, μ is a central subgroup. Set G_{red} = (∏_{i=1}^{m} Γ_{2n_{i}+1})/μ. Then, every normalized invariant in Inv^{3}(G_{red}) is ramified if either n_{i} ≥ 3 for some i with e_{i} ∈ R_{1} or n_{j} + n_{k} ≥ 3 for some j and k such that e_{j} + e_{k} ∈ R_{2}.

Proof. Let R_{3} be a complementary subspace of R_{1} + R_{2} in R. Then, by Proposition 6.3 any normalized invariant α in Inv^{3}(G_{red}) can be written as

α(ϕ) = e_{3}(ϕ[r_{1}]) + e_{3}(ϕ[r_{2}]) + e_{3}(ϕ[r_{3}])

for some r_{i} ∈ R_{i}, 1 ≤ i ≤ 3, where ϕ = (ϕ_{1}, …, ϕ_{m}) denotes a G_{red}-torsor.

Suppose that r_{1} ∈ R_{1} is nonzero. Then, we may assume that r_{1} = e_{1} with n_{1} ≥ 3. Choose a division quaternion algebra (x, y) over a field extension K/F. Find ϕ[e_{1}] = ϕ_{1} such that ϕ_{1} ⊥ ⟨1⟩ = ⟨⟨x, y, z⟩⟩ ⊥ h over the field of formal Laurent series K((z)) and set ϕ_{i} = h ⊥ ⟨1⟩ for all 2 ≤ i ≤ m, where h denotes a hyperbolic form. Then, we have ∂_{z}(α(ϕ)) = (x, y) ≠ 0, where ∂_{z} denotes the residue map, thus α(ϕ) ramifies.

Now we may assume that α(ϕ) = e_{3}(ϕ[r_{2}]) + e_{3}(ϕ[r_{3}]) with r_{2} ≠ 0. To show that α(ϕ) ramifies, we shall choose bases of R_{2} and a complementary subspace of R_{2}. For simplicity, we will write e(i_{1}, …, i_{k}) for e_{i_{1}} + ⋯ + e_{i_{k}}. We first select e(i_{p}, i_{p,q}) ∈ R_{2}, where i_{p}, i_{p,q} (1 ≤ p ≤ k, 1 ≤ q ≤ m_{p}) are all distinct integers for some m_{1}, …, m_{k}, so that B_{2} := {e(i_{p}, i_{p,q})} is a basis of R_{2}. In particular, if n_{i_{p,q}} = 1 for some p and q, say n_{i_{1,1}} = 1, then we replace the subset {e(i_{1}, i_{1,q}) | 1 ≤ q ≤ m_{1}} of B_{2} by {e(i_{1,1}, i_{1}), e(i_{1,1}, i_{1q}) | 2 ≤ q ≤ m_{1}} so that we may assume that n_{i_{p}} = 1. We set

I_{2} = {i_{p} | 1 ≤ p ≤ k} and I′_{2} = {i_{p}, i_{p,q} | 1 ≤ p ≤ k, 1 ≤ q ≤ m_{p}}.

We select a basis B_{3} of a complementary subspace of R_{2}. First, we find any basis D_{3} of R_{3}. Then, we modify each element d of D_{3} by adding e(i_{p}, i_{p,q}) to it whenever either e(i_{p,q}) or e(i_{p}, i_{p,q}) appears in d. Hence, we obtain a basis C_{3} := {e(k_{1}, …, k_{l})} of a complementary subspace of R_{2} such that the intersection

( ⋃{k_{1}, …, k_{l} | e(k_{1}, …, k_{l}) ∈ C_{3}}) ∩ I′_{2},

where the union is over all elements of C_{3}, is a subset of I_{2}. We denote by J_{2} the intersection. We can divide all elements of the basis C_{3} into two types: either e(i_{p}) for some i_{p} ∈ J_{2} appears in e(k_{1}, …, k_{l}) ∈ C_{3} (the first type) or not (the second type).

We first select basis elements from the first type elements as follows. We choose any element b(i_{1}) in C_{3} of the first type such that e(i_{1}) appears in the element (if there is no element of the first type, we skip the selection of elements of the first type). We write b(i_{1}) := e(i_{1}) + b′(i_{1}), where e(i_{1}) does not appear in b′(i_{1}). We modify every element of the first type by adding b(i_{1}) to the element whenever e(i_{1}) appears in the element. For simplicity, we shall use the same notation C_{3} for the modified basis of C_{3}. Then, e(i_{1}) appears only in b(i_{1}) among the elements of C_{3}. Now we choose another element b(i_{2}) of the first type in which e(i_{2}) appears for some i_{2} ∈ J_{2}. We write b(i_{2}) := e(i_{2}) + b′(i_{2}), where e(i_{2}) does not appear in b′(i_{2}). As e(i_{1}) appears only in b(i_{1}), both e(i_{1}) and e(i_{2}) do not appear in b′(i_{2}). Again, we modify every element of the first type by adding b(i_{2}) to the element whenever e(i_{2}) appears in the element. In particular, both e(i_{1}) and e(i_{2}) do not appear in the modified b′(i_{1}). We do the same procedure successively for all elements of the first type so that we have chosen basis elements b(i_{p}) := e(i_{p}) + b′(i_{p}) for all i_{p} in some subset J′_{2} ⊆ J_{2} such that all the terms e(i_{p}) do not appear in b′(i_{p}).

Similarly, we select basis elements from the second type elements. We choose any element b(j_{1}) of the second type with j_{1} ∉ J_{2}, so that we write b(j_{1}) := e(j_{1}) + b′(j_{1}), where e(j_{1}) does not appear in b′(j_{1}). We modify every element of C_{3} (i.e., b(i_{p}) and elements of the second type) by adding b(j_{1}) to the element whenever e(j_{1}) appears in the element. Then, in particular, all the terms e(i_{p}) and e(j_{1}) do not appear in the modified b′(i_{p}). Now we choose another element b(j_{2}) of the second type for some j_{2} ∉ J_{2}, so that we have b(j_{2}) := e(j_{2}) + b′(j_{2}), where both e(j_{1}) and e(j_{2}) do not appear in b′(j_{2}). Again we modify every element of C_{3} by adding b(j_{2}) to the element whenever e(j_{2}) appears in the element. Then, both e(j_{1}) and e(j_{2}) do not appear in modified b′(j_{2}) and all the terms e(i_{p}), e(j_{1}), and e(j_{2}) do not appear in the modified b′(i_{p}). Applying the same procedure to all elements of the second type, we obtain the following basis B_{3} of a complementary subspace of R_{2}:

b(i_{p}) := e(i_{p}) + b′(i_{p}), b(j_{1}) := e(j_{1}) + b′(j_{1}), ⋯, b(j_{s}) := e(j_{s}) + b′(j_{s})

for all i_{p} ∈ J′_{2} and some distinct j_{1}, …, j_{s} ∉ J_{2} such that all the terms e(i_{p}) and e(j_{r}) do not appear in b′(i_{p}), b′(j_{r}) for all 1 ≤ r ≤ s, thus

B_{3} = {b(i_{p}), b(j_{r}) | i_{p} ∈ J′_{2}, 1 ≤ r ≤ s}.

