VANISHING OF TRACE FORMS IN LOW CHARACTERISTICS

SKIP GARIBALDI

Abstract. A finite-dimensional representation of an algebraic group G gives a trace symmetric bilinear form on the Lie algebra of G. We give a criterion in terms of the root system data for this form to vanish. As a corollary, we show that a Lie algebra of type E_{8} over a field of characteristic 5 does not have a so-called “quotient trace form”, answering a question posed in the 1960s.

Let G be an algebraic group over a field F, acting on a finite-dimensional vector space V via a homomorphism ρ: G → GL(V). The differential dρ of ρ maps the Lie algebra Lie(G) of G into 𝔤𝔩(V), and we put Tr_{ρ} for the symmetric bilinear form

Tr_{ρ}(x, y) := trace(dρ(x) dρ(y)) for x, y ∈ Lie(G).

We call Tr_{ρ} a trace form of G. Such forms appear, for example, in the hypotheses for the Jacobson-Morozov Theorem [Ca, 5.3.1]. We prove:

Theorem A. Assume G is simply connected, split, and almost simple. Then the following are equivalent:

(a) The characteristic of F is a torsion prime for G.

(b) Every trace form of G is zero.

The set of torsion primes for G is given by the following table, cf. e.g. [St 75, 1.13]:

type of G  torsion primes
A_{n}, C_{n}  none
B_{n} (n ≥ 3), D_{n} (n ≥ 4), G_{2}  2
F_{4}, E_{6}, E_{7}  2, 3
E_{8}  2, 3, 5

A prime p is called a torsion prime for G if the corresponding group G(ℂ) over ℂ (or, equivalently, its compact form) is such that one of its homology groups, with coefficients in ℤ, contains an element of order p.

We also prove a generalization of Theorem A that removes the hypotheses “simply connected” and “split”; it is somewhat more complicated, so we leave the statement until Th. D (and Remark 4.6). Replacing the simply connected group G with a nontrivial quotient G′ changes the situation in two ways: the group G′ has “fewer” representations and the Lie algebras of G and G′ may be different. These two changes are reflected in the integers N(G) and E(G) defined below.

As a particular example of Th. A, for G of type E_{8} over a field of characteristic 2, 3, or 5, Tr_{ρ} is zero for every representation ρ of G. One may ask whether the same is true for the representations of the Lie algebra Lie(G). That is, for a representation ψ of Lie(G), we write Tr_{ψ} for the bilinear form (x, y) ↦ trace(ψ(x) ψ(y)), and ask if Tr_{ψ} is necessarily zero. We do not know if this holds in characteristic 2 or 3, but it does in characteristic 5:

2000 Mathematics Subject Classification. 20G05 (17B50, 17B25).

Corollary B. If F has characteristic 5 and G is of type E_{8}, then the trace form of every representation of Lie(G) is zero.

Indeed, we shall prove in §7 a slightly stronger statement, namely that Lie(G) has no “quotient trace form”, thus answering a question posed in the early 1960s, see e.g. [Bl, p. 554], [BlZ, p. 543], or [Se, p. 48].

Trace forms on Lie algebras seem to have been studied—motivated by results from Borel-Mostow [BoM, §3]—as a way of defining a class of Lie algebras that was small enough to be tractable but large enough to be interesting, see e.g. [Se, pp. 47–49] and [BlZ]. Since then, tremendous progress has been made: Block, Premet, Strade, Wilson, and others have classified the simple Lie algebras over algebraically closed fields of characteristic ≥ 5, see [Strade]. These algebras are of “classical”, Cartan, or Melikian type, and these types are distinct [Strade, §6.1]. Trace forms are only interesting for algebras of classical type—roughly, the simple algebras coming from Lie algebras of simple algebraic groups—because every trace form on a simple algebra of Cartan or Melikian type is zero by [Bl, Cor. 3.1]. Corollary B settles the last remaining question mark regarding the existence of nonzero trace forms on simple Lie algebras in characteristic ≥ 5, cf. [Bl, Cor. 3.1].

We do not use the Block-Premet-Strade-Wilson techniques here. Rather, we observe that it suffices to prove Theorem A, etc., for those representations “defined over ℤ” and we compute the trace form on those representations (over ℤ). We do need to compute the trace form and not just its discriminant; knowing the discriminant (over ℤ) only tells you if the form is degenerate in prime characteristic, whereas we want to know whether the form is zero.

Acknowledgments. It is a pleasure to thank Jean-Pierre Serre for helpful discussions and for pointing out the utility of the form b̃ (from [GN], see 3.1) early in this project. My research was partially supported by NSF grant DMS-0654502.

1. The number N(G) and the Dynkin index

1.1. Fix a simple root system R. We write P for its weight lattice and ⟨ , ⟩ for the canonical pairing between P and its dual. Fix a long root α ∈ R and write α^{∨} for the associated coroot. For each subset X of P that is invariant under the Weyl group, we put:

N(X) := \frac{1}{2} ∑_{x∈X} ⟨x, α^{∨}⟩^{2} ∈ ℤ[\frac{1}{2}] .

Note that N(X) does not depend on the choice of α because the long roots are conjugate under the Weyl group.

