The exterior algebra and ‘Spin’ of an orthogonal 𝔤-module

Dmitri I. Panyushev^{1}

Introduction

Let 𝔤 be a reductive algebraic Lie algebra over an algebraically closed field 𝕜 of characteristic zero and G is the corresponding connected and simply connected group. The symmetric algebra of a (finite-dimensional) 𝔤-module 𝕍 is the algebra of polynomial functions on the dual space 𝕍^{∗}. Therefore one can study the algebra of symmetric invariants using geometry of G-orbits in V^{∗}. In case of the exterior algebra, ∧^{•}𝕍, lack of such geometric picture results by now in absence of general structure theorems for the algebra of skew-invariants (∧^{•}𝕍)^{𝔤}. One may find in the literature several interesting results related to skew-symmetric invariants. We only mention Kostant’s computation for cohomology of the nilradical of a parabolic subalgebra in 𝔤 [Ko61] and R. Howe’s classification of “skew-multiplicity-free” 𝔤-modules [Ho95, ch. IV]. But the general situation still remains unsatisfactory, and developing of Invariant Theory in the skew-symmetric setting represents an attractive problem. In this paper, we begin with describing all irreducible orthogonal 𝔤-modules such that (∧^{•}𝕍)^{𝔤} is again an exterior algebra. It is shown that in this case either 𝕍 ≃ 𝔤 and hence 𝔤 is simple or 𝔤 ⊕ 𝕍 has a structure of simple ℤ_{2}-graded Lie algebra, which quickly leads to a short classification, see Table 1. Obviously, none of the symplectic representations (with dim 𝕍 > 2) can have an exterior algebra of skew-invariants. But the situation for the representations of “general type” is not yet clear. In case 𝕍 is orthogonal, a better understanding of the 𝔤-module structure of ∧^{•}𝕍 can be achieved through the notion of ‘Spin’ of 𝕍. This goes as follows. Let π : 𝔤 → 𝔰𝔬(𝕍) be the corresponding representation. Restricting the spinor representation of 𝔰𝔬(𝕍) to 𝔤 gives us a 𝔤-module, which is denoted by Spin(𝕍). The motivation came from Kostant’s result that Spin(𝔤) is a primary 𝔤-module; namely, Spin(𝔤) = 2^{[rk 𝔤/2]}𝕍_{ρ}, the highest weight ρ being the half-sum of the positive roots [Ko61, p. 358]. The main property of Spin(𝕍) is that, depending on parity of dim 𝕍, ∧^{•}𝕍 is isomorphic to either Spin(𝕍)^{⊗2} or 2·Spin(𝕍)^{⊗2}. It is thus interesting to find the orthogonal 𝔤-modules, where Spin(𝕍) has a simple structure. In general, Spin(𝕍), as element of the representation ring, has a numerical factor depending on the zero-weight multiplicity. Omitting this factor yields a 𝔤-module, which is called the reduced ‘Spin’ of 𝕍 and denoted by Spin_{0}(𝕍); e.g. Spin_{0}(𝔤) = 𝕍_{ρ}. In a sense, Spin_{0}(𝕍) behaves better than Spin(𝕍). For, regardless of parity of dim 𝕍, we have ∧^{•}𝕍 ≃ 2^{m(0)}·Spin_{0}(𝕍)^{⊗2}, where m(0) is the zero-weight multiplicity, and Spin_{0}(𝕍_{1} ⊕ 𝕍_{2}) = Spin_{0}(𝕍_{1}) ⊗ Spin_{0}(𝕍_{2}).

1supported in part by R.F.F.I. Grant � 98–01–00598

An orthogonal 𝔤-module 𝕍 is said to be co-primary, if Spin_{0}(𝕍) is irreducible. In sections 2 and 3, a classification of the co-primary modules is obtained. To this end, we give a geometric description of some highest weights of Spin_{0}(𝕍). These weights are called extreme. The assumption that Spin_{0}(𝕍) has a unique extreme weight imposes strong constraints on the weight structure of 𝕍. Using this, one shows that 𝔤 must be simple whenever 𝕍 is an irreducible faithful co-primary 𝔤-module, and that any reducible coprimary module is being obtained by iterating the “direct sum” procedure:

(𝔤_{i}, 𝕍_{i}), i = 1, 2 ↦ (𝔤_{1} ⊕ 𝔤_{2}, 𝕍_{1} ⊕ 𝕍_{2}).

It is thus sufficient to classify the irreducible co-primary modules. The resulting list appears to be rather short:

1. 𝔤 is simple and 𝕍 ≃ 𝔤;

2. 𝔤 is of type B_{n} or C_{n} or F_{4}, and 𝕍 is the little adjoint module;

3. 𝔤 = 𝔰𝔬(𝕎) and 𝕍 = {the Cartan component of S^{2}𝕎}; dim 𝕎 = 3, 5, 7, . . .. In case 2, 𝔤 has roots of two lengths and the 𝔤-module whose highest weight is the short dominant root is called little adjoint (l.a.). Actually, we give a unified proof for the fact that the l. a. module is co-primary whenever the ratio of root lengths is \sqrt{2}. Note that, for G_{2}, where this ratio is \sqrt{3}, the l. a. module is not co-primary. A true reason why this is so is that the l. a. module for G_{2} is not the isotropy representation of a symmetric space, whereas this is the case for B, C, and F. Furthermore, all representations listed above are the isotropy representations of symmetric spaces. A curious coincidence in this regard is the following. Let G̃/G be a symmetric space and 𝔤̃ = 𝔤 ⊕ 𝕍 the corresponding Lie algebra decomposition. Then the 𝔤-module 𝕍 is co-primary if and only if 𝔤 is nonhomologous to zero in 𝔤̃. Another by-product of our classifications is that any co-primary module has an exterior algebra of skew-invariants. Having observed that any co-primary representation is a very specific isotropy representation, one may suggest that Spin_{0}(𝕍) admits a nice description for all symmetric spaces. This is really the case, and a transparent formulation can be given for the inner involutory automorphisms. Let 𝔤 = 𝔤_{0} ⊕ 𝔤_{1} be a ℤ_{2}-grading of inner type, i.e., rk 𝔤_{0} = rk 𝔤. As the 𝔤_{0}-module 𝔤_{1} has no zero weight, Spin(𝔤_{1}) = Spin_{0}(𝔤_{1}). Choose a common Cartan subalgebra 𝔱 for 𝔤 and 𝔤_{0}, and consider the natural inclusion of the Weyl groups W_{0} ⊂ W. Although W_{0} is not necessary a parabolic subgroup of W, each coset wW_{0} contains a unique element of minimal length (see 4.1). Let W^{0} ⊂ W be the set of such elements. Then the irreducible constituents of the 𝔤_{0}-module Spin(𝔤_{1}) are parameterized by W^{0}. Namely, Spin(𝔤_{1}) = ⨁_{w∈W^{0}} 𝕍_{λ_{w}}, where λ_{w} = w^{−1}ρ−ρ_{0} is the highest weight, see section 5.

Moreover, the weights λ_{w} (w ∈ W^{0}) are distinct and hence Spin(𝔤_{1}) is a multiplicity free 𝔤_{0}-module. It is worth noting that the above expression for Spin(𝔤_{1}) is equivalent to an identity for root systems that seem to have not been observed before. Let Δ be the root system of (𝔤, 𝔱) and let Δ^{+} = Δ_{0}^{+} ⊔ Δ_{1}^{+} be the partition of the set of positive roots corresponding to the sum 𝔤 = 𝔤_{0} ⊕ 𝔤_{1}. In this situation, one can introduce the “cunning” parity τ : W → {1, −1}, determined by Δ_{0}^{+}. If w ∈ W_{0}, then τ(w) = (−1)^{l_{0}(w)}, where l_{0}(·) is the length in W_{0} relative to the set of positive roots Δ_{0}^{+}. To extend τ to W, one uses the aforementioned subset W^{0} (see section 4). Then the identity reads

∑_{w∈W} τ(w)e^{wρ} = ∏_{α∈Δ_{0}^{+}}(e^{α/2} − e^{−α/2}) ∏_{μ∈Δ_{1}^{+}}(e^{μ/2} + e^{−μ/2}) .

For the outer involutory automorphisms, the final description of Spin_{0}(𝔤_{1}) is almost identical to the previous one, see section 6. However, it requires much more preparations and its proof uses the classification of involutory automorphisms. Our arguments suggest that there should exist interesting connections between cohomology of symmetric spaces, twisted affine Kac–Moody algebras, and Spin(𝔤_{1}). The description of the highest weights of Spin(𝔤_{1}) (for all involutions!) shows that these weights are extreme. This also implies the following claim (see sect. 7):

Let Φ( , ) be an invariant bilinear form on 𝔤 and Φ( , )_{0} its restriction to 𝔤_{0}. Let c_{0} ∈ U(𝔤_{0}) be the Casimir element with respect to Φ( , )_{0}. Then c_{0} acts scalarly on Spin(𝔤_{1}); the value is (ρ, ρ) − (ρ_{0}, ρ_{0}), where ( , ) is the W -invariant bilinear form on 𝔱^{∗} induced by Φ( , ). A similar result holds for the isotropy representation 𝔥 → 𝔰𝔬(𝔪) of non-symmetric space G/H, if rk 𝔥 = rk 𝔤 and one considers the submodule of Spin(𝔪) generated by the extreme weight vectors. Recently, B. Kostant obtained a series of nice results for Spin(𝔤) [Ko97]. Since the adjoint representation is one of the isotropy representations of symmetric spaces, our results for Spin(𝔤_{1}) suggest that many parts of Kostant’s theory can be generalized to the setting of arbitrary symmetric spaces. Main notation. 𝔤 is a reductive Lie algebra with a fixed triangular decomposition: 𝔤 = 𝔲^{+} ⊕ 𝔱 ⊕ 𝔲^{−}. All 𝔤-modules are assumed to be finite-dimensional. Δ (resp. Δ^{+}) is the set of roots (resp. positive roots); Π ⊂ Δ^{+} is the set of simple roots; Π = {α_{i}}_{i∈I} and φ_{i} is the fundamental weight corresponding to α_{i}. For simple Lie algebras, we follow the numeration of the simple roots from [VO88] and [On95].

P – the lattice of integral weights, P_{+} – the monoid of dominant integral weights. W = N_{G}(𝔱)/Z_{G}(𝔱) = N_{G}(𝔱)/T – the Weyl group; for β ∈ Δ, s_{β} is the reflection in W.

P_{ℚ} = P ⊗_{ℤ} ℚ ⊂ 𝔱^{∗} and ( , ) is the W -invariant positive-definite scalar product in P_{ℚ} determined by a non-degenerate invariant bilinear form Φ( , ) on 𝔤. If M ⊂ P is any finite set of weights, then |M| = ∑_{m∈M} m; ρ := \frac{1}{2}|Δ^{+}|. If λ ∈ P_{+}, then 𝕍_{λ} stands for the irreducible 𝔤-module with highest weight λ.

Acknowledgements. I would like to thank A.L. Onishchik for conversations about cohomology of compact homogeneous spaces. I am indebted to B. Kostant for drawing my attention to Conlon’s paper [Co72]. Part of this work was done while I was visiting Université de Poitiers. I am grateful to this institution for the great hospitality I enjoyed there.

1 Orthogonal 𝔤-modules with an exterior algebra of skew- invariants

Let 𝕍 be a 𝔤-module. Study of the algebra (S^{•}𝕍)^{𝔤} of symmetric (or polynomial) invariants is the subject of a rich and well-developed theory. In contrast, little is known about the algebra (∧^{•}𝕍)^{𝔤} of skew-invariants. The skew-symmetric theory has some parallels to the symmetric case, and many interesting differences. We begin with two observations.

(1.1) Put n = dim 𝕍. Suppose 𝔤 ⊂ 𝔰𝔩(𝕍), e.g. 𝔤 is semisimple. Then ∧^{n}𝕍 is a trivial 𝔤-module. Therefore dim(∧^{•}𝕍)^{𝔤} ≥ 2, ∧^{i}𝕍 and ∧^{n−i}𝕍 are isomorphic 𝔤-modules, and the Poincaré polynomial of (∧^{•}𝕍)^{𝔤} is symmetric.

(1.2) Let P(𝕍) be the set of all weights of 𝕍 relative to 𝔱 ⊂ 𝔤 and 𝕍 = ⊕_{μ∈P(𝕍)}𝕍^{μ} the weight decomposition. Set m(μ) = dim 𝕍^{μ}. Recall that the character of 𝕍 is the element of the group algebra ℤ[P] defined by ch 𝕍 = ∑_{μ∈P(𝕍)} m(μ)e^{μ}. Then, t being an indeterminate, we have

∑_{i=1}^{n}(ch ∧^{i} 𝕍)t^{i} = ∏_{μ∈P(𝕍)}(1 + te^{μ})^{m(μ)} .

In particular, ch ∧^{•} 𝕍 = 2^{m(0)} ∏_{μ≠0}(1 + e^{μ})^{m(μ)}. Obviously, ∏_{μ≠0}(1 + e^{μ})^{m(μ)} must be the character of a 𝔤-module, say 𝕎. Hence 1 ≤ dim(∧^{•}𝕍)^{𝔤} = 2^{m(0)}· dim 𝕎^{𝔤} and therefore dim(∧^{•}𝕍)^{𝔤} ≥ 2^{m(0)}.

It is natural to first describe 𝔤-modules, where the algebra of skew-invariants has a simple structure. Definition. The algebra (∧^{•}𝕍)^{𝔤} is said to be free (or an exterior algebra), if there exists a graded subspace P ⊂ (∧^{•}𝕍)^{𝔤} such that (∧^{•}𝕍)^{𝔤} is the exterior algebra over P. Suppose 𝔤 ⊂ 𝔰𝔩(𝕍) and (∧^{•}𝕍)^{𝔤} is an exterior algebra. Let P = ⟨p_{1}, …, p_{l}⟩ with deg p_{i} = d_{i}. Then 0 ≠ p_{1} ∧ … ∧ p_{l} must be an element of ∧^{n}𝕍. Hence ∑_{i} d_{i} = n. It follows from the definition that all the d_{i}’s must be odd whenever l > 1. However, if l = 1, then d_{1} = n is allowed to be even. In other words, all 2-dimensional algebras of skew-invariants are proclaimed to be exterior. Example. Let 𝕍 = 𝔤. Then (∧^{•}𝔤)^{𝔤} is free. Here l = rk 𝔤 and d_{i} = 2m_{i} + 1, where m_{1}, …, m_{l} are the exponents of 𝔤. A purely algebraic proof of this result was given by Koszul [Kos50]. From now on, 𝕍 is an orthogonal 𝔤-module, i.e., we are given a representation π : 𝔤 → 𝔰𝔬(𝕍). In particular, 𝔤 ⊂ 𝔰𝔩(𝕍).

1.3 Lemma. Let 𝕍 be an irreducible orthogonal 𝔤-module with 𝕍^{𝔤} = 0. Suppose (∧^{•}𝕍)^{𝔤} is free. Then either (∧^{4}𝕍)^{𝔤} = 0 or dim 𝕍 = 4.