Using the basis B_{2} ∪ B_{3}, we rewrite the invariant α(ϕ) = e_{3}(ϕ[r_{2}]) + e_{3}(ϕ[r_{3}]) as

(51)  α(ϕ) = ∑_{b∈B′_{2}} e_{3}(ϕ[b]) + ∑_{b∈B′_{3}} e_{3}(ϕ[b])

for some subsets ∅ ≠ B′_{2} ⊆ B_{2} and B′_{3} ⊆ B_{3}. Now we show that the invariant α(ϕ) in (51) ramifies. It is convenient to split the proof into two cases.

Case 1: ∃ e(i_{p}, i_{p,q}) ∈ B′_{2} with n_{i_{p}} + n_{i_{p,q}} ≥ 3 such that i_{p} ∉ J′_{2}. Let e(i_{u}, i_{u,v}) ∈ B′_{2} be such an element for some 1 ≤ u ≤ k and 1 ≤ v ≤ m_{u} and let I = {1, …, m}. We take a division quaternion algebra (x, y) over a field extension K/F. Then, choose ϕ_{i} for all i ∈ I such that

(52)  ϕ[e(i_{u})] = ϕ[e(i_{u,q})] = ⟨x, y, xy⟩ ⊥ h, ϕ[e(i_{u,v})] = ⟨1, z, xz, yz, xyz⟩ ⊥ h

for all 1 ≤ q ≠ v ≤ m_{u},

ϕ[e(i_{p})] = ϕ[e(i_{p,q})] = \begin{cases}⟨x, y, xy⟩ ⊥ h & if e(i_{u}) appears in b(i_{p}), \\ ⟨1⟩ ⊥ h & otherwise, \end{cases}

for all i_{p} ∈ J′_{2} and all q with e(i_{p}, i_{p,q}) ∈ B_{2}, and ϕ_{i} = ⟨1⟩ ⊥ h for the remaining i ∈ I over K((z)), where h denotes a hyperbolic form depending on the dimension of each ϕ_{i}. Then, we have

(53)  ϕ[e(i_{u}, i_{u,v})] = ⟨⟨x, y, z⟩⟩, ϕ[e(i_{u}, i_{u,q})] = ⟨⟨x, y, 1⟩⟩

for all 1 ≤ q ≠ v ≤ m_{u},

(54)  ϕ[e(i_{p}, i_{p,q})] = ϕ[b(i_{p})] = ⟨⟨x, y, 1⟩⟩

for all p ∈ J′_{2} and all q with e(i_{p}, i_{p,q}) ∈ B_{2} such that e(i_{u}) appears in b(i_{p}), and ϕ[b] = 0 for all remaining b ∈ B_{2} ∪ B_{3} in the Witt ring of K((z)). Therefore, we obtain ∂_{z}(α(ϕ)) = (x, y) ≠ 0. Hence, α(ϕ) ramifies.

Case 2: ∃ e(i_{p}, i_{p,q}) ∈ B′_{2} with n_{i_{p}} + n_{i_{p,q}} ≥ 3 such that i_{p} ∈ J′_{2}. Let e(i_{u}, i_{u,v}) ∈ B′_{2} be such an element as in the previous case. Observe that by construction of B_{3} there exists

(55)  k_{1} ∈ I\{i_{p}, j_{r} | i_{p} ∈ J′_{2}, 1 ≤ r ≤ s}

such that e(k_{1}) appears in b′(i_{u}). We first choose ϕ[e(i_{u,v})] as in (52) and ϕ[e(k_{1})] = ⟨x, y, xy⟩ ⊥ h. Then, we choose ϕ_{i} for i ∈ I\{i_{u,v}, k_{1}} such that

ϕ[e(i)] = \begin{cases}⟨x, y, xy⟩ ⊥ h & if e(k_{1}) appears in b(i), \\ ⟨1⟩ ⊥ h & otherwise \end{cases}

for all i ∈ {i_{p}, j_{r} | i_{p} ∈ J′_{2}, 1 ≤ r ≤ s},

ϕ[e(i_{p,q})] = \begin{cases}⟨x, y, xy⟩ ⊥ h & if i_{p} = k_{1} or e(k_{1}) appears in b(i_{p}), \\ ⟨1⟩ ⊥ h & otherwise \end{cases}

for all q such that e(i_{p}, i_{p,q}) ∈ B_{2}, and ϕ_{i} = ⟨1⟩ ⊥ h for the remaining i ∈ I\{i_{u,v}, k_{1}} over K((z)). Therefore, we obtain (53),

(56)  ϕ[b(i)] = ϕ[e(i_{p}, i_{p,q})] = ⟨⟨x, y, 1⟩⟩

for all i ∈ {i_{p}, j_{r} | i_{p} ∈ J′_{2}, 1 ≤ r ≤ s} such that e(k_{1}) appears in b(i) and for all e(i_{p}, i_{p,q}) ∈ B_{2} such that i_{p} = k_{1} or e(k_{1}) appears in b(i_{p}), and ϕ[b] = 0 for all remaining b ∈ B_{2} ∪ B_{3} in the Witt ring of K((z)). Hence, ∂_{z}(α(ϕ)) = (x, y) ≠ 0, thus α(ϕ) ramifies. □

We present the second main result on the group of unramified degree 3 invariants for type B.

Theorem 6.5. Let G = (∏_{i=1}^{m} Spin_{2n_{i}+1})/μ defined over an algebraically closed field F, m, n_{i} ≥ 1, where μ is a central subgroup. Then, every unramified degree 3 invariant of G is trivial, i.e., Inv_{nr}^{3}(G) = 0.

