Stably Cayley Semisimple Groups

Mikhail Borovoi and Boris Kunyavskiĭ

Received: July 9, 2014

Abstract. A linear algebraic group G over a field k is called a Cayley group if it admits a Cayley map, i.e., a G-equivariant birational isomorphism over k between the group variety G and its Lie algebra Lie(G). A prototypical example is the classical “Cayley transform” for the special orthogonal group SO_{n} defined by Arthur Cayley in 1846. A linear algebraic group G is called stably Cayley if G × S is Cayley for some split k-torus S. We classify stably Cayley semisimple groups over an arbitrary field k of characteristic 0.

2010 Mathematics Subject Classification: 20G15, 20C10.

Keywords and Phrases: Linear algebraic group, stably Cayley group, quasi-permutation lattice.

To Alexander Merkurjev on the occasion of his 60th birthday

0 Introduction

Let k be a field of characteristic 0 and k̄ a fixed algebraic closure of k. Let G be a connected linear algebraic k-group. A birational isomorphism ϕ: G −−→^{≃} Lie(G) is called a Cayley map if it is equivariant with respect to the conjugation action of G on itself and the adjoint action of G on its Lie algebra Lie(G), respectively. A linear algebraic k-group G is called Cayley if it admits a Cayley map, and stably Cayley if G ×_{k} (𝔾_{m,k})^{r} is Cayley for some r ≥ 0. Here 𝔾_{m,k} denotes the multiplicative group over k. These notions were introduced by Lemire, Popov and Reichstein [LPR]; for a more detailed discussion and numerous classical examples we refer the reader to [LPR, Introduction]. The main results of [LPR] are the classifications of Cayley and stably Cayley simple groups over an algebraically closed field k of characteristic 0. Over an arbitrary field k of characteristic 0 stably Cayley simple k-groups, stably Cayley simply connected semisimple k-groups and stably Cayley adjoint semisimple k-groups were classified in the paper [BKLR] of Borovoi, Kunyavskiĭ, Lemire and Reichstein. In the present paper, building on results of [LPR] and [BKLR], we classify all stably Cayley semisimple k-groups (not necessarily simple, or simply connected, or adjoint) over an arbitrary field k of characteristic 0.

By a semisimple (or reductive) k-group we always mean a connected semisimple (or reductive) k-group. We shall need the following result of [BKLR] extending [LPR, Theorem 1.28].

Theorem 0.1 ([BKLR, Theorem 1.4]). Let k be a field of characteristic 0 and G an absolutely simple k-group. Then the following conditions are equivalent:

(a) G is stably Cayley over k;

(b) G is an arbitrary k-form of one of the following groups:

SL_{3}, PGL_{2}, PGL_{2n+1} (n ≥ 1), SO_{n} (n ≥ 5), Sp_{2n} (n ≥ 1), G_{2},

or an inner k-form of PGL_{2n} (n ≥ 2).

In this paper we classify stably Cayley semisimple groups over an algebraically closed field k of characteristic 0 (Theorem 0.2) and, more generally, over an arbitrary field k of characteristic 0 (Theorem 0.3). Note that Theorem 0.2 was conjectured in [BKLR, Remark 9.3].

Theorem 0.2. Let k be an algebraically closed field of characteristic 0 and G a semisimple k-group. Then G is stably Cayley if and only if G decomposes into a direct product G_{1} ×_{k} ⋯ ×_{k} G_{s} of its normal subgroups, where each G_{i} (i = 1, …, s) either is a stably Cayley simple k-group (i.e., isomorphic to one of the groups listed in Theorem 0.1) or is isomorphic to the stably Cayley semisimple k-group SO_{4}.

Theorem 0.3. Let G be a semisimple k-group over a field k of characteristic 0 (not necessarily algebraically closed). Then G is stably Cayley over k if and only if G decomposes into a direct product G_{1} ×_{k} ⋯ ×_{k} G_{s} of its normal ksubgroups, where each G_{i} (i = 1, …, s) is isomorphic to the Weil restriction R_{l_{i}/k}G_{i,l_{i}} for some finite field extension l_{i}/k, and each G_{i,l_{i}} is either a stably Cayley absolutely simple group over l_{i} (i.e., one of the groups listed in Theorem 0.1) or an l_{i}-form of the semisimple group SO_{4} (which is always stably Cayley, but is not absolutely simple and can be not l_{i}-simple).

Note that the “if” assertions in Theorems 0.2 and 0.3 follow immediately from the definitions.

The rest of the paper is structured as follows. In Section 1 we recall the definition of a quasi-permutation lattice and state some known results, in particular, an assertion from [LPR, Theorem 1.27] that reduces Theorem 0.2 to an assertion on lattices. In Sections 2 and 3 we construct certain families of nonquasi-permutation lattices. In particular, we correct an inaccuracy in [BKLR]; see Remark 2.5. In Section 4 we prove (in the language of lattices) Theorem 0.2 in the special case when G is isogenous to a direct product of simple groups of type A_{n−1} with n ≥ 3. In Section 5 we prove (again in the language of lattices) Theorem 0.2 in the general case. In Section 6 we deduce Theorem 0.3 from Theorem 0.2. In Appendix A we prove in terms of lattices only, that certain quasi-permutation lattices are indeed quasi-permutation.

1 Preliminaries on quasi-permutation groups and on character lattices

In this section we gather definitions and known results concerning quasipermutation lattices, quasi-invertible lattices and character lattices that we need for the proofs of Theorems 0.2 and 0.3. For details see [BKLR, Sections 2 and 10] and [LPR, Introduction].

1.1. By a lattice we mean a pair (Γ, L) where Γ is a finite group acting on a finitely generated free abelian group L. We say also that L is a Γ-lattice. A Γ-lattice L is called a permutation lattice if it has a ℤ-basis permuted by Γ. Following Colliot-Thélène and Sansuc [CTS], we say that two Γ-lattices L and L′ are equivalent, and write L ∼ L′, if there exist short exact sequences

0 → L → E → P → 0  and  0 → L′ → E → P′ → 0

with the same Γ-lattice E, where P and P′ are permutation Γ-lattices. For a proof that this is indeed an equivalence relation see [CTS, Lemma 8, p. 182] or [Sw, Section 8]. Note that if there exists a short exact sequence of Γ-lattices

0 → L → L′ → Q → 0

where Q is a permutation Γ-lattice, then, taking in account the trivial short exact sequence

0 → L′ → L′ → 0 → 0,

we obtain that L ∼ L′. If Γ-lattices L, L′, M, M′ satisfy L ∼ L′ and M ∼ M′, then clearly L ⊕ M ∼ L′ ⊕ M′.

Definition 1.2. A Γ-lattice L is called a quasi-permutation lattice if there exists a short exact sequence

0 → L → P → P′ → 0,  (1.1)

where both P and P′ are permutation Γ-lattices.

Lemma 1.3 (well-known). A Γ-lattice L is quasi-permutation if and only if L ∼ 0.

Proof. If L is quasi-permutation, then sequence (1.1) together with the trivial short exact sequence

0 → 0 → P → P → 0

shows that L ∼ 0. Conversely, if L ∼ 0, then there are short exact sequences

0 → L → E → P → 0  and  0 → 0 → E → P′ → 0,

where P and P′ are permutation lattices. From the second exact sequence we have E ≅ P′, hence E is a permutation lattice, and then the first exact sequence shows that L is a quasi-permutation lattice.

Definition 1.4. A Γ-lattice L is called quasi-invertible if it is a direct summand of a quasi-permutation Γ-lattice.

Note that if a Γ-lattice L is not quasi-invertible, then it is not quasipermutation.

Lemma 1.5 (well-known). If a Γ-lattice L is quasi-permutation (resp., quasiinvertible) and L′ ∼ L, then L′ is quasi-permutation (resp., quasi-invertible) as well.

Proof. If L is quasi-permutation, then using Lemma 1.3 we see that L′ ∼ L ∼ 0, hence L′ is quasi-permutation. If L is quasi-invertible, then L ⊕ M is quasipermutation for some Γ-lattice M, and by Lemma 1.3 we have L ⊕ M ∼ 0. We see that L′ ⊕ M ∼ L ⊕ M ∼ 0, and by Lemma 1.3 we obtain that L′ ⊕ M is quasi-permutation, hence L′ is quasi-invertible.

Let ℤ[Γ] denote the group ring of a finite group Γ. We define the Γ-lattice J_{Γ} by the exact sequence

0 → ℤ ⟶^{N} ℤ[Γ] → J_{Γ} → 0,

where N is the norm map, see [BKLR, before Lemma 10.4]. We refer to [BKLR, Proposition 10.6] for a proof of the following result, due to Voskresenskiĭ [Vo1, Corollary of Theorem 7]:

Proposition 1.6. Let Γ = ℤ/pℤ × ℤ/pℤ, where p is a prime. Then the Γlattice J_{Γ} is not quasi-invertible.

Note that if Γ = ℤ/2ℤ × ℤ/2ℤ, then rank J_{Γ} = 3.

We shall use the following lemma from [BKLR]:

Lemma 1.7 ([BKLR, Lemma 2.8]). Let W_{1}, …, W_{m} be finite groups. For each i = 1, …, m, let V_{i} be a finite-dimensional ℚ-representation of W_{i}. Set V := V_{1} ⊕ ⋯ ⊕ V_{m}. Suppose L ⊂ V is a free abelian subgroup, invariant under W := W_{1} × ⋯ × W_{m}. If L is a quasi-permutation W -lattice, then for each i = 1, …, m the intersection L_{i} := L ∩ V_{i} is a quasi-permutation W_{i}-lattice.

We shall need the notion, due to [LPR] and [BKLR], of the character lattice of a reductive k-group G over a field k. Let k̄ be a separable closure of k. Let T ⊂ G be a maximal torus (defined over k). Set \overline{T} = T ×_{k} k̄, \overline{G} = G ×_{k} k̄. Let X(\overline{T}) denote the character group of \overline{T} := T ×_{k} k̄. Let W = W (\overline{G}, \overline{T}) := N_{G}(\overline{T})/\overline{T} denote the Weyl group, it acts on X(\overline{T}). Consider the canonical Galois action on X(\overline{T}), it defines a homomorphism Gal(k̄/k) → Aut X(\overline{T}). The image im ρ ⊂ Aut X(\overline{T}) normalizes W, hence im ρ · W is a subgroup of Aut X(\overline{T}). By the character lattice of G we mean the pair X (G) := (im ρ · W, X(\overline{T})) (up to an isomorphism it does not depend on the choice of T). In particular, if k is algebraically closed, then X (G) = (W, X(T)).

We shall reduce Theorem 0.2 to an assertion about quasi-permutation lattices using the following result due to [LPR]:

Proposition 1.8 ([LPR, Theorem 1.27], see also [BKLR, Theorem 1.3]). A reductive group G over an algebraically closed field k of characteristic 0 is stably Cayley if and only if its character lattice X (G) is quasi-permutation, i.e., X(T) is a quasi-permutation W (G, T)-lattice.

We shall use the following result due to Cortella and Kunyavskiĭ [CK] and to Lemire, Popov and Reichstein [LPR].

