February 14, 2000

Isotropy representations, eigenvalues of a Casimir element, and commutative Lie subalgebras

Dmitri I. Panyushev^{1}

Introduction

In this paper, we work in the following setting. Let 𝔤 be a semisimple complex Lie algebra and 𝔥 a reductive algebraic subalgebra. The representation of 𝔥 in the complementary 𝔥-stable subspace 𝔪 of 𝔤 is called the isotropy representation of (𝔤, 𝔥). Let Φ be the Killing form on 𝔤 and Φ|_{𝔥} its restriction to 𝔥. Associated with Φ|_{𝔥}, one has the Casimir element C_{𝔥} ∈ U(𝔥).

Our first aim is to describe eigenvalues of C_{𝔥} on 𝔥 and 𝔪. A simple but useful observation is that interrelation between these eigenvalues depends on the Dynkin indices of 𝔪 as 𝔥-module. Recall that to every simple ideal 𝔥_{i} ⊂ 𝔥 one may attach a non-negative rational number ind_{𝔥_{i}}𝔪, the (Dynkin) index of the 𝔥_{i}-module 𝔪. If ind_{𝔥_{i}}𝔪 = μ_{i}, then the eigenvalue of C_{𝔥} on 𝔥_{i} is equal to 1/(1+μ_{i}). Because tr_{𝔤}(C_{𝔥}) = dim 𝔥, this is already sufficient for calculating the remaining eigenvalue of C_{𝔥} in the isotropy-irreducible case or, more generally, if C_{𝔥} acts scalarly on 𝔪. It is also shown that in some cases one can quickly find the eigenvalues making use of the dual Coxeter numbers for 𝔤 and 𝔥. In general, we have the following estimate. Let γ_{𝔪} be the maximal eigenvalue of C_{𝔥} on 𝔪 and 𝔪_{♣} the corresponding eigenspace of 𝔪.

0.1 Theorem. 1. γ_{𝔪} ≤ 1/2. Moreover, γ_{𝔪} = 1/2 if and only if 𝔥⊕𝔪_{♣} is a ℤ_{2}-graded reductive Lie subalgebra of 𝔤.

2. If the decomposition 𝔤 = 𝔥 ⊕ 𝔪 is a ℤ_{2}-grading, then C_{𝔥} always acts scalarly on 𝔪 (with eigenvalue 1/2).

With a mild constraint, this can be extended to periodic gradings of 𝔤 of higher order. To avoid technical details, assume that 𝔤 is simple in the next theorem.

0.2 Theorem. Let Θ be an automorphism of 𝔤 of order m and 𝔤 = ⊕_{ℤ_{m}}𝔤_{i} the corresponding periodic grading. Let C_{0} ∈ U(𝔤_{0}) be the Casimir element with respect to Φ|_{𝔤_{0}} . Suppose 𝔤_{0} is semisimple, if m ≥ 3. Then the eigenvalue of C_{0} on 𝔤_{i} is equal to 1/m, if g.c.d.(i, m) = 1, and (strictly) less than d/m, if g.c.d.(i, m) = d > 1.

The second aim is to study eigenvalues of C_{0} on ∧𝔤_{1}. This was inspired by an old result of B. Kostant on eigenvalues of the Casimir element C with respect to Φ on ∧𝔤 [Ko65]. The main result of that paper establishes a relation between eigenvalues of C on ∧^{k}𝔤 and commutative Lie subalgebras in 𝔤. In many aspects, the adjoint representation is convenient to regard as an isotropy representation of a symmetric space (= IRSS). This idea has led us to an extension of Kostant’s results to arbitrary ℤ_{2}-gradings and ℤ_{m}- gradings (m ≥ 3) with semisimple 𝔤_{0}. Adapting his proofs, we tried to argue in an “algebraically closed” fashion whenever possible. Still, at some crucial point we were forced to use (as Kostant did) a compact real form of 𝔤. One of the main results is:

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

0.3 Theorem. Let ℓ_{k} be the maximal eigenvalue of C_{0} on ∧^{k}𝔤_{1}. Then ℓ_{k} ≤ k/m. Moreover one has ℓ_{k} = k/m if and only if there exists a k-dimensional commutative Lie subalgebra of 𝔤_{1}. In such a case, the eigenspace of C_{0} in ∧^{k}𝔤_{1} corresponding to k/m is generated by ∧^{k}𝔞, where 𝔞 runs through all commutative k-dimensional subalgebras of 𝔤_{1}.

We also show that the 𝔤_{0}-submodule of ∧𝔤_{1} generated by ∧^{dim 𝔞}𝔞, where 𝔞 ranges over all commutative subalgebras, is multiplicity free.

Our third aim is to determine the maximal eigenvalue ℓ of C_{0} on ∧𝔤_{1}, if Θ is involutory, i.e., m = 2. Here we exploit the fact that 𝔤_{1} is an orthogonal 𝔤_{0}-module and therefore there exists a 𝔤_{0}-module, denoted by Spin_{0}(𝔤_{1}), such that ∧𝔤_{1} and Spin_{0}(𝔤_{1})^{⊗2} are equal up to a scalar factor. Using our description of Spin_{0}(𝔤_{1}) for all IRSS (see [Pa00] and Sect. 5), we give a formula for ℓ (see 5.4). Actually, we present a finite set of scalar products parameterized by some elements of W, ℓ being the maximal number in this set. We explicitly compute ℓ for the Hermitian involutions and show that, in this case, ℓ is attained on commutative subalgebras of maximal dimension.

Main Notation. Unless otherwise stated, 𝔤 is a semisimple Lie algebra. Then Φ is the Killing form on 𝔤 and G is the corresponding connected and simply-connected algebraic group. If 𝔥 is a reductive subalgebra of 𝔤, then Φ_{𝔥} is the Killing form on 𝔥 and Φ|_{𝔥} is the restriction of Φ to 𝔥. Note that Φ|_{𝔥} is non-degenerate, whereas Φ_{𝔥} is degenerate if 𝔥 is not semisimple. We fix a triangular decomposition 𝔤 = 𝔲^{+} ⊕𝔱⊕𝔲^{−}. Then Δ (resp. Δ^{+}) is the set of roots (resp. positive roots) of (𝔤, 𝔱) and Π ⊂ Δ^{+} is the set of simple roots. For any M ⊂ Δ, let |M| denote ∑_{α∈M} α, e.g. ρ = \frac{1}{2}|Δ^{+}|.

W is the Weyl group of (𝔤, 𝔱);

𝔱∗_{ℚ} is ℚ-vector subspace of 𝔱^{∗} generated by the lattice of integral weights and ( , )_{Φ} is the W -invariant positive-definite inner product on 𝔱∗_{ℚ} induced by Φ. The irreducible module with highest weight λ is denoted by V_{λ}. For the simple Lie algebras, we follow the numeration of the simple roots from [VO88] and [On95]; then φ_{i} denotes the fundamental weight corresponding to α_{i} ∈ Π; Λ is the highest root.

Acknowledgements. I am indebted to G. Röhrle for drawing my attention to [Ko65] and to E.B. Vinberg for interesting discussions on commutative subalgebras. Considerable part of this work was done while I was visiting the Max-Planck-Institut für Mathematik (Bonn). I am grateful to this institution for the hospitality and excellent working conditions.

1 Generalities on Casimir elements, the Dynkin index, and the dual Coxeter numbers

Most of the formulas in this section does not depend on the normalization of 𝔤-invariant bilinear form on 𝔤. To stress this fact, we shall also use an arbitrary 𝔤-invariant nondegenerate bilinear form ( , ) on 𝔤 and the induced form on 𝔱∗_{ℚ}.