Furthermore, N(X) is an integer. To see this, note that the reflection s in the hyperplane orthogonal to α satisfies ⟨sx, α^{∨}⟩ = ⟨x, sα^{∨}⟩ = −⟨x, α^{∨}⟩, so in the definition of N(X), the sum can be taken to run over those x satisfying x ≠ sx. For such x, we have ⟨x, α^{∨}⟩^{2} + ⟨sx, α^{∨}⟩^{2} = 2⟨x, α^{∨}⟩^{2}, proving the claim.

Example 1.2. The computations in [SpSt, pp. 180, 181] show that N(R) = 2h^{∨}, where h^{∨} denotes the dual Coxeter number of R, which is defined as follows. Fix a set of simple roots Δ of R. Write α̃ for the highest root; the corresponding coroot α̃^{∨} is

α̃^{∨} = ∑_{δ∈Δ} m_{δ}^{∨}δ^{∨}

for some natural numbers m_{δ}^{∨}. The dual Coxeter number h^{∨} is defined by

h^{∨} := 1 + ∑ m_{δ}^{∨}.

In case all the roots of R have the same length, it is the (usual) Coxeter number h and is given in the tables in [B Lie].

Suppose that there are two different root lengths in R; we write L for the set of long roots and S for the set of short roots. The arguments in [SpSt] are easily adapted to show that

N(L) = 2 [1 + ∑_{δ∈Δ∩L} m_{δ}^{∨}] and N(S) = 2 ∑_{δ∈Δ∩S} m_{δ}^{∨}.

We obtain the following numbers:

\begin{matrix}type of R & h & h^{∨} & N(L) & N(S) \\ B_{n} (n ≥ 2) & 2n & 2n − 1 & 4(n − 1) & 2 \\ C_{n} (n ≥ 2) & 2n & n + 1 & 4 & 2(n − 1) \\ G_{2} & 6 & 4 & 6 & 2 \\ F_{4} & 12 & 9 & 12 & 6 \end{matrix}

Definition 1.3. Fix a split almost simple linear algebraic group G over F. Fix also a pinning of G with respect to some maximal torus T; this includes a root system R and a set of simple roots Δ of G with respect to T. For a representation ρ of G over F, one defines

N(ρ) := ∑_{dominant weights λ} \binom{multiplicity of λ}{as a weight of ρ} · N(W λ) ∈ ℤ.

For example, the adjoint representation Ad has N(Ad) = 2h^{∨} by Example 1.2.

We put:

N(G) := gcd N(ρ),

where the gcd runs over the representations of G defined over F.

The map ρ ↦ N(ρ) is compatible with short exact sequences

(1.4)  0 ⟶ ρ′ ⟶ ρ ⟶ ρ/ρ′ ⟶ 0

in the sense that

N(ρ) = N(ρ′) + N(ρ/ρ′).

Writing RG for the representation ring of G, we obtain a homomorphism of abelian groups N: RG → ℤ with image N(G) · ℤ.

In the definition of N(G), it suffices to let the gcd run over generators of RG, e.g., the irreducible representations of G. For an irreducible representation ρ, the highest weight λ has multiplicity 1 and all the other weights of ρ are lower in the partial ordering. Inducting on the partial ordering, we find:

N(G) = gcd_{λ∈T^{∗}} N(W λ).

In particular, N(G) depends only on the root system R and the lattice T^{∗}, and not on the field F.

1.5. When G is simply connected, the number N(ρ) is the Dynkin index of the representation ρ defined in [D, p. 130] and studied in [Mer], and N(G) is the Dynkin index of the group G. The Dynkin index of G and of the fundamental irreducible representations of G (over ℂ) are listed in [LS, Prop. 2.6] or [MPR, pp. 36–44], correcting some small errors in Dynkin’s calculations. For G simply connected, the primes dividing N(G) are the torsion primes of G.

Example 1.6. Write Spin_{n} and SO_{n} for the spin and special orthogonal groups of an n-dimensional nondegenerate quadratic form of maximal Witt index. For n = 3 or n ≥ 5, these groups are split and almost simple of type B or D. The Dynkin index N(Spin_{n}) is 2; it obviously divides N(SO_{n}). On the other hand, the natural n-dimensional representation ρ of SO_{n} has N(ρ) = 2, so N(SO_{n}) = 2.

Example 1.7. Write PSp_{2n} for the split adjoint simple group of type C_{n}; it can be viewed as Sp_{2n}/μ_{2}. We claim that

N(PSp_{2n}) = \begin{cases}2 & if n is even \\ 4 & if n is odd \end{cases}

for n ≥ 2. The number N(PSp_{2n}) divides 4 and 2(n − 1) by Example 1.2. Further, N(PSp_{2n}) is even by [Mer, 14.2]. This shows that N(PSp_{2n}) is 2 or 4, and is 2 in case n is even.