Proposition 1.9 ([CK], [LPR]). Let D be a connected Dynkin diagram. Let R = R(D) denote the corresponding root system, W = W (D) denote the Weyl group, Q = Q(D) denote the root lattice, and P = P (D) denote the weight lattice. We say that L is an intermediate lattice between Q and P if Q ⊂ L ⊂ P (note that L = Q and L = P are possible). Then the following list gives (up to an isomorphism) all the pairs (D, L) such that L is a quasi-permutation intermediate W (D)-lattice between Q(D) and P (D):

Q(A_{n}), Q(B_{n}), P (C_{n}), X (SO_{2n}) (then D = D_{n}),

or D is any connected Dynkin diagram of rank 1 or 2 (i.e. A_{1}, A_{2}, B_{2}, or G_{2}) and L is any lattice between Q(D) and P (D), (i.e., either L = P (D) or L = Q(D)).

Proof. The positive result (the assertion that the lattices in the list are indeed quasi-permutation) follows from the assertion that the corresponding groups are stably Cayley (or that their generic tori are stably rational), see the references in [CK], Section 3. See Appendix A below for a proof of this positive result in terms of lattices only. The difficult part of Proposition 1.9 is the negative result (the assertion that all the other lattices are not quasi-permutation). This was proved in [CK, Theorem 0.1] in the cases when L = Q or L = P, and in [LPR, Propositions 5.1 and 5.2] in the cases when Q ⊊ L ⊊ P (this can happen only when D = A_{n} or D = D_{n}).

Remark 1.10. It follows from Proposition 1.9 that, in particular, the following lattices are quasi-permutation: Q(A_{1}), P (A_{1}), P (A_{2}), P (B_{2}), Q(C_{2}), Q(G_{2}) = P (G_{2}), Q(D_{3}) = Q(A_{3}), X (SL_{4}/μ_{2}) = X (SO_{6}).

2 A family of non-quasi-permutation lattices

In this section we construct a family of non-quasi-permutation (even non-quasi-invertible) lattices.

2.1. We consider a Dynkin diagram D ⊔ Δ (disjoint union). We assume that D = ⨆_{i∈I} D_{i} (a finite disjoint union), where each D_{i} is of type B_{l_{i}} (l_{i} ≥ 1) or D_{l_{i}} (l_{i} ≥ 2) (and where B_{1} = A_{1}, B_{2} = C_{2}, D_{2} = A_{1} ⊔ A_{1}, and D_{3} = A_{3} are permitted). We denote by m the cardinality of the finite index set I. We assume that Δ = ⨆_{ι=1}^{μ} Δ_{ι} (disjoint union), where Δ_{ι} is of type A_{2n_{ι}−1}, n_{ι} ≥ 2 (A_{3} = D_{3} is permitted). We assume that m ≥ 1 and μ ≥ 0 (in the case μ = 0 the diagram Δ is empty).

For each i ∈ I we realize the root system R(D_{i}) of type B_{l_{i}} or D_{l_{i}} in the standard way in the space V_{i} := ℝ^{l_{i}} with basis (e_{s})_{s∈S_{i}} where S_{i} is an index set consisting of l_{i} elements; cf. [Bou, Planche II] for B_{l} (l ≥ 2) (the relevant formulas for B_{1} are similar) and [Bou, Planche IV] for D_{l} (l ≥ 3) (again, the relevant formulas for D_{2} are similar). Let M_{i} ⊂ V_{i} denote the lattice generated by the basis vectors (e_{s})_{s∈S_{i}}. Let P_{i} ⊃ M_{i} denote the weight lattice of the root system D_{i}. Set S = ⨆_{i} S_{i} (disjoint union). Consider the vector space V = ⨁_{i} V_{i} with basis (e_{s})_{s∈S}. Let M_{D} ⊂ V denote the lattice generated by the basis vectors (e_{s})_{s∈S}, then M_{D} = ⨁_{i} M_{i}. Set P_{D} = ⨁_{i} P_{i}.

For each ι = 1, …, μ we realize the root system R(Δ_{ι}) of type A_{2n_{ι}−1} in the standard way in the subspace V_{ι} of vectors with zero sum of the coordinates in the space ℝ^{2n_{ι}} with basis ε_{ι,1}, …, ε_{ι,2n_{ι}}; cf. [Bou, Planche I]. Let Q_{ι} be the root lattice of R(Δ_{ι}) with basis ε_{ι,1} − ε_{ι,2}, ε_{ι,2} − ε_{ι,3}, …, ε_{ι,2n_{ι}−1} − ε_{ι,2n_{ι}}, and let P_{ι} ⊃ Q_{ι} be the weight lattice of R(Δ_{ι}). Set Q_{Δ} = ⨁_{ι} Q_{ι}, P_{Δ} = ⨁_{ι} P_{ι}.

Set

W := ∏_{i∈I} W (D_{i}) × ∏_{ι=1}^{μ} W (Δ_{ι}), L′ = M_{D} ⊕ Q_{Δ} = ⨁_{i∈I} M_{i} ⊕ ⨁_{ι=1}^{μ} Q_{ι},

then W acts on L′ and on L′ ⊗_{ℤ} ℝ. For each i consider the vector

x_{i} = ∑_{s∈S_{i}} e_{s} ∈ M_{i},

then \frac{1}{2}x_{i} ∈ P_{i}. For each ι consider the vector

ξ_{ι} = ε_{ι,1} − ε_{ι,2} + ε_{ι,3} − ε_{ι,4} + ⋯ + ε_{ι,2n_{ι}−1} − ε_{ι,2n_{ι}} ∈ Q_{ι},

then \frac{1}{2}ξ_{ι} ∈ P_{ι}; see [Bou, Planche I]. Write

ξ′_{ι} = ε_{ι,1} − ε_{ι,2}, ξ′′_{ι} = ε_{ι,3} − ε_{ι,4} + ⋯ + ε_{ι,2n_{ι}−1} − ε_{ι,2n_{ι}},

then ξ_{ι} = ξ′_{ι} + ξ′′_{ι}. Consider the vector

v = \frac{1}{2} ∑_{i∈I} x_{i} + \frac{1}{2} ∑_{ι=1}^{μ} ξ_{ι} = \frac{1}{2} ∑_{s∈S} e_{s} + \frac{1}{2} ∑_{ι=1}^{μ} ξ_{ι} ∈ P_{D} ⊕ P_{Δ}.

Set

L = ⟨L′, v⟩,  (2.1)

then [L : L′] = 2 because v ∈ \frac{1}{2}L′ ∖ L′. Note that the sublattice L ⊂ P_{D} ⊕ P_{Δ} is W -invariant. Indeed, the group W acts on (P_{D} ⊕ P_{Δ})/(M_{D} ⊕ Q_{Δ}) trivially.

Proposition 2.2. We assume that m ≥ 1, m + μ ≥ 2. If μ = 0, we assume that not all of D_{i} are of types B_{1} or D_{2}. Then the W -lattice L as in (2.1) is not quasi-invertible, hence not quasi-permutation.

Proof. We consider a group Γ = {e, γ_{1}, γ_{2}, γ_{3}} of order 4, where γ_{1}, γ_{2}, γ_{3} are of order 2. The idea of our proof is to construct an embedding

j: Γ → W

in such a way that L, viewed as a Γ-lattice, is equivalent to its Γ-sublattice L_{1}, and L_{1} is isomorphic to a direct sum of a Γ-sublattice L_{0} ≃ J_{Γ} of rank 3 and a number of Γ-lattices of rank 1. Since by Proposition 1.6 J_{Γ} is not quasiinvertible, this will imply that L_{1} and L are not quasi-invertible Γ-lattices, and hence L is not quasi-invertible as a W -lattice. We shall now fill in the details of this argument in four steps.

Step 1. We begin by partitioning each S_{i} for i ∈ I into three (non-overlapping) subsets S_{i,1}, S_{i,2} and S_{i,3}, subject to the requirement that

|S_{i,1}| ≡ |S_{i,2}| ≡ |S_{i,3}| ≡ l_{i} (mod 2) if D_{i} is of type D_{l_{i}}.  (2.2)

We then set U_{1} to be the union of the S_{i,1}, U_{2} to be the union of the S_{i,2}, and U_{3} to be the union of the S_{i,3}, as i runs over I.

Lemma 2.3. (i) If μ ≥ 1, the subsets S_{i,1}, S_{i,2} and S_{i,3} of S_{i} can be chosen, subject to (2.2), so that U_{1} ≠ ∅.

(ii) If μ = 0 (and m ≥ 2), the subsets S_{i,1}, S_{i,2} and S_{i,3} of S_{i} can be chosen, subject to (2.2), so that U_{1}, U_{2}, U_{3} ≠ ∅.

To prove the lemma, first consider case (i). For all i such that D_{i} is of type D_{l_{i}} with odd l_{i}, we partition S_{i} into three non-empty subsets of odd cardinalities. For all the other i we take S_{i,1} = S_{i}, S_{i,2} = S_{i,3} = ∅. Then U_{1} ≠ ∅ (note that m ≥ 1) and (2.2) is satisfied.

In case (ii), if one of the D_{i} is of type D_{l_{i}} where l_{i} ≥ 3 is odd, then we partition S_{i} for each such D_{i} into three non-empty subsets of odd cardinalities. We partition all the other S_{i} as follows:

S_{i,1} = S_{i,2} = ∅ and S_{i,3} = S_{i}.  (2.3)

Clearly U_{1}, U_{2}, U_{3} ≠ ∅ and (2.2) is satisfied.

If there is no D_{i} of type D_{l_{i}} with odd l_{i} ≥ 3, but one of the D_{i}, say for i = i_{0}, is D_{l} with even l ≥ 4, then we partition S_{i_{0}} into two non-empty subsets S_{i_{0},1} and S_{i_{0},2} of even cardinalities, and set S_{i_{0},3} = ∅. We partition the sets S_{i} for i ≠ i_{0} as in (2.3) (note that by our assumption m ≥ 2). Once again, U_{1}, U_{2}, U_{3} ≠ ∅ and (2.2) is satisfied.

If there is no D_{i} of type D_{l_{i}} with l_{i} ≥ 3 (odd or even), but one of the D_{i}, say for i = i_{0}, is of type B_{l} with l ≥ 2, we partition S_{i_{0}} into two non-empty subsets S_{i_{0},1} and S_{i_{0},2}, and set S_{i_{0},3} = ∅. We partition the sets S_{i} for i ≠ i_{0} as in (2.3) (again, note that m ≥ 2). Once again, U_{1}, U_{2}, U_{3} ≠ ∅ and (2.2) is satisfied.

Since by our assumption not all of D_{i} are of type B_{1} or D_{2}, we have exhausted all the cases. This completes the proof of Lemma 2.3.

Step 2. We continue proving Proposition 2.2. We construct an embedding Γ ↪ W.

For s ∈ S we denote by c_{s} the automorphism of L taking the basis vector e_{s} to −e_{s} and fixing all the other basis vectors. For ι = 1, …, μ we define τ_{ι}^{(12)} = Transp((ι, 1), (ι, 2)) ∈ W_{ι} (the transposition of the basis vectors ε_{ι,1} and ε_{ι,2}). Set

τ_{ι}^{>2} = Transp((ι, 3), (ι, 4)) · · · · · Transp((ι, 2n_{ι} − 1), (ι, 2n_{ι})) ∈ W_{ι}.

Write Γ = {e, γ_{1}, γ_{2}, γ_{3}} and define an embedding j: Γ ↪ W as follows:

j(γ_{1}) = ∏_{s∈S∖U_{1}} c_{s} · ∏_{ι=1}^{μ} τ_{ι}^{(12)}τ_{ι}^{>2} ;

j(γ_{2}) = ∏_{s∈S∖U_{2}} c_{s} · ∏_{ι=1}^{μ} τ_{ι}^{(12)} ;

j(γ_{3}) = ∏_{s∈S∖U_{3}} c_{s} · ∏_{ι=1}^{μ} τ_{ι}^{>2} .