Proof. Since (∧^{1}𝕍)^{𝔤} = (∧^{2}𝕍)^{𝔤} = 0, any nonzero element of (∧^{4}𝕍)^{𝔤} is a generator of (∧^{•}𝕍)^{𝔤}. □

Let μ : 𝕍×𝕍^{∗} → 𝔤^{∗} ≃ 𝔤 be the moment mapping associated with the standard symplectic structure on 𝕍 × 𝕍^{∗} ≃ T^{∗}(𝕍). Identifying 𝕍 and 𝕍^{∗}, one obtains an anti-commutative bilinear mapping μ̄ : 𝕍 × 𝕍 → 𝔤. Using the 𝔤-invariant symmetric bilinear forms Φ( , ) and ( , )_{𝕍}, one may explicitly define μ̄ by

Φ(μ̄(v_{1}, v_{2}), g) := (v_{2}, g·v_{1})_{𝕍} , (1.4)

where v_{1}, v_{2} ∈ 𝕍 and g·v_{1} is a shorthand for π(g)v_{1}. This μ̄ yields an anti-commutative multiplication, denoted by [ , ] , in 𝔤 ⊕ 𝕍:

[(g_{1}, v_{1}), (g_{2}, v_{2})] := ([g_{1}, g_{2}] + μ̄(v_{1}, v_{2}), g_{1}·v_{2} − g_{2}·v_{1}) .

The following assertion is stated in [Co72, p. 152], in the context of compact group representations, as the “Cartan-Kostant theorem”. It is an easy part of Kostant’s characterization of the isotropy representation of compact homogeneous spaces [loc. cit].

1.5 Proposition. The multiplication [ , ] satisfies the Jacobi identity if and only if the skew-symmetric 𝔤-invariant 4-form on 𝕍

(v_{1}, v_{2}, v_{3}, v_{4}) ↦^{κ} Φ(μ̄(v_{1}, v_{2}), μ̄(v_{3}, v_{4})) + Φ(μ̄(v_{2}, v_{3}), μ̄(v_{1}, v_{4})) + Φ(μ̄(v_{3}, v_{1}), μ̄(v_{2}, v_{4}))

is identically equal to zero.

Proof. By bilinearity of [ , ] , it suffices to verify the Jacobi identity for 4 sorts of triples: (i) (g_{1}, g_{2}, g_{3}), (ii) (g_{1}, g_{2}, v_{1}), (iii) (g_{1}, v_{1}, v_{2}), (iv) (v_{1}, v_{2}, v_{3}), where g_{i} ∈ 𝔤 and v_{i} ∈ 𝕍. The Jacobi identity is always satisfied for cases (i)–(iii), because, respectively, 𝔤 is a Lie algebra, 𝕍 is a 𝔤-module, and μ̄ is a homomorphism of 𝔤-modules. For v_{1}, v_{2}, v_{3} ∈ 𝕍, the identity means that

μ̄(v_{1}, v_{2})·v_{3} + μ̄(v_{2}, v_{3})·v_{1} + μ̄(v_{3}, v_{1})·v_{2} = 0 ∈ 𝕍

or

(μ̄(v_{1}, v_{2})·v_{3} + μ̄(v_{2}, v_{3})·v_{1} + μ̄(v_{3}, v_{1})·v_{2}, v_{4})_{𝕍} = 0

for any v_{4} ∈ 𝕍. Using Eq. (1.4), one rewrites the last equality as the condition that the mapping κ : 𝕍^{⊗4} → 𝕜 is zero. It is also easily seen that κ is skew-symmetric and 𝔤-invariant. □

1.6 Corollary. If (∧^{4}𝕍)^{𝔤} = 0, then 𝔤 ⊕ 𝕍, endowed with multiplication [ , ] , is a ℤ_{2}-graded Lie algebra.

Notice that the condition (∧^{4}𝕍)^{𝔤} = 0 is not necessary for 𝔤 ⊕ 𝕍 to be a ℤ_{2}-graded Lie algebra. We are going to list all irreducible orthogonal 𝔤-modules 𝕍 such that (∧^{•}𝕍)^{𝔤} is free.

1.7 Theorem. Let 𝔤 be semisimple and 𝕍 a faithful orthogonal irreducible 𝔤-module. Suppose (∧^{•}𝕍)^{𝔤} is free. Then either 𝔤 is simple and 𝕍 ≃ 𝔤 or 𝔤̃ := 𝔤 ⊕ 𝕍 is a \underline{simple} ℤ_{2}-graded Lie algebra.

Proof. 1. Assume that dim 𝕍 ≠ 4. By Lemma 1.3 and Corollary 1.6, it follows that [ , ] makes 𝔤̃ a ℤ_{2}-graded Lie algebra. Let 𝔞 ⊂ 𝔤̃ be an ideal. Then 𝔞 ∩ 𝕍 and 𝔞 ∩ 𝔤 are 𝔤-stable spaces.

(i) If 𝔞 ∩ 𝕍 = 𝕍, then 𝔞 also contains μ̄(𝕍, 𝕍) = [𝕍, 𝕍] . Since 𝕍 is faithful, μ̄(𝕍, 𝕍) meets all the simple components of 𝔤. Therefore 𝔞 = 𝔤̃.

(ii) If 𝔞 ∩ 𝔤 ≠ 0, then, since 𝕍 is faithful, (𝔞 ∩ 𝔤)·𝕍 ≠ 0. That is, 𝔞 ∩ 𝕍 ≠ 0 and we are back in part (i).

(iii) If 𝔞 ∩ 𝕍 = 0 and 𝔞 ∩ 𝔤 = 0, then the 𝔤-module 𝔞 is isomorphic to its projections to both 𝕍 and 𝔤. Hence pr_{𝔤}(𝔞) ≃ 𝔞 ≃ pr_{𝕍}(𝔞) ≃ 𝕍. Therefore pr_{𝔤}(𝔞) is a a simple component of 𝔤. As 𝕍 is a faithful 𝔤-module, we conclude that 𝔤 ≃ 𝕍 and therefore 𝔤 is simple in this case. Here 𝔤̃ is the sum of two isomorphic ideals, 𝔤̃ ≃ 𝔞⊕𝔞. The subalgebra 𝔤, which is isomorphic to 𝔞, is the diagonal in 𝔤̃, and 𝕍 = {(x, −x) | x ∈ 𝔞}.

2. Assume that dim 𝕍 = 4. Then 𝔤 ⊂ 𝔰𝔬_{4} = 𝔰𝔬(𝕍). Obviously, 𝔰𝔬_{4} ⊕ 𝕍 ≃ 𝔰𝔬_{5}, and one easily verifies that (∧^{•}𝕍)^{𝔤} is not free for any proper reductive subalgebra 𝔤 of 𝔰𝔬_{4}. □

As is mentioned above, (∧^{•}𝔤)^{𝔤} is free. Thus, all other irreducible orthogonal modules with free algebra of skew-invariants arise in connection with ℤ_{2}-gradings of simple Lie algebras. If 𝔤̃ = 𝔤 ⊕ 𝕍 is a simple ℤ_{2}-graded Lie algebra, then (∧^{•}𝕍)^{𝔤} is isomorphic to H^{∗}(G̃/G), the cohomology ring of the symmetric space G̃/G [On95, § 9, n.11]. The cases, where H^{∗}(G̃/G) is an exterior algebra, are well known, see [On95, § 13, Th. 1]. Note however that our interpretation of “exterior algebras” is a bit wider. In case dim H^{∗}(G̃/G) = 2, the generator is allowed to be of even degree. The resulting classification is presented in Table 1.

Table 1: The irreducible orthogonal representations with free algebra of skew-invariants

𝔤  𝕍  dim P Poincaré polynomial of (∧^{•}𝕍)^{𝔤}
any simple Lie algebra 𝔤  rk 𝔤  ∏_{i=1}^{rk 𝔤}(1 + t^{2m_{i}+1})
𝔰𝔭_{2n} (n ≥ 2) 𝕍_{φ_{2}}  n − 1 (1 + t^{5})(1 + t^{9}) … (1 + t^{4n−3})
𝔰𝔬_{2n+1} (n ≥ 2) 𝕍_{2φ_{1}}  n  (1 + t^{5})(1 + t^{9}) … (1 + t^{4n+1})
𝔰𝔩_{2}  𝕍_{4φ}  1  1 + t^{5}
𝔰𝔬_{n} (n ≥ 5) 𝕍_{φ_{1}}  1  1 + t^{n}
𝔰𝔩_{2} ⊕ 𝔰𝔩_{2}  𝕍_{φ} ⊗ 𝕍_{φ′}  1  1 + t^{4}
𝔣_{4}  𝕍_{φ_{1}}  2  (1 + t^{9})(1 + t^{17})

If 𝕍 is a symplectic 𝔤-module, then (∧^{2}𝕍)^{𝔤} ≠ 0. Thus, (∧^{•}𝕍)^{𝔤} cannot be free unless dim 𝕍 = 2. If 𝕍 is neither orthogonal nor symplectic, then all known instances of free algebras of skew-invariants are those with dim(∧^{•}𝕍)^{𝔤} = 2.

2 ‘Spin’ of an orthogonal 𝔤-module and its properties

Let 𝕍 be a 𝕜-vector space endowed with a non-degenerate quadratic form Q. Denote by 𝔰𝔬(𝕍) = 𝔰𝔬_{Q}(𝕍) the respective orthogonal Lie algebra and by C_{Q}(𝕍) the Clifford algebra of Q. Let 𝕎, 𝕎′ be maximal Q-isotropic subspaces of 𝕍 and 𝕎∩𝕎′ = 0. The following relations are well-known in the theory of Clifford algebras (see e.g. [FH96, § 20.1]) :

(i) C_{Q}(𝕍) ≃ End (∧^{•}𝕎), if dim 𝕍 is even,

(ii) C_{Q}(𝕍) ≃ End (∧^{•}𝕎) ⊕ End (∧^{•}𝕎′), if dim 𝕍 is odd. As ∧^{•}𝕎 (or ∧^{•}𝕎′) is the underlying space of the spin representation of 𝔰𝔬(𝕍) (in case (i) this representation is the sum of two half-spin representations), we shall write Spin(𝕍) in place of ∧^{•}𝕎. The above relations are thought of as isomorphisms of 𝔰𝔬(𝕍)-modules. It is well-known (and easily seen) that C_{Q}(𝕍) has an 𝔰𝔬(𝕍)-stable filtration such that the associated graded algebra is isomorphic to the exterior algebra of 𝕍. Since in both cases Spin(𝕍) is a self-dual module, we obtain the following isomorphisms of 𝔰𝔬(𝕍)-modules:

∧^{•}𝕍 ≃ Spin(𝕍) ⊗ Spin(𝕍), if dim 𝕍 is even,
∧^{•}𝕍 ≃ 2(Spin(𝕍) ⊗ Spin(𝕍)), if dim 𝕍 is odd.

(2.1)

Let 𝔤 be a reductive Lie algebra and π : 𝔤 → 𝔰𝔬(𝕍) an orthogonal representation. Using π, one may regard Spin(𝕍) as 𝔤-module. In this way, we obtain a mapping from the set of orthogonal 𝔤-modules to a set of 𝔤-modules: 𝕍 ↦ Spin(𝕍). Of course, the 𝔤-modules of the form Spin(𝕍) must satisfy some constraints; e.g. dim Spin(𝕍) is a power of 2. Equations (2.1), which can be treated as isomorphisms of 𝔤-modules, suggest that ‘Spin’ could be used for better understanding of 𝔤-module structure of the exterior algebra of an orthogonal module. The point of departure for our considerations is a simple formula for the character of the 𝔤-module Spin(𝕍). Fix some notation, which applies to arbitrary 𝔤-modules (i.e. not necessarily orthogonal ones). Let P(𝕍) (resp. Δ(𝕍)) denote the set of all (resp. all nonzero) weights of 𝕍. For instance, Δ(𝔤) = Δ. For μ ∈ P(𝕍), 𝕍^{μ} is the corresponding weight space and m(μ) = dim 𝕍^{μ}. If 𝕍 = 𝕍_{λ} is irreducible, then the multiplicity is denoted by m_{λ}(μ). Recall that 𝕍 is self-dual if and only if Δ(𝕍) = −Δ(𝕍) and m(μ) = m(−μ) for all μ ∈ Δ(𝕍). Given an orthogonal 𝔤-module 𝕍, let Δ(𝕍)^{+} denote an arbitrary subset such that Δ(𝕍) = Δ(𝕍)^{+} ⊔ (−Δ(𝕍)^{+}).

2.2 Lemma. ch Spin(𝕍) = 2^{[m(0)/2]} ∏_{μ∈Δ(𝕍)^{+}}(e^{μ/2} + e^{−μ/2})^{m(μ)}.

Proof. Using (1.2), one obtains ch (∧^{•}𝕍) = ∏_{μ∈P(𝕍)}(1 + e^{μ})^{m(μ)} =

2^{m(0)} ∏_{μ∈Δ(𝕍)^{+}}[(1 + e^{μ})(1 + e^{−μ})]^{m(μ)} = 2^{m(0)} ∏_{μ∈Δ(𝕍)^{+}}(e^{μ/2} + e^{−μ/2})^{2m(μ)} .

Since dim 𝕍 − m(0) is even, comparing with Eq. (2.1) completes the proof. □

Roughly speaking, Eq. (2.1) asserts that a “square root” of ∧^{•}𝕍 is again a 𝔤-module whenever 𝕍 is orthogonal. Lemma 2.2 gives a precise form for this. Notice that the transformation from the proof of Lemma can be performed for any self-dual 𝔤-module 𝕍. But the respective “square root” does not yield in general the character of a 𝔤-module. It is convenient to omit the numerical factor in ch Spin(𝕍). The remaining expression is still the character of a 𝔤-module. This module is said to be the reduced Spin of 𝕍 and we write Spin_{0}(𝕍) for it:

ch Spin_{0}(𝕍) = ∏_{μ∈Δ(𝕍)^{+}}(e^{μ/2} + e^{−μ/2})^{m(μ)} . (2.3)

Several easy properties of Spin_{0} are summarized below.

2.4 Proposition. Let 𝕍 = 𝕍^{(1)} and 𝕍^{(2)} be orthogonal 𝔤-modules. Then (i) dim Spin_{0}(𝕍) = 2^{(dim 𝕍−m(0))/2}; (ii) ∧^{•}𝕍 ≃ 2^{m(0)} · Spin_{0}(𝕍)^{⊗2}; (iii) Spin(𝕍) = Spin_{0}(𝕍) if and only if m(0) ≤ 1; (iv) Spin_{0}(𝕍^{(1)} ⊕ 𝕍^{(2)}) ≃ Spin_{0}(𝕍^{(1)}) ⊗ Spin_{0}(𝕍^{(2)});

(v) Spin_{0}(𝕍) is a self-dual 𝔤-module.

Proof. This immediately follows from (2.1), (2.2), and (2.3). □

2.5 Examples. 1. Our consideration of Spin(𝕍) was motivated by the following observation of Kostant, see [Ko61, p. 358] and [Ko97]. Suppose 𝕍 = 𝔤 and π = ad is the adjoint representation. Then ∧^{•}𝔤 ≃ 2^{rk 𝔤}(𝕍_{ρ} ⊗ 𝕍_{ρ}). This means that Spin(𝔤) = 2^{[rk 𝔤/2]} 𝕍_{ρ} and Spin_{0}(𝔤) = 𝕍_{ρ}.

2. 𝔤 = 𝔰𝔩_{2}. We shall write R_{d} in place of 𝕍_{dφ_{1}} . Recall that R_{2} = 𝔤, R_{d} = S^{d}R_{1}, and R_{d} is orthogonal if and only if d is even. Let P(R_{1}) = {ε, −ε}. Applying Prop. 2.4, we obtain Spin_{0}(R_{2d}) = Spin R_{2d} and ch Spin R_{2d} = ∏_{k=1}^{d}(e^{kε} + e^{−kε}). It is not hard to compute this character for small values of d. Here are first few formulas: Spin R_{2}=R_{1}, Spin R_{4}=R_{3}, Spin R_{6}=R_{6}+R_{0}, Spin R_{8}=R_{10}+R_{4}, Spin R_{10}=R_{15}+R_{9}+R_{5}. It is easily seen that if Spin R_{2d} = ⊕_{i∈I}R_{m_{i}}, then Spin R_{2(d+1)} ⊃ ⊕_{i∈I}R_{m_{i}+d+1}. Therefore the number of summands is a nondecreasing function of d.