Suppose that n is odd. We must show that N(W λ) is divisible by 4 for every element λ of the root lattice of PSp_{2n}. We use the same notation as [Mer, §14] for the weights of PSp_{2n}: they are a sum ∑_{i=1}^{n} x_{i}e_{i} such that ∑ x_{i} is even. The Weyl group W is a semidirect product of (ℤ/2ℤ)^{n} (acting by flipping the signs of the e_{i}) and the symmetric group on n letters (acting by permuting the e_{i}). Taking X for the (ℤ/2ℤ)^{n}-orbit of ∑ x_{i}e_{i}, we have

(1.8)  \frac{1}{2} ∑_{x∈X} ⟨∑_{i} x_{i}e_{i}, (2e_{n})^{∨}⟩^{2} = 2^{r−1}x_{n}^{2}

where r denotes the number of nonzero x_{i}’s, cf. [Mer, pf. of Lemma 14.2]. If r = 1, then the unique nonzero x_{i} is even, and we find that for r ≠ 2, the sum—hence also N(W ∑ x_{i}e_{i})—is divisible by 4. Suppose that x_{1}, x_{2} are the only nonzero x_{i}’s; then by (1.8) we have:

N(W (x_{1}e_{1} + x_{2}e_{2})) = \begin{cases}2(n − 1)(x_{1}^{2} + x_{2}^{2}) & if x_{1} ≠ ±x_{2} \\ 2(n − 1)x_{1}^{2} & if x_{1} = ±x_{2}. \end{cases}

As n is odd, N(W (x_{1}e_{1} + x_{2}e_{2})) is divisible by 4, which completes the proof of the claim.

Example 1.9. For G adjoint of type E_{7}, we have N(G) = 12. To see this, we note that N(G) is divisible by N of the universal covering G̃ of G (which is 12) and that N(G) divides 2h^{∨} = 36 by Example 1.2. For the minuscule representation ρ of G̃, we have dim ρ = 56 and N(ρ) = 12. The representation ρ^{⊗2} of G̃ factors through G and by the “derivation formula”

N(ρ_{1} ⊗ ρ_{2}) = (dim ρ_{1}) · N(ρ_{2}) + (dim ρ_{2}) · N(ρ_{1})

(see e.g. [Mer, p. 122]) we have

N(ρ^{⊗2}) = 2(dim ρ)N(ρ) = 2^{6} · 3 · 7.

It follows that N(G) equals 12, as claimed.

2. The Lie algebra of G

2.1. Let G be a split almost simple algebraic group over F. We fix a pinning for G; it gives a split form G_{ℤ} of G over Spec ℤ such that the base change ℤ → F sends G_{ℤ} to G. Similarly the pinning gives a ℤ-form T_{ℤ} of the maximal torus T. We have a Cartan decomposition of the Lie algebra of G_{ℤ}:

(2.2)  Lie(G_{ℤ}) = Lie(T_{ℤ}) ⊕ ⨁_{α∈R} ℤx_{α}

and

(2.3)  Lie(T_{ℤ}) = {h ∈ Lie(T_{ℂ}) | μ(h) ∈ ℤ for all μ ∈ T^{∗}}.

see [St 68, p. 64]. Note that Lie(G) is naturally identified with Lie(G_{ℤ}) ⊗_{ℤ} F and similarly for Lie(T).

2.4. Write G̃ for the universal covering of G; we use the obvious analogues of the notations in 2.1 for G̃. The group G acts on G̃ by conjugation, hence also on Lie(G̃). If the kernel of the map G̃ → G is étale, then the representation Lie(G̃) is equivalent to the adjoint representation on Lie(G). But in prime characteristic, this need not hold. In any case, the natural map Lie(G̃) → Lie(G) is an isomorphism on the F -span of the x_{α}’s.

2.5. We claim that Lie(G̃) is a Weyl module for G in the sense of [J, p. 183], i.e., its character is given by Weyl’s formula and it is generated as a G-module by a highest weight vector. The first condition holds by (2.2), so it suffices to check the second.

To check that the submodule Gxα̃ generated by the highest weight vector xα̃ is all of Lie(G̃), one quickly reduces to checking that Gxα̃ contains Lie(T̃_{ℤ}). Equation (2.3) gives a natural isomorphism ℤ[R^{∨}] ⟶^{∼} Lie(T̃_{ℤ}) where T̃_{ℤ} is the maximal torus in G̃_{ℤ} mapping onto T_{ℤ}. We write (as is usual) h_{α} for the image of α^{∨} under this map. As [x_{α}, x_{−α}] = h_{α}, the claim is proved.

See [Hi] or [Ho] for descriptions of the composition series of Lie(G̃).

3. The number E(G)

Definition 3.1. Maintain the notation of the preceding section. The Killing form on Lie(G̃_{ℤ}) is divisible by 2h^{∨} [GN] and dividing by 2h^{∨} gives an indivisible even symmetric bilinear form b̃ on Lie(G̃_{ℤ}) such that

(3.2)  b̃(h_{α}, h_{α}) = 2 and b̃(x_{α}, x_{−α}) = 1

for long roots α, see [SpSt, p. 181] or [B Lie, Lemma VIII.2.4.3]. For a short root β, we have: b̃(h_{β}, h_{β}) = 2c and b̃(x_{β}, x_{−β}) = c, where c is the square-length ratio of α to β. For example, G = SL_{n} has Lie algebra the trace zero n-by-n matrices, and the form b̃ is the usual trace bilinear form (x, y) ↦ trace(xy), cf. [B Lie, Exercise VIII.13.12].

The natural map Lie(G̃_{ℤ}) → Lie(G_{ℤ}) is an inclusion and extending scalars to ℚ gives an isomorphism. Therefore, b̃ gives a rational-valued symmetric bilinear form on Lie(G_{ℤ}). We define E(G) to be the smallest positive rational number such that E(G) · b̃ is integer-valued on Lie(G_{ℤ}); we write b for this form. Note that E(G) is an integer by (3.2).

Clearly, E(G) depends only on the root system of G and the character lattice T^{∗} viewed as a sublattice of the weight lattice, and not on the field F.