Note that if D_{i} is of type D_{l_{i}}, then by (2.2) for κ = 1, 2, 3 the cardinality #(S_{i} ∖ S_{i,κ}) is even, hence the product of c_{s} over s ∈ S_{i} ∖ S_{i,κ} is contained in W (D_{i}) for all such i, and therefore, j(γ_{κ}) ∈ W. Since j(γ_{1}), j(γ_{2}) and j(γ_{3}) commute, are of order 2, and j(γ_{1})j(γ_{2}) = j(γ_{3}), we see that j is a homomorphism. If μ ≥ 1, then, since 2n_{1} ≥ 4, clearly j(γ_{κ}) ≠ 1 for κ = 1, 2, 3, hence j is an embedding. If μ = 0, then the sets S ∖ U_{1}, S ∖ U_{2} and S ∖ U_{3} are nonempty, and again j(γ_{κ}) ≠ 1 for κ = 1, 2, 3, hence j is an embedding.

Step 3. We construct a Γ-sublattice L_{0} of rank 3. Write a vector x ∈ L as

x = ∑_{s∈S} b_{s}e_{s} + ∑_{ι=1}^{μ} ∑_{ν=1}^{2n_{ι}} β_{ι,ν}ε_{ι,ν},

where b_{s} , β_{ι,ν} ∈ \frac{1}{2}ℤ. Set n′ = ∑_{ι=1}^{μ}(n_{ι} − 1). Define a Γ-equivariant homomorphism

ϕ: L → ℤ^{n′}, x ↦ (β_{ι,2λ−1} + β_{ι,2λ})_{ι=1,...,μ, λ=2,...,n_{ι}}

(we skip λ = 1). We obtain a short exact sequence of Γ-lattices

0 → L_{1} → L ⟶^{ϕ} ℤ^{n′} → 0,

where L_{1} := ker ϕ. Since Γ acts trivially on ℤ^{n′}, we have L_{1} ∼ L. Therefore, it suffices to show that L_{1} is not quasi-invertible.

Recall that

v = \frac{1}{2} ∑_{s∈S} e_{s} + \frac{1}{2} ∑_{ι=1}^{μ} ξ_{ι}.

Set v_{1} = γ_{1} · v, v_{2} = γ_{2} · v, v_{3} = γ_{3} · v. Set

L_{0} = ⟨v, v_{1}, v_{2}, v_{3}⟩.

We have

v_{1} = \frac{1}{2} ∑_{s∈U_{1}} e_{s} − \frac{1}{2} ∑_{s∈U_{2}∪U_{3}} e_{s} − \frac{1}{2} ∑_{ι=1}^{μ} ξ_{ι},

whence

v + v_{1} = ∑_{s∈U_{1}} e_{s}.  (2.4)

We have

v_{2} = \frac{1}{2} ∑_{s∈U_{2}} e_{s} − \frac{1}{2} ∑_{s∈U_{1}∪U_{3}} e_{s} + \frac{1}{2} ∑_{ι=1}^{μ}(−ξ′_{ι} + ξ′′_{ι}),

whence

v + v_{2} = ∑_{s∈U_{2}} e_{s} + ∑_{ι=1}^{μ} ξ′′_{ι}.  (2.5)

We have

v_{3} = \frac{1}{2} ∑_{s∈U_{3}} e_{s} − \frac{1}{2} ∑_{s∈U_{1}∪U_{2}} e_{s} + \frac{1}{2} ∑_{ι=1}^{μ}(ξ′_{ι} − ξ′′_{ι}),

whence

v + v_{3} = ∑_{s∈U_{3}} e_{s} + ∑_{ι=1}^{μ} ξ′_{ι}.  (2.6)

Clearly, we have

v + v_{1} + v_{2} + v_{3} = 0.

Since the set {v, v_{1}, v_{2}, v_{3}} is the orbit of v under Γ, the sublattice L_{0} = ⟨v, v_{1}, v_{2}, v_{3}⟩ ⊂ L is Γ-invariant. If μ ≥ 1, then U_{1} ≠ ∅, and we see from (2.4), (2.5) and (2.6) that rank L_{0} ≥ 3. If μ = 0, then U_{1}, U_{2}, U_{3} ≠ ∅, and again we see from (2.4), (2.5) and (2.6) that rank L_{0} ≥ 3. Thus rank L_{0} = 3 and L_{0} ≃ J_{Γ}, whence by Proposition 1.6 L_{0} is not quasi-invertible.

Step 4. We show that L_{0} is a direct summand of L_{1}. Set m′ = |S|.

First assume that μ ≥ 1. Choose u_{1} ∈ U_{1} ⊂ S. Set S′ = S ∖ {u_{1}}. For each s ∈ S′ (i.e., s ≠ u_{1}) consider the one-dimensional (i.e., of rank 1) lattice X_{s} = ⟨e_{s}⟩. We obtain m′ − 1 Γ-invariant one-dimensional sublattices of L_{1}.

Denote by Υ the set of pairs (ι, λ) such that 1 ≤ ι ≤ μ, 1 ≤ λ ≤ n_{ι}, and if ι = 1, then λ ≠ 1, 2. For each (ι, λ) ∈ Υ consider the one-dimensional lattice

Ξ_{ι,λ} = ⟨ε_{ι,2λ−1} − ε_{ι,2λ}⟩.

We obtain −2 + ∑_{ι=1}^{μ} n_{ι} one-dimensional Γ-invariant sublattices of L_{1}.

We show that

L_{1} = L_{0} ⊕ ⨁_{s∈S′} X_{s} ⊕ ⨁_{(ι,λ)∈Υ} Ξ_{ι,λ}.  (2.7)

Set L′_{1} = ⟨ L_{0}, (X_{s})_{s≠u_{1}}, (Ξ_{ι,λ})_{(ι,λ)∈Υ} ⟩, then

rank L′_{1} ≤ 3 + (m′ − 1) − 2 + ∑_{ι=1}^{μ} n_{ι} = m′ + ∑_{ι=1}^{μ}(2n_{ι} − 1) − ∑_{ι=1}^{μ}(n_{ι} − 1) = rank L_{1}.

(2.8)

Therefore, it suffices to check that L′_{1} ⊃ L_{1}. The set

{v} ∪ {e_{s} | s ∈ S} ∪ {ε_{ι,2λ−1} − ε_{ι,2λ} | 1 ≤ ι ≤ μ, 1 ≤ λ ≤ n_{ι}}

is a set of generators of L_{1}. By construction v, v_{1}, v_{2}, v_{3} ∈ L_{0} ⊂ L′_{1}. We have e_{s} ∈ X_{s} ⊂ L′_{1} for s ≠ u_{1}. By (2.4) ∑_{s∈U_{1}} e_{s} ∈ L′_{1}, hence e_{u_{1}} ∈ L′_{1}. By construction

ε_{ι,2λ−1} − ε_{ι,2λ} ∈ L′_{1}, for all (ι, λ) ≠ (1, 1), (1, 2) .

From (2.6) and (2.5) we see that

∑_{ι=1}^{μ}(ε_{ι,1} − ε_{ι,2}) ∈ L′_{1}, ∑_{ι=1}^{μ} ξ′′_{ι} ∈ L′_{1} .

Thus

ε_{1,1} − ε_{1,2} ∈ L′_{1}, ε_{1,3} − ε_{1,4} ∈ L′_{1} .

We conclude that L′_{1} ⊃ L_{1}, hence L_{1} = L′_{1}. From dimension count (2.8) we see that (2.7) holds.

Now assume that μ = 0. Then for each κ = 1, 2, 3 we choose an element u_{κ} ∈ U_{κ} and set U′_{κ} = U_{κ} ∖{u_{κ}}. We set S′ = U′_{1} ∪U′_{2} ∪U′_{3} = S ∖{u_{1}, u_{2}, u_{3}}. Again for s ∈ S′ (i.e., s ≠ u_{1}, u_{2}, u_{3}) consider the one-dimensional lattice X_{s} = ⟨e_{s}⟩. We obtain m′−3 one-dimensional Γ-invariant sublattices of L_{1} = L. We show that

L_{1} = L_{0} ⊕ ⨁_{s∈S′} X_{s} .  (2.9)

Set L′_{1} = ⟨L_{0}, (X_{s})_{s∈S′}⟩, then

rank L′_{1} ≤ 3 + m′ − 3 = m′ = rank L_{1}.  (2.10)

Therefore, it suffices to check that L′_{1} ⊃ L_{1}. The set {v} ∪ {e_{s} | s ∈ S} is a set of generators of L_{1} = L. By construction v, v_{1}, v_{2}, v_{3} ∈ L′_{1} and e_{s} ∈ L′_{1} for s ≠ u_{1}, u_{2}, u_{3}. We see from (2.4), (2.5), (2.6) that e_{s} ∈ L′_{1} also for s = u_{1}, u_{2}, u_{3}. Thus L′_{1} ⊃ L_{1}, hence L′_{1} = L_{1}. From dimension count (2.10) we see that (2.9) holds.

We see that in both cases μ ≥ 1 and μ = 0, the sublattice L_{0} is a direct summand of L_{1}. Since by Proposition 1.6 L_{0} is not quasi-invertible as a Γlattice, it follows that L_{1} and L are not quasi-invertible as Γ-lattices. Thus L is not quasi-invertible as a W -lattice. This completes the proof of Proposition 2.2.

Remark 2.4. Since � ^{2}(Γ, J_{Γ}) ≅ ℤ/2ℤ (Voskresenskiĭ, see [BKLR, Section 10] for the notation and the result), our argument shows that � ^{2}(Γ, L) ≅ ℤ/2ℤ. Remark 2.5. The proof of [BKLR, Lemma 12.3] (which is a version with μ = 0 of Lemma 2.3 above) contains an inaccuracy, though the lemma as stated is correct. Namely, in [BKLR] we write that if there exists i such that Δ_{i} is of type D_{l_{i}} where l_{i} ≥ 3 is odd, then we partition S_{i} for one such i into three non-empty subsets S_{i,1}, S_{i,2} and S_{i,3} of odd cardinalities, and we partition all the other S_{i} as in [BKLR, (12.4)]. However, this partitioning of the sets S_{i} into three subsets does not satisfy [BKLR, (12.3)] for other i such that Δ_{i} is of type D_{l_{i}} with odd l_{i}. This inaccuracy can be easily corrected: we should partition S_{i} for each i such that Δ_{i} is of type D_{l_{i}} with odd l_{i} into three non-empty subsets of odd cardinalities.

3 More non-quasi-permutation lattices

In this section we construct another family of non-quasi-permutation lattices.

3.1. For i = 1, …, r let Q_{i} = ℤA_{n_{i}−1} and P_{i} = Λ_{n_{i}} denote the root lattice and the weight lattice of SL_{n_{i}}, respectively, and let W_{i} = 𝔖_{n_{i}} denote the corresponding Weyl group (the symmetric group on n_{i} letters) acting on P_{i} and Q_{i}. Set F_{i} = P_{i}/Q_{i}, then W_{i} acts trivially on F_{i}. Set

Q = ⨁_{i=1}^{r} Q_{i}, P = ⨁_{i=1}^{r} P_{i}, W = ∏_{i=1}^{r} W_{i},

then Q ⊂ P and the Weyl group W acts on Q and P. Set

F = P/Q = ⨁_{i=1}^{r} F_{i},

then W acts trivially on F.