3. Let 𝕎 be an arbitrary 𝔤-module. We may regard 𝕍 := 𝕎 ⊕ 𝕎^{∗} as orthogonal 𝔤-module equipped with the quadratic form Q((w, w^{∗})) := ⟨w|w^{∗}⟩, where (w, w^{∗}) ∈ 𝕍 and ⟨ | ⟩ is the canonical pairing of 𝕎 and 𝕎^{∗}. Assuming for simplicity that the weights of 𝕎 and 𝕎^{∗} are distinct, we see that P(𝕎) can be taken as Δ(𝕍)^{+}. Therefore

ch Spin(𝕍) = ∏_{μ∈P(𝕎)}(e^{μ/2} + e^{−μ/2})^{m(μ)} = e^{−ν} ∏_{μ∈P(𝕎)}(1 + e^{μ})^{m(μ)}, where ν =

\frac{1}{2} ∑_{μ∈P(𝕎)} m(μ)μ. Whence

Spin(𝕎 ⊕ 𝕎^{∗}) ≃ 𝕜_{−ν} ⊗ ∧^{•}𝕎 ≃ 𝕜_{ν} ⊗ ∧^{•}𝕎^{∗} ,

where 𝕜_{−ν} is 1-dimensional 𝔤-module with character −ν. Obviously, ν = 0 if and only if 𝔤 ⊂ 𝔰𝔩(𝕎), e.g. 𝔤 is semisimple. It is not hard to verify that the above formula for Spin(𝕎 ⊕ 𝕎^{∗}) remains true for all 𝕎.

Definition. An orthogonal 𝔤-module 𝕍 is said to be co-primary, if Spin_{0}(𝕍) is irreducible.

In this case Spin(𝕍) is a primary 𝔤-module. We are going to list all co-primary modules for the semisimple Lie algebras. At the moment, the following examples of such modules are known: 𝕍 = 𝔤, 𝔤 simple; 𝕍 = R_{4}, 𝔤 = 𝔰𝔩_{2}. As a step towards a classification, we describe another series of co-primary modules. Let 𝔤 be a simple Lie algebra having two root lengths. We use subscripts ‘s‘ and ‘l‘ to mark objects related to short and long roots, respectively. For instance, Δ_{s} is the set of short roots, Δ = Δ_{s} ⊔ Δ_{l}, and Π_{s} = Π ∩ Δ_{s}. Set ρ_{s} = \frac{1}{2}|Δ_{s}^{+}| and ρ_{l} = \frac{1}{2}|Δ_{l}^{+}|. As usual, s_{α} ∈ W is the reflection corresponding to α ∈ Δ and s_{i} := s_{α_{i}}.

2.6 Lemma. ρ_{s} = ∑_{α_{i}∈Π_{s}} φ_{i}.

Proof. It is easily seen that s_{i}(ρ_{s}) = \{ \begin{matrix}ρ_{s}, & if α_{i} ∈ Π_{l} \\ ρ_{s} − α_{i}, & if α_{i} ∈ Π_{s} \end{matrix} . □

Let θ ∈ Δ^{+} be the highest root and θ_{s} the short dominant root. Recall that Δ_{l} = W ·θ, Δ_{s} = W ·θ_{s}, and ||θ||^{2}/||θ_{s}||^{2} = 2 or 3. If μ ∈ Δ, then μ^{∨} := 2μ/||μ||^{2}.

2.7 Lemma. Suppose ||θ||^{2}/||θ_{s}||^{2} = 2 and μ ∈ Δ_{s}. Then (ρ + ρ_{s}, μ^{∨}) is even.

Proof. Let μ = ∑_{α_{i}∈Π_{s}} n_{i}α_{i} + ∑_{α_{j}∈Π_{l}} m_{j}α_{j}. Then μ^{∨} = ∑_{α_{i}∈Π_{s}} n_{i}α_{i}^{∨} + 2 ∑_{α_{j}∈Π_{l}} m_{j}α_{j}^{∨}. Therefore (ρ + ρ_{s}, μ^{∨}) = (2ρ_{s} + ρ_{l}, μ^{∨}) = (2ρ_{s}, ∑_{α_{i}∈Π_{s}} n_{i}α_{i}^{∨}) + (ρ_{l}, 2 ∑_{α_{j}∈Π_{l}} m_{j}α_{j}^{∨}) = 2(∑_{i} n_{i} + ∑_{j} m_{j}). □

The following assertion can be proved using classification, but we give a unified proof.

2.8 Proposition.

(i) dim 𝕍_{θ_{s}} = (h + 1)m_{θ_{s}}(0), where h is the Coxeter number of 𝔤; (ii) m_{θ_{s}}(0) = #Π_{s}.

Proof. (i) It is clear that P(𝕍_{θ_{s}}) = {0} ∪ Δ_{s}. Moreover, m_{θ_{s}}(α) = 1 for all α ∈ Δ_{s}. Applying Freudenthal’s multiplicity formula to m_{θ_{s}}(0), we obtain

(θ_{s} + 2ρ, θ_{s})m_{θ_{s}}(0) = 2 ∑_{α∈Δ^{+}} ∑_{t≥1} m_{θ_{s}}(tα)(tα, α) = 2 ∑_{α∈Δ_{s}^{+}} m_{θ_{s}}(α)(α, α) = 2 ∑_{α∈Δ_{s}^{+}}(α, α) .

Whence

(1 + (ρ, θ_{s}^{∨}))m_{θ_{s}}(0) = #Δ_{s} = dim 𝕍_{θ_{s}} − m_{θ_{s}}(0) .

As θ_{s}^{∨} is the highest root in the dual root system Δ^{∨}, we have (ρ, θ_{s}^{∨}) = h − 1.

(ii) By part (i), we have m_{θ_{s}}(0) = \frac{dim 𝕍_{θ_{s}} − m_{θ_{s}}(0)}{h} = \frac{#Δ_{s}}{h}. Let c ∈ W be a Coxeter element associated with Π. It is known that each orbit of c in Δ has cardinality h and contains a unique simple root, see [BOU, ch.VI, § 1, Prop. 33]. Hence #Δ_{s} = h(#Π_{s}). □

Some authors call 𝕍_{θ_{s}} the little adjoint module. To a great extent, properties of 𝕍_{θ_{s}} are similar with properties of 𝔤.

2.9 Theorem. Suppose ||θ||^{2}/||θ_{s}||^{2} = 2. Then ∧^{•}𝕍_{θ_{s}} ≃ 2^{#Π_{s}} · (𝕍_{ρ_{s}} ⊗ 𝕍_{ρ_{s}}).

Proof. By Proposition 2.8, we have ch 𝕍_{θ_{s}} = #Π_{s} + ∑_{α∈Δ_{s}} e^{α}. Therefore

ch ∧^{•} 𝕍_{θ_{s}} = 2^{#Π_{s}} ∏_{α∈Δ_{s}}(1 + e^{α}) = 2^{#Π_{s}} ∏_{α∈Δ_{s}^{+}}(e^{α/2} + e^{−α/2})^{2} .

Thus, the statement of theorem is equivalent to that

ch 𝕍_{ρ_{s}} = ∏_{α∈Δ_{s}^{+}}(e^{α/2} + e^{−α/2}) = e^{ρ_{s}} ∏_{α∈Δ_{s}^{+}}(1 + e^{−α}). (2.10)

By Weyl’s character formula

ch 𝕍_{ρ_{s}} = \frac{∑_{w∈W} ε(w)e^{w(ρ+ρ_{s})}}{∑_{w∈W} ε(w)e^{wρ}} = \frac{∑_{w∈W} ε(w)e^{w(ρ+ρ_{s})}}{∏_{α∈Δ^{+}}(e^{α/2} − e^{−α/2})} . (2.11)

Here ε(w) = (−1)^{l(w)}, where l(w) is the length of w with respect to Δ^{+}. Take α ∈ Δ_{s}^{+}. We are going to prove that 1 + e^{−α} divides ch 𝕍_{ρ_{s}} in ℤ[P]. Since ch 𝕍_{ρ_{s}} is W -invariant, it is enough to consider the case in which α is simple, i.e., α ∈ Π_{s}. Actually, we shall prove that 1 + e^{−α} divides the numerator in Eq. (2.11). For this, we show how to group together the summands of the numerator. Let W^{α} = {w ∈ W | w^{−1}α ∈ Δ^{+}}. Then W is the disjoint union of pairs {s_{α}w, w} (w ∈ W^{α}). Consider the corresponding pairs of summands in the numerator of (2.11). Since α ∈ Π_{s}, we have ε(s_{α}w) = −ε(w) and

ε(w)e^{w(ρ+ρ_{s})} + ε(s_{α}w)e^{s_{α}w(ρ+ρ_{s})} = ε(w)e^{w(ρ+ρ_{s})}(1 − e^{−nα}) ,

where n = (w(ρ + ρ_{s}), α^{∨}) = (ρ + ρ_{s}, (w^{−1}α)^{∨}). By the definition of W^{α}, n is positive. The divisibility will follow from the fact that n is even. But this is just Lemma 2.7. Since ℤ[P] is factorial and the factors 1 + e^{−α} (α ∈ Δ_{s}^{+}) are coprime (see [BOU, ch. VI, § 3, Lemma 1]), e^{ρ_{s}} ∏_{α∈Δ_{s}^{+}}(1 + e^{−α}) divides ch 𝕍_{ρ_{s}}. The quotient is a W -invariant element of ℤ[P]. Comparing the maximal terms in both expressions, we see that the quotient must be equal to 1. □

2.12 Corollary. 1. Spin_{0}(𝕍_{θ_{s}}) = 𝕍_{ρ_{s}}; 2. dim(∧^{•}𝕍_{θ_{s}})^{𝔤} = 2^{#Π_{s}}.

2.13 Examples. To realize the scope of Theorem 2.9, we look at all simple Lie algebras with two root lengths.

1. 𝔤 = 𝔰𝔭_{2n}. Here θ = 2φ_{1}, θ_{s} = φ_{2}, and ρ_{s} = φ_{1} + … + φ_{n−1}. Thus

∧^{•}𝕍_{φ_{2}} = 2^{n−1}·(𝕍_{φ_{1}+...+φ_{n−1}})^{⊗2} and Spin_{0}(𝕍_{φ_{2}}) = 𝕍_{φ_{1}+...+φ_{n−1}} .

2. 𝔤 = 𝔣_{4}. Here θ = φ_{4}, θ_{s} = φ_{1}, and ρ_{s} = φ_{1} + φ_{2}. Thus

∧^{•}𝕍_{φ_{1}} = 4(𝕍_{φ_{1}+φ_{2}})^{⊗2} and Spin_{0}(𝕍_{φ_{1}}) = 𝕍_{φ_{1}+φ_{2}} .

3. 𝔤 = 𝔰𝔬_{2n+1}. Here θ = φ_{2}, θ_{s} = φ_{1}, and ρ_{s} = φ_{n}. In this case Spin_{0}(𝕍_{φ_{1}}) = Spin(𝕍_{φ_{1}}) = 𝕍_{φ_{n}} and the formula of Theorem 2.9 is nothing but the second equality in Eq. (2.1). Hence the theorem also yields another approach to defining ‘Spin’ of an orthogonal representation.

4. 𝔤 = 𝔤_{2}. Here ||θ||^{2}/||θ_{s}||^{2} = 3 and Theorem 2.9 does not apply. In this case ρ_{s} = θ_{s} = φ_{1} and θ = φ_{2}. An explicit (easy) computation with characters shows that

∧^{•}𝕍_{φ_{1}} = 2·(𝕍_{φ_{1}} ⊕ I1)^{⊗2}, i.e., Spin(𝕍_{φ_{1}}) = 𝕍_{φ_{1}} ⊕ I1. Hence 𝕍_{θ_{s}} is not co-primary. Here I1 stands for the trivial 1-dimensional module.

(2.14) Another proof of Theorem 2.9. Making use of Weyl’s character formula, we interpret Eq. (2.10) as Weyl’s denominator identity for the dual root system. Recall that Δ = Δ_{l} ⊔ Δ_{s} and we assume that ||θ||^{2}/||θ_{s}||^{2} = 2. The dual root system is therefore isomorphic to Δ̃ := Δ_{l} ⊔2Δ_{s}. Here (Δ̃)_{l} = 2Δ_{s} and (Δ̃)_{s} = Δ_{l}. Since W̃ ≃ W, Weyl’s denominator identity for Δ̃ reads

∑_{w∈W} ε(w)e^{wρ̃} = ∏_{α∈Δ̃^{+}}(e^{α/2} − e^{−α/2}) .

We have ρ̃ = ρ + ρ_{s} on the left hand side and

∏_{α∈Δ_{l}^{+}}(e^{α/2} − e^{−α/2})· ∏_{μ∈Δ_{s}^{+}}(e^{μ} − e^{−μ}) = ∏_{α∈Δ^{+}}(e^{α/2} − e^{−α/2}) ∏_{α∈Δ_{s}^{+}}(e^{μ/2} + e^{−μ/2})

on the right hand side. Hence dividing Weyl’s identity by ∏_{α∈Δ^{+}}(e^{α/2} − e^{−α/2}) yields ch 𝕍_{ρ_{s}} = ∏_{μ∈Δ_{s}^{+}}(e^{μ/2} + e^{−μ/2}).

Remark. The previous argument suggests a proper analogue of (2.10) for the exceptional Lie algebra 𝔤_{2}. Here the dual root system is isomorphic to Δ̃ := Δ_{l} ⊔ 3Δ_{s} and a similar transformation proves that ch 𝕍_{2ρ_{s}} = ∏_{μ∈Δ_{s}^{+}}(e^{μ} + 1 + e^{−μ}).

3 Classification of co-primary 𝔤-modules

In this section, 𝔤 is a semisimple Lie algebra and 𝕍 an orthogonal 𝔤-module. From Eq. (2.3) it is clear that 𝕍^{𝔤} has no affect on Spin_{0}(𝕍). We may therefore assume that 𝕍^{𝔤} = 0.

3.1 Proposition. Suppose 𝕍 is co-primary. Then there exist decompositions 𝔤 = 𝔤_{1} ⊕ … ⊕ 𝔤_{s}, 𝕍 = 𝕍_{1} ⊕ … ⊕ 𝕍_{s} such that (i) Each 𝔤_{i} is a (semisimple) ideal of 𝔤, (ii) 𝔤_{i} acts trivially on 𝕍_{j} (i ≠ j), (iii) 𝕍_{i} is an irreducible orthogonal co-primary 𝔤_{i}-module.