\underline{Casimir elements.} Let e_{1}, …, e_{n} be a basis for 𝔤 and \{e′_{i}\}_{i} the dual basis relative to ( , ). The Casimir element with respect to ( , ) is C = ∑_{i=1}^{n} e_{i}e′_{i} ∈ U(𝔤). It is well-known that C does not depend on the choice of basis and that C is a scalar operator in any irreducible representation. If π_{λ} : 𝔤 → 𝔤𝔩(V_{λ}) is irreducible, then

π_{λ}(C) = (λ, λ + 2ρ)·id_{V_{λ}} . (1.1)

Of course, all this holds for any reductive Lie algebra equipped with a non-degenerate invariant symmetric bilinear form. It is easily seen that the Casimir element with respect to the Killing form gives rise to the identity operator on 𝔤. Therefore, if 𝔤 is simple, which will be assumed until the end of the section, then (Λ, Λ + 2ρ)_{Φ} = 1 . (1.2)

\underline{The dual Coxeter number.} Let O_{min} ⊂ 𝔤 be the unique adjoint G-orbit of minimal nonzero dimension. As usual, Λ^{∨} = 2Λ/(Λ, Λ).

1.3 Lemma. The following numbers are equal

1 + (ρ, Λ^{∨}), \frac{1}{2} dim O_{min} + 1, \frac{1}{(Λ, Λ)_{Φ}} .

Proof. 1. Noting that the first number does not depend on the choice of an invariant bilinear form, we deduce the equality 1 + (ρ, Λ^{∨})_{Φ} = \frac{1}{(Λ, Λ)_{Φ}} from Eq. (1.2)

2. Since dim O_{min} = 1 + #{α ∈ Δ^{+} | (α, Λ) ≠ 0} and (α, Λ^{∨}) ∈ {0, 1} for α ∈ Δ^{+} \ Λ, we obtain

dim O_{min} = 1 + ( ∑_{(α,Λ^{∨})=1}(α, Λ^{∨})) + 1 = ∑_{α∈Δ^{+}}(α, Λ^{∨}) = 2(ρ, Λ^{∨}). □

This common number is called the dual Coxeter number of 𝔤, denoted by h^{∗}(𝔤). In the simply-laced case, h^{∗}(𝔤) = h(𝔤)—the usual Coxeter number. For the other simple Lie algebras we have h^{∗}(B_{n}) = 2n−1, h^{∗}(C_{n}) = n+1, h^{∗}(F_{4}) = 9, h^{∗}(G_{2}) = 4. Some interesting properties of the dual Coxeter numbers are found in [Su98].

\underline{The Dynkin index.} Let V be a ℂ-vector space. We denote by tr_{V} the trace function on End V . If V is a 𝔤-module, then the function x ↦ tr_{V} (x^{2}) (x ∈ 𝔤) is a 𝔤-invariant quadratic form on 𝔤, hence a multiple of the Killing form x ↦ tr_{𝔤}(x^{2}). The multiplication factor is called the (Dynkin) index of V , denoted by ind_{𝔤}V . The index was introduced by E.B. Dynkin in the seminal paper [Dy52]. (He used however another normalization of invariant form on 𝔤. In place of the Killing form, he exploited the form such that (α, α) = 2 for a long root α.) Substantial applications of the index to Invariant Theory were demonstrated in [AVE] and then in the work of A.G. Elashvili on “stabilizers in general position”. Clearly, the index is an additive function of V . It is therefore sufficient to know the indices for the irreducible representations. The following result is essentially contained in [Dy52, Thm. 2.5]:

(1.4) ind_{𝔤}V_{λ} = \frac{dim V_{λ}}{dim 𝔤} ·\frac{(λ, λ + 2ρ)}{(Λ, Λ + 2ρ)} .

Thus, the index of any non-trivial representation is a positive rational number and the adjoint representation has index 1. Remark. In [On95, §3.10], A.L. Onishchik uses another normalization for the index, which goes back to Dynkin’s paper. In that interpretation, the index is always an integer. In view of Lemma 1.3, it is not hard to realize that the transition factor between the two versions is 2h^{∗}(𝔤).

2 On a Casimir element and isotropy representations

Let 𝔥 be a reductive subalgebra of 𝔤. Then Φ|_{𝔥} is non-degenerate and 𝔤 = 𝔥 ⊕ 𝔪, where 𝔪 = 𝔥^{⊥}. Then 𝔪 ≃ 𝔤/𝔥 is an orthogonal 𝔥-module and the representation 𝔥 → 𝔰𝔬(𝔪) is called the isotropy representation of the pair (𝔤, 𝔥). It is assumed that

(⋄) 𝔥 does not contain simple ideals of 𝔤. Choose a basis (e_{i})_{i=1}^{n} for 𝔤 so that ⟨e_{1}, …, e_{s}⟩ = 𝔥 and ⟨e_{s+1}, …, e_{n}⟩ = 𝔪. Let (e′_{i})_{i=1}^{n} be the dual basis relative to Φ. Consider the following elements in U(𝔤):

C_{𝔥} = ∑_{i=1}^{s} e_{i}e′_{i}, C_{𝔪} = ∑_{i=s+1}^{n} e_{i}e′_{i} .

Then C_{𝔥} ∈ U(𝔥) is the Casimir element with respect to Φ|_{𝔥}, and C := C_{𝔥} + C_{𝔪} is the Casimir element for 𝔤. Therefore

C(x) = x for all x ∈ 𝔤 .

Our goal is to obtain some relations for the eigenvalues of C_{𝔥}. If x, y ∈ 𝔤 and V ⊂ 𝔤 is an (ad x·ad y)-stable subspace, then we set tr_{V} (x·y) := tr (ad x·ad y |_{V}).

2.1 Lemma.

1. tr_{𝔤}(C_{𝔥}) = dim 𝔥;

2. Let x, y ∈ 𝔥. Then (i) Φ(C_{𝔥}(x), y) = tr_{𝔥}(x·y) = Φ_{𝔥}(x, y); (ii) Φ(C_{𝔪}(x), y) = tr_{𝔪}(x·y).

Proof. 1. tr_{𝔤}(C_{𝔥})= ∑_{j=1}^{n} Φ(C_{𝔥}(e_{j}), e′_{j})= ∑_{i=1}^{s} ∑_{j=1}^{n} Φ([e′_{i}, e_{j}], [e′_{j}, e_{i}])= ∑_{i=1}^{s} Φ(e′_{i}, C(e_{i}))= dim 𝔥.

2. (i) Φ(C_{𝔥}(x), y) = ∑_{i=1}^{s} Φ([e_{i}, [e′_{i}, x]], y]) = ∑_{i=1}^{s} Φ((ad y·ad x)e′_{i}, e_{i}) = tr_{𝔥}(y·x). (ii) Similar. □

Let 𝔥 = 𝔥_{1} ⊕…⊕𝔥_{k} ⊕𝔷, where 𝔷 is the centre and the 𝔥_{i}’s are the simple ideals. Using an appropriate basis for 𝔥, we may further decompose C_{𝔥} as C_{𝔥_{1}} +…+C_{𝔥_{k}} +C_{𝔷}. If x, y ∈ 𝔥_{i}, then C_{𝔥}(x) = C_{𝔥_{i}}(x) and Φ_{𝔥}(x, y) = Φ_{𝔥_{i}}(x, y). It then follows from Lemma 2.1(2) that the ratio Φ(C_{𝔪}(x), y)/Φ(C_{𝔥}(x), y) is nothing but the Dynkin index of the 𝔥_{i}-module 𝔪. Note that the 𝔥_{i}-modules 𝔪 and 𝔤/𝔥_{i} have the same non-trivial summands and therefore equal indices. Let γ_{i} be the eigenvalue of C_{𝔥} on 𝔥_{i} (i = 1, …, k). Then C_{𝔥}(x) = γ_{i}x, C_{𝔪}(x) = (1 − γ_{i})x for x ∈ 𝔥_{i}. Hence

ind_{𝔥_{i}}(𝔤/𝔥_{i}) = ind_{𝔥_{i}}𝔪 = \frac{1 − γ_{i}}{γ_{i}} =: μ_{i} . (2.2)

Hypothesis (⋄) guarantees us that 𝔤/𝔥_{i} is a non-trivial 𝔥_{i}-module. Whence μ_{i} > 0 and γ_{i} < 1. It follows that

\frac{Φ(x, y)}{Φ_{𝔥_{i}}(x, y)} = \frac{Φ(C_{𝔥}(x), y) + Φ(C_{𝔪}(x), y)}{Φ_{𝔥}(x, y)} = 1 + μ_{i} = \frac{1}{γ_{i}} . (2.3)

Let us now give practical formulas for computing the index of isotropy representations and related eigenvalues of Casimir elements. Without loss of generality, one may assume that 𝔱 ∩ 𝔥_{i} is a Cartan subalgebra of 𝔥_{i}. Then, identifying 𝔱 and 𝔱^{∗}, we may regard the roots for 𝔥_{i} as elements of 𝔱^{∗}. Thus, we can compare length of roots for 𝔤 and 𝔥_{i}.