We regard Q_{i} = ℤA_{n_{i}−1} and P_{i} = Λ_{n_{i}} as the lattices described in Bourbaki [Bou, Planche I]. Then we have an isomorphism F_{i} ≅ ℤ/n_{i}ℤ. Note that for each 1 ≤ i ≤ r, the set {α_{κ,i} | 1 ≤ κ ≤ n_{i} − 1} is a ℤ-basis of Q_{i}.

Set c = gcd(n_{1}, …, n_{r}); we assume that c > 1. Let d > 1 be a divisor of c. For each i = 1, …, r, let ν_{i} ∈ ℤ be such that 1 ≤ ν_{i} < d, gcd(ν_{i}, d) = 1, and assume that ν_{1} = 1. We write ν = (ν_{i})_{i=1}^{r} ∈ ℤ^{r}. Let \overline{ν} denote the image of ν in (ℤ/dℤ)^{r}. Let S_{ν} ⊂ (ℤ/dℤ)^{r} ⊂ ⨁_{i=1}^{r} ℤ/n_{i}ℤ = F denote the cyclic subgroup of order d generated by \overline{ν}. Let L_{ν} denote the preimage of S_{ν} ⊂ F in P under the canonical epimorphism P ↠ F, then Q ⊂ L_{ν} ⊂ P.

Proposition 3.2. Let W and the W -lattice L_{ν} be as in Subsection 3.1. In the case d = 2^{s} we assume that ∑ n_{i} > 4. Then L_{ν} is not quasi-permutation.

This proposition follows from Lemmas 3.3 and 3.8 below.

Lemma 3.3. Let p|d be a prime. Then for any subgroup Γ ⊂ W isomorphic to (ℤ/pℤ)^{m} for some natural m, the Γ-lattices L_{ν} and L_{1} := L_{(1,...,1)} are equivalent for any ν = (ν_{1}, …, ν_{r}) as above (in particular, we assume that ν_{1} = 1).

Note that this lemma is trivial when d = 2.

3.4. We compute the lattice L_{ν} explicitly. First let r = 1. We have Q = Q_{1}, P = P_{1}. Then P_{1} is generated by Q_{1} and an element ω ∈ P_{1} whose image in P_{1}/Q_{1} is of order n_{1}. We may take

ω = \frac{1}{n_{1}} [(n_{1} − 1)α_{1} + (n_{1} − 2)α_{2} + ⋯ + 2α_{n_{1}−2} + α_{n_{1}−1}],

where α_{1}, …, α_{n_{1}−1} are the simple roots, see [Bou, Planche I]. There exists exactly one intermediate lattice L between Q_{1} and P_{1} such that [L : Q_{1}] = d, and it is generated by Q_{1} and the element

w = \frac{n_{1}}{d} ω = \frac{1}{d}[(n_{1} − 1)α_{1} + (n_{1} − 2)α_{2} + ⋯ + 2α_{n_{1}−2} + α_{n_{1}−1}].

Now for any natural r, the lattice L_{ν} is generated by Q and the element

w_{ν} = \frac{1}{d} ∑_{i=1}^{r} ν_{i}[(n_{i} − 1)α_{1,i} + (n_{i} − 2)α_{2,i} + ⋯ + 2α_{n_{i}−2,i} + α_{n_{i}−1,i}].

In particular, L_{1} is generated by Q and

w_{1} = \frac{1}{d} ∑_{i=1}^{r}[(n_{i} − 1)α_{1,i} + (n_{i} − 2)α_{2,i} + ⋯ + 2α_{n_{i}−2,i} + α_{n_{i}−1,i}].

3.5. Proof of Lemma 3.3. Recall that L_{ν} = ⟨Q, w_{ν}⟩ with

Q = ⟨α_{κ,i}⟩ , where i = 1, …, r, κ = 1, …, n_{i} − 1.

Set Q_{ν} = ⟨ν_{i}α_{κ,i}⟩. Denote by 𝔗_{ν} the endomorphism of Q that acts on Q_{i} by multiplication by ν_{i}. We have Q_{1} = Q, Q_{ν} = 𝔗_{ν}Q_{1}, w_{ν} = 𝔗_{ν}w_{1}. Consider

𝔗_{ν}L_{1} = ⟨Q_{ν}, w_{ν}⟩.

Clearly the W -lattices L_{1} and 𝔗_{ν}L_{1} are isomorphic. We have an embedding of W -lattices Q ↪ L_{ν}, in particular, an embedding Q ↪ L_{1}, which induces an embedding 𝔗_{ν}Q ↪ 𝔗_{ν}L_{1}. Set M_{ν} = L_{ν}/𝔗_{ν}L_{1}, then we obtain a homomorphism of W -modules Q/𝔗_{ν}Q → M_{ν}, which is an isomorphism by Lemma 3.6 below.

Now let p|d be a prime. Let Γ ⊂ W be a subgroup isomorphic to (ℤ/pℤ)^{m} for some natural m. As in [LPR, Proof of Proposition 2.10], we use Roiter’s version [Ro, Proposition 2] of Schanuel’s lemma. We have exact sequences of Γ-modules

0 → 𝔗_{ν}L_{1} → L_{ν} → M_{ν} → 0,

0 → Q ⟶^{𝔗_{ν}} Q → M_{ν} → 0.

Since all ν_{i} are prime to p, we have |Γ| · M_{ν} = p^{m}M_{ν} = M_{ν}, and by [Ro, Corollary of Proposition 3] the morphisms of ℤ[Γ]-modules L_{ν} → M_{ν} and Q → M_{ν} are projective in the sense of [Ro, § 1]. Now by [Ro, Proposition 2] there exists an isomorphism of Γ-lattices L_{ν} ⊕ Q ≃ 𝔗_{ν}L_{1} ⊕ Q. Since Q is a quasi-permutation W -lattice, it is a quasi-permutation Γ-lattice, and by Lemma 3.7 below, L_{ν} ∼ 𝔗_{ν}L_{1} as Γ-lattices. Since 𝔗_{ν}L_{1} ≃ L_{1}, we conclude that L_{ν} ∼ L_{1}.

Lemma 3.6. With the above notation L_{ν}/𝔗_{ν}L_{1} ≃ Q/𝔗_{ν}Q = ⨁_{i=2}^{r} Q_{i}/ν_{i}Q_{i}.

Proof. We have 𝔗_{ν}L_{1} = ⟨S_{ν}⟩, where S_{ν} = {ν_{i}α_{κ,i}}_{i,κ} ∪ {w_{ν}}. Note that

dw_{ν} = ∑_{i=1}^{r} ν_{i}[(n_{i} − 1)α_{1,i} + (n_{i} − 2)α_{2,i} + ⋯ + 2α_{n_{i}−2,i} + α_{n_{i}−1,i}].

We see that dw_{ν} is a linear combination with integer coefficients of ν_{i}α_{κ,i} and that α_{n_{1}−1,1} appears in this linear combination with coefficient 1 (because ν_{1} = 1). Set B′_{ν} = S_{ν} ∖{α_{n_{1}−1,1}}, then ⟨B′_{ν}⟩ ∋ α_{n_{1}−1,1}, hence ⟨B′_{ν}⟩ = ⟨S_{ν}⟩ = 𝔗_{ν}L_{1}, thus B′_{ν} is a basis of 𝔗_{ν}L_{1}. Similarly, the set B_{ν} := {α_{κ,i}}_{i,κ}∪{w_{ν}}∖{α_{n_{1}−1,1}} is a basis of L_{ν}. Both bases B_{ν} and B′_{ν} contain α_{1,1}, …, α_{n_{1}−2,1} and w_{ν}. For all i = 2, …, r and all κ = 1, …, n_{i} − 1, the basis B_{ν} contains α_{κ,i}, while B′_{ν} contains ν_{i}α_{κ,i}. We see that the homomorphism of W -modules Q/𝔗_{ν}Q = ⨁_{i=2}^{r} Q_{i}/ν_{i}Q_{i} → L_{ν}/𝔗_{ν}L_{1} is an isomorphism.

Lemma 3.7. Let Γ be a finite group, A and A′ be Γ-lattices. If A⊕B ∼ A′ ⊕B′, where B and B′ are quasi-permutation Γ-lattices, then A ∼ A′.

Proof. Since B and B′ are quasi-permutation, by Lemma 1.3 they are equivalent to 0, and we have

A = A ⊕ 0 ∼ A ⊕ B ∼ A′ ⊕ B′ ∼ A′ ⊕ 0 = A′.

This completes the proof of Lemma 3.7 and hence of Lemma 3.3.

To complete the proof of Proposition 3.2 it suffices to prove the next lemma.

Lemma 3.8. Let p|d be a prime. Then there exists a subgroup Γ ⊂ W isomorphic to (ℤ/pℤ)^{m} for some natural m such that the Γ-lattice L_{1} := L_{(1,...,1)} is not quasi-permutation.

3.9. Denote by U_{i} the space ℝ^{n_{i}} with canonical basis ε_{1,i} , ε_{2,i} , …, ε_{n_{i},i}. Denote by V_{i} the subspace of codimension 1 in U_{i} consisting of vectors with zero sum of the coordinates. The group W_{i} = 𝔖_{n_{i}} (the symmetric group) permutes the basis vectors ε_{1,i} , ε_{2,i} , …, ε_{n_{i},i} and thus acts on U_{i} and V_{i}. Consider the homomorphism of vector spaces

χ_{i}: U_{i} → ℝ, ∑_{λ=1}^{n_{i}} β_{λ,i} ε_{λ,i} ↦ ∑_{λ=1}^{n_{i}} β_{λ,i}

taking a vector to the sum of its coordinates. Clearly this homomorphism is W_{i}-equivariant, where W_{i} acts trivially on ℝ. We have short exact sequences

0 → V_{i} → U_{i} ⟶^{χ_{i}} ℝ → 0.

Set U = ⨁_{i=1}^{r} U_{i}, V = ⨁_{i=1}^{r} V_{i}. The group W = ∏_{i=1}^{r} W_{i} naturally acts on U and V, and we have an exact sequence of W -spaces

0 → V → U ⟶^{χ} ℝ^{r} → 0,  (3.1)

where χ = (χ_{i})_{i=1,...,r} and W acts trivially on ℝ^{r}.

Set n = ∑_{i=1}^{r} n_{i}. Consider the vector space \overline{U} := ℝ^{n} with canonical basis \overline{ε}_{1}, \overline{ε}_{2}, …, \overline{ε}_{n}. Consider the natural isomorphism

φ: U = ⨁_{i} U_{i} →^{∼} \overline{U}

that takes ε_{1,1}, ε_{2,1}, …, ε_{n_{1},1} to \overline{ε}_{1}, \overline{ε}_{2}, …, \overline{ε}_{n_{1}}, takes ε_{1,2}, ε_{2,2}, …, ε_{n_{2},2} to \overline{ε}_{n_{1}+1}, \overline{ε}_{n_{1}+2}, …, \overline{ε}_{n_{1}+n_{2}}, and so on. Let \overline{V} denote the subspace of codimension 1 in \overline{U} consisting of vectors with zero sum of the coordinates. Sequence (3.1) induces an exact sequence of W -spaces

0 → φ(V) → \overline{V} ⟶^{ψ} ℝ^{r} ⟶^{Σ} ℝ → 0.  (3.2)

Here ψ = (ψ_{i})_{i=1,...,r}, where ψ_{i} takes a vector ∑_{j=1}^{n} β_{j} \overline{ε}_{j} ∈ \overline{V} to ∑_{λ=1}^{n_{i}} β_{n_{1}+···+n_{i−1}+λ}, and the map Σ takes a vector in ℝ^{r} to the sum of its coordinates. Note that W acts trivially on ℝ^{r} and ℝ.