Proof. Let G_{red} = (∏_{i=1}^{m} Γ_{2n_{i}+1})/μ. Since the classifying space BG is stably birational to the classifying space BG_{red}, by (1) we have Inv_{nr}^{3}(G) = Inv_{nr}^{3}(G_{red}). We shall show that Inv_{nr}^{3}(G_{red}) = 0. Let G′ = (Spin_{3})^{m}/μ and G′_{red} = (Γ_{3})^{m}/μ. Then, the standard embeddings Spin_{3} → Spin_{2n_{i}+1} and Γ_{3} → Γ_{2n_{i}+1} induce morphisms G′ → G and G′_{red} → G_{red}, thus we have

(57)  \xymatrix{Inv^{3}(G) \ar[r] & Inv^{3}(G′) \\ Inv^{3}(G_{red}) \ar@{^{(}->}[u] \ar[r] & Inv^{3}(G′_{red}) \ar@{^{(}->}[u]}

By (50) in Proposition 6.3 and Lemma 6.4 we may assume that the bottom map in (57) is an isomorphism. By [14, Lemma 4.3] and the exceptional isomorphism A_{1} = B_{1}, we have Inv_{nr}^{3}(G′_{red}) = 0, thus every invariant of G_{red} is ramified. □

6.2. Type C.

Lemma 6.6. Let G = (∏_{i=1}^{m} Sp_{2n_{i}})/μ, m, n_{i} ≥ 1, where μ is a central subgroup. Let R be the subgroup of (μ_{2}^{m})^{∗} = (ℤ/2ℤ)^{m} whose quotient is the character group μ^{∗}. Set G_{red} = (∏_{i=1}^{m} GSp_{2n_{i}})/μ, where GSp_{2n_{i}} denotes the group of symplectic similitudes. Then, for any field extension K/F the first Galois cohomology set H^{1}(K, G_{red}) is bijective to the set of m-tuples ((A_{1}, σ_{1}), …, (A_{m}, σ_{m})) of pairs of central simple Kalgebra A_{i} of degree 2n_{i} with symplectic involution σ_{i} such that for all r = (r_{i}) ∈ R

r_{1}A_{1} + ⋯ + r_{m}A_{m} = 0 in Br(K),

where Br(K) denotes the Brauer group of K.

Proof. Let G_{red} = (∏_{i=1}^{m} GSp_{2n_{i}})/μ, where GSp_{2n_{i}} denotes the group of symplectic similitudes. Consider the exact sequence

1 → (𝔾_{m})^{m}/μ → G_{red} → ∏_{i=1}^{m} PGSp_{2n_{i}} → 1.

Then, by the same argument as in the proof of Lemma 6.1 the set H^{1}(F, G_{red}) is bijective to the kernel of following map

H^{1}(F, ∏_{i=1}^{m} PGSp_{2n_{i}}) → Br_{2}(F)^{m} →^{τ} H^{2}(F, (μ_{2})^{m}/μ),

where the first map sends an m-tuple ((A_{1}, σ_{1}), …, (A_{m}, σ_{m})) of simple algebra A_{i} of degree 2n_{i} with symplectic involution σ_{i} to the m-tuple (A_{1}, …, A_{m}) and the map τ is induced by the natural surjection (μ_{2})^{m} → (μ_{2})^{m}/μ. Since (A_{1}, …, A_{m}) ∈ Ker(τ) if and only if it is contained in the kernel of the map in (48) for all r ∈ R, thus we have

(58)  H^{1}(F, G_{red}) ≃ {((A_{1}, σ_{1}), …, (A_{m}, σ_{m})) | deg A_{i} = 2n_{i}, ∑_{i=1}^{m} r_{i}A_{i} = 0}

for all r = (r_{i}) ∈ R. □

Let (A, σ) be a pair of central simple F -algebra A of degree 2n with involution σ of the first kind. The trace form T_{σ} : A → F is given by T_{σ}(a) = Trd(σ(a)a), where Trd denotes the reduced trace. We denote by T_{σ}^{+} the restriction of T_{σ} to Sym(A, σ). Set

(59)  ϕ[r] :=⊥_{i=1}^{m} r_{i}ϕ_{i}, where ϕ_{i} = \begin{cases}T_{σ_{i}} & if n_{i} ≡ 1 mod 2, \\ T_{σ_{i}}^{+} & if n_{i} ≡ 2 mod 4 \end{cases}

for all r = (r_{i}) ∈ R ∩ (⨁_{4∤n_{i}}(ℤ/2ℤ)e_{i}). For all i ∈ I such that 4|n_{i}, we simply write Δ for the Garibaldi-Parimala-Tignol invariant Δ(A_{i}, σ_{i}) defined in [11, Theorem A]. Then, this degree 3 invariant induces the following invariants of G_{red}

(60)  Δ_{i} : H^{1}(K, G_{red}) → H^{1}(K, PGSp_{2n_{i}}) →^{Δ} H^{3}(K),

where the first map in (60) is the projection and K/F is a field extension. We show that every invariant of semisimple group of type C is generated by the Arason invariants associated to ϕ[r] and the Garibaldi-Parimala-Tignol invariants Δ_{i}.

Proposition 6.7. Let G = (∏_{i=1}^{m} Sp_{2n_{i}})/μ defined over an algebraically closed field F, where m, n_{i} ≥ 1, μ is a central subgroup. Let R be the subgroup of (μ_{2}^{m})^{∗} whose quotient is the character group μ^{∗}. Set G_{red} = (∏_{i=1}^{m} GSp_{2n_{i}})/μ. Then, every normalized invariant in Inv^{3}(G_{red}) is of the form

(61)  ∑_{r∈R′} e_{3}(ϕ[r]) + ∑_{i∈I′} Δ_{i}

for some R′ ⊆ R ∩ (⨁_{4∤n_{i}}(ℤ/2ℤ)e_{i}) and some subset I′ ⊆ {i ∈ I | 4|n_{i}}, where ϕ[r] denotes the quadratic form defined in (59) and e_{3} : I^{3}(K) → H^{3}(K) denotes the Arason invariant over a field extension K/F. Moreover, we have

(62) Inv^{3}(G_{red})_{norm} ≃ \frac{⨁_{4 |n_{i}}(ℤ/2ℤ)e_{i} ⨁ (R ∩ (⨁_{4∤n_{i}}(ℤ/2ℤ)e_{i}))}{⟨e_{i}, e_{j} + e_{k} ∈ R | e_{j}, e_{k} ∉ R, n_{j} ≡ n_{k} ≡ 1 mod 2⟩}.