Proof. Assume that 𝕍 = 𝕍_{1} ⊕ 𝕍_{2}, where 𝕍_{1} and 𝕍_{2} are orthogonal 𝔤-modules. It follows from the assumptions and Proposition 2.4(iv) that the 𝔤-module Spin_{0}(𝕍_{1}) ⊗ Spin_{0}(𝕍_{2}) is irreducible. Since both factors are non-trivial, the only possibility for this is that 𝔤 = 𝔤_{1} ⊕ 𝔤_{2}, where 𝔤_{i} acts trivially on 𝕍_{j} (i ≠ j) and 𝕍_{i} is a co-primary 𝔤_{i}-module (i = 1, 2). Repeating this procedure, we obtain a decomposition satisfying (i) and (ii), where each 𝕍_{i} is orthogonal co-primary and is not a sum of two proper orthogonal 𝔤_{i}-submodules. Then either 𝕍_{i} is irreducible or 𝕍_{i} = 𝕎_{i} ⊕ 𝕎_{i}^{∗}, where 𝕎_{i} is already irreducible. In the second case, we have Spin(𝕍_{i}) ≃ ∧^{•}𝕎_{i} (see Example 2.5(3)). It is easily seen that the 𝔤_{i}-module ∧^{•}𝕎_{i} is never primary, i.e., Spin_{0}(𝕍_{i}) can not be irreducible here. □

Whenever (𝔤, 𝕍) admits a decomposition satisfying conditions (i) and (ii) of the Proposition, this will be denoted by (𝔤, 𝕍) = (𝔤_{1}, 𝕍_{1}) ⊕ … ⊕ (𝔤_{s}, 𝕍_{s}). Notice that if each 𝕍_{i} is irreducible, then all the summands in the above decomposition are uniquely determined.

3.2 Lemma. If 𝕍_{λ} is an irreducible co-primary 𝔤-module, then m_{λ}(0) ≠ 0.

Proof. If m_{λ}(0) = 0, then ∧^{•}𝕍_{λ} ≃ Spin_{0}(𝕍_{λ})^{⊗2}, see 2.4(ii). Since dim(∧^{•}𝕍_{λ})^{𝔤} ≥ 2, the Schur lemma shows that Spin_{0}(𝕍_{λ}) cannot be irreducible. □

It follows from the above two assertions that P(𝕍) lies in the root lattice whenever 𝕍 is co-primary. Let us present an explicit way for finding some irreducible constituents of Spin_{0}(𝕍). To write an expression for ch Spin_{0}(𝕍) in (2.3), we exploited an arbitrary ‘half’ Δ(𝕍)^{+} of Δ(𝕍). However a clever choice of Δ(𝕍)^{+} will provide us with a maximal term in ch Spin_{0}(𝕍) and hence with a highest weight. Take ν ∈ P_{+} such that (ν, μ) ≠ 0 for all μ ∈ Δ(𝕍). Put Δ(𝕍)_{ν}^{+} = {μ ∈ Δ(𝕍) | (μ, ν) > 0}. A subset of such form is said to be a dominant half of Δ(𝕍). Set Λ_{ν} := \frac{1}{2} ∑_{μ} m(μ)μ, where μ ranges over Δ(𝕍)_{ν}^{+}.

3.3 Lemma. Λ_{ν} is a highest weight of Spin_{0}(𝕍).

Proof. We show that Λ_{ν} is dominant and it is a maximal element in P(Spin_{0}(𝕍)). Note that the first part is not tautological. We exploit formula 2.3 with Δ(𝕍)_{ν}^{+}:

ch Spin_{0}(𝕍) = ∏_{μ∈Δ(𝕍)_{ν}^{+}}(e^{μ/2} + e^{−μ/2})^{m(μ)} .

This shows that e^{Λ_{ν}} occurs in ch Spin_{0}(𝕍) with coefficient 1, (ν, Λ_{ν}) = max_{μ∈P(Spin_{0}(𝕍))}(ν, μ), and Λ_{ν} is the unique element of P(Spin_{0}(𝕍)), where the maximal value is attained. Let Λ′_{ν} be the dominant representative in W ·Λ_{ν}. Then Λ′_{ν} ∈ P(Spin_{0}(𝕍)) and Λ′_{ν} − Λ_{ν} = ∑_{α_{i}∈Π} n_{i}α_{i} with n_{i} ≥ 0. Therefore (ν, Λ′_{ν}) ≥ (ν, Λ_{ν}) and hence Λ′_{ν} = Λ_{ν}. □

(3.4) The highest weights of Spin_{0}(𝕍) of the form Λ_{ν} are said to be extreme. It is easy to describe all dominant halfs of Δ(𝕍) and hence all extreme weights of Spin_{0}(𝕍). Consider the Weyl chamber C := ℚ_{+}P_{+} ⊂ P_{ℚ} and its interior C^{o}. Let H_{μ} denote the hyperplane in P_{ℚ} orthogonal to μ ∈ P. Recall that C^{o} is the connected component^{2} of P_{ℚ} \∪_{γ∈Δ}H_{γ}, containing dominant weights. Then the hyperplanes H_{μ} (μ ∈ Δ(𝕍)) cut C in smaller chambers. When ν varies inside of such a ‘small’ chamber the corresponding extreme weight does not change. We thus obtain a bijection

{extreme weights of Spin(𝕍)} ↔ {connected components of C^{o} \ ⋃_{μ∈Δ(𝕍)} H_{μ}} .

In particular, Spin_{0}(𝕍) has a unique extreme weight if and only if Δ(𝕍) has a unique dominant half if and only if none of the hyperplanes H_{μ} cuts C^{o}.

3.5 Lemma. Suppose Δ(𝕍) lies in the root lattice. Then:

none of the hyperplanes H_{μ} (μ ∈ Δ(𝕍)) cuts C^{o} ⟺ Δ(𝕍) ⊂ ∪_{α∈Δ}ℤα.

Proof. “⇐” This is obvious. “⇒” Assume that M := Δ(𝕍) \ ∪_{α∈Δ}ℤα ≠ ∅. Let μ ∈ M ∩ P_{+} be an element closest to 0. Write μ as sum of positive roots with positive integral coefficients μ = ∑_{i=1}^{d} k_{i}γ_{i} (γ_{i} ≠ γ_{j}) and so that ∑_{i} k_{i} is minimal over all such presentations. Then γ_{i} + γ_{j} is not a root, i.e., (γ_{i}, γ_{j}) ≥ 0. Therefore (μ, γ_{1}) > 0 and hence μ−γ_{1} ∈ Δ(𝕍). As ||μ−γ_{1}|| < ||μ||, we obtain μ − γ_{1} ∈ ∪_{α∈Δ^{+}}ℕα. Thus, k_{1} = 1, d = 2 and, by symmetry, μ = γ_{1} + γ_{2}. Since (γ_{1}, γ_{2}) ≥ 0 and γ_{1} + γ_{2} is not a multiple of a root, it is easily seen that (γ_{1}, −γ_{2}) is a basis of the root system Δ ∩ (ℚγ_{1} + ℚγ_{2}). Therefore (γ_{1}, −γ_{2}) is W -conjugate to a pair of simple roots (α_{i}, α_{j}) (see [BOU, ch. VI, § 1,Prop. 24]). Thus, α_{i} − α_{j} ∈ Δ(𝕍) and H_{α_{i}−α_{j}} cuts C^{o}. □

3.6 Proposition. Let 𝕍 be a co-primary faithful irreducible 𝔤-module. Then Δ(𝕍) ⊂ ∪_{α∈Δ}ℤα and 𝔤 is simple.

Proof. By Lemma 3.2, Δ(𝕍) lies in the root lattice. Therefore the first claim readily follows from (3.4) and Lemma 3.5. Assume that 𝔤 = 𝔤_{1} ⊕ 𝔤_{2} is a sum of two ideals. Then 𝕍 = 𝕍_{1} ⊗ 𝕍_{2}, where 𝕍_{i} is a non-trivial 𝔤_{i}-module. Obviously, if μ_{i} ∈ Δ(𝕍_{i}) (i = 1, 2), then μ_{1} + μ_{2} ∈ Δ(𝕍) and it is not a multiple of a root of 𝔤. □

Now, we are ready to state a classification.

3.7 Theorem. (i) Let 𝔤 be semisimple and 𝕍 a faithful orthogonal 𝔤-module with 𝕍^{𝔤} = 0. Suppose 𝕍 is co-primary. Then

(𝔤, 𝕍) = (𝔤_{1}, 𝕍_{1}) ⊕ … ⊕ (𝔤_{s}, 𝕍_{s}) ,

2Strictly speaking, use of the term “connected component” is correct only for the real vector space P_{ℝ}.

where each 𝔤_{i} is simple and 𝕍_{i} is irreducible and co-primary. Each weight of 𝕍 is a multiple of a root of 𝔤. (ii) If 𝔤 is simple and 𝕍 = 𝕍_{λ} is irreducible and co-primary, then the pair (𝔤, λ) is one of the following:

(a) 𝔤 is any and λ = θ.

(b) 𝔤 ∈ {𝔰𝔬_{2n+1}, 𝔰𝔭_{2n}, 𝔣_{4}} and λ = θ_{s}.

(c) 𝔤 = 𝔰𝔬_{2n+1}, λ = 2θ_{s} = 2φ_{1} (n ≥ 2).

(d) 𝔤 = 𝔰𝔩_{2}, λ = 4φ_{1}.

Proof. (i) By Proposition 3.1, such a decomposition with irreducible and co-primary summands 𝕍_{i} exists. The other assertions are proved in Proposition 3.6. (ii) By part (i), we have λ ∈ {kθ, kθ_{s} | k ∈ ℕ}.

Let rk 𝔤 = 1. It follows from Example 2.5(2) that the only co-primary 𝔰𝔩_{2}-modules are 𝔰𝔩_{2} = R_{2} and R_{4}.

Let rk 𝔤 ≥ 2. Consider the following possibilities.

• λ = 2θ. Take α_{i} ∈ Π such that (α_{i}, θ) ≠ 0. Then 2θ − α_{i} is a weight of 𝕍_{2θ}, which is not a multiple of a root. Thus, 𝕍_{2θ} is not co-primary.

• 𝔤 = 𝔰𝔭_{2n} or 𝔣_{4} and λ = 2θ_{s}. If α_{i} is the unique simple root such that (α_{i}, θ_{s}) ≠ 0, then 2θ_{s} − α_{i} ∈ Δ(𝕍_{2θ_{s}}) is not a multiple of a root.

• 𝔤 = 𝔰𝔬_{2n+1} and λ = 3θ_{s} = 3φ_{1}. Here 3φ_{1} − α_{1} is not proportional to a root.

• 𝔤 = 𝔤_{2}. We have already shown in Example 2.13(4) that 𝕍_{θ_{s}} is not co-primary. Obviously, if 𝕍_{kλ} is not co-primary, then the same holds for any m ≥ k. Thus, comparing with results of section 2, we see that the only unclear case is ii(c). Our proof that this module is co-primary is similar to the first proof of Theorem 2.9. It will be given in the next proposition, where we also compute the reduced Spin of 𝕍_{2θ_{s}}. □

3.8 Proposition. Let 𝔤 = 𝔰𝔬_{2n+1}. Then 1. Spin_{0}(𝕍_{2φ_{1}}) = 𝕍_{ρ+2φ_{n}};

2. ∧^{•}𝕍_{2φ_{1}} = 2^{n}·(𝕍_{ρ+2φ_{n}})^{⊗2}.

Proof. First, we describe the weight structure of the 𝔤-module 𝕍_{2φ_{1}} . This is easy, since 𝕍_{2φ_{1}} is the Cartan (highest) component in S^{2}𝕍_{φ_{1}} . Here P(𝕍_{2φ_{1}}) = {0} ∪ Δ ∪ 2Δ_{s}. Hence Δ(𝕍)^{+} = Δ^{+} ∪ 2Δ_{s}^{+}. The non-zero weights are of multiplicity 1, and m_{2φ_{1}}(0) = n. Therefore, making use of Eq. (2.3), we obtain

ch Spin_{0}(𝕍_{2φ_{1}})= ∏_{α∈Δ^{+}}(e^{α/2} +e^{−α/2}) ∏_{α∈Δ_{s}^{+}}(e^{α} +e^{−α})=e^{ρ+2φ_{n}} ∏_{α∈Δ^{+}}(1+e^{−α}) ∏_{α∈Δ_{s}^{+}}(1+e^{−2α}) =

= e^{ρ+2φ_{n}} ∏_{α∈Δ_{l}^{+}}(1 + e^{−α}) ∏_{α∈Δ_{s}^{+}}(1 + e^{−α} + e^{−2α} + e^{−3α}) .

On the other hand,

ch 𝕍_{ρ+2φ_{n}} = \frac{∑_{w∈W} ε(w)e^{w(ρ+ρ+2φ_{n})}}{∏_{α∈Δ^{+}}(e^{α/2} − e^{−α/2})} . (3.9)

Since ch Spin_{0}(𝕍_{2φ_{1}}) and ch 𝕍_{ρ+2φ_{n}} have the same maximal term e^{ρ+2φ_{n}}, it suffices to prove that each factor in the last expression for ch Spin_{0}(𝕍_{2φ_{1}}) divides ch 𝕍_{ρ+2φ_{n}}, i.e., the numerator in Eq. (3.9).

The same procedure, as in the proof of Theorem 2.9, reduces the problem to proving that, for any w ∈ W^{α},

(w(2ρ + 2φ_{n}), α^{∨}) \{ \begin{matrix}is even & , & if α ∈ Π_{l} \\ is divisible by 4 & , & if α ∈ Π_{s} \end{matrix} .

That is, we need actually to verify that (w(ρ + φ_{n}), α^{∨}) is even whenever α is short. As φ_{n} = ρ_{s} for our 𝔤, this is just Lemma 2.7.

2. This is a formal consequence of part 1, see Proposition 2.4. □

Having obtained the list of all irreducible co-primary modules in Theorem 3.7(ii), it is worth looking it through again in order to find out common features and latent regularities for the representations in question. First, item (ii)d in (3.7) can be thought of as starting point for the series in (ii)c. Indeed, V_{2φ_{1}} is the Cartan component in S^{2}𝕍_{φ_{1}} and 𝕍_{φ_{1}} is the tautological module for 𝔰𝔬_{2n+1} (n ≥ 2), whereas 𝔰𝔩_{2}-module R_{4} is the Cartan component in S^{2}R_{2} and R_{2} is the tautological module for 𝔰𝔬_{3}. Thus, the list consists of three groups of representations:

1. (𝔤, 𝕍_{θ} = 𝔤);

2. (𝔤, 𝕍_{θ_{s}}), where 𝔤 is of type B, C, or F;

3. 𝔤 = 𝔰𝔬(𝕎) and 𝕍 = S_{0}^{2}(𝕎), where dim 𝕎 = 3, 5, 7, …

The second (more interesting) observation is that, for all items (𝔤, 𝕍) in the list, 𝔤 → 𝔰𝔬(𝕍) is the isotropy representation of an irreducible symmetric space. In other words, 𝔤̃ := 𝔤⊕𝕍 has a structure of irreducible ℤ_{2}-graded semisimple Lie algebra. More precisely, 𝔤̃ is simple for items 2 and 3, and 𝔤̃ ≃ 𝔤 ⊕ 𝔤 for item 1. Furthermore, it follows from the well-known classification of symmetric spaces that items 1–3 correspond exactly to the cases, where 𝔤 is non-homologous to zero^{3} in 𝔤̃. The class of homogeneous spaces G̃/G (not necessarily symmetric ones) such that G̃, G are connected and 𝔤 is non-homologous to zero in 𝔤̃ has many nice descriptions. We refer the reader to [On95, § 13, n.2] for a thorough treatment in the context of homogeneous spaces of compact Lie groups. In the symmetric case, yet another characterization is that this happens if and only if 𝔤 is determined by a diagram involutory automorphism of 𝔤̃. An explicit description of the diagram automorphisms of simple Lie algebras is found in [Ka90, § 7.9, 7.10]. The third observation is that any irreducible co-primary module occurs in Table 1 in section 1, i.e., it has a free algebra of skew-invariants.

These observations give us some hope that the reduced Spin of the isotropy representation of an arbitrary symmetric spaces might have some interesting properties. This is really the case and we turn to such considerations in the following sections.