3.3. Write Ḡ for the adjoint group of G; we use the obvious analogues of the notations in 2.1 for Ḡ. We have a commutative diagram

Q^{∨} ⟶^{∼} Lie(T̃_{ℤ})
↓  ↓
P^{∨} ⟶^{∼} Lie(T_{ℤ})

where Q^{∨} and P^{∨} are the root and weight lattices of the dual root system. The form b̃ restricts to be an inner product on Q^{∨} such that the square-length of a short coroot α^{∨} is 2. This inner product extends to a rational-valued inner product on P^{∨}, and E(Ḡ) is the smallest positive integer such that E(Ḡ) · b̃ is integer-valued on P^{∨}.

Example 3.4. Consider the case where G is PSp_{2n} for some n ≥ 2, i.e., adjoint of type C_{n}. In the notation of the tables in [B Lie], the form b̃ is twice the usual scalar product, i.e., b̃(e_{i}, e_{j}) = 2δ_{ij} (Kronecker delta). The fundamental weight ω_{n} has b̃(ω_{n}, ω_{n}) = n/2. Checking b̃(ω_{i}, ω_{j}) for all i, j, shows that E(Ḡ) is 1 if n is even and 2 if n is odd.

Example 3.5. Suppose that all the roots of G have the same length, so that we may identify the root system R with its dual and normalize lengths so that ⟨ , ⟩ is symmetric and equals b̃.

(1): E(Ḡ) is the exponent of P/Q, the weight lattice modulo the root lattice. Indeed, the natural isomorphism between P and Lie(T̄_{ℤ}) shows that E(Ḡ) is the smallest natural number such that E(Ḡ)·⟨ , ⟩ is integer-valued on P ×P, equivalently, the smallest natural number e such that eP is contained in Q; this is the exponent of P/Q.

(2): The bilinear form

b̃: Lie(G̃_{ℤ}) × Lie(Ḡ_{ℤ}) → ℚ

has image ℤ and identifies Lie(Ḡ_{ℤ}) with Hom_{ℤ}(Lie(G̃_{ℤ}), ℤ). (On the span of the x_{α}’s, this is clear from (3.2). On the Cartan subalgebras, it amounts to the statement that ⟨ , ⟩ identifies P with Hom(Q, ℤ).) It follows that Lie(Ḡ), as a G-module, is the dual of Lie(G̃), i.e., Lie(Ḡ) is the module denoted by H^{0}(α̃) in [J].

Example 3.6. For n = 3 or n ≥ 5, we claim that E(SO_{n}) = 1.

For n odd, SO_{n} is adjoint of type B_{ℓ} for ℓ = (n − 1)/2, and we compute as in 3.3 and Example 3.4. The dual root system is of type C_{ℓ}, and the form b̃ is the usual scalar product, i.e., b(e_{i}, e_{j}) = δ_{ij}. The fundamental weight ω_{i} is e_{1} + e_{2} + ⋯ + e_{i}, so E(SO_{2ℓ+1}) = 1.

For n even, SO_{n} has type D_{ℓ} for ℓ = n/2. The character group T^{∗} of a maximal torus in SO_{n} consists of the weights whose restriction to the center of Spin_{n} is 0 or agrees with the vector representation, i.e., the weights ∑ c_{i}ω_{i} such that c_{ℓ−1} + c_{ℓ} is even. It follows that that the cocharacter lattice T_{∗} is generated by the (co)root lattice and

ω_{ℓ−1} + ω_{ℓ} = α_{1} + 2α_{2} + ⋯ + (ℓ − 2)α_{ℓ−2} + \frac{ℓ − 1}{2}(α_{ℓ−1} + α_{ℓ}).

We have:

b̃(ω_{ℓ−1} + ω_{ℓ}, ω_{ℓ−1} + ω_{ℓ}) = ⟨ω_{ℓ−1} + ω_{ℓ}, ω_{ℓ−1} + ω_{ℓ}⟩ = ℓ − 1 ∈ ℤ,

so the form b̃ is integer-valued on T_{∗} and E(SO_{2ℓ}) = 1.

Example 3.7. For n ≥ 3, write HSpin_{4n} for the nontrivial quotient of Spin_{4n} that is neither SO_{4n} nor adjoint. A calculation like the one in Example 3.6 gives:

E(HSpin_{4n}) = \begin{cases}1 & if n is even \\ 2 & if n is odd. \end{cases}

4. Main results

The integer-valued symmetric bilinear form b on Lie(G_{ℤ}) defined in 3.1 gives by scalar extension a symmetric bilinear form on Lie(G) which we denote by b_{/F}.

Proposition 4.1. Let ρ be a representation of a split and almost simple algebraic group G over F. Then:

(1) E(G) divides N(ρ).

(2) Tr_{ρ} = \frac{N(ρ)}{E(G)}b_{/F}.

The following example is really the crux of the proof of the proposition.

Example 4.2. Suppose that ρ is a Weyl module of G. There is a ℤ-form ρ_{ℤ} of ρ and composing ρ_{ℤ} with the natural homomorphism G̃_{ℤ} → G_{ℤ} gives a representation ρ̃_{ℤ} of G̃_{ℤ}.