Proof. Since F is algebraically closed, we get Inv^{3}(G_{red})_{norm} = Inv^{3}(G_{red})_{ind}. Let i be an integer such that n_{i} ≡ 0 mod 4. If e_{i} ∈ R, then, as every symplectic involution on a split algebra is hyperbolic, by Lemma 6.6 and [11, Theorem A] the invariant Δ_{i} defined in (60) vanishes. Now assume that e_{i} ∉ R. Let Q = (x, y) be a division quaternion algebra over a field extension K/F and let b = ⟨1, z⟩ ⊥ h be a symmetric bilinear form on E^{n_{i}}, where h denotes a hyperbolic form and E = K((z)). Consider the linear system as in (58) with the coefficients given by a basis of R. As e_{i} ∉ R, it follows by the rank theorem (or Rouché-Capelli theorem) that there exists a G_{red}- torsor η = ((A_{1}, σ_{1}), …, (A_{m}, σ_{m})) over E such that

(63) (A_{i}, σ_{i}) = (M_{n_{i}}(Q), σ_{b} ⊗ γ) and (A_{j}, σ_{j}) = (M_{2n_{j}}(E), σ_{ω}) or (M_{n_{j}}(Q), t ⊗ γ)

for all 1 ≤ j ≠ i ≤ m, where γ denotes the canonical involution on Q, σ_{b} denotes the adjoint involution on End(E^{n_{i}}) = M_{n_{i}}(E) with respect to b, σ_{ω} denotes the adjoint involution with respect to the standard symplectic bilinear form ω, and t denotes the transpose involution on M_{n_{j}}(E). Then, by [11, Example 3.1] we have

(64)  Δ_{i}(η) = (Q) ∪ (z),

thus, ∂_{z}(α(η)) = (x, y) ≠ 0. Therefore, we have a nontrivial invariant Δ_{i} of order 2 for any i such that n_{i} ≡ 0 mod 4 and e_{i} ∉ R.

Let r ∈ R ∩ (⨁_{4∤n_{i}}(ℤ/2ℤ)e_{i}). Since each quadratic form ϕ_{i} in (59) has even dimension and trivial discriminant, we obtain ϕ[r] ∈ I^{2}(K) for each r. By [20, Theorem 1] the Hasse invariant of ϕ_{i} in (59) coincides with the class of A_{i} in Br(K), thus by the relation in (58), we have ϕ[r] ∈ I^{3}(K) for each r ∈ R ∩ (⨁_{4∤n_{i}}(ℤ/2ℤ)e_{i}). Therefore, the Arason invariant induces a normalized invariant e_{3}(ϕ[r]) of order dividing 2 that sends an m-tuple in (58) to e_{3}(ϕ[r]) ∈ H^{3}(K).

Let r ∈ R′′_{1}+R′′_{2}, where R′′_{1} = ⟨e_{i} ∈ R | n_{i} ≢ 0 mod 4⟩ and R′′_{2} denotes the subgroup of R defined in Proposition 4.2. For any e_{i} ∈ R′′_{1} and any e_{j} + e_{k} ∈ R′′_{2}, we have

(65)  ϕ_{i} = T_{σ_{i}} = h and ϕ_{j} ⊥ ϕ_{k} = T_{σ_{j}} ⊥ T_{σ_{k}} = ⟨⟨a, b, 1⟩⟩ ⊥ h′,

where A_{j} = A_{k} = (a, b) in Br(K), h and h′ denote hyperbolic forms, thus both invariants e_{3}(ϕ[e_{i}]) and e_{3}(ϕ[e_{j} + e_{k}]) vanish. Therefore, the invariant e_{3}(ϕ[r]) vanishes.

To complete the proof, by Theorem 5.5 it suffices to show that the invariant e_{3}(ϕ[r]) is nontrivial for any r ∈ R ∩ (⨁_{4∤n_{i}}(ℤ/2ℤ)e_{i})\(R′′_{1} + R′′_{2}). Let G′_{red} = (GSp_{2})^{m}/μ. Then, the rest of the proof of Proposition 6.3 still works if we replace the exceptional isomorphism A_{1} = B_{1}, the standard embedding Γ_{3} → Γ_{2n_{i}+1}, and Lemma 6.4 in the proof of Proposition 6.3 by the exceptional isomorphism A_{1} = C_{1}, the standard embedding GSp_{2} → GSp_{2n_{i}}, and Lemma 6.9, respectively. □

Remark 6.8. If m = 2, n_{1} ≡ n_{2} ≡ 0 mod 2, and μ ⊆ μ_{2}^{2} is the diagonal subgroup, then the invariant in Proposition 6.7 coincides with the invariant defined in [3].

We present the following analogue of Lemma 6.4, which plays the same role for the triviality of unramified invariants as Lemma 6.4 plays for the groups of type B.

Lemma 6.9. Let G = (∏_{i=1}^{m} Sp_{2n_{i}})/μ defined over an algebraically closed field F, where m, n_{i} ≥ 1, μ is a central subgroup. Set G_{red} = (∏_{i=1}^{m} GSp_{2n_{i}})/μ. Then, every normalized invariant in Inv^{3}(G_{red}) is ramified if either n_{i} is divisible by 4 for some i with e_{i} ∉ R_{1} or n_{j}n_{k} ≢ 1 mod 2 for some j and k such that e_{j} + e_{k} ∈ R ∩ (⨁_{4∤n_{i}}(ℤ/2ℤ)e_{i}).

Proof. Let α be a normalized invariant in Inv^{3}(G_{red}) be written as in (61) for some subspace R′ ⊆ R ∩ (⨁_{4∤n_{i}}(ℤ/2ℤ)e_{i}) and subset I′ ⊆ {i ∈ I | n_{i} ≡ 0 mod 4, e_{i} ∉ R}.

Assume that there exist i ∈ I′. Let η = ((A_{1}, σ_{1}), …, (A_{m}, σ_{m})) be a G_{red}-torsor as in the proof of Proposition 6.7. Then, by (63), [11, Example 3.1], and [11, Theorem A] we have

Δ_{j}(η) = 0

for all j ≠ i such that n_{j} ≡ 0 mod 4. Since

ϕ_{j} = \begin{cases}h & if (A_{j}, σ_{j}) = (M_{2n_{j}}(E), σ_{ω}), \\ ⟨⟨x, y⟩⟩ ⊥ h & if (A_{j}, σ_{j}) = (M_{n_{j}}(Q), t ⊗ γ), \end{cases}

where h denotes a hyperbolic form and the pairs of the form (M_{n_{j}}(Q), t ⊗ γ) appear an even number of times in the relation of (58) for any r ∈ R′, we have e_{3}(ϕ[r]) = 0 for any r ∈ R′. Therefore, by (64) we have ∂_{z}(α(η)) = (x, y) ≠ 0, thus the invariant α ramifies.

We may assume that n_{i} ≢ 0 mod 4 for all 1 ≤ i ≤ m, thus

α(η) = e_{3}(ϕ[r_{2}]) + e_{3}(ϕ[r_{3}])

for some nonzero r_{2} ∈ R_{2} and some r_{3} ∈ R_{3}, where R_{1} and R_{2} denote the subspaces of R in (12), R_{3} is a complementary subspace of R_{1} + R_{2} in R, and η is a G_{red}- torsor. Then, we choose bases B_{2} = {e(i_{p}, i_{p,q})} of R_{2} with n_{i_{p,q}} ≥ n_{i_{p}} and B_{3} of a complementary subspace of R_{2} as in Lemma 6.4 so that the invariant α is written as in (51). We show that the invariant α(η) ramifies following the proof of Lemma 6.4.