We have a lattice Q_{i} ⊂ V_{i} for each i = 1, …, r, a lattice Q = ⨁_{i} Q_{i} ⊂ ⨁_{i} V_{i}, and a lattice \overline{Q} := ℤA_{n−1} in \overline{V} with basis \overline{ε}_{1} − \overline{ε}_{2}, …, \overline{ε}_{n−1} − \overline{ε}_{n}. The isomorphism φ induces an embedding of Q = ⨁_{i} Q_{i} into \overline{Q}. Under this embedding

α_{1,1} ↦ \overline{α}_{1}, α_{2,1} ↦ \overline{α}_{2}, …, α_{n_{1}−1,1} ↦ \overline{α}_{n_{1}−1},
α_{1,2} ↦ \overline{α}_{n_{1}+1}, α_{2,2} ↦ \overline{α}_{n_{1}+2}, …, α_{n_{2}−1,2} ↦ \overline{α}_{n_{1}+n_{2}−1},
. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
α_{1,r} ↦ \overline{α}_{n_{1}+n_{2}+···+n_{r−1}+1}, …, α_{n_{r}−1,r} ↦ \overline{α}_{n−1},

while \overline{α}_{n_{1}}, \overline{α}_{n_{1}+n_{2}}, …, \overline{α}_{n_{1}+n_{2}+···+n_{r−1}} are skipped.

3.10. We write L for L_{1} and w for w_{1} ∈ \frac{1}{d}Q, where Q = ⨁_{i} Q_{i}. Then

w = ∑_{i=1}^{r} w_{i}, w_{i} = \frac{1}{d}[(n_{i} − 1)α_{1,i} + ⋯ + α_{n_{i}−1,i}].

Recall that

Q_{i} = ℤA_{n_{i}−1} = {(a_{j}) ∈ ℤ^{n_{i}} | ∑_{j=1}^{n_{i}} a_{j} = 0}.

Set

\overline{w} = \frac{1}{d} ∑_{j=1}^{n−1}(n − j)\overline{α}_{j}.

Set Λ_{n}(d) = ⟨\overline{Q}, \overline{w}⟩. Note that Λ_{n}(d) = Q_{n}(n/d) with the notation of [LPR, Subsection 6.1]. Set

N = φ(Q ⊗_{ℤ} ℝ) ∩ Λ_{n}(d) = φ(V) ∩ Λ_{n}(d).

Lemma 3.11. φ(L) = N.

Proof. Write j_{1} = n_{1}, j_{2} = n_{1} + n_{2}, …, j_{r−1} = n_{1} + ⋯ + n_{r−1}. Set J = {1, 2, …, n − 1} ∖ {j_{1}, j_{2}, …, j_{r−1}}. Set

μ = \frac{1}{d} ∑_{j∈J}(n − j)\overline{α}_{j} = \overline{w} − ∑_{i=1}^{r−1} \frac{n − j_{i}}{d} \overline{α}_{j_{i}}.

Note that d|n and d|j_{i} for all i, hence the coefficients (n − j_{i})/d are integral, and therefore μ ∈ Λ_{n}(d). Since also μ ∈ φ(Q ⊗_{ℤ} ℝ), we see that μ ∈ N.

Let y ∈ N. Then

y = b\overline{w} + ∑_{j=1}^{n−1} a_{j}\overline{α}_{j}

where b, a_{j} ∈ ℤ, because y ∈ Λ_{n}(d). We see that in the basis \overline{α}_{1}, …, \overline{α}_{n−1} of Λ_{n}(d) ⊗_{ℤ} ℝ, the element y contains \overline{α}_{j_{i}} with coefficient

b\frac{n − j_{i}}{d} + a_{j_{i}}.

Since y ∈ φ(Q ⊗_{ℤ} ℝ), this coefficient must be 0:

b\frac{n − j_{i}}{d} + a_{j_{i}} = 0.

Consider

y − bμ = y − b (\overline{w} − ∑_{i=1}^{r−1} \frac{n − j_{i}}{d} \overline{α}_{j_{i}}) =y − b\overline{w} + ∑_{i=1}^{r−1} \frac{b(n − j_{i})}{d}\overline{α}_{j_{i}}

= ∑_{j=1}^{n−1}a_{j}\overline{α}_{j} + ∑_{i=1}^{r−1} \frac{b(n − j_{i})}{d}\overline{α}_{j_{i}} = ∑_{j∈J} a_{j}\overline{α}_{j} ,

where a_{j} ∈ ℤ. We see that y ∈ ⟨\overline{α}_{j} (j ∈ J), μ⟩ for any y ∈ N, hence N ⊂ ⟨\overline{α}_{j} (j ∈ J), μ⟩. Conversely, μ ∈ N and \overline{α}_{j} ∈ N for j ∈ J, hence ⟨\overline{α}_{j} (j ∈ J), μ⟩ ⊂ N, thus

N = ⟨\overline{α}_{j} (j ∈ J), μ⟩.  (3.3)

Now

φ(w) = \frac{1}{d} [ ∑_{j=1}^{n_{1}−1}(n_{1} − j)\overline{α}_{j} + ∑_{j=1}^{n_{2}−1}(n_{2} − j)\overline{α}_{n_{1}+j} + ⋯ + ∑_{j=1}^{n_{r}−1}(n_{r} − j)\overline{α}_{j_{r−1}+j}]

while

μ = \frac{1}{d} [ ∑_{j=1}^{n_{1}−1}(n − j)\overline{α}_{j} + ∑_{j=1}^{n_{2}−1}(n − n_{1} − j)\overline{α}_{n_{1}+j} + ⋯ + ∑_{j=1}^{n_{r}−1}(n_{r} − j)\overline{α}_{j_{r−1}+j}] .

Thus

μ = φ(w) + \frac{n − n_{1}}{d} ∑_{j=1}^{n_{1}−1} \overline{α}_{j} + \frac{n − n_{1} − n_{2}}{d} ∑_{j=1}^{n_{2}−1} \overline{α}_{n_{1}+j} + ⋯ + \frac{n_{r}}{d} ∑_{j=1}^{n_{r}−1} \overline{α}_{j_{r−1}+j} ,

where the coefficients

\frac{n − n_{1}}{d} , \frac{n − n_{1} − n_{2}}{d} , …, \frac{n_{r}}{d}

are integral. We see that

⟨\overline{α}_{j} (j ∈ J), μ⟩ = ⟨\overline{α}_{j} (j ∈ J), φ(w)⟩.  (3.4)

From (3.3) and (3.4) we obtain that

N = ⟨\overline{α}_{j} (j ∈ J), μ⟩ = ⟨\overline{α}_{j} (j ∈ J), φ(w)⟩ = φ(L).

3.12. Now let p| gcd(n_{1}, …, n_{r}). Recall that W = ∏_{i=1}^{r} 𝔖_{n_{i}}. Since p|n_{i} for all i, we can naturally embed (𝔖_{p})^{n_{i}/p} into 𝔖_{n_{i}}. We obtain a natural embedding

Γ := (ℤ/pℤ)^{n/p} ↪ (𝔖_{p})^{n/p} ↪ W.

In order to prove Lemma 3.8, it suffices to prove the next Lemma 3.13. Indeed, if n has an odd prime factor p, then by Lemma 3.13 L is not quasi-permutation. If n = 2^{s}, then we take p = 2. By the assumptions of Proposition 3.2, n > 4 = 2^{2}, and again by Lemma 3.13 L is not quasi-permutation. This proves Lemma 3.8.

Lemma 3.13. If either p odd or n > p^{2}, then L is not quasi-permutation as a Γ-lattice.

Proof. By Lemma 3.11 it suffices to show that N is not quasi-permutation. Since N = Λ_{n}(d) ∩ φ(V), we have an embedding

Λ_{n}(d)/N ↪ \overline{V} /φ(V).

By (3.2) \overline{V} /φ(V) ≃ ℝ^{r−1} and W acts on \overline{V} /φ(V) trivially. Thus Λ_{n}(d)/N ≃ ℤ^{r−1} and W acts on ℤ^{r−1} trivially. We have an exact sequence of W -lattices

0 → N → Λ_{n}(d) → ℤ^{r−1} → 0,

with trivial action of W on ℤ^{r−1}. We obtain that N ∼ Λ_{n}(d) as a W -lattice, and hence, as a Γ-lattice. Therefore, it suffices to show that Λ_{n}(d) = Q_{n}(n/d) is not quasi-permutation as a Γ-lattice if either p is odd or n > p^{2}. This is done in [LPR] in the proofs of Propositions 7.4 and 7.8. This completes the proof of Lemma 3.13 and hence those of Lemma 3.8 and Proposition 3.2.

4 Quasi-permutation lattices – case A_{n−1}

In this section we prove Theorem 0.2 in the special case when G is isogenous to a direct product of groups of type A_{n−1} for n ≥ 3.

We maintain the notation of Subsection 3.1. Let L be an intermediate lattice between Q and P, i.e., Q ⊂ L ⊂ P ( L = Q are L = P are possible). Let S denote the image of L in F, then L is the preimage of S ⊂ F in P. Since W acts trivially on F, the subgroup S ⊂ F is W -invariant, and therefore, the sublattice L ⊂ P is W -invariant.

Theorem 4.1. With the notation of Subsection 3.1 assume that n_{i} ≥ 3 for all i = 1, 2, …, r. Let L between Q and P be an intermediate lattice, and assume that L∩P_{i} = Q_{i} for all i such that n_{i} = 3 or n_{i} = 4. If L is a quasi-permutation W -lattice, then L = Q.

Proof. We prove the theorem by induction on r. The case r = 1 follows from our assumptions if n_{1} = 3 or n_{1} = 4, and from Proposition 1.9 if n_{1} > 4.

We assume that r > 1 and that the assertion is true for r − 1. We prove it for r.

For i between 1 and r we set

Q′_{i} = ⨁_{j≠i} Q_{j}, P′_{i} = ⨁_{j≠i} P_{j}, F′_{i} = ⨁_{j≠i} F_{j}, W′_{i} = ∏_{j≠i} W_{j},

then Q′_{i} ⊂ Q, P′_{i} ⊂ P, F′_{i} ⊂ F and W′_{i} ⊂ W. If L is a quasi-permutation W -lattice, then by Lemma 1.7 L ∩ P′_{i} is a quasi-permutation W′_{i}-lattice, and by the induction hypothesis L ∩ P′_{i} = Q′_{i}.

Now let Q ⊂ L ⊂ P, and assume that L ∩ P′_{i} = Q′_{i} for all i = 1, …, r. We shall show that if L ≠ Q then L is not a quasi-permutation W -lattice. This will prove Theorem 4.1.

Assume that L ≠ Q. Set S = L/Q ⊂ F, then S ≠ 0. We first show that (L ∩ P′_{i})/Q′_{i} = S ∩ F′_{i}. Indeed, clearly (L ∩ P′_{i})/Q′_{i} ⊂ L/Q ∩ P′_{i}/Q′_{i} = S ∩ F′_{i}. Conversely, let f ∈ S∩F′_{i}, then f can be represented by some l ∈ L and by some p ∈ P′_{i}, and q := l − p ∈ Q. Since L ⊃ Q, we see that p = l − q ∈ L ∩ P′_{i}, hence f ∈ (L∩P′_{i})/Q′_{i}, and therefore S∩F′_{i} ⊂ (L∩P′_{i})/Q′_{i}. Thus (L∩P′_{i})/Q′_{i} = S∩F′_{i}.