We first compute Trρ̃ over ℂ. If we decompose the representation ρ with respect to the action of T̃ and write V_{μ} for the eigenspace relative to the weight μ, then h_{α} acts on V_{μ} by scalar multiplication by ⟨μ, α^{∨}⟩, hence Tr_{ρ}(h_{α}, h_{α}) = ∑ dim(V_{μ})⟨μ, α^{∨}⟩^{2}. By putting together the μ in an orbit W λ (where λ is dominant) and taking α to be a long root, one gets:

(4.3)  Trρ̃(h_{α}, h_{α}) = 2 N(ρ).

The represenation Lie(G̃_{ℤ})⊗ℂ is irreducible and has a nondegenerate G̃_{/ℂ}-invariant symmetric bilinear form, so by Schur’s Lemma we have:

HomG̃_{/ℂ}(Lie(G̃) ⊗ ℂ, (Lie(G̃)^{∗}) ⊗ ℂ) = ℂ.

In particular, Trρ̃ equals z b̃ for some complex number z and

2N(ρ) = Trρ̃(h_{α}, h_{α}) = z b̃(h_{α}, h_{α}) = 2z.

Hence Trρ̃ = N(ρ) b̃.

Now Lie(G_{ℤ}) ⊗ ℂ is naturally identified with Lie(G̃_{ℤ}) ⊗ ℂ, so the form Tr_{ρ} on Lie(G_{ℤ}) is exactly N(ρ) b̃, i.e.,

(4.4)  Tr_{ρ} = \frac{N(ρ)}{E(G)} b.

As the representation ρ of G is defined over ℤ, the form Tr_{ρ} is integer-valued. But b is indivisible, and it follows that E(G) divides N(ρ).

By extending scalars, equation (4.4) holds with b replaced by b_{/F}. This verifies Prop. 4.1 for the Weyl module ρ.

Proof of Prop. 4.1. The number N(ρ) depends only on the class of ρ in the representation ring RG. As the Weyl modules generate RG as an abelian group and E(G) divides N(ψ) for every Weyl module ψ by Example 4.2, (1) follows.

For (2), we note that the map ρ ↦ Tr_{ρ} −(N(ρ)/E(G)) b_{/F} is compatible with exact sequences like (1.4) in the sense that Tr_{ρ} = Tr_{ρ′} + Tr_{ρ/ρ′}. We obtain a homomorphism of abelian groups

RG → symmetric bilinear forms on Lie(G)

that vanishes on the Weyl modules by Example 4.2, hence is zero. □

The form b_{/F} is not zero, because b is indivisible (as a form over ℤ). Proposition 4.1 immediately gives:

Theorem C. Let ρ be a representation of a split and almost simple algebraic group G over F. Then Tr_{ρ} is zero if and only if the characteristic of F divides N(ρ)/E(G). □

Theorem D. Assume G is split and almost simple. Then the following are equivalent:

(a) The characteristic of F divides the integer N(G)/E(G).

(b) Every trace form of G is zero.

Proof of Th. D. The number N(G) is defined to be gcd N(ρ) as ρ varies over the representations ρ of G defined over F. Therefore,

N(G)/E(G) = gcd_{ρ}{N(ρ)/E(G)}.

The theorem now follows from Th. C. □

Theorem A is the special case of Th. D where G is simply connected. (Indeed, for G simply connected, E(G) is 1 and the primes dividing N(G) are the torsion primes of G by 1.5.) Although Th. A is weaker, it has a much simpler condition (a).

Example 4.5. Suppose that the characteristic of F is an odd prime p, and let n be a natural number divisible by char F.

(1): The groups SL_{n} and PGL_{n} act naturally by conjugation on the n-by-n matrices M_{n}(F). For this representation ρ, the number N(ρ) is 2n (by Example 1.2), hence the corresponding trace form is zero on Lie(SL_{n}). But the trace form is nonzero on Lie(PGL_{n}), as one finds by checking directly or looking ahead to Prop. 5.1 below; the radical is the (codimension 1) image of Lie(SL_{n}).

(2): If p^{2} divides n, then p divides N(G)/E(G) for G = SL_{n} /μ_{p} by Prop. 5.1. Theorem D says that every trace form of G is zero, even though the universal covering SL_{n} and adjoint group PGL_{n} have representations with nonzero trace forms. (I thank George McNinch for suggesting Lie(G) as an interesting example. The representation Lie(G) of G is not only reducible, it is a direct sum of the image of Lie(SL_{n}) and a 1-dimensional subspace.)

Remark 4.6 (Non-split groups). One can extend Prop. 4.1 and Theorems A, C, and D to the case where G is nonsplit as follows. Assume G is absolutely almost simple and fix a pinning of G over a separable closure F_{sep} of F. For a representation ρ of G (over F), define N(ρ) to be the integer calculated as in 1.3 relative to the pinning over F_{sep}. We define N(G) to be gcd N(ρ) as ρ varies over the representations of G defined over F. We take E(G) to be the number given by the pinning over F_{sep} as in §3. (Note that with these extended definitions, N(G) now depends on the field F, but E(G) does not.) Proposition 4.1 (applied to the split group over F_{sep}) implies that E(G) divides N(G). Similarly, Theorems A, C, and D hold with the hypothesis “absolutely almost simple” instead of “split and almost simple”.