Case 1: ∃ e(i_{p}, i_{p,q}) ∈ B′_{2} with n_{i_{p}}n_{i_{p,q}} ≢ 1 mod 2 such that i_{p} ∉ J′_{2}. Let e(i_{u}, i_{u,v}) ∈ B′_{2} be such an element for some 1 ≤ u ≤ k and 1 ≤ v ≤ m_{u}. Let Q = (x, y) be a division quaternion algebra over K/F and let Q_{1} = (x, z) and Q_{2} = (x, yz) be quaternions over E. We denote by γ, γ_{1}, γ_{2} the canonical involutions on Q, Q_{1}, Q_{2}, respectively. For the sake of simplicity, we shall write the symbol d for the corresponding degree of the matrix algebras in the rest of the proof. Now we choose η = ((A_{i}, σ_{i})) for i ∈ I such that

(66) (A_{i}, σ_{i}) = (M_{d}(Q), t ⊗ γ), (A_{i_{u,v}}, σ_{i_{u,v}}) = (M_{d}(Q_{1} ⊗ Q_{2}), t ⊗ γ′_{1} ⊗ γ_{2})

for i = i_{u}, i_{u,q} and all 1 ≤ q ≠ v ≤ m_{u}, where t denotes the transpose involution on a matrix algebra and γ′_{1} denotes an orthogonal involution on Q_{1} given by the composition of γ_{1} and the inner automorphism induced by one of the generators of pure quaternions in Q_{1},

(67) (A_{i_{p}}, σ_{i_{p}}), (A_{i_{p,q}}, σ_{i_{p,q}}) = \begin{cases}(M_{d}(Q), t ⊗ γ) & if e(i_{u}) appears in b(i_{p}), \\ (M_{d}(E), σ_{ω}) & otherwise, \end{cases}

for all i_{p} ∈ J′_{2} and all q with e(i_{p}, i_{p,q}) ∈ B_{2}, and

(68)  (A_{i}, σ_{i}) = (M_{d}(E), σ_{ω})

for the remaining i ∈ I. Then, we have

ϕ[e(i_{u})] = ϕ[e(i_{u,q})] = ⟨⟨x, y⟩⟩ ⊥ h, ϕ[e(i_{u,v})] = ⟨z, xz, yz, xyz⟩ ⊥ h

for all 1 ≤ q ≠ v ≤ m_{u}, thus we obtain (53), (54), and ϕ[b] = 0 for all remaining b ∈ B_{2} ∪ B_{3} in the Witt ring of E. Hence, ∂_{z}(α(η)) = (x, y) ≠ 0, i.e., α ramifies.

Case 2: ∃ e(i_{p}, i_{p,q}) ∈ B′_{2} with n_{i_{p}}n_{i_{p,q}} ≢ 1 mod 2 such that i_{p} ∈ J′_{2}. Let e(i_{u}, i_{u,v}) ∈ B′_{2} be such an element. We choose k_{1} as in (55) and then choose (A_{k_{1}}, σ_{k_{1}}) and (A_{i_{u,v}}, σ_{i_{u,v}}) as in (66). Then, we choose (A_{i}, σ_{i}) for i ∈ I\{i_{u,v}, k_{1}} such that

(69)  (A_{i}, σ_{i}) = \begin{cases}(M_{d}(Q), t ⊗ γ) & if e(k_{1}) appears in b(i), \\ (M_{d}(E), σ_{ω}) & otherwise \end{cases}

for all i ∈ {i_{p}, j_{r} | i_{p} ∈ J′_{2}, 1 ≤ r ≤ s},

(70) (A_{i_{pq}}, σ_{i_{p,q}}) = \begin{cases}(M_{d}(Q), t ⊗ γ) & if i_{p} = k_{1} or e(k_{1}) appears in b(i_{p}), \\ (M_{d}(E), σ_{ω}) & otherwise \end{cases}

for all q such that e(i_{p}, i_{p,q}) ∈ B_{2}, and

(71)  (A_{i}, σ_{i}) = (M_{d}(E), σ_{ω})

for the remaining i ∈ I\{i_{u,v}, k_{1}}. Therefore, we obtain (53), (56), and ϕ[b] = 0 for all remaining b ∈ B_{2} ∪ B_{3} in the Witt ring of E. Therefore, ∂_{z}(α(η)) = (x, y) ≠ 0, thus α ramifies. □

We show that the same result in Theorem 6.5 holds for the groups of type C.

Theorem 6.10. Let G = (∏_{i=1}^{m} Sp_{2n_{i}})/μ defined over an algebraically closed field F, m, n_{i} ≥ 1, where μ is a central subgroup. Then, every unramified degree 3 invariant of G is trivial, i.e., Inv_{nr}^{3}(G) = 0.

Proof. Let G_{red} = (∏_{i=1}^{m} GSp_{2n_{i}})/μ, G′_{red} = (GSp_{2})^{m}/μ, and G′ = (Sp_{2})^{m}/μ. Then, the proof of Theorem 6.5 still works if we replace Proposition 6.3, Lemma 6.4, and the exceptional isomorphism A_{1} = B_{1} in the proof by Proposition 6.7, Lemma 6.9, and the exceptional isomorphism A_{1} = C_{1}, respectively. □

6.3. Type D.

Lemma 6.11. Let G = (∏_{i=1}^{m} Spin_{2n_{i}})/μ, m, ≥ 1, n_{i} ≥ 3, where μ is a central subgroup.Let R be the subgroup of the character group Z defined in (17) such that μ^{∗} = Z/R. Set G_{red} = (∏_{i=1}^{m} Ω_{2n_{i}})/μ, where Ω_{2n_{i}} denotes the extended Clifford group. Then, for any field extension K/F the first Galois cohomology set H^{1}(K, G_{red}) is bijective to the set of m-tuples ((A_{1}, σ_{1}, f_{1}), …, (A_{m}, σ_{m}, f_{m})) of triples consisting of a central simple K-algebra A_{i} of degree 2n_{i} with orthogonal involution σ_{i} of trivial discriminant and a K-algebra isomorphism f_{i} : Z(C(A_{i}, σ_{i})) ≃ K × K, where Z(C(A_{i}, σ_{i})) denotes the center of the Clifford algebra C(A_{i}, σ_{i}), satisfying

B_{1} + ⋯ + B_{m} = 0 in Br(K)

for all ∑_{i=1}^{m} r′_{i} ∈ R with

r′_{i} = \begin{cases}r_{i}e_{i} & if n_{i} odd, \\ r_{i,1}e_{i,1} + r_{i,2}e_{i,2} & if n_{i} even, \end{cases}

where

B_{i} := \begin{cases}r_{i}C_{i,1} or r_{i}C_{i,2} & if n_{i} odd, \\ r_{i,1}C_{i,1} + r_{i,2}C_{i,2} or r_{i,1}C_{i,2} + r_{i,2}C_{i,1} & if n_{i} even \end{cases}

depending on the choice of two isomorphisms f_{i} for each triple (A_{i}, σ_{i}, f_{i}), C_{i,1} and C_{i,2} denote simple K-algebras such that C(A_{i}, σ_{i}) = C_{i,1} × C_{i,2}, and Br(K) denotes the Brauer group of K.