2.4 Proposition. Suppose 𝔤 is simple. Let r be the ratio of the square of length of a long root for 𝔤 to that for 𝔥_{i}. Then the eigenvalue of C_{𝔥_{i}} on 𝔥_{i} is equal to \frac{h^{∗}(𝔥_{i})}{r·h^{∗}(𝔤)} and

ind_{𝔥_{i}}𝔪 = r·\frac{h^{∗}(𝔤)}{h^{∗}(𝔥_{i})} − 1.

Proof. In view of Eq. (2.2), it is enough to deduce one of the two relations. By Eq. (2.3), 1 + ind_{𝔥_{i}}𝔪 = Φ(x, x)/Φ_{𝔥_{i}}(x, x) for all x ∈ 𝔥_{i}, i.e., (1 + ind_{𝔥_{i}}𝔪)Φ_{𝔥_{i}} = Φ|_{𝔥_{i}}. On the other hand, let α be a long root for 𝔥_{i}. By the definition of r and the third expression for the dual Coxeter number (see 1.3), we have (α, α)_{Φ_{𝔥_{i}}} = 1/h^{∗}(𝔥_{i}) and (α, α)_{Φ} = 1/r·h^{∗}(𝔤). Whence 1 + ind_{𝔥_{i}}𝔪 = r·h^{∗}(𝔤)/h^{∗}(𝔥_{i}). □

Let γ_{𝔪} be the maximal eigenvalue of C_{𝔥} on 𝔪. If C_{𝔥} acts scalarly on 𝔪, then γ_{𝔪} can be computed using Lemma 2.1(1): dim 𝔥 = tr (C_{𝔥}) = ∑_{i=1}^{k} γ_{i} dim 𝔥_{i} + γ_{𝔪} dim 𝔪. Therefore

(2.5) γ_{𝔪} dim 𝔪 = ∑_{j=1}^{k}(1 − γ_{j}) dim 𝔥_{j} + dim 𝔷 = ∑_{j=1}^{k} \frac{μ_{j}}{1 + μ_{j}} dim 𝔥_{j} + dim 𝔷 .

A sufficient (but not necessary) condition for C_{𝔥} to act scalarly on 𝔪 is that 𝔪 has an irreducible real form. Let 𝔤_{c} be a compact form of 𝔤 such that 𝔤_{c} ∩ 𝔥 =: 𝔥_{c} is a compact form of 𝔥. Set 𝔪_{c} = 𝔪 ∩ 𝔤_{c}. Then the 𝔥_{c}-module 𝔪_{c} can be irreducible, whereas the 𝔥-module 𝔪 is not. Let 𝔪_{♣} denote the eigenspace of C_{𝔥} in 𝔪 belonging to γ_{𝔪}.

2.6 Theorem. 1. γ_{𝔪} ≤ 1/2. Moreover, γ_{𝔪} = 1/2 if and only if 𝔥⊕𝔪_{♣} is a ℤ_{2}-graded reductive Lie algebra.

2. If the decomposition 𝔤 = 𝔥 ⊕ 𝔪 is a ℤ_{2}-grading (i.e. [𝔪, 𝔪] ⊂ 𝔥), then C_{𝔥} always acts scalarly on 𝔪 (with eigenvalue 1/2).

Proof. 1. Let u ∈ 𝔪. Using the above basis for 𝔤, we obtain the matrix presentation

ad u = ( \begin{matrix}0 & M \\ N & X \end{matrix} ), where M is an s × (n − s)-matrix, N an (n − s) × s-matrix, and X

an (n − s) × (n − s)-matrix. Then

(ad u)^{2} = ( \begin{matrix}MN & MX \\ XN & NM + X^{2} \end{matrix} ) .

Since tr (MN) = tr (NM), we conclude that Φ(u, u) = tr ((ad u)^{2}) = 2tr (MN)+tr (X^{2}). As in Lemma 2.1, we obtain

Φ(C_{𝔥}(u), u) = ∑_{i=1}^{s} Φ((adu)^{2}e′_{i}, e_{i}) = tr (MN) .

Now, if u ∈ 𝔪_{♣}, then C_{𝔥}(u) = γ_{𝔪}u. Therefore tr (MN) = γ_{𝔪}Φ(u, u) and

(1 − 2γ_{𝔪})Φ(u, u) = tr (X^{2}) .

Without loss of generality, one may assume that the basis (e_{i}) is chosen so that e_{1}, …, e_{n} ∈ 𝔤_{c} and Φ(e_{i}, e_{j}) = −δ_{i,j}. Taking 0 ≠ u ∈ 𝔪_{c} ∩ 𝔪_{♣}, we see that Φ(u, u) < 0 and X is a real skew-symmetric matrix. Hence tr (X^{2}) ≤ 0 and 1 − 2γ_{𝔪} ≥ 0.

If γ_{𝔪} = 1/2, then tr (X^{2}) = 0 for all u ∈ 𝔪_{♣}. Whence X = 0 for all u ∈ 𝔪_{c} ∩ 𝔪_{♣}. This implies that X = 0 for all u ∈ 𝔪_{♣} and hence [𝔪_{♣}, 𝔪] ⊂ 𝔥. In particular, 𝔥 ⊕ 𝔪_{♣} is a ℤ_{2}-grading. Since Φ|_{𝔥⊕𝔪_{♣}} is non-degenerate, this Lie algebra is reductive.

2. If 𝔤 = 𝔥 ⊕ 𝔪 is a ℤ_{2}-grading, then X = 0. Therefore Φ(u, u) = 2tr (MN) = 2Φ(C_{𝔥}(u), u). Since this holds for all u ∈ 𝔪, we are done. □

2.7 Corollary. Let 𝔥 be a maximal reductive subalgebra of 𝔤. Then either 𝔥 is symmetric or γ_{𝔪} < 1/2.

Under a mild constraint, Theorem 2.6(2) can be generalized to higher periodic gradings. Let Θ be an automorphism of 𝔤 of order m. Letting 𝔤_{i} = {x ∈ 𝔤 | Θ(x) = ζ^{i}x}, where ζ is a fixed primitive m^{th} root of unity, we obtain a periodic grading 𝔤 = ⊕_{i∈ℤ_{m}}𝔤_{i}. Here 𝔤_{0} is a reductive Lie subalgebra and each 𝔤_{i} is a 𝔤_{0}-module. Consider the Casimir element C_{0} ∈ U(𝔤_{0}), with respect to Φ|_{𝔤_{0}} . For 𝔥 = 𝔤_{0}, hypothesis (⋄) means that Θ is non-trivial on each simple ideal of 𝔤.

2.8 Theorem. Suppose 𝔤_{0} is semisimple. Then the eigenvalue of C_{0} on 𝔤_{i} is equal to 1/m whenever i and m are coprime.

Proof. It clearly suffices to handle the case i = 1. Take y ∈ 𝔤_{1} and z ∈ 𝔤_{m−1}. Having chosen a basis for 𝔤 compatible with the decomposition 𝔤 = 𝔤_{0} ⊕ 𝔤_{1} ⊕ … ⊕ 𝔤_{m−1}, we obtain the following block presentations:

ad y = ( \begin{matrix}0 & … & 0 & A_{0} \\ A_{1} & … & 0 & 0 \\ ⋯ & ... & ⋯ & ⋯ \\ 0 & … & A_{m−1} & 0 \end{matrix} ) , ad z = ( \begin{matrix}0 & B_{1} & … & 0 \\ ⋯ & ⋯ & ... & ⋯ \\ 0 & 0 & … & B_{m−1} \\ B_{0} & 0 & … & 0 \end{matrix} ) .