By assumption we have L ∩ P′_{i} = Q′_{i}, and we obtain that S ∩ F′_{i} = 0 for all i = 1, …, r. Let S_{(i)} denote the image of S under the projection F → F_{i}. We have a canonical epimorphism p_{i}: S → S_{(i)} with kernel S∩F′_{i}. Since S∩F′_{i} = 0, we see that p_{i}: S → S_{(i)} is an isomorphism. Set q_{i} = p_{i} ∘ p_{1}^{−1} : S_{(1)} → S_{(i)}, it is an isomorphism.

We regard Q_{i} = ℤA_{n_{i}−1} and P_{i} = Λ_{n_{i}} as the lattices described in [Bou, Planche I]. Then we have an isomorphism F_{i} ≅ ℤ/n_{i}ℤ. Since S_{(i)} is a subgroup of the cyclic group F_{i} ≅ ℤ/n_{i}ℤ and S ≅ S_{(i)}, we see that S is a cyclic group, and we see also that |S| divides n_{i} for all i, hence d := |S| divides c := gcd(n_{1}, …, n_{r}).

We describe all subgroups S of order d in ⨁_{i=1}^{r} ℤ/n_{i}ℤ such that S ∩ (⨁_{j≠i} ℤ/n_{j}ℤ) = 0 for all i. The element a_{i} := n_{i}/d + n_{i}ℤ is a generator of S_{(i)} ⊂ F_{i} = ℤ/n_{i}ℤ. Set b_{i} = q_{i}(a_{1}). Since b_{i} is a generator of S_{(i)}, we have b_{i} = \overline{ν}_{i}a_{i} for some \overline{ν}_{i} ∈ (ℤ/dℤ)^{×}. Let ν_{i} ∈ ℤ be a representative of \overline{ν}_{i} such that 1 ≤ ν_{i} < d, then gcd(ν_{i}, d) = 1. Moreover, since q_{1} = id, we have b_{1} = a_{1}, hence \overline{ν}_{1} = 1 and ν_{1} = 1. We obtain an element ν = (ν_{1}, …, ν_{r}). With the notation of Subsection 3.1 we have S = S_{ν} and L = L_{ν}.

By Proposition 3.2 L_{ν} is not a quasi-permutation W -lattice. Thus L is not quasi-permutation, which completes the proof of Theorem 4.1.

5 Proof of Theorem 0.2

Lemma 5.1 (well-known). Let P_{1} and P_{2} be abelian groups. Set P = P_{1} ⊕P_{2} = P_{1} × P_{2}, and let π_{1} : P → P_{1} denote the canonical projection. Let L ⊂ P be a subgroup. If π_{1}(L) = L ∩ P_{1}, then

L = (L ∩ P_{1}) ⊕ (L ∩ P_{2}).

Proof. Let x ∈ L. Set x_{1} = π_{1}(x) ∈ π_{1}(L). Since π_{1}(L) = L ∩ P_{1}, we have x_{1} ∈ L ∩ P_{1}. Set x_{2} = x − x_{1}, then x_{2} ∈ L ∩ P_{2}. We have x = x_{1} + x_{2}. This completes the proof of Lemma 5.1.

5.2. Let I be a finite set. For any i ∈ I let D_{i} be a connected Dynkin diagram. Let D = ⨆_{i} D_{i} (disjoint union). Let Q_{i} and P_{i} be the root and weight lattices of D_{i}, respectively, and W_{i} be the Weyl group of D_{i}. Set

Q = ⨁_{i∈I} Q_{i}, P = ⨁_{i∈I} P_{i}, W = ∏_{i∈I} W_{i}.

5.3. We construct certain quasi-permutation lattices L such that Q ⊂ L ⊂ P.

Let { {i_{1}, j_{1}}, …, {i_{s}, j_{s}} } be a set of non-ordered pairs in I such that D_{i_{l}} and D_{j_{l}} for all l = 1, …, s are of type B_{1} = A_{1} and all the indices i_{1}, j_{1}, …, i_{s}, j_{s} are distinct. Fix such an l. We write {i, j} for {i_{l}, j_{l}} and we set D_{i,j} := D_{i}⊔D_{j}, Q_{i,j} := Q_{i} ⊕ Q_{j}, P_{i,j} := P_{i} ⊕ P_{j}. We regard D_{i,j} as a Dynkin diagram of type D_{2}, and we denote by M_{i,j} the intermediate lattice between Q_{i,j} and P_{i,j} isomorphic to X (SO_{4}), the character lattice of the group SO_{4}; see Section 1, after Lemma 1.7. Let f_{i} be a generator of the lattice Q_{i} of rank 1, and let f_{j} be a generator of Q_{j}, then P_{i} = ⟨\frac{1}{2}f_{i}⟩ and P_{j} = ⟨\frac{1}{2}f_{j}⟩. Set e_{1}^{(l)} = \frac{1}{2}(f_{i} + f_{j}), e_{2}^{(l)} = \frac{1}{2}(f_{i} − f_{j}), then \{e_{1}^{(l)}, e_{2}^{(l)}\} is a basis of M_{i,j} , and

M_{i,j} = ⟨Q_{i,j}, e_{1}^{(l)}⟩ ,  P_{i,j} = ⟨M_{i,j}, \frac{1}{2}(e_{1}^{(l)} + e_{2}^{(l)}) ⟩ .  (5.1)

We have M_{i,j} ∩ P_{i} = Q_{i}, M_{i,j} ∩ P_{j} = Q_{j}, and [M_{i,j} : Q_{i,j}] = 2. Concerning the Weyl group, we have

W (D_{i,j}) = W (D_{i}) × W (D_{j}) = W (D_{2}) = 𝔖_{2} × {±1},

where the symmetric group 𝔖_{2} permutes the basis vectors e_{1}^{(l)} and e_{2}^{(l)} of M_{i,j} , while the group {±1} acts on M_{i,j} by multiplication by scalars. We say that M_{i,j} is an indecomposable quasi-permutation lattice (it corresponds to the semisimple Cayley group SO_{4} which does not decompose into a direct product of its normal subgroups).

Set I′ = I ∖ ⋃_{l=1}^{s}{i_{l}, j_{l}}. For i ∈ I′ let M_{i} be any quasi-permutation intermediate lattice between Q_{i} and P_{i} (such an intermediate lattice exists if and only if D_{i} is of one of the types A_{n}, B_{n}, C_{n}, D_{n}, G_{2}, see Proposition 1.9). We say that M_{i} is a simple quasi-permutation lattice (it corresponds to a stably Cayley simple group). We set

L = ⨁_{l=1}^{s} M_{i_{l},j_{l}} ⊕ ⨁_{i∈I′} M_{i}.  (5.2)

We say that a lattice L as in (5.2) is a direct sum of indecomposable quasipermutation lattices and simple quasi-permutation lattices. Clearly L is a quasipermutation W -lattice.

Theorem 5.4. Let D, Q, P, W be as in Subsection 5.2. Let L be an intermediate lattice between Q and P, i.e., Q ⊂ L ⊂ P (where L = Q and L = P are possible). If L is a quasi-permutation W -lattice, then L is as in (5.2). Namely, then L is a direct sum of indecomposable quasi-permutation lattices M_{i,j} for some set of pairs { {i_{1}, j_{1}}, …, {i_{s}, j_{s}} } and some family of simple quasi-permutation intermediate lattices M_{i} between Q_{i} and P_{i} for i ∈ I′.

Remark 5.5. The set of pairs { {i_{1}, j_{1}}, …, {i_{s}, j_{s}} } in Theorem 5.4 is uniquely determined by L. Namely, a pair {i, j} belongs to this set if and only if the Dynkin diagrams D_{i} and D_{j} are of type B_{1} = A_{1} and

L ∩ P_{i} = Q_{i} , L ∩ P_{j} = Q_{j} , while L ∩ (P_{i} ⊕ P_{j}) ≠ Q_{i} ⊕ Q_{j} .

Proof of Theorem 5.4. We prove the theorem by induction on m = |I|, where I is as in Subsection 5.2. The case m = 1 is trivial.

We assume that m ≥ 2 and that the theorem is proved for all m′ < m. We prove it for m. First we consider three special cases.

Split case. Assume that for some subset A ⊂ I, A ≠ I, A ≠ ∅, we have π_{A}(L) = L ∩ P_{A}, where P_{A} = ⨁_{i∈A} P_{i} and π_{A}: P → P_{A} is the canonical projection. Then by Lemma 5.1 we have L = (L ∩ P_{A}) ⊕ (L ∩ P_{A′}), where A′ = I ∖ A. By Lemma 1.7 L ∩ P_{A} is a quasi-permutation W_{A}-lattice, where W_{A} = ∏_{i∈A} W_{i}. By the induction hypothesis the lattice L∩P_{A} is a direct sum of indecomposable quasi-permutation lattices and simple quasi-permutation lattices. Similarly, L ∩ P_{A′} is such a direct sum. We conclude that L = (L ∩ P_{A}) ⊕ (L ∩ P_{A′}) is such a direct sum, and we are done.

A_{n−1}-case. Assume that all D_{i} are of type A_{n_{i}−1}, where n_{i} ≥ 3 (so A_{1} is not permitted). We assume also that when n_{i} = 3 and when n_{i} = 4 (that is, for A_{2} and for A_{3} = D_{3}) we have L ∩ P_{i} = Q_{i} (for n_{i} > 4 this is automatic because L ∩ P_{i} is a quasi-permutation W_{i}-lattice, see Proposition 1.9). In this case by Theorem 4.1 we have L = Q = ⨁ Q_{i}, hence L is a direct sum of simple quasi-permutation lattices, and we are done.

A_{1}-case. Assume that all D_{i} are of type A_{1}. Then by [BKLR, Theorem 18.1] the lattice L is a direct sum of indecomposable quasi-permutation lattices and simple quasi-permutation lattices, and we are done.

Now we shall show that these three special cases exhaust all the quasipermutation lattices. In other words, we shall show that if Q ⊂ L ⊂ P and L is not as in one of these three cases, then L is not quasi-permutation. This will complete the proof of the theorem.

For the sake of contradiction, let us assume that Q ⊂ L ⊂ P, that L is not in one of the three special cases above, and that L is a quasi-permutation W -lattice.

We shall show in three steps that L is as in Proposition 2.2. By Proposition 2.2, L is not quasi–permutation, which contradicts our assumptions. This contradiction will prove the theorem.

Step 1. For i ∈ I consider the intersection L ∩ P_{i}, it is a quasi-permutation W_{i}-lattice (by Lemma 1.7), hence D_{i} is of one of the types A_{n−1}, B_{n}, C_{n}, D_{n}, G_{2} (by Proposition 1.9). Note that π_{i}(L) ≠ L ∩ P_{i} (otherwise we are in the split case).