Proof. Let G_{red} = (∏_{i=1}^{m} Ω_{2n_{i}})/μ, where Ω_{2n_{i}} denotes the extended Clifford group ([12, §13]). Consider the exact sequence

1 → (𝔾_{m})^{2m}/μ → G_{red} → ∏_{i=1}^{m} PGO_{2n_{i}}^{+} → 1,

where PGO_{2n_{i}}^{+} denotes the projective orthogonal group. Applying the same argument as in the proof of Lemma 6.1 we see that the set H^{1}(K, G_{red}) is bijective to the kernel of following map

H^{1}(K, ∏_{i=1}^{m} PGO_{2n_{i}}^{+}) →^{β} Br (Z(∏_{i=1}^{m} Spin_{2n_{i}})) →^{τ} H^{2}(K, Z(∏_{i=1}^{m} Spin_{2n_{i}})/μ),

where the map β sends an m-tuple ((A_{i}, σ_{i}, f_{i})) of triples consisting of a central simple K-algebra A_{i} of degree 2n_{i} with orthogonal involution σ_{i} of trivial discriminant and a K-algebra isomorphism f_{i} : Z(C(A_{i}, σ_{i})) ≃ K × K to the m-tuple (B′_{1}, …, B′_{m}) with

B′_{i} := \begin{cases}C_{i,1} or C_{i,2} & if n_{i} odd, \\ (C_{i,1}, C_{i,2}) or (C_{i,2}, C_{i,1}) & if n_{i} even, \end{cases}

depending on the choice of two isomorphisms f_{i} for each triple (A_{i}, σ_{i}, f_{i}) (i.e., For odd (resp. even) n_{i}, the image of (A_{i}, σ_{i}, f_{i}) under β is C_{i,1} (resp. (C_{i,1}, C_{i,2})) if and only if the image of (A_{i}, σ_{i}, f′_{i}) for another isomorphism f′_{i} : Z(C(A_{i}, σ_{i})) ≃ K × K under β is C_{i,2} (resp. (C_{i,2}, C_{i,1}))) and the map τ is induced by the natural surjection Z(∏_{i=1}^{m} Spin_{2n_{i}}) → Z(∏_{i=1}^{m} Spin_{2n_{i}})/μ. As (B′_{1}, …, B′_{m}) ∈ Ker(τ) if and only if it is contained in the kernel of the composition

H^{2}(K, Z(∏_{i=1}^{m} Spin_{2n_{i}}) →^{τ} H^{2}(K, Z(∏_{i=1}^{m} Spin_{2n_{i}})/μ) →^{r_{∗}} H^{2}(K, 𝔾_{m})

for all r ∈ R = (Z(∏_{i=1}^{m} Spin_{2n_{i}})/μ)^{∗}, we obtain

(72)  H^{1}(K, G_{red}) ≃ {((A_{i}, σ_{i}, f_{i})) | ∑_{i=1}^{m} B_{i} = 0 in Br(K)}

for all ∑_{i=1}^{m} r′_{i} ∈ R. □

Recall from Theorem 5.6 the following subsets

I_{1} = {i | Z_{i} = R_{1,i} or R′_{1,i}, n_{i} ≠ 3} and I_{2} = {i | R′_{1,i} = 0, 4|n_{i}} ∪ {i | R′_{1,i} = (ℤ/2ℤ)e_{i,1} or (ℤ/2ℤ)e_{i,2}, n_{i} ≥ 6, 4|n_{i}} =: I_{21} ∪ I_{22}.

Let i ∈ I_{1}. Then, from Lemma 6.11, we see that both K-algebras A_{i} and C(A_{i}, σ_{i}) split, thus we have (A_{i}, σ_{i}, f_{i}) ≃ (M_{2n_{i}}(K), σ_{ψ_{i}}) for some adjoint involution σ_{ψ_{i}} with respect to a quadratic form ψ_{i} such that ψ_{i} ∈ I^{3}(K). Hence, the Arason invariant e_{3} induces the following invariant

(73)  e_{3,i} : H^{1}(K, G_{red}) → H^{3}(K)

given by e_{3,i}((A_{1}, σ_{1}, f_{1}), …, (A_{m}, σ_{m}, f_{m})) = e_{3}(ψ_{i}). This invariant is obviously nontrivial.

Now let i ∈ I_{2}. Then, the invariant Δ′ of PGO_{2n}^{+} ([19, Theorem 4.7]) gives the following invariant of G_{red} (74)

Δ′_{i} : \begin{cases}H^{1}(K, G_{red}) → H^{1}(K, PGO_{2n_{i}}^{+}) →^{Δ′} H^{3}(K) & if i ∈ I_{21}, \\ H^{1}(K, G_{red}) → H^{1}(K, HSpin_{2n_{i}}) → H^{1}(K, PGO_{2n_{i}}^{+}) →^{Δ′} H^{3}(K) & if i ∈ I_{22}, \end{cases}

where HSpin_{2n_{i}} denotes the half-spin group and the first map in (74) is the projection for each case.

We shall need the following analogue of [11, Example 3.1].

Lemma 6.12. Let Q be a quaternion algebra over F and let (A, σ, f) ∈ H^{1}(F, PGO_{2n}^{+}) such that n ≡ 0 mod 2 and (A, σ) = (M_{n}(F) ⊗ Q, σ_{1} ⊗ σ_{2}) for some orthogonal involutions σ_{1} and σ_{2} on M_{n}(F) and Q, respectively. Then, we have

Δ′(A, σ, f) = Q ∪ (disc σ_{1}).