Then

ad y·ad z = ( \begin{matrix}A_{0}B_{0} & 0 & ⋯ \\ 0 & A_{1}B_{1} & 0 & ⋯ \\ ⋯ & ⋯ & ... & ⋯ \\ & ⋯ & 0 & A_{m−1}B_{m−1} \end{matrix} ) , ad z·ad y = ( \begin{matrix}B_{1}A_{1} & 0 & ⋯ \\ 0 & B_{2}A_{2} & 0 \\ ⋯ & ⋯ & ... & ⋯ \\ ⋯ & ⋯ & 0 & B_{0}A_{0} \end{matrix} ) .

Since [y, z] ∈ 𝔤_{0} and 𝔤_{0} is semisimple, tr_{𝔤_{i}}[y, z] = 0 for all i. Hence tr (A_{i}B_{i}) = tr (B_{i+1}A_{i+1}) ∀i ∈ ℤ_{m}. On the other hand, tr (A_{i}B_{i}) = tr (B_{i}A_{i}). Thus, tr (A_{0}B_{0}) = tr (A_{1}B_{1}) = … = tr (A_{m−1}B_{m−1}) and tr (A_{i}B_{i}) = \frac{1}{m}tr (ad y·ad z) = \frac{1}{m}Φ(y, z). Finally,

Φ(C_{0}(y), z) = tr_{𝔤_{0}}(y·z) = tr (A_{0}B_{0}) = \frac{1}{m}Φ(y, z) .

Whence C_{0}(y) = \frac{1}{m}y. □

Remark. For m = 2, the semisimplicity of 𝔤_{0} is not important, see Theorem 2.6. However simple examples show that it cannot be dropped, if m ≥ 3.

2.9 Corollary. If m is prime and 𝔤_{0} is semisimple, then C_{0} acts scalarly on 𝔤_{1} ⊕ … ⊕ 𝔤_{m−1}, with eigenvalue 1/m, and

tr_{𝔤_{0}}(C_{0}) = \frac{(m + 1) dim 𝔤_{0} − dim 𝔤}{m} .

In particular, (m + 1) dim 𝔤_{0} > dim 𝔤.

Proof. To get the equality, use Lemma 2.1(1) with 𝔥 = 𝔤_{0}. □

Given a finite order automorphism Θ of 𝔤, let us say that a Θ-stable ideal of 𝔤 is Θindecomposable, if it is not a sum of two proper Θ-stable ideals. It is easily seen that 𝔤 is (uniquely) the sum of Θ-indecomposable ideals, and every such ideal is either simple, or is the sum of isomorphic simple ideals that are cyclically permuted by Θ. Consider the second possibility in more details.

(2.10) Let 𝔤 = \underbrace{𝔥 ⊕ … ⊕ 𝔥}_{m} and Θ(x_{1}, …, x_{m}) = (x_{m}, x_{1}, …, x_{m−1}). Then 𝔤 = ⨁_{i∈ℤ_{m}} 𝔤_{i},

where 𝔤_{0} ≃ 𝔥 is the diagonal in 𝔤 and 𝔤_{i} ≃ 𝔥 as 𝔥-module for all i. Here Φ|_{𝔤_{0}} = mΦ_{𝔥}. Therefore C_{0} acts scalarly, with eigenvalue 1/m, on the whole 𝔤. That is, the condition that i, m should be coprime does not arise in this case.

If 𝔤 is not Θ-indecomposable, then it may happen that the restriction of Θ to some Θ-indecomposable ideals has order < m. To avoid reservations related to this, it is convenient to give Definition. Θ ∈ Aut 𝔤 is said to be of pure order m, if the restriction of Θ to any Θ-indecomposable ideal of 𝔤 has order m.

2.11 Theorem. Let Θ be an automorphism of 𝔤 of pure order m. Suppose 𝔤_{0} is semisimple, if m ≥ 3. Assume that g.c.d.(k, m) =: d > 1. Then all the eigenvalues of C_{0} on 𝔤_{k} are less than d/m.

Proof. It follows from (2.10) that we may restrict ourselves with the case, where 𝔤 is simple.

(a) \underline{k = d = m}. Here 𝔤_{m} = 𝔤_{0}. The eigenvalue of C_{0} on the centre of 𝔤_{0} (if any) is equal to 0. If 𝔥_{i} is a simple ideal of 𝔤_{0}, then the eigenvalue of C_{0} on 𝔥_{i} is equal to 1/(1 + ind_{𝔥_{i}}(𝔤/𝔤_{0})), see Eq. (2.3). Since 𝔤/𝔤_{0} is a faithful 𝔤_{0}-module, ind_{𝔥_{i}}(𝔤/𝔤_{0}) > 0.

(b) \underline{1 < d < m}. For the reductive “Veronese” subalgebra 𝔤(k) = ⨁_{i} 𝔤_{ki}, we have Θ|_{𝔤(k)} is an automorphism of order m/d, 𝔤(k)_{0} = 𝔤_{0}, and 𝔤(k)_{1} = 𝔤_{k}. Because 𝔤(k) can be not semisimple, take 𝔰 = [𝔤(k), 𝔤(k)]. This step does not change 𝔤_{0} = 𝔰_{0} and eventually kills some trivial 𝔤_{0}-submodules of 𝔤_{k}. But the nonzero eigenvalues of C_{0} on 𝔰_{1} and 𝔤_{k} remains the same. We are thus reduced to considering a semisimple subalgebra 𝔰 together with \overline{Θ} := Θ|_{𝔰}. Consider \overline{Θ}-indecomposable ideals of 𝔰. Of course, we are only interested in those of them that have nonzero intersection with 𝔰_{1}.

Let 𝔰^{(j)} be a \overline{Θ}-indecomposable ideal of 𝔰. The condition 𝔰_{1}^{(j)} ≠ 0 means that \overline{Θ}|_{𝔰^{(j)}} still has the order m/d. Since 𝔰_{0}^{(j)} is an ideal of 𝔰_{0}, it is also semisimple. Clearly, the Casimir element C_{0} has the j -component C_{0}^{(j)}, determined by 𝔰_{0}^{(j)}, which is the only component responsible for the eigenvalue on 𝔰_{1}^{(j)}. If C_{0}^{(j)} were defined with respect to the Killing form on 𝔰^{(j)}, then, by Theorem 2.8, the eigenvalue on 𝔰_{1}^{(j)} was equal to 1/(m/d) = d/m. But C_{0}^{(j)} is defined with respect to the Killing form coming from 𝔤, and the above quantity gains a correcting factor. This factor is just the ratio of the two Killing forms in question and, by Eq. (2.3), it is equal to 1/(1 + ind_{𝔰^{(j)}}(𝔤/𝔰^{(j)}). Thus the

true eigenvalue is \frac{1}{1 + ind_{𝔰^{(j)}}(𝔤/𝔰^{(j)})}·\frac{d}{m} < \frac{d}{m}. □

This results in the following general statement:

2.12 Corollary. Let Θ be an automorphism of order m of a semisimple Lie algebra 𝔤 which acts non-trivially on each simple ideal of 𝔤. Suppose 𝔤_{0} is semisimple, if m > 2. Then the eigenvalues of C_{0} on 𝔤 are less than 1. □

2.13 Examples. Recall that the number r = r(𝔥_{i}) is defined in Prop. 2.4.

1. 𝔤 = E_{8}, 𝔥 = 𝔰𝔩_{8}. Here 𝔥 is the fixed-point subalgebra for an automorphism of order 3 and the 𝔥-module 𝔪 is V_{φ_{3}} ⊕ V_{φ_{3}}^{∗}. Since rk 𝔤 = rk 𝔥 and 𝔤 is simply-laced, r = 1.

By Prop. 2.4, γ_{𝔥}, the eigenvalue of C_{𝔥} on 𝔥 is 30/9 and ind_{𝔰𝔩_{8}}𝔪 = 30/9 − 1. Hence ind_{𝔰𝔩_{8}}(V_{φ_{3}}) = 7/6. Using Eq. (2.5), one computes that γ_{𝔪} = 1/3, which is not surprising in view of Theorem 2.8.