Remark 4.7 (char. 2). Readers familiar with characteristic 2 might prefer to consider the quadratic form

s_{ρ}: x ↦ − trace (∧^{2}dρ(x))

instead of the symmetric bilinear form Tr_{ρ}. The form s_{ρ} gives the negative of the “degree 2” coefficient of the characteristic polynomial of dρ(x). (Because dρ(Lie(G)) consists of trace zero matrices, s_{ρ} is the map x ↦ trace(dρ(x)^{2})/2; our definition has the advantage that it obviously makes sense also in characteristic 2.) The bilinear form derived from s_{ρ}—i.e., (x, y) ↦ s_{ρ}(x + y) − s_{ρ}(x) − s_{ρ}(y)—is Tr_{ρ}.

Theorem A is easy to extend. In case G is simply connected, Lie(G) is a Weyl module by 2.5 and s_{ρ} is zero if and only if Tr_{ρ} is zero by [Ga, Prop. 6.4(1)]. That is, conditions (a) and (b) in Th. A are equivalent to:

(c) For every representation ρ of G, the quadratic form s_{ρ} is zero.

Alternatively, one can proceed as follows. The bilinear form b̃ on Lie(G̃_{ℤ}) is even [GN, Prop. 4], so it is the bilinear form derived from a unique quadratic form q̃ on Lie(G̃_{ℤ}). The form q̃ extends to a rational-valued quadratic form on Lie(G_{ℤ}) and we write E_{q}(G) for the smallest positive rational number such that E_{q}(G) q̃ is integer-valued on Lie(G_{ℤ}). It is easy to see that E_{q}(G) is E(G) or 2E(G), and both cases can occur. (E.g., take G = SO_{2ℓ} with ℓ odd or even, respectively.) The statements and proofs of Theorems C and D go through if we replace Tr_{ρ}, E(G), and b with s_{ρ}, E_{q}(G), and E_{q}(G) q̃ respectively.

5. The ratio N(G)/E(G) for G = SL_{n} /μ_{m}

With Theorem D in hand, it remains to determine the primes dividing N(G)/E(G) for each group G. In this section, we fix natural numbers m and n with m dividing n, and we prove:

Proposition 5.1. For G = SL_{n} /μ_{m}, the primes dividing N(G)/E(G) are precisely the primes dividing

\begin{cases}gcd(m, n/m) & if m is odd \\ 2 gcd(m, n/m) & if m is even. \end{cases}

Here μ_{m} denotes the group scheme of m-th roots of unity, identified with the corresponding scalar matrices in SL_{n}.

In the important special cases where G is simply connected (m = 1), G is adjoint (m = n), or n is square-free, the gcd in the proposition is 1, and we have that N(G)/E(G) is 1 if m is odd and 2 if m is even.

Lemma 5.2.

E(SL_{n} /μ_{m}) = \frac{m}{gcd(m, n/m)} .

Proof. Use the notation of [B Lie] for the simple roots and fundamental weights of the root system A_{n−1} of SL_{n}. Let Λ denote the lattice generated by the root lattice Q and

β := \frac{n}{m} ω_{n−1} = \frac{1}{m} (α_{1} + 2α_{2} + ⋯ + (n − 1)α_{n−1}) .

We claim that Λ is identified with the cocharacter lattice T_{∗} for a pinning of SL_{n} /μ_{m}. Certainly, Λ/Q is cyclic of order m, so it suffices to check that the set of inner products (Λ, T^{∗}) consists of integers. But T^{∗} is the collection of weights ∑ c_{i}ω_{i} with c_{i} ∈ ℤ such that ∑_{i=1}^{n−1} ic_{i} is divisible by m. We have:

⟨β, ∑ c_{i}ω_{i}⟩ = ∑_{i} \frac{1}{m}ic_{i} ∈ ℤ (∑ c_{i}ω_{i} ∈ T^{∗}) ,

which proves that T_{∗} = Λ as claimed.

Finally, we compute:

⟨β, α_{n−1}⟩ = \frac{n}{m} ∈ ℤ and ⟨β, β⟩ = ⟨\frac{1}{m} ∑ iα_{i}, \frac{n}{m}ω_{n−1}⟩ = \frac{n(n − 1)}{m^{2}}.

Since m divides n, it is relatively prime to n − 1, so the minimum multiplier of ⟨ , ⟩ that takes integer values on T_{∗} is m/gcd(m, n/m), as claimed. □

5.3. Weights of representations of SL_{n} /μ_{m}. Fix the “usual” pinning of SL_{n}, where the torus T consists of diagonal matrices and the dominant weights are the maps

( ..._{t_{n}}^{t_{1}} ) ↦ ∏_{i=1}^{n−1} t_{i}^{e_{i}}

where e_{1} ≥ e_{2} ≥ ⋯ ≥ e_{n−1} ≥ 0. Such a weight restricts to x ↦ x^{∑ e_{i}} on the center of SL_{n}; in particular, m divides ∑ e_{i} for every dominant weight λ of a representation of SL_{n} /μ_{m}. The proof of [Mer, Lemma 11.4] shows that m divides N(W λ), hence m divides N(SL_{n} /μ_{m}).

We recall how to compute N(W λ) from [Mer, p. 136]. Write a_{1} > a_{2} > ⋯ > a_{k−1} > a_{k} = 0 for the distinct values of the exponents e_{i} in λ, where a_{i} appears r_{i} times, so that n = ∑ r_{i}. We have:

(5.4)  N(W λ) = \frac{(n − 2)!}{r_{1}! r_{2}! ⋯ r_{k}!} [n (∑_{i} r_{i}a_{i}^{2}) − (∑_{i} r_{i}a_{i})^{2}] .