3this means that the canonical map of homology spaces H_{∗}(𝔤) → H_{∗}(𝔤^{}˜ ) is injective.

4 Some auxiliary results

In this section, we prove an auxiliary result on Weyl groups and recall some standard facts on involutions of simple Lie algebras.

Let W be the Weyl corresponding to a reduced root system Δ with a set of positive roots Δ^{+}. Let w ↦ l(w) be the length function on W determined by Δ^{+}. Recall that l can be defined as l(w) = #{α ∈ Δ^{+} | w(α) ∈ Δ^{−}}. Consider an arbitrary subset Δ_{0} ⊂ Δ which is a root system in its own right, but is not necessarily closed in Δ. That is, it is allowed that α + β ∈ Δ \ Δ_{0} for some α, β ∈ Δ_{0}. It is easily seen that such a phenomenon can only occur if Δ has roots of different length. As a sample of such non-closed subset, we mention Δ_{0} = Δ_{s}. Nevertheless, W_{0}, the Weyl group of Δ_{0}, is always identified with a subgroup of W. Clearly, Δ_{0}^{+} := Δ_{0} ∩ Δ^{+} can be taken as set of positive roots for Δ_{0}.

4.1 Proposition. 1. Any coset wW_{0} ⊂ W contains a unique representative of minimal length. Denoting by W^{0} the set of minimal length representatives, we have W^{0} = {w ∈ W | w(Δ_{0}^{+}) ⊂ Δ^{+}}.

2. The mapping W^{0} × W_{0} → W ((w^{o}, w_{o}) ↦ w_{o}(w^{o})^{−1}) is a bijection.

Proof. 1. Set W′ = {w ∈ W | w(Δ_{0}^{+}) ⊂ Δ^{+}}. Then W^{0} ⊂ W′. Indeed, assume that w ∈ W^{0} and w(β) ∈ Δ^{−} for some β ∈ Δ_{0}^{+}. Then ws_{β}(β) ∈ Δ^{+} and it follows from [BGG, 2.3] that l(ws_{β}) < l(w). But this contradicts the fact that w ∈ W_{0}. Obviously, each coset contains elements of minimal length and hence elements from W′. Assume that u, v ∈ W′ ∩ vW_{0}. Then u = vw for some w ∈ W_{0}. If w ≠ e, then w(β) ∈ Δ_{0}^{−} for some β ∈ Δ_{0}^{+}. Whence u(β) = v(w(β)) ∈ Δ^{−}, which contradicts the assumption. Thus, each coset contains a unique element of W′, W^{0} = W′, and we are done.

2. Obvious. □

Remark. If Δ_{0} is generated by a part of the basis Π ⊂ Δ^{+} (i.e., Δ_{0} ∩ Π is a basis of Δ_{0}), then W_{0} is a parabolic subgroup of W. In this case the Proposition is well known and, moreover, the relation l(w^{o}w_{0}) = l(w^{o}) + l(w_{o}) holds, see e.g. [Hu95, 1.10]. However this relation does not hold in general. For w_{o} ∈ W_{0}, let l_{0}(w_{o}) denote the length of w_{o} in W_{0}. That is, l_{0}(w_{o}) = #{μ ∈ Δ_{0}^{+} | w_{o}(μ) ∈ Δ_{0}^{−}}. If W_{0} is a parabolic subgroup, then l_{0}(w_{o}) = l(w_{o}), but in general we have only “≤”. The usual determinant or parity for the elements of W is defined by ε(w) = (−1)^{l(w)}. Making use of the above bijection, one may introduce a parity depending on Δ_{0}. By Prop. 4.1(2), each element w ∈ W has a unique presentation w = w_{o}(w^{o})^{−1}, where w_{o} ∈ W_{0} and w^{o} ∈ W^{0}. Set l_{0}(w) := l_{0}(w_{o}) and τ(w) := (−1)^{l_{0}(w)}. So, if w = w_{o}, then τ(w_{o}) is nothing but the usual parity on W_{0}, which will be denoted by ε_{0}(w_{o}). Therefore one may say that τ is the extension of the parity ε_{0} to W determined by the ‘section’ W^{0}. The function w ∈ W ↦ τ(w) ∈ {1, −1} is said to be the cunning parity on W, determined by Δ_{0}^{+} (or W_{0}). It is convenient to give an expression for l_{0}(w), and hence for τ(w), where w_{o} is not explicitly mentioned.

4.2 Lemma. l_{0}(w) = #{α ∈ Δ^{−} | w(α) ∈ Δ_{0}^{+}}.

Proof. Let w = w_{o}(w^{o})^{−1}, as above. Consider the subsets

M_{1} = {α ∈ Δ^{−} | w(α) ∈ Δ_{0}^{+}} and M_{2} = {μ ∈ Δ^{−} | w_{o}(μ) ∈ Δ_{0}^{+}}. Since Δ_{0} is W_{0}-stable, M_{2} ⊂ Δ_{0}^{−} and therefore l_{0}(w_{o}) = #M_{2}. By Prop. 4.1(1), we have (w^{o})^{−1}M_{1} ∩ Δ_{0}^{+} = ∅. Since w_{o}((w^{o})^{−1}M_{1}) ⊂ Δ_{0}^{+}, we see that (w_{o})^{−1}M_{1} ⊂ Δ_{0}^{−}. Thus (w^{o})^{−1}M_{1} ⊂ M_{2}. Similarly, one proves the opposite containment. Thus, l_{0}(w) = #M_{2} = #M_{1}, and we are done. □

(4.3) Classes of involutory automorphisms. Here 𝔤 is a simple Lie algebra.

Given an involutory automorphism Θ of 𝔤, consider the ℤ_{2}-grading 𝔤 = 𝔤_{0} ⊕ 𝔤_{1}, where 𝔤_{i} = {x ∈ 𝔤 | Θ(x) = (−1)^{i}x}. The reductive subalgebra 𝔤_{0} is called symmetric. The involutory automorphisms fall into three classes:

a) rk 𝔤 = rk 𝔤_{0} and 𝔤_{0} is semisimple;

b) rk 𝔤 = rk 𝔤_{0} and 𝔤_{0} has 1-dimensional centre;

c) rk 𝔤 > rk 𝔤_{0}. In cases a) and b), Θ is inner and, accordingly, both 𝔤_{0} and the ℤ_{2}-grading are said to be of inner type. It is well known that the 𝔤_{0}-module 𝔤_{1} is irreducible in cases a) and c), and is the sum of two dual submodules in case b). However, 𝔤_{1} is orthogonal in all three cases and one may consider the 𝔤_{0}-module Spin(𝔤_{1}). An important feature of this situation is that all nonzero weights of 𝔤_{1} are of multiplicity 1. This is clear in the equal rank cases, and can also be proved for c). An invariant theoretic proof of this uses Lemma 3.4 in [Ka80] and the fact that the linear group G_{0} → GL(𝔤_{1}) is visible.

5 Spin(𝔤_{1}) for the inner involutory automorphisms

In this section, 𝔤 is simple and 𝔤_{0} is a symmetric subalgebra of inner type. Retain for 𝔤 the previous notation such as 𝔱, Δ, Δ^{+}, ρ, C, etc. Since rk 𝔤 = rk 𝔤_{0}, we may assume that 𝔱 is a Cartan subalgebra in both 𝔤 and 𝔤_{0}. Let Δ_{0} be the root system of (𝔤_{0}, 𝔱) and Δ_{1} the set of weights of the 𝔤_{0}-module 𝔤_{1}. Then Δ = Δ_{0} ⊔ Δ_{1} and Δ_{0} is a closed subset of Δ. We regard Δ_{0}^{+} := Δ^{+} ∩ Δ_{0} as set of positive roots for 𝔤_{0}. Note also that Δ_{1} contains a distinguished ‘half’ Δ_{1}^{+} = Δ^{+} ∩ Δ_{1}. Then Prop. 4.1 applies to the Weyl groups W_{0} ⊂ W and one obtains the “minimal length” subset W^{0} ⊂ W. Our aim is to describe the 𝔤_{0}-module Spin_{0}(𝔤_{1}). As 𝔤_{1} has no zero weight, we have Spin(𝔤_{1}) = Spin_{0}(𝔤_{1}). As a first step, we find all extreme weights of Spin(𝔤_{1}). Recall from (3.3), (3.4) that each dominant half of Δ_{1} determines an extreme weight for Spin(𝔤_{1}). According to that discussion, one has to take the dominant chamber C_{0} for 𝔤_{0} and cut it up by the hyperplanes orthogonal to the roots of Δ_{1}. Clearly, each small chamber is isomorphic to C. Since there are #W chambers for 𝔤 and #W_{0} chambers for 𝔤_{0}, we obtain the partition of C_{0} in #(W/W_{0}) small chambers. Then any weight inside of a small chamber determines a dominant half of Δ_{1} and an extreme weight. In the following proposition we give a formula for these extreme weights.

5.1 Proposition.

1. The set of hyperplanes H_{μ} (μ ∈ Δ_{1}) cuts C_{0} in #(W/W_{0}) small chambers;

2. The collection of (𝔤_{0}-dominant) weights w^{−1}ρ (w ∈ W^{0}) contains representatives of all small chambers in C_{0}.

3. The extreme weight of Spin(𝔤_{1}) corresponding to w^{−1}ρ is λ_{w} := w^{−1}ρ − ρ_{0}.

Proof. 1. This is proved in the previous paragraph. 2 & 3. If α ∈ Δ_{0}^{+} and w ∈ W^{0}, then wα ∈ Δ^{+}, see Prop. 4.1(1). Therefore w^{−1}ρ is 𝔤_{0}-dominant. Since the number of these weights is #(W/W_{0}), as required, it suffices to verify that the corresponding dominant halfs are different. By definition, the dominant half of Δ_{1} associated with w^{−1}ρ is

(Δ_{1})_{w}^{+} = {μ ∈ Δ_{1} | (w^{−1}ρ, μ) > 0} = {μ ∈ Δ_{1} | wμ ∈ Δ^{+}} .

Because all weight multiplicities in 𝔤_{1} are equal to 1, the corresponding extreme weight is λ_{w} := \frac{1}{2}|(Δ_{1})_{w^{−1}ρ}^{+}|. Set M_{w} = \{μ ∈ Δ_{1}^{+} | wμ ∈ Δ^{+}\} and \overline{M}_{w} = Δ_{1}^{+} \ M_{w}. Then Δ^{+} = Δ_{0}^{+} ⊔ M_{w} ⊔ \overline{M}_{w} and

ρ = ρ_{0} + \frac{1}{2}|M_{w}| + \frac{1}{2}|\overline{M}_{w}| .

Since w ∈ W^{0}, we obtain

w^{−1}ρ = ρ_{0} + \frac{1}{2}|M_{w}| − \frac{1}{2}|\overline{M}_{w}| .

Note also that |(Δ_{1})_{w}^{+}| = |M_{w}|−|\overline{M}_{w}|. Whence λ_{w} = w^{−1}ρ−ρ_{0}. Thus, we have obtained the required number of different extreme weights. □

In the remainder of the section, notation 𝕍_{λ} refers to a 𝔤_{0}-module.

5.2 Theorem. Let 𝔤 = 𝔤_{0} ⊕ 𝔤_{1} be a ℤ_{2}-grading of inner type. Then

Spin_{0}(𝔤_{1}) = Spin(𝔤_{1}) = ⨁_{w∈W^{0}} 𝕍_{λ_{w}} .

Proof. It follows from the preceding exposition that

⨁_{w∈W^{0}} 𝕍_{λ_{w}} ⊂ Spin_{0}(𝔤_{1}) = Spin(𝔤_{1}) .

Since Spin(𝔤_{1}) is self-dual, dim(Spin(𝔤_{1})^{⊗2})^{𝔤_{0}} is greater than or equal to the number of irreducible summands of Spin(𝔤_{1}). Therefore the desired equality is equivalent to that dim(Spin(𝔤_{1})^{⊗2})^{𝔤_{0}} = #W^{0}. Recall the main property of ‘Spin’ in our situation:

∧^{•}𝔤_{1} ≃ Spin(𝔤_{1})^{⊗2} .

Hence the question about 𝔤_{0}-invariants is being translated in the setting of exterior algebras. Assuming that 𝕜 = ℂ, we can exploit de Rham cohomology with complex coefficients. It is well known that (∧^{•}𝔤_{1})^{𝔤_{0}} is isomorphic to H^{∗}(G/G_{0}), the cohomology ring of the symmetric space G/G_{0} [On95, §9 n.11], and that dim H^{∗}(G/G_{0}) = #(W/W_{0}) [On95, §13 n.3]. This completes the proof. □

5.3 Example. Let Θ be a ‘Hermitian’ involutory automorphism, i.e., 𝔤_{0} has a 1dimensional centre and 𝔤_{1} ≃ 𝕎 ⊕ 𝕎^{∗}, where 𝕎 is a faithful irreducible 𝔤_{0}-module. This is just case 4.3(b). Then Δ(𝕎) = P(𝕎) = Δ_{1}^{+} and, according to Example 2.5(3), Spin(𝔤_{1}) ≃ 𝕜_{ρ_{1}} ⊗ ∧^{•}𝕎^{∗}, where ρ_{1} = \frac{1}{2}|Δ_{1}^{+}|. It then follows from Theorem 5.2 that

∧^{•}𝕎^{∗} = 𝕜_{−ρ_{1}} ⊗ Spin(𝔤_{1}) = ⨁_{w∈W^{0}} 𝕍_{wρ−ρ} .

Or, equivalently, {ρ − wρ | w ∈ W^{0}} is the set of all highest weights for the 𝔤_{0}-module ∧^{•}𝕎. This result was obtained by Kostant (see [Ko61, 8.2]) as application of his results on the cohomology of the nilpotent radical of a parabolic subalgebra of 𝔤. In this situation, 𝕎 is the Abelian nilpotent radical of the parabolic subalgebra 𝔤_{0} ⊕ 𝕎. So, the concept of ‘Spin’ and Theorem 5.2 yield another generalization of Kostant’s result.

Purists may condemn the above proof of Theorem 5.2, since it invokes the cohomology theory of compact Lie groups over ℂ. Fortunately, there exists also a rather simple and purely algebraic proof. We shall show that the equality in 5.2 is equivalent to an identity in ℤ[P], which is a variation of the Weyl denominator formula. Recall from section 4 the cunning parity τ(w) for w ∈ W, determined by W_{0}.

5.4 Theorem. Let Δ = Δ_{0} ⊔ Δ_{1} be the partition corresponding to a ℤ_{2}-grading of 𝔤 of inner type. Then

∑_{w∈W} τ(w)e^{wρ} = ∏_{α∈Δ_{0}^{+}}(e^{α/2} − e^{−α/2}) ∏_{μ∈Δ_{1}^{+}}(e^{μ/2} + e^{−μ/2}) .

Proof. The fact that Δ_{0} and Δ_{1} originate from an inner involutory automorphism can alternatively be stated as follows:

(∗)  if α ∈ Δ_{i}, β ∈ Δ_{j}, and α + β ∈ Δ, then α + β ∈ Δ_{i+j},

where, of course, i, j ∈ ℤ/2ℤ. Let Q ⊂ P be the root lattice. For ℤ[Q], with basis e^{α} (α ∈ Q), one has a version of Weyl’s denominator identity:

∑_{w∈W} ε(w)e^{wρ−ρ} = ∏_{α∈Δ^{+}}(1 − e^{−α}) .