2. 𝔤 = 𝔰𝔭_{2n}, 𝔥 = 𝔰𝔩_{n}. Here 𝔪 = V_{2φ_{1}} ⊕ V_{2φ_{1}}^{∗} ⊕ V_{0} and r = 2. Hence 1 + ind_{𝔰𝔩_{n}}(𝔪) = 2(n + 1)/n and ind_{𝔰𝔩_{n}}(V_{2φ_{1}}) = (n + 2)/2n. Since the eigenvalue of C_{𝔥} on 𝔥 is \frac{n}{2(n + 1)} , one computes, using Lemma 2.1(1), that the eigenvalue of C_{𝔥} on V_{2φ_{1}} is

\frac{1}{2} − \frac{1}{n(n + 1)} . It is worth noting that 𝔥̃ := 𝔫_{𝔤}(𝔥) ≃ 𝔤𝔩_{n} is a symmetric subalgebra of 𝔰𝔭_{2n}. This small modification results in a modification for the eigenvalues. The eigenvalue of C𝔥̃ on V_{2φ_{1}} to be 1/2.

3. 𝔤 = 𝔰𝔬_{7}, 𝔥 = G_{2}. Here 𝔪 = V_{φ_{1}} . Making use of Dynkin’s formula (see (1.4)), we obtain ind_{G_{2}}V_{φ_{1}} = 1/4. One then calculates that γ_{𝔥} = 4/5 and γ_{𝔪} = 2/5. Since G/H is not symmetric here, γ_{𝔪} < 1/2. Note that the equality \frac{1}{4} = r·\frac{5}{4} − 1 implies that r = 1. However one can begin with proving that r = 1 and therefore avoid using Eq. (1.4).

4. The exceptional algebra 𝔤 = E_{6} has an outer automorphism of order 4 such that 𝔤_{0} = 𝔰𝔬_{7}×𝔰𝔩_{2}. This easily can be detected using Kac’s description of finite order automorphisms (see [Ka90, ch. 8] or [Vi76, § 8]). The automorphism is determined by the labelled twisted

Dynkin diagram E_{6}^{(2)}: [figure: 1; <] .

Here 𝔤_{1} ≃ 𝔤_{3} = V_{φ_{3}} ⊗V_{φ′} and 𝔤_{2} = V_{φ_{1}} ⊗V_{2φ′} ⊕V_{0} (φ′ refers to the fundamental weight of 𝔰𝔩_{2}). Having stared at the above diagram, it is not hard to realize that r_{1} := r(𝔰𝔬_{7}) = 1 and r_{2} := r(𝔰𝔩_{2}) = 2. Therefore the eigenvalue of C_{0} on 𝔰𝔬_{7} is h^{∗}(B_{3})/h^{∗}(E_{6}) = 5/12 and on 𝔰𝔩_{2} is h^{∗}(A_{1})/2h^{∗}(E_{6}) = 1/12. By Theorem 2.8, the eigenvalues on 𝔤_{1} and 𝔤_{3} are equal to 1/4. To compute the eigenvalue of C_{0} on the non-trivial representation in 𝔤_{2}, one may exploit Lemma 2.1(1). We proceed however in a round-about way. Since 𝔤_{0} ⊕ 𝔤_{2} is a symmetric subalgebra of E_{6}, of dimension 46, it is nothing but 𝔰𝔬_{10} ⊕ {1-dim torus}. Hence V_{φ_{1}} ⊗ V_{2φ′} is just the isotropy representation for the symmetric space SO(10)/SO(7) × SO(3). Therefore, if C_{0} were defined w.r.t. the Killing form on 𝔰𝔬_{10}, the eigenvalue was equal to 1/2. But since we are exploiting the restriction of the Killing form of E_{6}, the correcting factor h^{∗}(D_{5})/h^{∗}(E_{6}) = 8/12 is needed (we have used here the obvious fact that r(𝔰𝔬_{10}) = 1). Thus, the eigenvalue on V_{φ_{1}} ⊗ V_{2φ′} is 1/3. Now, one can check that the trace of C_{0} on 𝔤 is 24 = dim 𝔤_{0}, as required.

3 Preliminaries on exterior algebras

A nice exposition of the standard facts on ∧𝔤 is found Koszul’s paper [Kos50]. We identify 𝔤 with 𝔤^{∗}, using Φ, and consider ∧𝔤 with the usual differentials: d : ∧^{l}𝔤 → ∧^{l+1}𝔤 and ∂ : ∧^{l}𝔤 → ∧^{l−1}𝔤. Recall that

∂(x_{1} ∧ … ∧ x_{l}) := ∑_{i<j}(−1)^{i+j−1}[x_{i}, x_{j}]x_{1} ∧ … x̂_{i} … x̂_{j} … ∧ x_{l}

for l ≥ 2 and ∂(x_{1}) = 0. In particular, ∂(x_{1} ∧ x_{2}) = [x_{1}, x_{2}]. We regard Φ as having been extended, in the usual way, via determinants, from 𝔤 to ∧𝔤. Then d = −∂^{t}. More precisely, denoting the extension of Φ to ∧^{l}𝔤 by Φ^{(l)}, we have

Φ^{(l)}(x_{1} ∧ … ∧ x_{l}, y_{1} ∧ … ∧ y_{l}) := det ∥Φ(x_{i}, y_{j})∥ .

For x ∈ 𝔤, ε(x) is the exterior product operator and i(x) is the interior product operator in ∧𝔤. That is,

ε(x)·x_{1} ∧ … ∧ x_{l} := x ∧ x_{1} ∧ … ∧ x_{l},

i(x)·x_{1} ∧ … ∧ x_{l} := ∑_{i=1}^{l}(−1)^{i−1}Φ(x, x_{i})x_{1} ∧ … x̂_{i} … ∧ x_{l} .

Denote by θ the natural extension of the adjoint representation of 𝔤 to ∧𝔤:

θ(x)·x_{1} ∧ … ∧ x_{l} := ∑_{i=1}^{l} x_{1} ∧ … ∧ [x, x_{i}] … ∧ x_{l} .

These operators satisfy the following relations:

(3.1) \begin{matrix}[d, θ(x)] = 0, \\ [∂, θ(x)] = 0, \\ ε(x)∂ + ∂ε(x) = θ(x) \end{matrix} \} for all x ∈ 𝔤.

After Koszul [Kos50, 3.4], it is known that

d = \frac{1}{2} ∑_{i=1}^{n} ε(e′_{i})θ(e_{i}) . (3.2)

Combining Eq. (3.1) and (3.2) yields

2(d∂ + ∂d) = ∑_{i=1}^{n} θ(e′_{i})θ(e_{i}) = θ(C) .

Below, we write merely C in place of θ(C). Let m_{k} be the maximal eigenvalue of C on ∧^{k}𝔤. The following result was obtained by B. Kostant.

3.3 Theorem ([Ko65, § 4], see also [Ko98, 1.1]). For k = 0, …, n, one has m_{k} ≤ k. Moreover, m_{k} = k if and only if there exists a commutative k-dimensional Lie subalgebra of 𝔤. Furthermore, in such a case, the k-eigenspace of C in ∧^{k}𝔤 is spanned by the decomposable k-vectors corresponding to k-dimensional commutative subalgebras. (m_{k} here is twice the value assigned to m_{k} in [Ko65] and the same as in [Ko98].)

The adjoint representation is only a member of the large family of the isotropy representations of symmetric spaces. It is therefore natural to look at the same problem for all symmetric spaces. In the next section, we show that Theorem 3.3 can be generalized to this and even more general setting.

In [Ko65, § 4], Kostant proved an assertion concerning the eigenspace of C in ∧^{k}𝔤 belonging to the maximal eigenvalue. It was an essential step for his proof of Theorem 3.3. Actually, his argument applies to the exterior algebra of any 𝔤-module. Since this more general formulation is needed below, we recall the necessary setup and outline a proof. In the remainder of the section, 𝔤 is a reductive Lie algebra and C is a Casimir element with respect to a non-degenerate invariant bilinear form ( , ) on 𝔤. Let V be a vector space. An element u ∈ ∧^{k}V is called decomposable, if u = v_{1} ∧ … ∧ v_{k} for some v_{i} ∈ V . If V is a 𝔤-module, then a submodule W ⊂ ∧^{k}V is called decomposably-generated, if it is spanned by decomposable elements.

3.4 Theorem (Kostant). 1. Suppose W ⊂ ∧^{k}V is irreducible; then W is decomposably-generated if and only if a (any) highest weight vector is decomposable.