Example 5.5. Let λ be a weight of G and let r_{i}, a_{i} be as in 5.3. Suppose that

v_{2} (∑ r_{i}a_{i}) ≥ v_{2}(n) > 0,

where v_{2}(x) is the 2-adic valuation of x, i.e., the exponent of the largest power of 2 dividing x. We claim that

(5.6)  v_{2}(N(W λ)) > v_{2}(n).

Write ∑ r_{i}a_{i} = 2^{θ}t and n = 2^{ν}u where θ = v_{2}(∑ r_{i}a_{i}) and ν = v_{2}(n). Our hypothesis is that 0 < ν ≤ θ. We may rewrite (5.4) as:

(5.7)  N(W λ) = \frac{(n − 2)!}{r_{1}! r_{2}! ⋯ r_{k}!} [u (∑_{i} r_{i}a_{i}^{2}) − 2^{2θ−ν}t^{2}] · 2^{ν}.

Write ℓ for the minimum of v_{2}(r_{i}), and fix an index j such that v_{2}(r_{j}) = ℓ. Note that since ∑ r_{i} = n, we have ℓ ≤ ν ≤ 2θ − ν.

The first term on the right side of (5.7) has 2-adic value ≥ −ℓ [Mer, p. 137]. The term in brackets has value ≥ ℓ. Therefore, to prove claim (5.6), it suffices to consider the case where v_{2}(∑ r_{i}a_{i}^{2}) = ℓ and the first term on the right side of (5.7) has value −ℓ; this latter condition implies that

(5.8) s_{2}(n − 1) = s_{2}(r_{1}) + ⋯ + s_{2}(r_{j−1}) + s_{2}(r_{j} − 1) + s_{2}(r_{j+1}) + ⋯ + s_{2}(r_{k}),

where s_{2} denotes the number of 1’s appearing in the binary representation of the integer [Mer, p. 137]. That is, when adding up the numbers r_{1}, …, r_{j−1}, r_{j} − 1, r_{j+1}, …, r_{k} in base 2 (to get n − 1), there are no carries.

Suppose first that ℓ < ν. Equation (5.8) implies that there are exactly two indices, say, j, j′ with v_{2}(r_{j}) = v_{2}(r_{j′}) = ℓ. As 2^{ℓ+1} divides ∑ r_{i}a_{i}, it also divides r_{j}a_{j} + r_{j′}a_{j′}, hence a_{j} and a_{j′} have the same parity. It follows that 2^{ℓ+1} divides r_{j}a_{j}^{2}+r_{j′}a_{j′}^{2} and the term in brackets in (5.7) has 2-adic valuation > ℓ and we are done in this case.

We are left with the case where ℓ = ν. By (5.8), r_{j} is the unique r_{i} with 2-adic valuation ℓ. As v_{2}(∑ r_{i}a_{i}^{2}) = ℓ, the number a_{j} is odd and we have:

ℓ = v_{2} (∑ r_{i}a_{i}) = θ ≥ ν = ℓ.

Hence both u(∑ r_{i}a_{i}^{2}) and 2^{2θ−ν}t have 2-adic valuation ℓ. It follows that the term in brackets in (5.7) has 2-adic valuation strictly greater than ℓ, and claim (5.6) is proved.

Proof of Prop. 5.1. We write G for SL_{n} /μ_{m}. For an upper bound, N(G) divides 2n by Example 1.2. Also, the dominant weight λ with e_{1} = m and e_{i} = 0 for i > 1 belongs to T^{∗} and has N(W λ) = m^{2} by (5.4), so N(G) divides m^{2}. Applying Lemma 5.2 gives:

gcd(m, n/m) divides N(G)/E(G) divides gcd(m, n/m) gcd(m, 2n/m).

This completes the proof for m odd.

Clearly, an odd prime divides N(G)/E(G) if and only if it divides gcd(m, n/m). So suppose that m is even and 2 does not divide gcd(m, n/m), i.e., v_{2}(m) = v_{2}(n). Then every weight of a representation of G satisfies the hypotheses of Example 5.5, hence v_{2}(N(G)) > v_{2}(n) = v_{2}(m). By Lemma 5.2, v_{2}(E(G)) = v_{2}(m), so 2 divides N(G)/E(G). This completes the proof of Prop. 5.1. □

6. The ratio N(G)/E(G) for simple G

The purpose of this section is to compute the primes dividing N(G)/E(G) for all almost simple split groups G. The results are given in Table I. We write PSO_{n} for the adjoint group of SO_{n}; when n is odd it is the same as SO_{n}. 6.1. Justification of Table I. We now justify the claims about N(G)/E(G) given in Table I. For G simply connected, E(G) is 1 and N(G) is divisible precisely

G  primes dividing N(G)/E(G)
SL_{n} /μ_{m}  see Prop. 5.1
Sp_{2n}  none
SO_{n}, Spin_{n}, and PSO_{n} for n = 3 and n ≥ 5 2
HSpin_{4n} for n ≥ 3, PSp_{2n}, E_{6} adjoint 2
E_{6} simply connected, E_{7}, F_{4}, G_{2} 2, 3
E_{8}  2, 3, 5

Table I. The primes dividing N(G)/E(G).

by the torsion primes of G, see 1.5. We assume that G is not simply connected and write G̃ for the universal covering of G; obviously N(G̃) divides N(G).