Proof. Let t be the transpose involution on M_{n}(F). Since σ_{1} = Int(x) ∘ t for some tsymmetric invertible element x, where Int(x) denotes the inner automorphism induced by x, we have

disc(σ_{1}) = Nrd_{M_{n}(F)}(x) = \sqrt{Nrd_{A}(x ⊗ 1)}

and σ = Int(x ⊗ 1) ∘ (t ⊗ σ_{2}), where Nrd denotes the reduced norm. As x ⊗ 1 is a σ-symmetric invertible element, the result follows from [19, §4b]. □

Proposition 6.13. Let G = (∏_{i=1}^{m} Spin_{2n_{i}})/μ defined over an algebraically closed field F, where m ≥ 1, n_{i} ≥ 3, μ is a central subgroup. Set G_{red} = (∏_{i=1}^{m} Ω_{2n_{i}})/μ, where Ω_{2n_{i}} is the extended Clifford group. Then, every normalized invariant in Inv^{3}(G_{red}) is of the form

(75)  ∑_{i∈I′_{1}} e_{3,i} + ∑_{i∈I′_{2}} Δ′_{i} + ∑_{r∈R′′} e_{3}(ϕ[r])

for some subsets I′_{1} ⊆ I_{1}, I′_{2} ⊆ I_{2}, and R′′ ⊆ R′, where R′ denotes the group as defined in Theorem 5.6, ϕ[r] is the quadratic form defined in (59) and e_{3} : I^{3}(K) → H^{3}(K) denotes the Arason invariant for a field extension K/F. Moreover, we have

Inv^{3}(G_{red})_{norm} ≃ \frac{⨁_{i∈I_{1}∪I_{2}}(ℤ/2ℤ)ē_{i} ⨁ R′}{⟨ē_{i}, ē_{j} + ē_{k} ∈ R′ | ē_{j}, ē_{k} ∉ R′, n_{j} ≡ n_{k} ≡ 1 mod 2⟩}.

Proof. Since F is algebraically closed, we obtain Inv^{3}(G_{red})_{norm} = Inv^{3}(G_{red})_{ind}. We first show that the invariant Δ′_{j} is nontrivial for all j ∈ I_{2}. Choose a field extension K/F containing variables x_{i,1}, x_{i,2}, x_{i}, y_{i}, division quaternion K-algebras

Q_{i,1} = (x_{i,1}, y_{i}), Q_{i,2} = (x_{i,2}, y_{i})

for all i ∈ I such that n_{i} ≡ 0 mod 2, and cyclic division K-algebras

P_{i} = (x_{i}, y_{i})_{4}

of exponent 4 for all i ∈ I such that n_{i} ≡ 1 mod 2. Let

Q_{i} = \begin{cases}(x_{i,1}x_{i,2}, y_{i}) & if n_{i} even, \\ (x_{i}, y_{i}) & if n_{i} odd, \end{cases} so that Q_{i} = \begin{cases}Q_{i,1} + Q_{i,2} & if n_{i} even, \\ 2P_{i} & if n_{i} odd \end{cases}

in Br(K). For r ∈ R, let

D_{1,r} = ⨂_{2|n_{i}}(Q_{i,1}^{r_{i,1}} ⊗ Q_{i,2}^{r_{i,2}}), D_{2,r} = ⨂_{2∤n_{i}} P_{i}^{r_{i}}, and D_{r} = D_{1,r} ⊗ D_{2,r}.

Let L be the function field of the product ∏_{r∈R} SB(D_{r}) of Severi-Brauer varieties SB(D_{r}) of D_{r} over K. For all i such that n_{i} ≡ 1 mod 2, consider the exterior square λ^{2}P_{i} of P_{i} with its canonical involution ρ_{i} [12, §10]. By the exceptional isomorphism A_{3} = D_{3} ([12, 15.32]) we have

(76)  C(λ^{2}P_{i}, ρ_{i}) = P_{i} × Pi^{op},

where Pi^{op} denotes the opposite algebra of P_{i}. Let χ_{i} be a skew-hermitian form over Q_{i} such that (M_{3}(Q_{i}), σ_{χ_{i}}) = (λ^{2}P_{i}, ρ_{i}), where σ_{χ_{i}} is the adjoint involution with respect to χ_{i}. Let ψ_{i} = χ_{i} ⊥ h be a skew-hermitian form over Q_{i} of rank n_{i}, where h denotes a hyperbolic form (if n_{i} = 3, then ψ_{i} = χ_{i}). We denote by σ_{ψ_{i}} the adjoint involution on M_{n_{i}}(Q_{i}) with respect to ψ_{i}. Let

(A_{i}, σ_{i}) = \begin{cases}(M_{n_{i}}(L) ⊗ Q_{i}, σ_{i,1} ⊗ σ_{i,2}) & if n_{i} even, \\ (M_{n_{i}}(Q_{i}), σ_{ψ_{i}}) & if n_{i} odd \end{cases}

for some orthogonal involutions σ_{i,1} on M_{n_{i}}(L) and σ_{i,2} on Q_{i} such that disc(σ_{i,1}) = x_{i,1} and disc(σ_{i,2}) = y_{i}. Then, by [8, Theorem 1.1] and [7, Corollary 3] together with (76) we obtain

C(A_{i}, σ_{i}) = \begin{cases}M_{2^{n_{i}−2}}(Q_{i,1}) × M_{2^{n_{i}−2}}(Q_{i,2}) & if n_{i} even, \\ M_{2^{n_{i}−3}}(P_{i}) × M_{2^{n_{i}−3}}(P_{i})^{op} & if n_{i} odd, \end{cases}

thus by a theorem of Amitsur we have a G_{red}(L)-torsor η = ((A_{i}, σ_{i}, f_{i})). Finally, by Lemma 6.12 we get Δ′_{j}(η) = (x_{j,1}, x_{j,2}, y_{j}) ≠ 0.

Now, let r = (r̄_{1}, …, r̄_{m}) ∈ R′. Then, from Lemma 6.11 we have

B_{i} = A_{i} = \begin{cases}2r̄_{i}C_{i,1} = 2r̄_{i}C_{i,2} & if n_{i} odd, \\ r̄_{i}C_{i,1} + r̄_{i}C_{i,2} & if n_{i} even, \end{cases}

in Br(K), thus the relation in (72) is equivalent to

(77)  r̄_{1}A_{1} + ⋯ + r̄_{m}A_{m} = 0.

As each quadratic form ϕ_{i} in (59) has even dimension and trivial discriminant, we have ϕ[r] ∈ I^{2}(K) for each r ∈ R′. By [20, Theorem 1] the Hasse invariant of ϕ_{i} in (59) coincides with the class of A_{i} in Br(K), thus by the relation in (77), we have ϕ[r] ∈ I^{3}(K) for each r ∈ R′. Therefore, the Arason invariant induces a normalized invariant e_{3}(ϕ[r]) of order dividing 2 that sends an m-tuple in (72) to e_{3}(ϕ[r]) ∈ H^{3}(K).

Let r ∈ R̄′′_{1} + R̄′′_{2}, where R̄′′_{1} = ⟨ē_{i} ∈ R′⟩ and R̄′′_{2} = ⟨ē_{j} + ē_{k} ∈ R′ | ē_{j}, ē_{k} ∉ R′, n_{j} ≡ n_{k} ≡ 1 mod 2⟩. Then, by (65) both invariants e_{3}(ϕ[ē_{i}]) and e_{3}(ϕ[ē_{j} + ē_{k}]) vanish for any ē_{i} ∈ R̄′′_{1} and any ē_{j} + ē_{k} ∈ R̄′′_{2}, thus e_{3}(ϕ[r]) vanishes.