2. Let m_{k} be the maximal eigenvalue of C on ∧^{k}V and M_{k} the corresponding eigenspace. Then M_{k} is a sum of irreducible decomposably-generated 𝔤-submodules.

Proof. 1. Let u ∈ W be a decomposable element. Consider the line ⟨u⟩ := ℂu as element of the Grassmannian Gr_{k}(W). The closure of the orbit G⟨u⟩ in Gr_{k}(W) contains a closed orbit, say O = G⟨w⟩. Since O is projective, the stabilizer of ⟨w⟩ is a parabolic subgroup of G and hence w ∈ W is a decomposable highest weight vector.

2. Let P be the set of all weights of ∧^{k}V . Given μ ∈ P, let (∧^{k}V)^{μ} denote the corresponding weight space. Using Eq (1.1) and the standard properties of weights of representations, one sees that m_{k} = max_{μ∈P} (μ, μ+2ρ). Set P^{max} = {μ ∈ P | (μ, μ+2ρ) = m_{k}} and (∧^{k}V)^{max} = ⊕_{μ∈P^{max}}(∧^{k}V)^{μ}. It follows from the construction that each weight vector in (∧^{k}V)^{max} is a highest weight vector and that these are exactly the highest weight vectors for M_{k}. Since each (∧^{k}V)^{μ} has a basis consisting of decomposable elements, we are done. □

It follows that in order to determine the maximal eigenvalue of C on some exterior power of V , it suffices to test C only on the decomposable elements.

4 On eigenvalues of C_{0} on ∧𝔤_{1} and commutative subalgebras

Throughout this section, Θ is an automorphism of 𝔤 of pure order m and 𝔤 = 𝔤_{0} ⊕ … ⊕ 𝔤_{m−1} is the corresponding ℤ_{m}-grading. The indices are regarded as elements of ℤ_{m}, so that 𝔤_{−i} = 𝔤_{m−i}. Let {e_{i}} be a basis for 𝔤 compatible with ℤ_{m}-grading, i.e., consisting of Θ-eigenvectors. Consider the respective decomposition for the coboundary operator: d = d_{0} + … + d_{m−1}, where

d_{i} = \frac{1}{2} ∑_{j: e_{j}∈𝔤_{i}} ε(e_{j})θ(e′_{j}) . (4.1)

Notice that here e′_{j} ∈ 𝔤_{−i}. Similarly, we have the decomposition C = C_{0} + … + C_{m−1}, where

C_{i} = ∑_{j: e_{j}∈𝔤_{i}} e_{j}e′_{j} . (4.2)

In particular, C_{0} is the Casimir element for 𝔤_{0}, with respect to Φ|_{𝔤_{0}} . It is easily verified that d_{i} and C_{i} do not depend on a choice of basis for 𝔤_{i}. Combining Equations (4.1) and (4.2) with (3.1), we obtain

θ(C_{i}) = 2(d_{i}∂ + ∂d_{i}), i = 0, 1, …, m − 1.

In what follows, we shall write C_{i} in place of θ(C_{i}). Notice that

d_{i}(𝔤_{j}) ⊂ \{ \begin{matrix}𝔤_{i} ⊗ 𝔤_{j−i}, & if i ≠ j/2 \\ ∧^{2}(𝔤_{i}), & if i = j/2 \end{matrix} \} ⊂ ∧^{2}𝔤 . (4.3)

We shall repeatedly use the fact that Φ(𝔤_{i}, 𝔤_{j}) = 0 unless i + j = 0, and consequences of this relation for higher exterior powers of 𝔤. The following assertion is a refinement of Lemma 2 in [Ko65].

4.4 Proposition. Let y, z ∈ 𝔤_{1} be arbitrary. Then

∑_{i: e_{i}∈𝔤_{0}}[e_{i}, y] ∧ [e′_{i}, z] = −d_{1}([y, z]).

Proof. Let L denote the left hand side. Take arbitrary u, v ∈ 𝔤_{−1}. Using the extension of Φ to ∧𝔤, we obtain

Φ^{(2)}(L, u ∧ v) = ∑_{e_{i}∈𝔤_{j}} det ( \begin{matrix}Φ([e_{i}, y], u) & Φ([e′_{i}, z], u) \\ Φ([e_{i}, y], v) & Φ([e′_{i}, z], v) \end{matrix} ) =

= ∑_{e_{i}∈𝔤_{j}}(Φ(e_{i}, [y, u])Φ(e′_{i}, [z, v]) − Φ(e_{i}, [y, v])Φ(e′_{i}, [z, u])) .

Since [y, u], …, [z, v] ∈ 𝔤_{0}, the last sum is equal to

Φ([y, u], [z, v]) − Φ([y, v], [z, u]) =^{Jacobi} Φ([y, z], [u, v]) = Φ([y, z], ∂(u ∧ v)) .

Hence Φ^{(2)}(L, u ∧ v) = Φ([y, z], ∂(u ∧ v)) = Φ^{(2)}(−d([y, z]), u ∧ v). Thus, L is equal to the component of −d([y, z]) lying in ∧^{2}𝔤_{1}. Since [y, z] ∈ 𝔤_{2}, it follows from Eq. (4.3) with j = 2 that the component in question is −d_{1}([y, z]). □

With a little bit more care, one can prove that ∑_{i: e_{i}∈𝔤_{j}}[e_{i}, y] ∧ [e′_{i}, z] = −d_{j+1}([y, z]). But

we do not need this equality. For a future use, we record a by-product of the proof. If x ∈ 𝔤_{2} and u, v ∈ 𝔤_{−1}, then

Φ^{(2)}(d_{1}(x), u ∧ v) = −Φ(x, ∂(u ∧ v)) . (4.5)

4.6 Proposition. Suppose 𝔤_{0} is semisimple whenever m ≥ 3. Then the eigenvalues of C_{0} on ∧^{2}𝔤_{1} are less than or equal to 2/m. The eigenspace belonging to 2/m is spanned by the bi-vectors y ∧ z such that [y, z] = 0.

Proof. Take any y ∧ z ∈ ∧^{2}𝔤_{1}. Recall from Theorems 2.6(2) and 2.8 that C_{0}(y) = y/m for all y ∈ 𝔤_{1}. Using the definition of C_{0} and Prop. 4.4, we obtain:

C_{0}(y∧z) = C_{0}(y)∧z+y∧C_{0}(z)+∑_{i=1}^{s}[e_{i}, y]∧[e′_{i}, z]+∑_{i=1}^{s}[e′_{i}, y]∧[e_{i}, z] = \frac{2}{m}y∧z−2d_{1}([y, z]) .

This already implies that if [y, z] = 0, then C_{0}(y ∧ z) = \frac{2}{m}y ∧ z. Assume now that y ∧ z is an eigenvector of C_{0} and [y, z] ≠ 0. Since [y, z] = ∂(y ∧ z) and ∂ : ∧^{2}𝔤_{1} → 𝔤_{2} is 𝔤_{0}-equivariant, the C_{0}-eigenvalues of y ∧ z and [y, z] are equal. Then Theorem 2.11 implies that the eigenvalue of y ∧ z is less than 2/m. Thus, we can conclude the proof by invoking Theorem 3.4. □

Before proceeding to the general case, we draw a conclusion (cf. [Ko65, Cor. 5.1]). Let A_{2} ⊂ ∧^{2}𝔤_{1} be the linear span of the decomposable bi-vectors with commuting components.

4.7 Proposition. Under the assumptions of Prop. 4.6, we have 1. A_{2} = Ker (∂|_{∧^{2}𝔤_{1}});

2. ∧^{2}𝔤_{1} = d_{1}(𝔤_{2}) ⊕ A_{2}.

Proof. 1. Let u = ∑_{i} y_{i} ∧ z_{i}. Then, as in the above proof, we obtain

C_{0}(u) = \frac{2}{m}u − 2 ∑_{i} d_{1}([y_{i}, z_{i}]) = \frac{2}{m}u − 2d_{1}∂(u) .

Hence if ∂(u) = 0, then u ∈ A_{2} by Prop. 4.6. The opposite containment is obvious.