Now assume that for some i ∈ I, the Dynkin diagram D_{i} is of type G_{2} or C_{n} for some n ≥ 3, or D_{i} is of type A_{2} and L ∩ P_{i} ≠ Q_{i}. Then L ∩ P_{i} is a quasipermutation W_{i}-lattice (by Lemma 1.7), hence L ∩ P_{i} = P_{i} (by Proposition 1.9). Since P_{i} ⊃ π_{i}(L) ⊃ L ∩ P_{i}, we obtain that π_{i}(L) = L ∩ P_{i}, which is impossible. Thus no D_{i} can be of type G_{2} or C_{n}, n ≥ 3, and if D_{i} is of type A_{2} for some i, then L ∩ P_{i} = Q_{i}.

Thus all D_{i} are of types A_{n−1}, B_{n} or D_{n}, and if D_{i} is of type A_{2} for some i ∈ I, then L∩P_{i} = Q_{i}. Since L is not as in the A_{n−1}-case, we may assume that one of the D_{i}, say D_{1}, is of type B_{n} for some n ≥ 1 (B_{1} = A_{1} is permitted), or of type D_{n} for some n ≥ 4, or of type D_{3} with L ∩ P_{1} ≠ Q_{1}. Indeed, otherwise all D_{i} are of type A_{n_{i}−1} for n_{i} ≥ 3, and in the cases A_{2} (n_{i} = 3) and A_{3} (n_{i} = 4) we have L ∩ P_{i} = Q_{i}, i.e., we are in the A_{n−1}-case, which contradicts our assumptions.

Step 2. In this step, using the Dynkin diagram D_{1} of type B_{n} or D_{n} from the previous step, we construct a quasi-permutation sublattice L′ ⊂ L of index 2 such that L′ is as in (5.2). First we consider the cases B_{n} and D_{n} separately.

Assume that D_{1} is of type B_{n} for some n ≥ 1 (B_{1} = A_{1} is permitted). We have [P_{1} : Q_{1}] = 2. Since P_{1} ⊃ π_{1}(L) ⊋ L ∩ P_{1} ⊃ Q_{1}, we see that π_{1}(L) = P_{1} and L ∩ P_{1} = Q_{1}. Set M_{1} = Q_{1}. We have π_{1}(L) = P_{1}, L ∩ P_{1} = M_{1}, and [P_{1} : M_{1}] = 2.

Now assume that D_{1} is of type D_{n} for some n ≥ 4, or of type D_{3} with L∩P_{1} ≠ Q_{1}. Set M_{1} = L∩P_{1}, then M_{1} is a quasi-permutation W_{1}-lattice by Lemma 1.7, and it follows from Proposition 1.9 that (W_{1}, M_{1}) ≃ X (SO_{2n}), where X (SO_{2n}) denotes the character lattice of SO_{2n}; see Section 1, after Lemma 1.7. It follows that [M_{1} : Q_{1}] = 2 and [P_{1} : M_{1}] = 2. Since P_{1} ⊃ π_{1}(L) ⊋ L∩P_{1} = M_{1}, we see that π_{1}(L) = P_{1}. Again we have π_{1}(L) = P_{1}, L ∩ P_{1} = M_{1}, and [P_{1} : M_{1}] = 2.

Now we consider together the cases when D_{1} is of type B_{n} for some n ≥ 1 and when D_{1} is of type D_{n} for some n ≥ 3, where in the case D_{3} we have L ∩ P_{1} ≠ Q_{1}. Set

L′ := ker[L ⟶^{π_{1}} P_{1} → P_{1}/M_{1}].

Since π_{1}(L) = P_{1}, and [P_{1} : M_{1}] = 2, we have [L : L′] = 2. Clearly we have π_{1}(L′) = M_{1}. Set

L_{1}^{†} := ker[π_{1} : L → P_{1}] = L ∩ P′_{1},

where P′_{1} = ⨁_{i≠1} P_{i}. Since L is a quasi-permutation W -lattice, by Lemma 1.7 the lattice L_{1}^{†} is a quasi-permutation W′_{1}-lattice, where W′_{1} = ∏_{i≠1} W_{i}. By the induction hypothesis, L_{1}^{†} is a direct sum of indecomposable quasi-permutation lattices and simple quasi-permutation lattices as in (5.2). Since M_{1} = L ∩ P_{1}, we have M_{1} ⊂ L′ ∩ P_{1}, and L′ ∩ P_{1} ⊂ L ∩ P_{1} = M_{1}, hence L′ ∩ P_{1} = M_{1} = π_{1}(L′), and by Lemma 5.1 we have L′ = M_{1} ⊕ L_{1}^{†}. Since M_{1} is a simple quasipermutation lattice, we conclude that L′ is a direct sum of indecomposable quasi-permutation lattices and simple quasi-permutation lattices as in (5.2), and [L : L′] = 2.

Step 3. In this step we show that L is as in Proposition 2.2. We write

L′ = ⨁_{l=1}^{s}(L′ ∩ P_{i_{l},j_{l}}) ⊕ ⨁_{i∈I′}(L′ ∩ P_{i}),

where P_{i_{l},j_{l}} = P_{i_{l}} ⊕P_{j_{l}}, the Dynkin diagrams D_{i_{l}} and D_{j_{l}} are of type A_{1} = B_{1}, and L′ ∩ P_{i_{l},j_{l}} = M_{i_{l},j_{l}} as in (5.1). For any i ∈ I′, we have [π_{i}(L) : π_{i}(L′)] ≤ 2, because [L : L′] = 2. Furthermore, for i ∈ I′ we have

π_{i}(L′) = L′ ∩ P_{i} ⊂ L ∩ P_{i} ⊊ π_{i}(L),

hence [π_{i}(L) : (L∩P_{i})] = 2 and L′ ∩P_{i} = L∩P_{i}. Similarly, for any l = 1, …, s, if we write i = i_{l}, j = j_{l}, then we have

M_{i,j} = L′ ∩ P_{i,j} ⊂ L ∩ P_{i,j} ⊊ π_{i,j}(L) ⊂ P_{i,j}, [P_{i,j} : M_{i,j}] = 2,

whence π_{i,j}(L) = P_{i,j}, L ∩ P_{i,j} = M_{i,j}, and therefore [π_{i,j}(L) : (L ∩ P_{i,j})] = [P_{i,j} : M_{i,j}] = 2 and L′ ∩ P_{i,j} = M_{i,j} = L ∩ P_{i,j}.

We view the Dynkin diagram D_{i_{l}} ⊔ D_{j_{l}} of type A_{1} ⊔ A_{1} corresponding to the pair {i_{l}, j_{l}} (l = 1, …, s) as a Dynkin diagram of type D_{2}. Thus we view L′ as a direct sum of indecomposable quasi-permutation lattices and simple quasi-permutation lattices corresponding to Dynkin diagrams of type B_{n}, D_{n} and A_{n}.

We wish to show that L is as in Proposition 2.2. We change our notation in order to make it closer to that of Proposition 2.2.

As in Subsection 2.1, we now write D_{i} for Dynkin diagrams of types B_{l_{i}} and D_{l_{i}} only, appearing in L′, where B_{1} = A_{1}, B_{2} = C_{2}, D_{2} = A_{1} ⊔ A_{1} and D_{3} = A_{3} are permitted, but for D_{l_{i}} with l_{i} = 2, 3 we require that

L ∩ P_{i} = M_{i} := X (SO_{2l_{i}}).

We write L′_{i} := L∩P_{i} = L′∩P_{i}. We have [π_{i}(L) : L′_{i}] = 2, hence [P_{i} : L′_{i}] ≥ 2. If D_{i} is of type B_{l_{i}}, then [P_{i} : L′_{i}] = 2. If D_{i} is of type D_{l_{i}}, then L′_{i} = L∩P_{i} ≠ Q_{i}, for D_{2} and D_{3} by our assumption and for D_{l_{i}} with l_{i} ≥ 4 because L ∩ P_{i} is a quasi-permutation W_{i}-lattice (see Proposition 1.9); again we have [P_{i} : L′_{i}] = 2.

We see that for all i we have [P_{i} : L′_{i}] = 2, π_{i}(L) = P_{i}, and the lattice L′_{i} = M_{i} is as in Subsection 2.1. We realize the root system R(D_{i}) of type B_{l_{i}} or D_{l_{i}} in the standard way (cf. [Bou, Planches II, IV]) in the space V_{i} := ℝ^{l_{i}} with basis (e_{s})_{s∈S_{i}}, then L′_{i} is the lattice generated by the basis vectors (e_{s})_{s∈S_{i}} of V_{i}, and we have P_{i} = ⟨L′_{i}, \frac{1}{2}x_{i}⟩, where

x_{i} = ∑_{s∈S_{i}} e_{s} ∈ L′_{i} .

In particular, when D_{i} is of type D_{2} we have x_{i} = e_{1}^{(l)} + e_{2}^{(l)} with the notation of formula (5.1).

As in Subsection 2.1, we write Δ_{ι} for Dynkin diagrams of type A_{n′_{ι}−1} appearing in L′, where n′_{ι} ≥ 3 and for A_{3} = D_{3} we require that L ∩ P_{ι} = Q_{ι}. We write L′_{ι} := L ∩ P_{ι} = L′ ∩ P_{ι}. Then L′_{ι} = Q_{ι} for all ι, for A_{2} by Step 1, for A_{3} by our assumption, and for other A_{n′_{ι}−1} because L′_{ι} is a quasi-permutation W_{ι}- lattice; see Proposition 1.9. We have π_{ι}(L) ⊋ L ∩ P_{ι} = L′_{ι} and [π_{ι}(L) : L′_{ι}] = [π_{ι}(L) : π_{ι}(L′)] ≤ 2 (because [L : L′] = 2). It follows that [π_{ι}(L) : L′_{ι}] = 2, i.e., [π_{ι}(L) : Q_{ι}] = 2. We know that P_{ι}/Q_{ι} is a cyclic group of order n′_{ι}. Since it has a subgroup π_{ι}(L)/Q_{ι} of order 2, we conclude that n′_{ι} is even, n′_{ι} = 2n_{ι} (where 2n_{ι} ≥ 4), and π_{ι}(L)/Q_{ι} is the unique subgroup of order 2 of the cyclic group P_{ι}/Q_{ι} of order 2n_{ι}. As in Subsection 2.1, we realize the root system Δ_{ι} of type A_{2n_{ι}−1} in the standard way (cf. [Bou, Planche I]) in the subspace V_{ι} of vectors with zero sum of the coordinates in the space ℝ^{2n_{ι}} with basis ε_{ι,1}, …, ε_{ι,2n_{ι}}. We set

ξ_{ι} = ε_{ι,1} − ε_{ι,2} + ε_{ι,3} − ε_{ι,4} + ⋯ + ε_{ι,2n_{ι}−1} − ε_{ι,2n_{ι}},

then ξ_{ι} ∈ L′_{ι} and \frac{1}{2}ξ_{ι} ∈ π_{ι}(L) ∖ L′_{ι} (cf. [Bou, Planche I, formula (VI) ] ), hence π_{ι}(L) = ⟨L′_{ι}, \frac{1}{2}ξ_{ι}⟩.

Now we set

v = \frac{1}{2} ∑_{i∈I} x_{i} + \frac{1}{2} ∑_{ι=1}^{μ} ξ_{ι} .

We claim that

L = ⟨L′, v⟩.