Consider the second copy of ℤ[Q], with basis q^{α} (α ∈ Q), and the equality in ℤ[Q]⊗ℤ[Q]:

∑_{w∈W} ε(w)q^{wρ−ρ}e^{wρ−ρ} = ∏_{α∈Δ^{+}}(1 − q^{−α}e^{−α}) . (5.5)

Take the specialization of this identity such that q^{α} → \{ \begin{matrix}1, & α ∈ Δ_{0} \\ −1, & α ∈ Δ_{1} \end{matrix} . It has to be verified

that one obtains a well-defined homomorphism (Q, +) → {1, −1} ≃ ℤ/2ℤ. In other words, if ν = ∑_{i∈I} μ_{i} is a sum of roots then the number of summands lying in Δ_{1} should have the same parity for all such presentations. Indeed, assume that ∑_{i∈I} μ_{i} = ∑_{j∈J} β_{j}. We argue by induction on #I+#J. Since (∑_{i∈I} μ_{i}, ∑_{j∈J} β_{j}) > 0, there exist i_{0}, j_{0} such that (μ_{i_{0}}, β_{j_{0}}) > 0. Hence μ_{i_{0}} − β_{j_{0}} is a root and ∑_{i∈I\{i_{0}}} μ_{i} + (μ_{i_{0}} − β_{j_{0}}) = ∑_{j∈J\{j_{0}}} β_{j}. We conclude by applying the inductive hypothesis to this equality and using (∗). Thus, the specialization is well-defined and we obtain ∏_{α∈Δ_{0}^{+}}(1 − e^{−α}) ∏_{μ∈Δ_{1}^{+}}(1 + e^{−μ}) at the

right hand side of Eq. (5.5). It is easily seen that wρ − ρ = −|Δ(w)|, where Δ(w) = {α ∈ Δ^{+} | w^{−1}α ∈ Δ^{−}} = Δ^{+} ∩ w(Δ^{−}). Therefore q^{wρ−ρ} specializes to (−1)^{n}, where n = #(Δ_{1}^{+} ∩ wΔ^{−}). Recall that ε(w) = (−1)^{l(w)} and l(w) = #(Δ^{+} ∩ wΔ^{−}). Thus the resulting sign on the left hand side is (−1)^{#(Δ_{0}∩wΔ^{−})}, which is just τ(w) by Lemma 4.2. This completes the proof of the theorem. □

(5.6) Another proof of theorem 5.2. By Weyl’s character formula for 𝔤_{0}- modules and Prop. 5.1(3),

ch 𝕍_{λ_{w}} = \frac{∑_{w̃∈W_{0}} ε_{0}(w̃)e^{w̃(ρ_{0}+λ_{w})}}{∏_{α∈Δ_{0}^{+}}(e^{α/2} − e^{−α/2})} = \frac{∑_{w̃∈W_{0}} ε_{0}(w̃)e^{w̃w^{−1}ρ}}{∏_{α∈Δ_{0}^{+}}(e^{α/2} − e^{−α/2})} .

Hence

ch ( ⨁_{w∈W^{0}} 𝕍_{λ_{w}}) = \frac{∑_{w∈W^{0}} ∑_{w̃∈W_{0}} ε_{0}(w̃)e^{w̃w^{−1}ρ}}{∏_{α∈Δ_{0}^{+}}(e^{α/2} − e^{−α/2})} .

By the very definition of τ(w) (see section 4) and Prop. 4.1(2), it follows that the numerator is equal to ∑_{w∈W} τ(w)e^{wρ}. Whence, by Theorem 5.4,

ch ( ⨁_{w∈W^{0}} 𝕍_{λ_{w}}) = ∏_{μ∈Δ_{1}^{+}}(e^{μ/2} + e^{−μ/2}) = ch Spin(𝔤_{1}) .

5.7 Examples. 1. 𝔤 = 𝔰𝔬_{2n+1}, 𝔤_{0} = 𝔰𝔬_{2n}. Here 𝔤_{1} ≃ 𝕍_{φ_{1}} is the tautological 𝔰𝔬_{2n}- module and #(W/W_{0}) = 2. Let {ε_{1}, …, ε_{n}} be the standard basis of 𝔱^{∗} so that Δ = {±ε_{i} ± ε_{j}, ±ε_{i} | 1 ≤ i, j ≤ n, i ≠ j}. Here Δ_{0}^{+} = {ε_{i} ± ε_{j} (i < j), ε_{i}} and Δ_{0} = Δ_{l}. Then W^{0} = {id, w_{n}}, where w_{n}(ε_{i}) = ε_{i} (i ≤ n − 1) and w_{n}(ε_{n}) = −ε_{n}. Since Δ_{1} = Δ(𝕍_{φ_{1}}) = {±ε_{1}, …, ±ε_{n}}, the corresponding dominant halfs are (Δ_{1})_{id}^{+} = {ε_{1}, …, ε_{n}} and (Δ_{1})_{w_{n}}^{+} = {ε_{1}, …, ε_{n−1}, −ε_{n}}, and the corresponding extreme weights are φ_{n} and φ_{n−1}. Thus, Spin(𝕍_{φ_{1}}) = 𝕍_{φ_{n−1}} ⊕ 𝕍_{φ_{n}} and ∧^{•}𝕍_{φ_{1}} = (𝕍_{φ_{n−1}} ⊕ 𝕍_{φ_{n}})^{⊗2}. Notice that the last equality is nothing but the first equality in Eq. (2.1).

2. 𝔤 = 𝔣_{4}, 𝔤_{0} = 𝔰𝔬_{9}. Here 𝔤_{1} ≃ 𝕍_{φ_{4}} and #(W/W_{0}) = 3. In the standard notation for 𝔣_{4}, we have Δ^{+} = \{ε_{i} ± ε_{j} (i<j), ε_{i}, \frac{1}{2}(ε_{1} ± ε_{2} ± ε_{3} ± ε_{4})\}. Then Δ_{0}^{+} = Δ_{l}^{+} ⊔ \{ε_{1}, ε_{2}, ε_{3}, ε_{4}\}. An explicit computation shows that W^{0} = {id, w′, w′′}, where

w′ : \{ \begin{matrix}ε_{1} ↦ \frac{1}{2}(ε_{1} + ε_{2} + ε_{3} + ε_{4}) \\ ε_{2} ↦ \frac{1}{2}(ε_{1} + ε_{2} − ε_{3} − ε_{4}) \\ ε_{3} ↦ \frac{1}{2}(ε_{1} − ε_{2} + ε_{3} − ε_{4}) \\ ε_{4} ↦ \frac{1}{2}(ε_{1} − ε_{2} − ε_{3} + ε_{4}) \end{matrix} and w′′ : \{ \begin{matrix}ε_{1} ↦ \frac{1}{2}(ε_{1} + ε_{2} + ε_{3} − ε_{4}) \\ ε_{2} ↦ \frac{1}{2}(ε_{1} + ε_{2} − ε_{3} + ε_{4}) \\ ε_{3} ↦ \frac{1}{2}(ε_{1} − ε_{2} + ε_{3} + ε_{4}) \\ ε_{4} ↦ \frac{1}{2}(ε_{1} − ε_{2} − ε_{3} − ε_{4}) \end{matrix} .

(One may notice that any w ∈ W^{0} must preserve (Δ_{0})_{l}^{+} = Δ_{l}^{+}, Δ_{l} being the root system of type D_{4}. Hence w takes ε_{2} − ε_{3} to itself and permutes somehow ε_{1} − ε_{2}, ε_{3} − ε_{4}, and ε_{3} + ε_{4}.) Whence

λ_{id} = ρ − ρ_{0} = 2ε_{1},

λ_{w′} = (w′)^{−1}ρ − ρ_{0} = ε_{1} + ε_{2} + ε_{3},

λ_{w′′} = (w′′)^{−1}ρ − ρ_{0} = (3ε_{1} + ε_{2} + ε_{3} + ε_{4})/2. Thus

Spin(𝕍_{φ_{4}}) = 𝕍_{2φ_{1}} ⊕ 𝕍_{φ_{3}} ⊕ 𝕍_{φ_{1}+φ_{4}} .

6 Spin(𝔤_{1}) for the outer involutory automorphisms

In this section 𝔤 is simple and Θ is outer. Since rk 𝔤_{0} < rk 𝔤, there is no clear relation between roots and Weyl groups of the two algebras, and the approach of section 5 seems to fail completely. Yet, it appears to be possible to describe Spin(𝔤_{1}) in a similar fashion, but with some complications. Another price is that we have to exploit case-by-case arguments several times.

(6.1) Associated diagram involutory automorphism of 𝔤. By a result of Steinberg, Θ keeps stable a Borel subalgebra and a Cartan subalgebra in it. Therefore we may (and shall) assume that Θ𝔱 = 𝔱 and Θ𝔲^{+} = 𝔲^{+}. Then Θ also preserves Δ^{+} and Π, as subsets of 𝔱^{∗}. In particular, Θ induces an involution of the Dynkin diagram. Associated with this involution, one has the specific involutory automorphism of 𝔤, which is called the diagram involutory automorphism and denoted by \overline{Θ}. Roughly speaking, \overline{Θ} performs the same involution on Π, as Θ, and transforms ‘well’ the Chevalley generators of 𝔤^{4}. We are going to compare properties of the ℤ_{2}-gradings

𝔤 = 𝔤_{0} ⊕ 𝔤_{1} and 𝔤 = 𝔤\overline{0} ⊕ 𝔤\overline{1}

arising from Θ and \overline{Θ}. By construction, Θ|_{𝔱} = \overline{Θ}|_{𝔱}. Therefore Θ and \overline{Θ} act identically on Δ^{+} and 𝔱_{0} := 𝔱^{Θ} is a Cartan subalgebra for both 𝔤_{0} and 𝔤\overline{0}. Let us organize notation for roots and weights of the symmetric subalgebras in question:

• Δ_{0} (resp. Δ\overline{0}) is the root system of 𝔤_{0} (resp. 𝔤\overline{0}) relative to 𝔱_{0};

• Δ_{1} (resp. Δ\overline{1}) is the set of non-zero weights of the 𝔤_{0}-module 𝔤_{1} (resp. 𝔤\overline{0}-module 𝔤\overline{1}) relative to 𝔱_{0}. Since all these sets are defined with respect to a common Cartan subalgebra, Δ_{0} ∪ Δ_{1} = Δ\overline{0}∪Δ\overline{1} and, more precisely, the totality of weights occurring in \{Δ_{0}, Δ_{1}\} is the same as in \{Δ\overline{0}, Δ\overline{1}\}. Because 𝔱_{0} contains regular elements of 𝔤 (see e.g. [Ka90, 8.1(b)]), none of the roots of 𝔤 vanishes on 𝔱_{0}. Therefore the above totality of weights consists of all restricted roots. Moreover, since the non-zero weights of 𝔱_{0} in 𝔤_{1} (or 𝔤\overline{1}) are of multiplicity 1,

#Δ = #Δ_{0} + #Δ_{1} = #Δ\overline{0} + #Δ\overline{1} .

4Explicit formulas for the diagram automorphisms of all simple Lie algebras are written in [Ka90, § 7.9,7.10].

Warning. Unlike section 5, elements of Δ_{0} and Δ_{1} have not much in common with roots of 𝔤. Actually, we do not need Δ in this section. The next assertion follows from the classification.

Fact. The fixed point subalgebra of an outer involutory automorphism always has roots of different length, with ||long||^{2}/||short||^{2} = 2.

This applies to both 𝔤_{0} and 𝔤\overline{0} and, as in section 2, we use the subscripts ‘s’ and ‘l’ to denote the objects related to short and long roots in Δ_{0} and Δ\overline{0}. A close look to the classification list reveals important features of this situation.

(6.2) Suppose Θ ≠ \overline{Θ}. Then \{ \begin{matrix}𝔤\overline{0} is simple and 𝔤\overline{1} is the little adjoint module for 𝔤\overline{0}; \\ Δ\overline{1} = (Δ\overline{0})_{s}; \\ Δ_{0} ⊂ Δ\overline{0} and (Δ_{0})_{s} = (Δ\overline{0})_{s}. \end{matrix}

In fact, there are 7 series of outer involutory automorphisms of simple Lie algebras. They form three pairs (Θ, \overline{Θ}) and one “isolated” diagram involutory automorphism, where (6.2) is not satisfied. The relevant data for all these series are presented in Table 2.

Table 2: The outer involutory automorphisms

Θ  \overline{Θ}
𝔤  𝔤_{0}  𝔤_{1}  𝔤\overline{0}  𝔤\overline{1} #W\overline{0}/W_{0}
𝔰𝔩_{2n}  𝔰𝔬_{2n}  𝕍_{2φ_{1}}  𝔰𝔭_{2n}  𝕍_{φ_{2}}  2
𝔰0_{2n+2m+2} 𝔰𝔬_{2n+1} ⊕ 𝔰𝔬_{2m+1} 𝕍_{φ_{1}} ⊗ 𝕍′_{φ_{1}} 𝔰𝔬_{2n+2m+1} 𝕍_{φ_{1}}  \binom{n+m}{m}
𝔢_{6}  𝔰𝔭_{8}  𝕍_{φ_{4}}  𝔣_{4}  𝕍_{φ_{1}}  3
𝔰𝔩_{2n+1}  —  —  𝔰𝔬_{2n+1} 𝕍_{2φ_{1}}  —

It follows from (6.2) that Δ\overline{0} ∪ Δ\overline{1} = Δ\overline{0}. Thus, everything lies in Δ\overline{0}. Therefore a choice of the set of positive roots Δ+ \overline{0} determines Δ+ \overline{1}, Δ_{0}^{+}, and Δ_{1}^{+} as well. Of course, we choose Δ+ \overline{0} so that it is the image of Δ^{+} under the projection 𝔱^{∗} → (𝔱_{0})^{∗}. Then \{Δ_{0}^{+}, Δ_{1}^{+}\} and \{Δ+ \overline{0}, Δ+ \overline{1}\} are two presentations for the totality of all restricted positive roots of 𝔤. Set ρ_{i} = \frac{1}{2}|Δ_{i}^{+}| and ρ\overline{ı} = \frac{1}{2}|Δ+ ; \overline{ı}| (i = 0, 1). Recall that ρ = \frac{1}{2}|Δ^{+}| and therefore Θρ = ρ. That is, ρ ∈ (𝔱^{∗})^{Θ} ≃ 𝔱_{0}^{∗}. It then follows from the above discussion that ρ = ρ_{0} + ρ_{1} = ρ\overline{0} + ρ\overline{1} = ρ\overline{0} + (ρ\overline{0})_{s} . (6.3)

Let W_{0} and W\overline{0} be the Weyl groups of 𝔤_{0} and 𝔤\overline{0}, respectively. Although Δ_{0} ⊂ Δ\overline{0}, 𝔤_{0} is not a subalgebra of 𝔤\overline{0} (if Θ ≠ \overline{Θ}). In other words, Δ_{0} is a non-closed subset of Δ\overline{0}. Nevertheless, Prop. 4.1 applies to W_{0} ⊂ W\overline{0} and one obtains the subset W′ ⊂ W\overline{0} consisting of the elements of minimal length in the cosets {wW_{0}}. Equivalently, W′ = \{w ∈ W\overline{0} | w(Δ_{0}^{+}) ⊂ Δ+ \overline{0}\}. Below, we consider the Weyl chambers C_{0} and C\overline{0}, and the hyperplanes H_{μ} (μ ∈ Δ_{1}). They are regarded as subsets of the rational span of Δ\overline{0} in 𝔱_{0}^{∗}.