2. Clearly, the map d_{1} : 𝔤_{2} → ∧^{2}𝔤_{1} is 𝔤_{0}-equivariant. Therefore the eigenvalues of C_{0} on d_{1}(𝔤_{2}) are less than 2/m and hence the sum of d_{1}(𝔤_{2}) and A_{2} is direct. Assume ∧^{2}𝔤_{1} = d_{1}(𝔤_{2}) ⊕ A_{2} ⊕ W for some W. Then, by the first part, ∂(∧^{2}𝔤_{1}) = ∂d_{1}(𝔤_{2}) ⊕ ∂W and dim ∂d_{1}(𝔤_{2}) = dim d_{1}(𝔤_{2}). Using Eq. (4.5), we obtain dim d_{1}(𝔤_{2}) = dim ∂(∧^{2}𝔤_{−1}) = dim ∂(∧^{2}𝔤_{1}). Therefore dim ∂W = 0 and hence W = 0. □

Our aim is to demonstrate an analogue of Prop. 4.6 for ∧^{k}𝔤_{1} with arbitrary k. Unfortunately, I was unable to come up with an “algebraically closed” proof (as for the case k = 2). Therefore it will merely be shown that Kostant’s argument (see [Ko65]) can be adapted to the ℤ_{m}-graded setting. Let 𝔤_{c} be a compact form of 𝔤. Define a ⋆-conjugation in 𝔤, relative to 𝔤_{c}, by the following formula: if x = u+\sqrt{−1} v (u, v ∈ 𝔤_{c}), then x^{⋆} = −u+\sqrt{−1} v. Then \{x, y\} := Φ(x, y^{⋆}) is a positive definite Hermitian inner product on 𝔤. Therefore x ∈ 𝔤_{i} if and only if x^{⋆} ∈ 𝔤_{−i}. Note also that

[x^{⋆}, y^{⋆}] = −[x, y]^{⋆} for all x, y ∈ 𝔤.

4.8 Theorem. Suppose 𝔤_{0} is semisimple whenever m ≥ 3. Let ℓ_{k} be the maximal eigenvalue of C_{0} on ∧^{k}𝔤_{1}. Then ℓ_{k} ≤ k/m. Moreover one has ℓ_{k} = k/m if and only if 𝔤_{1} contains a k-dimensional commutative Lie subalgebra. In such a case, the eigenspace of C_{0} in ∧^{k}𝔤_{1} corresponding to k/m is generated by ∧^{k}𝔞, where 𝔞 runs through all commutative k-dimensional subalgebras of 𝔤_{1}.

Proof. Let u = y_{1} ∧ … ∧ y_{k} be an arbitrary decomposable element in ∧^{k}𝔤_{1}. Set 𝔞 = ⟨y_{1}, …, y_{k}⟩. We may (and shall) assume that y_{1}, …, y_{k} is an orthonormal basis for 𝔞 relative to { , }, i.e., Φ(y_{i}, y_{j}^{⋆}) = δ_{ij}.

A straightforward computation (cf. the proof of Prop. 4.6) shows that

C_{0}(u) = \frac{k}{m}u − 2 ∑_{i<j}(−1)^{i+j−1}d_{1}([y_{i}, y_{j}]) ∧ y_{1} … ŷ_{i} … ŷ_{j} … ∧ y_{k} =: \frac{k}{m}u − 2 ∑_{i<j} v_{ij} .

Put u^{⋆} = y_{1}^{⋆} ∧ … ∧ y_{k}^{⋆}. Then Φ^{(k)}(u, u^{⋆}) = 1 and

Φ^{(k)}(v_{ij}, u^{⋆}) = Φ^{(2)}(d_{1}([y_{i}, y_{j}]), y_{i}^{⋆} ∧ y_{j}^{⋆}) = −Φ([y_{i}, y_{j}], [y_{i}^{⋆}, y_{j}^{⋆}]) = {[y_{i}, y_{j}], [y_{i}, y_{j}]} ≥ 0 .

Hence

Φ^{(k)}(C_{0}(u), u^{⋆}) = \frac{k}{m} − 2 ∑_{i<j}\{[y_{i}, y_{j}], [y_{i}, y_{j}]\} ≤ \frac{k}{m} .

Thus, if u is an eigenvector of C_{0}, then the respective eigenvalue ≤ k/m. Moreover, the eigenvalue is equal to k/m if and only if all [y_{i}, y_{j}] vanish, i.e., 𝔞 is commutative. By Theorem 3.4, the eigenspace belonging to the maximal eigenvalue is generated by decomposable highest weight polyvectors. This completes the proof. □

Remark. Theorem 3.3 is a formal consequence of the last result. Indeed, take 𝔤̃ = 𝔤 ⊕ 𝔤 and Θ ∈ Aut 𝔤̃ such that Θ(x, y) = (y, x). Then 𝔤̃_{0} is the diagonal (≃ 𝔤) and 𝔤̃_{1} ≃ 𝔤 as 𝔤-module. Let C̃ be the Casimir operator for 𝔤̃ and C̃ = C̃_{0} + C̃_{1} the decomposition corresponding to Θ. By Theorem 4.8, the maximal eigenvalue of C̃_{0} on ∧^{k}𝔤̃_{1} ≃ ∧^{k}𝔤 is at most k/2. It remains to observe that C̃_{0} = \frac{1}{2}C and the isomorphism 𝔤̃_{1} ≃ 𝔤 respects commutative subalgebras.

Let A_{k} be the linear span in ∧^{k}𝔤_{1} of the lines ∧^{k}𝔞, where 𝔞 ranges over all commutative k-dimensional Lie subalgebras of 𝔤_{1}. In other words, A_{k} is the eigenspace of C_{0} in ∧^{k}𝔤_{1} belonging to the eigenvalue k/m. Our next goal is to get more information on A := ⊕_{k}A_{k} as 𝔤_{0}-module. To this end, fix a Cartan subalgebra 𝔱_{0} ⊂ 𝔤_{0}. Recall some properties of 𝔱_{0}.

4.9 Lemma. 1. 𝔷_{𝔤}(𝔱_{0}) is a Cartan subalgebra of 𝔤; 2. The nonzero weights of 𝔱_{0} on each 𝔤_{i} are of multiplicity 1.

Proof. 1. By [Ka90, 8.1(b)], 𝔱_{0} contains a regular semisimple element of 𝔤.

2. Let G_{0} be the connected subgroup of G corresponding to 𝔤_{0}. According to [Vi76], the linear group G_{0} → GL(𝔤_{i}) is visible, i.e. each fibre of the quotient map 𝔤_{i} → 𝔤_{i}//G_{0} contains finitely many G_{0}-orbits. We then conclude by [Ka80, 3.4]. □

Without loss, one may assume that 𝔷_{𝔤}(𝔱_{0}) is 𝔱, the fixed Cartan subalgebra of 𝔤. Set 𝔱_{i} := 𝔱 ∩ 𝔤_{i}. Then 𝔱 = 𝔱_{0} ⊕ … ⊕ 𝔱_{m−1} and 𝔱_{i} is exactly the zero weight space in 𝔤_{i}. Let Δ_{i} be the set of nonzero weights of 𝔱_{0} on 𝔤_{i}. This yields the weight decompositions

𝔤_{i} = 𝔱_{i} ⊕ ( ⨁_{α∈Δ_{i}} 𝔤_{i}^{α}) ,

where dim 𝔤_{i}^{α} = 1 (i ∈ ℤ_{m}). The totality of weights ⋃_{i} Δ_{i} lies in (𝔱_{0}^{∗})_{ℚ}, and the last space will be identified with a subspace of 𝔱∗_{ℚ}. Note that Δ_{−i} = −Δ_{i}. There are two different possibilities for {Δ_{i}}:

• Θ is inner. Then 𝔱_{0} = 𝔱 and 𝔱_{i} = 0 for i ≠ 0. Here Δ_{i} ∩ Δ_{j} = ∅ (i ≠ j), and

⨆_{i=0}^{m−1} Δ_{i} = Δ is the whole root system for 𝔤;

• Θ is outer. Then 𝔱_{0} ≠ 𝔱 and it may happen that Δ_{i} ∩ Δ_{j} ≠ ∅. Here \overline{Δ} := ⋃_{i} Δ_{i}

consists of all restricted roots γ̄ (γ ∈ Δ), where γ̄ := γ|_{𝔱_{0}} .