As before, by Theorem 5.6 it is enough to show that the invariant e_{3}(ϕ[r]) is nontrivial for any r ∈ R′\(R̄′′_{1} +R̄′′_{2}). Let G′_{red} = (Ω_{6})^{m}/μ. Then, the same arguments as in the proof of Proposition 6.3 work if we replace [14, Lemma 4.3], the exceptional isomorphism A_{1} = B_{1}, the standard embedding Γ_{3} → Γ_{2n_{i}+1}, and Lemma 6.4 in the proof of Proposition 6.3 by [14, Lemma 4.2], the exceptional isomorphism A_{3} = D_{3}, the standard embedding Ω_{6} → Ω_{2n_{i}}, and Lemma 6.14, respectively. □

We shall present the following analogue of Lemmas 6.4 and 6.9.

Lemma 6.14. Let G = (∏_{i=1}^{m} Spin_{2n_{i}})/μ defined over an algebraically closed field F, where m ≥ 1, n_{i} ≥ 3, μ is a central subgroup. Set G_{red} = (∏_{i=1}^{m} Ω_{2n_{i}})/μ. Then, every normalized invariant in Inv^{3}(G_{red}) is ramified if either n_{i} ≥ 4 for some i ∈ I_{1} ∪ I_{2} or n_{j}n_{k} ≢ 1 mod 2 for some j and k such that ē_{j} + ē_{k} ∈ R′.

Proof. Let α be a normalized invariant in Inv^{3}(G_{red}) be written as in (75) for some subsets I′_{1} ⊆ I_{1}, I′_{2} ⊆ I_{2} and R′′ ⊆ R′.

First, assume that there exists j ∈ I′_{1}. Let Q = (x, y) be a division quaternion algebra over a field extension K/F and let ψ_{j} = ⟨⟨x, y, z⟩⟩ ⊥ h be a quadratic form over E := K((z)), where h denotes a hyperbolic form. Choose a G_{red}-torsor η = ((A_{1}, σ_{1}, f_{1}), …, (A_{m}, σ_{m}, f_{m})) such that

(A_{j}, σ_{j}, f_{j}) = (M_{2n_{j}}(E), σ_{ψ_{j}}) and (A_{i}, σ_{i}, f_{i}) = (M_{2n_{i}}(E), t)

for all 1 ≤ i ≠ j ≤ m, where σ_{ψ_{j}} denotes the adjoint involution on M_{2n_{j}}(E) with respect to ψ_{j} and t denotes the transpose involution on M_{2n_{i}}(E). Then, we have

∑_{i∈I′_{1}} e_{3,i}(η) = (x, y, z), ∑_{i∈I′_{2}} Δ′_{i}(η) = ∑_{r∈R′′} e_{3}(ϕ[r])(η) = 0.

Therefore, we have ∂_{z}(α(η)) = (x, y) ≠ 0. Hence, the invariant α ramifies.

We assume that I′_{1} = ∅ and I′_{2} ≠ ∅, i.e., α(η) = ∑_{i∈I′_{2}} Δ′_{i} + ∑_{r∈R′′} e_{3}(ϕ[r]). Let j ∈ I′_{2} and let η = ((A_{1}, σ_{1}, f_{1}), …, (A_{m}, σ_{m}, f_{m})) be a G_{red}-torsor over L as in the proof of Proposition 6.13. Then, over L((y_{j})) we have

∂_{y_{j}}(α(η)) = ∂_{y_{j}}(Δ′_{j}(η)) = ∂_{y_{j}}((x_{j,1}, x_{j,2}, y_{j})) = (x_{j,1}, x_{j,2}) ≠ 0,

thus the invariant α ramifies.

Now we may assume that n_{i} ≢ 0 mod 4 and R′_{1,i}, R_{1,i} ≠ Z_{i} for all 1 ≤ i ≤ m, thus

α(η) = e_{3}(ϕ[r_{2}]) + e_{3}(ϕ[r_{3}])

for some nonzero r_{2} ∈ R_{2} and r_{3} ∈ R_{3}, where R_{2} denotes the subspace of R̄ generated by ē_{i} + ē_{j} for all 1 ≤ i ≠ j ≤ m, R_{3} denotes a complementary subspace of R_{2} in R̄, and η is a G_{red}-torsor. For simplicity, we write e(i_{1}, …, i_{k}) for ē_{i_{1}} + ⋯ + ē_{i_{k}}. Choose bases B_{2} = {e(i_{p}, i_{p,q})} of R_{2} with n_{i_{p,q}} ≥ n_{i_{p}} and B_{3} of a complementary subspace of R_{2} as in Lemma 6.4 so that the invariant α is written as in (51).

To show that the invariant α(η) ramifies, we now proceed as in the proof of Lemma 6.9, with the following simple modifications. Let (Q, γ), (Q_{1}, γ_{1}), (Q_{2}, γ_{2}) be the quaternions with canonical involutions as in the proof of Lemma 6.9 and let σ be an orthogonal involution on Q given by the composition of γ and the inner automorphism induced by one of the generators of pure quaternions in Q. Then, the same proof as in Lemma 6.9 still works if we choose η = ((A_{i}, σ_{i}, f_{i})) satisfying (66), (67), (68) for Case 1 and (69), (70), and (71) for Case 2, after replacing the involutions γ′_{1}, γ, and σ_{ω} in those equations by γ_{1}, σ, and t, respectively. □

Finally, we prove the second main result on the group of unramified degree 3 invariants for type D.

Theorem 6.15. Let G = (∏_{i=1}^{m} Spin_{2n_{i}})/μ defined over an algebraically closed field F, m ≥ 1, n_{i} ≥ 3, where μ is a central subgroup. Then, every unramified degree 3 invariant of G is trivial, i.e., Inv_{nr}^{3}(G) = 0.

Proof. Let G_{red} = (∏_{i=1}^{m} Ω_{2n_{i}})/μ, G′_{red} = (Ω_{6})^{m}/μ, and G′ = (SL_{4})^{m}/μ. Then, by the same argument as in the proof of Theorem 6.5 together with Proposition 6.13 and Lemma 6.14 we may assume that the bottom map in (57) is an isomorphism. By [14, Lemma 4.2], we have Inv_{nr}^{3}(G′_{red}) = 0. Hence, every invariant of G_{red} is ramified. □

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(Sanghoon Baek) Department of Mathematical Sciences, KAIST, 291 Daehak-ro, Yuseong-gu, Daejeon 305-701, Republic of Korea

  E-mail address: sanghoonbaek@kaist.ac.kr
  URL: http://mathsci.kaist.ac.kr/~sbaek/