For G = PSp_{2n}, SO_{n}, or adjoint of type E_{7}, one combines Examples 1.7 and 3.4; 1.6 and 3.6; or 1.9 and 3.5, respectively.

For G adjoint of type D_{n}, we have E(G) = 2 by Example 3.5. Also, 4 divides N(G) by [Mer, 15.2]. On the other hand, the spinor representations of G̃ have Dynkin index 2^{n−3} [LS], and it is easy to use this as in Example 1.9 to construct a representation ρ of G with N(ρ) a power of 2. This shows that N(G)/E(G) is a power of 2 and is not 1.

Now let G = HSpin_{4n} for some n ≥ 3. The dual of the center of Spin_{4n} is the Klein four-group, and we write χ for the unique element that vanishes on the kernel of the map Spin_{4n} → HSpin_{4n}. The gcd of N(W λ) as λ varies over the weights that restrict to χ (respectively, 0) on the center of Spin_{4n} is 2^{2n−3} (resp., divisible by 4) by [Mer, p. 146], hence N(G) is a power of 2 and at least 4. On the other hand, E(HSpin_{4n}) is 1 or 2. We conclude that N(G)/E(G) is a power of 2 and is not 1.

For G adjoint of type E_{6}, the number N(G) is divisible by N(G̃) = 6 and divides 2h^{∨} = 24 by Example 1.2. By Example 3.5, N(G)/E(G) is 2, 4, or 8.

7. Trace forms and Lie algebras

We assume in this section that G is absolutely almost simple, split, and simply connected and that the characteristic of F is neither 2 nor 3. Write 𝔠 for the center of Lie(G); the quotient \overline{𝔤} := Lie(G)/𝔠 is a simple Lie algebra [St 61, 2.6(5)]. Over an algebraically closed field, the algebras \overline{𝔤} arising in this way are sometimes called “simple Lie algebras of classical type” (even when the root system R is exceptional).

Proposition 7.1. If \overline{𝔤} has a representation ψ over F with Tr_{ψ} nonzero, then G has an irreducible representation ρ over F such that Tr_{ρ} is not zero and whose differential vanishes on 𝔠.

Proof. Replacing ψ with one of the irreducible quotients in its composition series, we may assume that ψ is irreducible. Then ψ is restricted by [Bl, Th. 5.1] (using that F has characteristic ≠ 2, 3). Because the projection Lie(G) → \overline{𝔤} is restricted, the composition gives a restricted irreducible representation of Lie(G), which is the differential of a representation of G by [Cu] and [St 63]. (These references only give a representation of G defined over an algebraic closure of F, but G is split, so the irreducible representations of G over F are in natural one-to-one correspondence with those over an algebraic closure.) □

Because of our hypothesis on the characteristic, Lie(G) is not simple only for groups of type A_{n−1} where n is divisible by the characteristic of F. In that case, Lie(G) is 𝔰𝔩_{n} and its center 𝔠 consists of the scalar matrices F · 1.

Corollary 7.2 (Block [Bl, Th. 6.2]). If char F divides n, then every representation of 𝔰𝔩_{n}/𝔠 has zero trace form.

Proof. Suppose that 𝔰𝔩_{n}/𝔠 has a representation with a nonzero trace form. Then SL_{n} has an irreducible representation ρ such that Tr_{ρ} is not zero and dρ vanishes on the scalar matrices. Identifying the center of SL_{n} with the (non-reduced) group scheme μ_{n} identifies the restriction of ρ to μ_{n} with a map x ↦ x^{ℓ}. Our hypothesis on dρ says that ℓ is divisible by the characteristic p of F, hence ρ factors through the natural map SL_{n} → SL_{n} /μ_{p}. It follows from 5.3 that N(ρ) is divisible by p. Hence Tr_{ρ} vanishes by Th. C, a contradiction. □

We close by proving a stronger version of Cor. B from the introduction. For a Lie algebra L over F and a representation ψ of L, write rad ψ for the radical of the trace bilinear form Tr_{ψ}; it is an ideal of L. We prove:

Corollary B′. For every representation ψ of every Lie algebra L over a field of characteristic 5, the quotient L/ rad ψ is not isomorphic to the Lie algebra of an algebraic group of type E_{8}.

That is, over a field of characteristic 5, the Lie algebra of a group of type E_{8} “has no quotient trace form”.

Proof of Cor. B′. Suppose the corollary is false. That is, suppose that there is a group G of type E_{8} and a Lie algebra L with a representation ψ and a surjection π: L → Lie(G) with kernel the radical of Tr_{ψ}.

By [Bl, Lemma 2.1], we may assume that the radical of Tr_{ψ} is contained in the center of L, i.e., L is a central extension of Lie(G). It follows that there is a map f : Lie(G) → L such that πf is the identity [St 62, Th. 6.1(c)]. Clearly, the representation ψf of Lie(G) has nonzero trace form.

As Lie(G) is simple, we can apply Prop. 7.1 and deduce that the algebraic group of type E_{8} over F has a representation ρ such that Tr_{ρ} is not zero, but this is impossible by Theorem A. □

Note that in the course of proving Corollary B′ we have also proved Cor. B.

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Department of Mathematics & Computer Science, Emory University, Atlanta, GA 30322, USA

E-mail address: skip@member.ams.org
URL: http://www.mathcs.emory.edu/~skip/