Proof of the claim. Let w ∈ L ∖ L′, then L = ⟨L′, w⟩, because [L : L′] = 2. Set z_{i} = \frac{1}{2}x_{i} −π_{i}(w), then z_{i} ∈ L′_{i} ⊂ L′, because \frac{1}{2}x_{i}, π_{i}(w) ∈ π_{i}(L)∖L′_{i}. Similarly, we set ζ_{ι} = \frac{1}{2}ξ_{ι} − π_{ι}(w), then ζ_{ι} ∈ L′_{ι} ⊂ L′. We see that

v = w + ∑_{i} z_{i} + ∑_{ι} ζ_{ι},

where ∑_{i} z_{i} + ∑_{ι} ζ_{ι} ∈ L′, and the claim follows.

It follows from the claim that L is as in Proposition 2.2 (we use the assumption that we are not in the A_{1}-case). Now by Proposition 2.2 L is not quasiinvertible, hence not quasi-permutation, which contradicts our assumptions. This contradiction proves Theorem 5.4.

Proof of Theorem 0.2. Theorem 0.2 follows immediately from Theorem 5.4 by virtue of Proposition 1.8.

6 Proof of Theorem 0.3

In this section we deduce Theorem 0.3 from Theorem 0.2.

Let G be a stably Cayley semisimple k-group. Then \overline{G} := G ×_{k} k̄ is stably Cayley over an algebraic closure k̄ of k. By Theorem 0.2, Gk̄ = ∏_{j∈J} G_{j,k̄} for some finite index set J, where each G_{j,k̄} is either a stably Cayley simple group or is isomorphic to SO_{4,k̄}. (Recall that SO_{4,k̄} is stably Cayley and semisimple, but is not simple.) Here we write G_{j,k̄} for the factors in order to emphasize that they are defined over k̄. By Remark 5.5 the collection of direct factors G_{j,k̄} is determined uniquely by \overline{G}. The Galois group Gal(k̄/k) acts on Gk̄, hence on J. Let Ω denote the set of orbits of Gal(k̄/k) in J. For ω ∈ Ω set Gω ; k̄ = ∏_{j∈ω} G_{j,k̄}, then \overline{G} = ∏_{ω∈Ω} Gω ; k̄. Each Gω ; k̄ is Gal(k̄/k)-invariant, hence it defines a k-form G_{k}^{ω} of Gω ; k̄. We have G = ∏_{ω∈Ω} G_{k}^{ω}.

For each ω ∈ Ω choose j = j_{ω} ∈ ω. Let l_{j}/k denote the Galois extension in k̄ corresponding to the stabilizer of j in Gal(k̄/k). The subgroup G_{j,k̄} is Gal(k̄/l_{j})-invariant, hence it comes from an l_{j}-form G_{j,l_{j}}. By the definition of Weil’s restriction of scalars (see e.g. [Vo2, Subsection 3.12]) G_{k}^{ω} ≅ R_{l_{j}/k}G_{j,l_{j}}, hence G ≅ ∏_{ω∈Ω} R_{l_{j}/k}G_{j,l_{j}}. Each G_{j,l_{j}} is either absolutely simple or an l_{j}-form of SO_{4}.

We complete the proof using an argument from [BKLR, Proof of Lemma 11.1]. We show that G_{j,l_{j}} is a direct factor of G_{l_{j}} := G ×_{k} l_{j}. It is clear from the definition that G_{j,k̄} is a direct factor of Gk̄ with complement G′ ; k̄ = ∏_{i∈J∖{j}} G_{i,k̄}. Then G′ ; k̄ is Gal(k̄/l_{j})-invariant, hence it comes from some l_{j}-group G′_{l_{j}}. We have G_{l_{j}} = G_{j,l_{j}} ×_{l_{j}} G′_{l_{j}}, hence G_{j,l_{j}} is a direct factor of G_{l_{j}}.

Recall that G_{j,l_{j}} is either a form of SO_{4} or absolutely simple. If it is a form of SO_{4}, then clearly it is stably Cayley over l_{j}. It remains to show that if G_{j,l_{j}} is absolutely simple, then G_{j,l_{j}} is stably Cayley over l_{j}. The group Gk̄ is stably Cayley over k̄. Since G_{j,k̄} is a direct factor of the stably Cayley k̄- group Gk̄ over the algebraically closed field k̄, by [LPR, Lemma 4.7] G_{j,k̄} is stably Cayley over k̄. Comparing [LPR, Theorem 1.28] and [BKLR, Theorem 1.4], we see that G_{j,l_{j}} is either stably Cayley over l_{j} (in which case we are done) or an outer form of PGL_{2n} for some n ≥ 2. Thus assume by the way of contradiction that G_{j,l_{j}} is an outer form of PGL_{2n} for some n ≥ 2. Then by [BKLR, Example 10.7] the character lattice of G_{j,l_{j}} is not quasi-invertible, and by [BKLR, Proposition 10.8] the group G_{j,l_{j}} cannot be a direct factor of a stably Cayley l_{j}-group. This contradicts the fact that G_{j,l_{j}} is a direct factor of the stably Cayley l_{j}-group G_{l_{j}}. We conclude that G_{j,l_{j}} cannot be an outer form of PGL_{2n} for any n ≥ 2. Thus G_{j,l_{j}} is stably Cayley over l_{j}, as desired.

A Appendix: Some quasi-permutation character lattices

The positive assertion of Proposition 1.9 above is well known. It is contained in [CK, Theorem 0.1] and in [BKLR, Theorem 1.4]. However, [BKLR] refers to [CK, Theorem 0.1], and [CK] refers to a series of results on rationality (rather than only stable rationality) of corresponding generic tori. In this appendix for the reader’s convenience we provide a proof of the following positive result in terms of lattices only.

Proposition A.1. Let G be any form of one of the following groups

SL_{3}, PGL_{n} (n odd), SO_{n} (n ≥ 3), Sp_{2n}, G_{2}

or an inner form of PGL_{n} (n even). Then the character lattice of G is quasipermutation.

Proof. SO_{2n+1}. Let L be the character lattice of SO_{2n+1} (including SO_{3}). Then the Dynkin diagram is D = B_{n}. The Weyl group is W = 𝔖_{n} ⋉ (ℤ/2ℤ)^{n}. Then L = ℤ^{n} with the standard basis e_{1}, …, e_{n}. The group 𝔖_{n} naturally permutes e_{1}, …, e_{n}, while (ℤ/2ℤ)^{n} acts by sign changes. Since W permutes the basis up to ± sign, the W -lattice L is quasi-permutation, see [Lo, § 2.8].

SO_{2n}, any form, inner or outer. Let L be the character lattice of SO_{2n} (including SO_{4}). Then the Dynkin diagram is D = D_{n}, with root system R = R(D). We consider the pair (A, L) where A = Aut(R, L), then (A, L) is isomorphic to the character lattice of SO_{2n+1}, hence is quasi-permutation.

Sp_{2n}. The character lattice of Sp_{2n} is isomorphic to the character lattice of SO_{2n+1}, hence is quasi-permutation.

PGL_{n}, inner form. The character lattice of PGL_{n} is the root lattice L = Q of A_{n−1}. It is a quasi-permutation 𝔖_{n}-lattice, cf. [Lo, Example 2.8.1].

PGL_{n}, outer form, n odd. Let P be the weight lattice of A_{n−1}, where n ≥ 3 is odd. Then P is generated by elements e_{1}, …, e_{n} subject to the relation

e_{1} + ⋯ + e_{n} = 0.

The automorphism group A = Aut(A_{n−1}) is the product of 𝔖_{n} and 𝔖_{2}. The group A acts on P as follows: 𝔖_{n} permutes e_{1}, …, e_{n}, and the nontrivial element of 𝔖_{2} takes each e_{i} to −e_{i}.

We denote by M the A-lattice of rank 2n + 1 with basis s_{1}, …, s_{n}, t_{1}, …, t_{n}, u. The group 𝔖_{n} permutes s_{i} and permutes t_{i} (i = 1, …, n), and the nontrivial element of 𝔖_{2} permutes s_{i} and t_{i} for each i. The group A acts trivially on u. Clearly M is a permutation lattice.

We define an A-epimorphism π: M → P as follows:

π: s_{i} ↦ e_{i}, t_{i} ↦ −e_{i}, u ↦ 0.

Set M′ = ker π, it is an A-lattice of rank n+2. We show that it is a permutation lattice. We write down a set of n + 3 generators of M′:

ρ_{i} = s_{i} + t_{i}, σ = s_{1} + ⋯ + s_{n}, τ = t_{1} + ⋯ + t_{n}, u.

There is a relation

ρ_{1} + ⋯ + ρ_{n} = σ + τ.

We define a new set of n + 2 generators:

ρ̃_{i} = ρ_{i} + u, σ̃ = σ + \frac{n−1}{2}u, τ̃ = τ + \frac{n−1}{2}u,

where \frac{n−1}{2} is integral because n is odd. We have

ρ̃_{1} + ⋯ + ρ̃_{n} − σ̃ − τ̃ = u,

hence this new set indeed generates M′, hence it is a basis. The group 𝔖_{n} permutes ρ̃_{1}, …, ρ̃_{n}, while 𝔖_{2} permutes σ̃ and τ̃. Thus A permutes our basis, and therefore M′ is a permutation lattice. We have constructed a left resolution of P:

0 → M′ → M → P → 0,

(with permutation lattices M and M′), which by duality gives a right resolution of the root lattice Q ≅ P^{∨} of A_{n−1}:

0 → Q → M^{∨} → (M′)^{∨} → 0

with permutation lattices M^{∨} and (M′)^{∨}. Thus the character lattice Q of PGL_{n} is a quasi-permutation A-lattice for odd n.

The assertion that the character lattice of G is quasi-permutation in the remaining cases SL_{3} and G_{2} follows from the next Lemma A.2.

Lemma A.2 ([BKLR, Lemma 2.5]). Let Γ be a finite group and L be any Γlattice of rank r = 1 or 2. Then L is quasi-permutation.

This lemma, which is a version of [Vo2, § 4.9, Examples 6 and 7], was stated in [BKLR] without proof. For the sake of completeness we supply a short proof here.

We may assume that Γ is a maximal finite subgroup of GL_{r}(ℤ). If r = 1, then GL_{1}(ℤ) = {±1}, and the lemma reduces to the case of the character lattice of SO_{3} treated above.

Now let r = 2. Up to conjugation there are two maximal finite subgroups of GL_{2}(ℤ), they are isomorphic to the dihedral groups D_{8} (of order 8) and to D_{12} (of order 12), resp., see e.g. [Lo, § 1.10.1, Table 1.2]. The group D_{8} is the group of symmetries of a square, and in this case it suffices to show that the character lattice of SO_{5} is quasi-permutation, which we have done above. The group D_{12} is the group of symmetries of a regular hexagon, and in this case it suffices to show that the character lattice of PGL_{3} (outer form) is quasi-permutation, which we have done above as well. This completes the proofs of Lemma A.2 and Proposition A.1.

Acknowledgements. The authors are very grateful to the anonymous referee for prompt and thorough refereeing the paper and for noticing a (correctable) error in Theorem 4.1 and in the proof of Theorem 5.4. The authors thank Rony A. Bitan for his help in proving Lemma 3.8. The first-named author was supported in part by the Hermann Minkowski Center for Geometry. The second-named author was supported in part by the Israel Science Foundation, grant 1207/12, and by the Minerva Foundation through the Emmy Noether Institute for Mathematics.

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Mikhail Borovoi Raymond and Beverly Sackler School of Mathematical Sciences, Tel Aviv University 6997801 Tel Aviv Israel borovoi@post.tau.ac.il Boris Kunyavskiĭ Department of Mathematics Bar-Ilan University 5290002 Ramat Gan Israel kunyav@macs.biu.ac.il