6.4 Proposition.

1. The set of hyperplanes H_{μ} (μ ∈ Δ_{1}) cuts C_{0} in #(W\overline{0}/W_{0}) small chambers;

2. The collection of (𝔤_{0}-dominant) weights w^{−1}ρ\overline{0} (w ∈ W′) contains representatives of all small chambers in C_{0}.

3. The extreme weight of Spin_{0}(𝔤_{1}) corresponding to w^{−1}ρ\overline{0} is λ_{w} := w^{−1}(ρ_{0} + ρ_{1}) − ρ_{0} = w^{−1}ρ − ρ_{0}.

Proof. To a great extent, the proof is parallel to the proof of Prop. 5.1.

1. The union Δ_{0} ∪ Δ_{1} coincides with Δ\overline{0}. Therefore each small chamber is isomorphic to C\overline{0}. Comparing the total number of chambers, we see that C_{0} splits into #(W\overline{0}/W_{0}) small chambers.

2 & 3. By the definition of W′, it follows that w^{−1}ρ\overline{0} is 𝔤_{0}-dominant. So, we have the required number of dominant weights and it suffices to verify that the corresponding extreme weights of Spin_{0}(𝔤_{1}) are different.

Given w ∈ W′, the dominant half of Δ_{1} associated with w^{−1}ρ\overline{0} is

(Δ_{1})_{w}^{+} := \{μ ∈ Δ_{1} | (w^{−1}ρ\overline{0}, μ) > 0\} = \{μ ∈ Δ_{1} | wμ ∈ Δ+ \overline{0}\}

and the corresponding extreme weight is λ_{w} := \frac{1}{2}|(Δ_{1})_{w}^{+}|. Set M_{w} = \{μ ∈ Δ_{1}^{+} | wμ ∈ Δ+ \overline{0}\} and \overline{M}_{w} = Δ_{1}^{+} \ M_{w}. Then

ρ\overline{0} + ρ\overline{1} = ρ_{0} + ρ_{1} = ρ_{0} + \frac{1}{2}|M_{w}| + \frac{1}{2}|\overline{M}_{w}| .

Since w ∈ W′, we have

w^{−1}(ρ\overline{0} + ρ\overline{1}) = ρ_{0} + \frac{1}{2}|M_{w}| − \frac{1}{2}|\overline{M}_{w}| .

Noting that |(Δ_{1})_{w}^{+}| = |M_{w}| − |\overline{M}_{w}|, we obtain λ_{w} = w^{−1}(ρ\overline{0} + ρ\overline{1}) − ρ_{0}. Obviously, these weights are different, and we are done. □

In the next theorem, 𝕍_{λ} denotes a 𝔤_{0}-module.

6.5 Theorem. Let 𝔤 = 𝔤_{0} ⊕ 𝔤_{1} be a ℤ_{2}-grading of outer type and 𝔤 = 𝔤\overline{0} ⊕ 𝔤\overline{1} the associated diagram ℤ_{2}-grading. Let W′ be the set of representatives of minimal length for W\overline{0}/W_{0}. Then

Spin_{0}(𝔤_{1}) = ⨁_{w∈W′} 𝕍_{λ_{w}} .

Proof. First, note that if Θ is a diagram involutory automorphism, then W\overline{0} = W_{0}. Here the theorem claims that Spin_{0}(𝔤_{1}) is irreducible, with highest weight (ρ\overline{0})_{s}. This was already demonstrated in Theorem 2.9 and Prop. 3.8. In the general case, we proceed as follows. By Prop. 6.4, ⨁_{w∈W′} 𝕍_{λ_{w}} ⊂ Spin_{0}(𝔤_{1}), and the equality will follow from the fact that dim(Spin_{0}(𝔤_{1})^{⊗2})^{𝔤_{0}} = #W′. As in the proof of Theorem 5.2, a crucial step in the next argument is of “cohomological” nature. Since 𝔱_{0} contains regular elements, dim(𝔤_{1})^{𝔱_{0}} = dim 𝔱 − dim 𝔱_{0}, i.e., the multiplicity of the zero weight in 𝔤_{1} is equal to rk 𝔤 − rk 𝔤_{0}. By Prop. 2.4(ii),

∧^{•}𝔤_{1} ≃ 2^{rk𝔤−rk𝔤_{0}}·Spin_{0}(𝔤_{1})^{⊗2}

and hence

dim(∧^{•}𝔤_{1})^{𝔤_{0}} = 2^{rk𝔤−rk𝔤_{0}}· dim(Spin_{0}(𝔤_{1})^{⊗2})^{𝔤_{0}} .

At the rest of the proof, 𝕜 = ℂ. Inspecting the list of the symmetric spaces of outer type and their cohomology rings over ℂ (see e.g. [Ta62, § 4]) yields the equality

dim H^{∗}(G/G_{0}) = 2^{rk𝔤−rk𝔤_{0}}·#(W\overline{0}/W_{0}) .

Since (∧^{•}𝔤_{1})^{𝔤_{0}} ≃ H^{∗}(G/G_{0}), we are done. □

The following proof, although also being not free of case-by-case arguments, does not appeal to ℂ.

(6.6) Another proof of Theorem 6.5. Arguing as in (5.6) and using Prop. 6.4(3), we obtain

ch ( ⨁_{w∈W′} 𝕍_{λ_{w}}) = ∑_{w∈W′} \frac{∑_{w̃∈W_{0}} ε_{0}(w̃)e^{w̃(ρ_{0}+λ_{w})}}{∏_{α∈Δ_{0}^{+}}(e^{α/2} − e^{−α/2})} = (6.7)

= \frac{∑_{w∈W′} ∑_{w̃∈W_{0}} ε_{0}(w̃)e^{w̃w^{−1}ρ}}{∏_{α∈Δ_{0}^{+}}(e^{α/2} − e^{−α/2})} = \frac{∑_{w∈W\overline{0}} τ(w)e^{wρ}}{∏_{α∈Δ_{0}^{+}}(e^{α/2} − e^{−α/2})}.

Here τ(w) is the cunning parity for w ∈ W\overline{0}, relative to the subgroup W_{0}. To get another expression for the numerator, we exploit the following observation concerning the pairs (𝔤_{0}, 𝔤\overline{0}) in Table 2. Although Δ_{0} is not closed in Δ\overline{0}, the dual root system Δ̃_{0} is closed in Δ̃\overline{0} and, moreover, it is a “symmetric” subset. That is, the partition Δ̃\overline{0} = Δ̃_{0} ⊔ (Δ̃\overline{0} \ Δ̃_{0}) arises from an inner involutory automorphism of the “dual” Lie algebra. (E.g. the pair (C_{4}, F_{4}) inverts in (B_{4}, F_{4}).) Here Δ̃\overline{0} = (Δ\overline{0})_{l} ⊔ 2(Δ\overline{0})_{s}. Recall from (6.2) that Δ\overline{1} = (Δ\overline{0})_{s} = (Δ_{0})_{s}. This means in particular that Δ\overline{0} \ Δ_{0} consists of long roots and these are exactly the roots constituting (Δ_{1})_{l}. Hence Δ̃\overline{0} \ Δ̃_{0} = (Δ_{1})_{l}. After these preparations, write out the identity from Theorem 5.4 for the partition Δ̃\overline{0} = Δ̃_{0} ⊔ (Δ_{1})_{l}:

∑_{w∈W\overline{0}} τ(w)e^{wρ̃} = ∏_{α∈_{Δ_{0}^{+}}}(e^{α/2} − e^{−α/2}) ∏_{μ∈(Δ_{1}^{+})_{l}}(e^{μ/2} + e^{−μ/2}). (6.8)

Here ρ̃ := \frac{1}{2}|Δ̃+ \overline{0}| = \frac{1}{2}|(Δ+ \overline{0})_{l}| + |(Δ+ \overline{0})_{s}| = (ρ\overline{0})_{l} + 2(ρ\overline{0})_{s} = ρ\overline{0} + (ρ\overline{0})_{s} = ρ (see Eq. 6.3). Transforming the first factor on the right hand side of Eq. (6.8) yields

∏_{α∈_{Δ_{0}^{+}}}(e^{α/2} − e^{−α/2}) = ∏_{α∈(Δ_{0}^{+})_{s}}(e^{α} − e^{−α}) ∏_{β∈(Δ_{0}^{+})_{l}}(e^{β/2} − e^{−β/2}) =

= ∏_{α∈(Δ_{0}^{+})_{s}}(e^{α/2} − e^{−α/2})(e^{α/2} + e^{−α/2}) ∏_{β∈(Δ_{0}^{+})_{l}}(e^{β/2} − e^{−β/2}) =

= ∏_{α∈Δ_{0}^{+}}(e^{α/2} − e^{−α/2}) ∏_{μ∈(Δ_{0}^{+})_{s}}(e^{μ/2} + e^{−μ/2}).

Since (Δ_{0}^{+})_{s} = (Δ_{1}^{+})_{s}, the whole expression on the right hand side of Eq. (6.8) is equal to

∏_{α∈Δ_{0}^{+}}(e^{α/2} − e^{−α/2}) ∏_{α∈Δ_{1}^{+}}(e^{μ/2} + e^{−μ/2}).

Substituting this expression for ∑_{w∈W\overline{0}} τ(w)e^{wρ} in Eq. (6.7), we obtain

ch ( ⨁_{w∈W′} 𝕍_{λ_{w}}) = ∏_{α∈Δ_{1}^{+}}(e^{μ/2} + e^{−μ/2}) = ch Spin(𝔤_{1}). q.e.d.

(6.9) Connexion with cohomology of symmetric spaces. Using classical structure results on H^{∗}(G/G_{0}) (see e.g. [On95, ch.3]), one may notice some interesting coincidences.

1. Recall that the image of the canonical map η : H^{∗}(G/G_{0}) → H^{∗}(G) is generated by primitive elements. Actually, there exists a subalgebra of H^{∗}(G/G_{0}) that is mapped isomorphically onto Im (η). It is called Samelson and denoted by Sam(G/G_{0}). There exists also another subalgebra of H^{∗}(G/G_{0}), which is called characteristic and denoted by _{0}H^{∗}(G/G_{0}). It is the zero component for a natural grading in H^{∗}(G/G_{0}). Then H^{∗}(G/G_{0}) ≃ Sam(G/G_{0}) ⊗ _{0}H^{∗}(G/G_{0}). This holds not only for the symmetric spaces but for a wider class of formal homogeneous spaces, see [loc. cit., § 12, Th.2]. A general fact for the formal homogeneous spaces is that dim Sam(G/G_{0}) = 2^{rk𝔤−rk𝔤_{0}} . Hence dim Spin_{0}(𝔤_{1}) = dim _{0}H^{∗}(G/G_{0}), which suggests that Spin_{0}(𝔤_{1}) might somehow be related to the characteristic subalgebra of H^{∗}(G/G_{0}).

2. Note that H^{∗}(G/G_{0}) = _{0}H^{∗}(G/G_{0}) if and only if Θ is inner, and H^{∗}(G/G_{0}) = Sam(G/G_{0}) if and only if Θ is a diagram involutory automorphism. In the mixed case, the associated diagram involutory automorphism \overline{Θ} seems to yield a splitting for H^{∗}(G/G_{0}). Namely, one has dim Sam(G/G_{0}) = dim H^{∗}(G/G\overline{0}) and dim _{0}H^{∗}(G/G_{0}) = dim H^{∗}(G̃\overline{0}/G̃_{0}), where G̃\overline{0} and G̃_{0} are the groups corresponding to the dual root systems Δ̃\overline{0} and Δ̃_{0}. Moreover, these two pairs of graded algebras have equal Poincaré polynomials and it is likely that they are naturally isomorphic. I think that a better understanding of this situation as well as elimination of the case-by-case arguments in section 6 can be achieved through application of the theory of twisted affine Kac–Moody algebras.

3. In the above exposition, \overline{Θ} has appeared as deus ex machina. But the Kac-Moody theory provides some explanation for this. Namely, the outer automorphism Θ determines the twisted affine Kac-Moody algebra L̂(𝔤, Θ, 2) = L̂(𝔤) [Ka90, ch. 8]. Next, L̂(𝔤) has the standard ℤ-grading associated with the special vertex of the Dynkin diagram of L̂(𝔤). If 𝔤 ≠ 𝔰𝔬_{2n+1}, this ℤ-grading determines another outer automorphism of 𝔤, which is just \overline{Θ}. 6.10 Examples. 1. 𝔤 = 𝔰𝔩_{2n}, 𝔤_{0} = 𝔰𝔬_{2n}. Then 𝔤_{1} ≃ 𝕍_{2φ_{1}} . As indicated in Table 2, 𝔤\overline{0} = 𝔰𝔭_{2n} and therefore #(W\overline{0}/W_{0}) = 2. Here Δ_{0} = \{±ε_{i} ± ε_{j} | 1 ≤ i, j ≤ n, i ≠ j\} and Δ\overline{0} = \{±ε_{i} ± ε_{j}, ±2ε_{i}\}. This example is a kind of outer version of Example 5.7(1). Indeed, taking the “dual” Lie algebras for (𝔤_{0}, 𝔤\overline{0}) yields the symmetric pair considered there. In our case, Δ_{1} = Δ\overline{0} and W′ = \{id, w_{n}\}, where w_{n}(ε_{i}) = ε_{i} (i ≤ n − 1) and w_{n}(ε_{n}) = −ε_{n}. Therefore (Δ_{1})_{id}^{+} = Δ+ \overline{0} = Δ_{0}^{+} ∪ \{2ε_{1}, …, 2ε_{n}\} and (Δ_{1})_{w_{n}}^{+} = Δ_{0}^{+} ∪ {2ε_{1}, …, 2ε_{n−1}, −2ε_{n}}. Hence the extreme weights are ρ_{0} + 2φ_{n} and ρ_{0} + 2φ_{n−1}. Thus, Spin_{0}(𝕍_{2φ_{1}}) = 𝕍_{ρ_{0}+2φ_{n−1}} ⊕ 𝕍_{ρ+2φ_{n}} and, because m_{2φ_{1}}(0) = n − 1, ∧^{•}𝕍_{2φ_{1}} = 2^{n−1}·(𝕍_{ρ+2φ_{n−1}} ⊕ 𝕍_{ρ+2φ_{n}})^{⊗2}.

2. 𝔤 = 𝔢_{6}, 𝔤_{0} = 𝔰𝔭_{8}. Then 𝔤_{1} ≃ 𝕍_{φ_{4}} . As indicated in Table 2, 𝔤\overline{0} = 𝔣_{4} and hence #(W\overline{0}/W_{0}) = 3. This is the outer version of Example 5.7(2). Here Δ\overline{0} = \{±ε_{i} ± ε_{j}, ±ε_{i}, (±ε_{1} ±ε_{2} ±ε_{3} ±ε_{4})/2 | 1 ≤ i, j ≤ 4, i ≠ j\} and Δ_{0} = (Δ\overline{0})_{s} ∪\{±ε_{1} ±ε_{2}, ±ε_{3} ±ε_{4}\}. The standard set of simple roots for Δ\overline{0} is α′_{1} = \frac{1}{2}(ε_{1} −ε_{2} −ε_{3} −ε_{4}), α′_{2} = ε_{4}, α′_{3} = ε_{3} −ε_{4}, α′_{4} = ε_{2} − ε_{4}. The roots for 𝔰𝔭_{8} have non standard presentation, but it is not hard to find that the simple roots in Δ+ \overline{0} ∩ Δ_{0} are α_{1} = ε_{2}, α_{2} = \frac{1}{2}(ε_{1} − ε_{2} − ε_{3} − ε_{4}), α_{3} = ε_{4}, α_{4} = ε_{3} − ε_{4}. Therefore the fundamental weights for 𝔰𝔭_{8} are φ_{1} = \frac{1}{2}(ε_{1} + ε_{2}), φ_{2} = ε_{1}, φ_{3} = ε_{1} + \frac{1}{2}(ε_{3} + ε_{4}), φ_{4} = ε_{1} + ε_{3}. Then an explicit verification shows that W′ = {id, w′, w′′}, where w′, w′′ are permutations of {ε_{1}, ε_{2}, ε_{3}, ε_{4}} determined by the cycles (23) and (432), respectively. (E.g. w′′(ε_{4}) = ε_{3}.) The direct computation of the extreme weights gives:

λ_{id} = ρ\overline{0} + ρ\overline{1} − ρ_{0} = \frac{1}{2}(9ε_{1} + 5ε_{2} + ε_{3} + ε_{4}) = 5φ_{1} + φ_{2} + φ_{3},

λ_{w′} = (w′)^{−1}(ρ\overline{0} + ρ\overline{1}) − ρ_{0} = \frac{1}{2}(9ε_{1} + 3ε_{2} + 3ε_{3} + ε_{4}) = 3φ_{1} + φ_{2} + φ_{3} + φ_{4} = ρ_{0} + 2φ_{1}, λ_{w′′} = (w′′)^{−1}(ρ\overline{0} + ρ\overline{1}) − ρ_{0} = \frac{1}{2}(9ε_{1} + ε_{2} + 3ε_{3} + 3ε_{4}) = φ_{1} + φ_{2} + 3φ_{3}.

Whence

Spin_{0}(𝕍_{φ_{4}}) = 𝕍_{5φ_{1}+φ_{2}+φ_{3}} ⊕ 𝕍_{φ_{1}+φ_{2}+3φ_{3}} ⊕ 𝕍_{ρ_{0}+2φ_{1}}

and

∧^{•}𝕍_{φ_{4}} = 4 (𝕍_{5φ_{1}+φ_{2}+φ_{3}} ⊕ 𝕍_{φ_{1}+φ_{2}+3φ_{3}} ⊕ 𝕍_{ρ_{0}+2φ_{1}})^{⊗2} .

7 Decomposably-generated ‘Spin’ modules and a Casimir ele- ment

Recall that we have given a geometric description of the extreme weights of Spinrepresentations in (3.4). Definition. Given an orthogonal 𝔤-module 𝕍, the 𝔤-submodule of Spin_{0}(𝕍) generated by the extreme weight vectors is denoted by Spin_{0}^{dg}(𝕍); Spin_{0}(𝕍) is called decomposablygenerated, if it is equal to Spin_{0}^{dg}(𝕍), i.e., if all its highest weights are extreme.

Since the extreme weights are of multiplicity 1, “decomposably-generated” implies “multiplicity free”. As a consequence of previous development, we have

7.1 Proposition. Let 𝔤 = 𝔤_{0} ⊕ 𝔤_{1} be a ℤ_{2}-graded semisimple Lie algebra. Then the 𝔤_{0}-module Spin_{0}(𝔤_{1}) is decomposably-generated (and multiplicity free).

Proof. The problem immediately reduces to the case in which 𝔤 is an irreducible ℤ_{2}- graded algebra. Then either 𝔤 is simple or 𝔤 ≃ 𝔥 ⊕ 𝔥, where 𝔥 is simple and Θ(h_{1}, h_{2}) = (h_{2}, h_{1}). In the second case, 𝔤_{0} ≃ 𝔥 is the diagonal in 𝔤, and 𝔤_{1} ≃ 𝔤_{0} as 𝔤_{0}-module. Here the conclusion follows by Kostant’s result, see Example 2.5(1). In the first case, for 𝔤_{0} of inner type, use Prop. 5.1(3) and Theorem 5.2; for 𝔤_{0} of outer type, use Prop. 6.4(3) and Theorem 6.5. □

An explanation of the term “decomposably-generated” comes from Example 2.5(3). If 𝕍 = 𝕎 ⊕ 𝕎^{∗}, then Spin(𝕍) ≃ 𝕜_{−ν} ⊗ ∧^{•}𝕎 and each extreme weight vector is represented by a decomposable vector in the exterior algebra. I think that the property of being “decomposably-generated” characterizes the representations of the form Spin_{0}(𝔤_{1}), i.e.,

7.2 Conjecture. Let 𝕍 be an orthogonal 𝔤-module. Then Spin_{0}(𝕍) is decomposably-generated if and only if 𝔤̃ := 𝔤 ⊕ 𝕍 is a ℤ_{2}-graded semisimple Lie algebra.

The conjecture will be proved in a particular case. Until the end of the section, the following situation is being considered: 𝔤 is semisimple, 𝔥 is a reductive subalgebra of 𝔤, and 𝔪 := 𝔥^{⊥} ⊂ 𝔤. Then 𝔪 is an orthogonal 𝔥-module and 𝔤 = 𝔥 ⊕ 𝔪 is a vector space sum. Clearly, this decomposition is a ℤ_{2}-grading if and only if [𝔪, 𝔪] ⊂ 𝔥. The representation 𝔥 → 𝔰𝔬(𝔪) is the isotropy representation of the affine homogeneous space G/H. Our aim is to study Spin_{0}(𝔪) and Spin_{0}^{dg}(𝔪) in the equal rank case. That is, it is assumed from now on that rk 𝔥 = rk 𝔤 and, more precisely, 𝔱 ⊂ 𝔥. Then Δ^{+} = Δ_{𝔥}^{+} ⊔ Δ_{𝔪}^{+}, 𝔪 has no zero weight and ∧^{•}𝔪 ≃ (Spin_{0}(𝔪))^{⊗2}. Denoting by W_{𝔥} the Weyl group of (𝔥, 𝔱), one may consider the minimal length “section” W^{𝔥} for W → W/W_{𝔥} and the cunning parity τ : W → {1, −1}, determined by Δ_{𝔥}^{+}. The proof of Prop. 5.1 applies in the present situation as well. This yields exactly #W^{𝔥} extreme weights of Spin_{0}(𝔪). Hence

dim(∧^{•}𝔪)^{𝔥} = dim(Spin_{0}(𝔪)^{⊗2})^{𝔥} ≥ #W^{𝔥} .

For w ∈ W^{𝔥}, the corresponding extreme weight is λ_{w} = w^{−1}ρ − ρ_{𝔥}, where ρ_{𝔥} = \frac{1}{2}|Δ_{𝔥}^{+}|. Therefore, arguing as in (5.6), we obtain

ch (Spin_{0}^{dg}(𝔪)) = ch ( ⨁_{w∈W^{𝔥}} 𝕍_{λ_{w}}) = \frac{∑_{w∈W} τ(w)e^{wρ}}{∏_{α∈Δ_{𝔥}^{+}}(e^{α/2} − e^{−α/2})} .

Recall that ch Spin_{0}(𝔪) = ∏_{μ∈Δ_{𝔪}^{+}}(e^{μ/2} + e^{−μ/2}). On the other hand, dim H^{∗}(G/H) = #W^{𝔥} [On95, § 13, Th. 2] and H^{∗}(G/H) can be computed via the complex of G-invariant exterior forms on G/H, i.e., the complex ((∧^{•}(𝔤/𝔥)^{∗})^{𝔥}, d), where d is the usual Lie algebra coboundary operator. We shall identify the 𝔥-modules (𝔤/𝔥)^{∗} and 𝔪. Having compared the previous expressions, we obtain

7.3 Proposition. Let 𝔥 ⊂ 𝔤 be a reductive subalgebra of maximal rank and 𝔥 → 𝔰𝔬(𝔪) the isotropy representation. Then the following conditions are equivalent:

(i) d is trivial on (∧^{•}𝔪)^{𝔥};

(ii) dim(∧^{•}𝔪)^{𝔥} = #W^{𝔥};

(iii) Spin_{0}(𝔪) = Spin_{0}^{dg}(𝔪);

(iv) ∑_{w∈W} τ(w)e^{wρ} = ∏_{α∈Δ_{𝔥}^{+}}(e^{α/2} − e^{−α/2}) ∏_{μ∈Δ_{𝔪}^{+}}(e^{μ/2} + e^{−μ/2}). □

Thus, conjecture 7.2 claims that neither of these conditions holds unless G/H is symmetric.

7.4 Theorem. Let 𝔥 be a reductive subalgebra of 𝔤, with rk 𝔥 = rk 𝔤, and 𝔪 := 𝔥^{⊥}. Suppose [𝔪, 𝔪] ⊄ 𝔥; then d is non-trivial on (∧^{•}𝔪)^{𝔥}. More precisely, d((∧^{3}𝔪)^{𝔥}) ≠ 0.

Proof. For any x ∈ 𝔤, let x_{𝔥} and x_{𝔪} denote its components in 𝔥 and 𝔪, respectively. Given x, y ∈ 𝔪, consider the decomposition [x, y] = [x, y]_{𝔥} + [x, y]_{𝔪}. We regard [ , ]_{𝔪} as mapping from 𝔪 × 𝔪 to 𝔪, and likewise for [ , ]_{𝔥}. By assumption, [ , ]_{𝔪} ≢ 0. On the other hand, applying construction from section 1 (see (1.4) and around) to 𝔥 and 𝔪 in place of 𝔤 and 𝕍, we see that [x, y]_{𝔥} = μ̄(x, y) for any x, y ∈ 𝔪.

Define the 3-form Ψ : ∧^{3}𝔪 → 𝕜 by Ψ(x, y, z) = Φ([x, y], z). Obviously, Ψ is 𝔥invariant. The assumption [𝔪, 𝔪] ⊄ 𝔥 precisely means that Ψ ≢ 0. We shall prove dΨ ≠ 0. To compute dΨ, we regard Ψ as 𝔥-invariant 3-form on 𝔤, orthogonal to 𝔥, and use the standard formula for d. The resulting expression is

dΨ(x, y, z, u) = 2(Φ([x, y]_{𝔪}, [z, u]_{𝔪}) + Φ([y, z]_{𝔪}, [x, u]_{𝔪}) + Φ([z, x]_{𝔪}, [y, u]_{𝔪})) .

Since [x, y] = μ̄(x, y) + [x, y]_{𝔪}, 𝔥 is orthogonal to 𝔪, and ([x, y], [z, u]) + ([y, z], [x, u]) + ([z, x], [y, u]) = 0 (because of the Jacobi identity), we have

dΨ(x, y, z, u) = −2(Φ(μ̄(x, y), μ̄(z, u)) + Φ(μ̄(y, z), μ̄(x, u)) + Φ(μ̄(z, x), μ̄(y, u))) .

In the notation of Prop. 1.5, for 𝔥 and 𝔪 in place of 𝔤 and 𝕍, this means that dΨ = −2κ. Assume that κ = 0. Then 𝔥⊕𝔪 equipped with the modified multiplication [ , ] becomes a ℤ_{2}-graded Lie algebra (see Prop. 1.5). Clearly, the multiplication does change only for pairs of elements in 𝔪: [m_{1}, m_{2}] := [m_{1}, m_{2}]_{𝔥}, whereas the structure of Lie algebra on 𝔥 and the 𝔥-module structure on 𝔪 remain undisturbed. Let 𝔤̃ denote the Lie algebra with modified multiplication and Θ the corresponding involutory automorphism of 𝔤̃. It is easily seen that 𝔤̃ is semisimple (use the proof of Theorem 1.7) and Θ is inner (because 𝔱 ⊂ 𝔥 remains a Cartan subalgebra in 𝔤̃). For the symmetric space G̃/H, we have H^{3}(G̃/H) = (∧^{3}𝔪)^{𝔥} ≠ 0. But H^{odd}(·) = 0 for the symmetric spaces of inner type [On95, § 13, n.3]. This contradiction proves κ = dΨ ≠ 0. □

7.5 Corollary. Conjecture 7.2 is true for the isotropy representations of affine homogeneous spaces G/H with rk 𝔤 = rk 𝔥. □

Remark. Theorem 7.4 is true even if rk 𝔥 < rk 𝔤 and some mild conditions are satisfied (e.g. 𝔤 is simple). However this has no immediate relation to Conjecture 7.2. For a reductive Lie algebra 𝔥, the Casimir element in U(𝔥) is determined by the choice of an invariant bilinear form on 𝔥. If 𝔥 is not simple, then the choice is essentially non unique. But for the isotropy representations one has a preferred choice of the bilinear form. In the above setting, let Φ( , )_{𝔥} be the restriction of Φ( , ) to 𝔥. Notice that even if 𝔥 is semisimple and we begin with the Killing form on 𝔤, thenΦ( , )_{𝔥} is not necessarily proportional to the Killing form on 𝔥. Let c_{𝔥} ∈ U(𝔥) be the Casimir element with respect to Φ( , )_{𝔥}. Recall that the W -invariant scalar product on P_{ℚ} is determined by Φ( , ).

7.6 Proposition. Suppose rk 𝔥 = rk 𝔤. Then the Casimir element c_{𝔥} acts scalarly on Spin_{0}^{dg}(𝔪). Its eigenvalue is equal to (ρ, ρ) − (ρ_{𝔥}, ρ_{𝔥}).

Proof. As is indicated above, Spin_{0}^{dg}(𝔪) = ⊕_{w∈W^{𝔥}}𝕍_{λ_{w}} and λ_{w} = w^{−1}ρ − ρ_{𝔥}. Therefore the eigenvalue of c_{𝔥} on 𝕍_{λ_{w}} is (λ_{w} +2ρ_{𝔥}, λ_{w}) = (w^{−1}ρ, w^{−1}ρ)−(ρ_{𝔥}, ρ_{h}) = (ρ, ρ)−(ρ_{𝔥}, ρ_{𝔥}). □

7.7 Theorem. Let 𝔤 = 𝔤_{0} ⊕ 𝔤_{1} be a ℤ_{2}-graded semisimple Lie algebra. Define the Casimir element c_{0} for 𝔤_{0} using the restriction of Φ( , ) to 𝔤_{0}. Then c_{0} acts on Spin_{0}(𝔤_{1}) scalarly, with value (ρ, ρ) − (ρ_{0}, ρ_{0}).

Proof. 1. If Θ is inner, then rk 𝔤_{0} = rk 𝔤; we conclude by Propositions 7.1, 7.6.

2. If Θ is outer, some accuracy is needed, since rk 𝔤_{0} < rk 𝔤. We use notation and information from section 6. Since Spin_{0}(𝔤_{1}) = Spin_{0}^{dg}(𝔤_{1}) = ⊕_{w∈W′}𝕍_{λ_{w}} and λ_{w} = w^{−1}ρ− ρ_{0}, the value of c_{0} on 𝕍_{λ_{w}} is equal to (w^{−1}ρ, w^{−1}ρ) − (ρ_{0}, ρ_{0}). Here W′ ⊂ W\overline{0}, where W\overline{0} is the Weyl group of 𝔤\overline{0}. Recall that 𝔱_{0} = 𝔱^{Θ} is a Cartan subalgebra for both 𝔤\overline{0} and 𝔤_{0}. As 𝔱_{0} contains regular elements of 𝔱, we have N_{G\overline{0}}(𝔱_{0}) ⊂ N_{G}(𝔱). Furthermore, since G\overline{0} is connected and 𝔱_{0} is Cartan, we have G\overline{0} ∩ T = T_{0}. Therefore W\overline{0} = N_{G\overline{0}}(𝔱_{0})/T_{0} can be identified with a subgroup of W = N_{G}(𝔱)/T. Hence (w^{−1}ρ, w^{−1}ρ) = (ρ, ρ). □

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ul. Akad. Anokhina d.30, kor.1, kv.7 Moscow 117602 Russia E-mail: dmitri@panyushev.mccme.ru