To avoid ambiguities, especially in the second case, we let v_{α}^{(i)} denote a nonzero weight vector for α ∈ Δ_{i}. The following lemma contains two different auxiliary results on periodic gradings.

4.10 Lemma.

1. Let α ∈ Δ_{i}, β ∈ Δ_{j} and (α, β)_{Φ} < 0. Then [v_{α}^{(i)}, v_{β}^{(j)}] ≠ 0 and [[v_{α}^{(i)}, v_{β}^{(j)}], v_{−β}^{(−j)}] ≠ 0.

2. If h ∈ 𝔤_{1} is semisimple and [𝔤_{0}, h] = 0, then h = 0.

Proof. 1. If Θ is inner, this is a standard property of the root decomposition. A general argument goes as follows. Since Φ|_{𝔤_{j}⊕𝔤_{−j}} is non-degenerate, there exist v_{−β}^{(−j)} such that Φ(v_{β}^{(j)}, v_{−β}^{(−j)}) = 1. Then the equality Φ(h, [v_{β}^{(j)}, v_{−β}^{(−j)}]) = β(h)Φ(v_{β}^{(j)}, v_{−β}^{(−j)}) for h ∈ 𝔱_{0} shows that h_{β} := [v_{β}, v_{−β}] is a nonzero element of 𝔱_{0}. It is easily verified that ⟨v_{β}^{(j)}, h_{β}, v_{−β}^{(−j)}⟩ is a three-dimensional simple subalgebra of 𝔤. Since [h_{β}, v_{α}^{(i)}] = (α, β)_{Φ}·v_{α}^{(i)}, we see that v_{α}^{(i)} cannot be a highest weight vector, i.e., [v_{α}^{(i)}, v_{β}^{(j)}] ≠ 0. It also follows from the theory of 𝔰𝔩_{2}-representations that since [v_{α}^{(i)}, v_{β}^{(j)}] ∈ Im (ad v_{β}^{(j)}), it cannot belong to Ker (ad v_{−β}^{(−j)}).

2. It is enough to show that [𝔤_{i}, h] = 0 for all i. We argue by induction on i, the case i = 0 being given by the hypothesis. Assume that [𝔤_{i}, h] = 0. The equality Φ([𝔤_{−i−1}, h], 𝔤_{i}) = Φ(𝔤_{−i−1}, [𝔤_{i}, h]) = 0 shows that [𝔤_{−i−1}, h] = 0. Since the centralizer of h is reductive, one also must have [𝔤_{i+1}, h] = 0, which completes the inductive step. □

Choose a system of positive roots Δ_{0}^{+} ⊂ Δ_{0}, and let 𝔟_{0} denote the corresponding Borel subalgebra of 𝔤_{0}. The following is obvious.

4.11 Lemma. y_{1} ∧ … ∧ y_{k} ∈ ∧^{k}𝔤_{1} is a 𝔟_{0}-eigenvector (i.e., a highest weight vector) if and only if ⟨y_{1}, …, y_{k}⟩ is a 𝔟_{0}-stable subspace of 𝔤_{1}. □

Let 𝔞 ⊂ 𝔤_{1} be a 𝔟_{0}-stable subspace. In particular, 𝔞 is 𝔱_{0}-stable. Therefore

𝔞 = (𝔞 ∩ 𝔱_{1}) ⊕ ( ⨁_{α∈Δ_{𝔞}} 𝔤_{1}^{α})

for some subset Δ_{𝔞} ⊂ Δ_{1}.

4.12 Proposition. Suppose 𝔞 is 𝔟_{0}-stable and [𝔞, 𝔞] = 0. Then 1. 𝔞 contains no semisimple elements. In particular, 𝔞 ∩ 𝔱_{1} = 0;

2. Δ_{𝔞} is contained in an open half-space of (𝔱_{0}^{∗})_{ℚ}.

Proof. 1. Suppose 0 ≠ h ∈ 𝔞 is semisimple. The equality [𝔟_{0}, h] = 0 cannot hold, since we then would also have [𝔤_{0}, h] = 0, which implies that h = 0, see Lemma 4.10(2). So, take e ∈ 𝔟_{0} such that [h, e] ≠ 0. Since 𝔞 is 𝔟_{0}-stable, [e, h] ∈ 𝔞. On the other hand, h is semisimple, and [h, e] ≠ 0 forces that [h, [h, e]] ≠ 0. Thus, 𝔞 is not commutative.

2. From Lemma 4.10(1) it follows that (α, β)_{Φ} ≥ 0 for any pair α, β ∈ Δ_{𝔞}. Letting |Δ_{𝔞}| := ∑_{α∈Δ_{𝔞}} α, we obtain (|Δ_{𝔞}|, α)_{Φ} > 0 for all α in Δ_{𝔞}. Thus, |Δ_{𝔞}| determines a required open-half space. □

4.13 Corollary. Let Θ be an involutory automorphism. Suppose 𝔞 is a commutative Lie subalgebra of 𝔤_{1}. Then

dim 𝔞 ≤ (dim 𝔤_{1} − dim 𝔱_{1})/2.

Proof. For any commutative subalgebra 𝔞 of 𝔤_{1} there exists a 𝔟_{0}-stable commutative subalgebra of the same dimension (consider ∧^{k}𝔞 as element in A_{k}, k = dim 𝔞, and take the subalgebra corresponding to a highest weight vector). Assume therefore that 𝔞 is 𝔟_{0}-stable. In case m = 2, we have Δ_{1} = −Δ_{1}. Since Δ_{𝔞} lies in an open half-space, #Δ_{𝔞} ≤ \frac{1}{2}#Δ_{1}. Because also 𝔞 ∩ 𝔱_{1} = 0, the assertion follows. □

Remarks. 1. Notice that Lemma 4.10(1) and Prop. 4.12(2) are not really needed for the proof of the Corollary. The only required thing is the relation Δ_{𝔞} ∩ (−Δ_{𝔞}) = ∅, which follows from the fact that [v_{β}^{(1)}, v_{−β}^{(1)}] ≠ 0.

2. A weaker inequality dim 𝔞 ≤ dim 𝔤_{1}/2 was proved by J.A. Carlson and D. Toledo [CT89, 3.1].

4.14 Proposition. Let 𝔞, 𝔞′ be 𝔟_{0}-stable commutative subalgebras of 𝔤_{1}. If |Δ_{𝔞}| = |Δ_{𝔞′}|, then Δ_{𝔞} = Δ_{𝔞′} and 𝔞 = 𝔞′.

Proof. Obviously, we also have |Δ_{𝔞} \ Δ_{𝔞′}| = |Δ_{𝔞′} \ Δ_{𝔞}|. Assume Δ_{𝔞} \ Δ_{𝔞′} ≠ ∅. Then |Δ_{𝔞} \ Δ_{𝔞′}| ≠ 0, because Δ_{𝔞} lies in an open half-space. Since (|Δ_{𝔞} \ Δ_{𝔞′}|, |Δ_{𝔞′} \ Δ_{𝔞}|)_{Φ} > 0, one may conclude that there exist α ∈ Δ_{𝔞} \ Δ_{𝔞′} and β ∈ Δ_{𝔞′} \ Δ_{𝔞} such that (α, β)_{Φ} > 0. Applying Lemma 4.10 to (α, −β) shows that γ := α − β ∈ Δ_{0} and 0 ≠ [e_{γ}^{(0)}, v_{β}^{(1)}] ∈ 𝔤_{1}^{α}. Without loss of generality, one may assume that γ ∈ Δ_{0}^{+}. (Otherwise consider (−α, β).) As 𝔞′ is 𝔟_{0}-stable, we must then have α ∈ Δ_{𝔞′}. A contradiction! □

Now, we can summarize our considerations in the following 4.15 Theorem. 1. Let S = {α_{1}, …, α_{k}} be any set of (distinct) weights in Δ_{1} and 𝔞_{S} := ⟨v_{α_{1}}^{(1)}, …, v_{α_{k}}^{(1)}⟩ the corresponding subspace of 𝔤_{1}. Set 2ρ_{0} := |Δ_{0}^{+}|. Then one always has the inequality

(α_{1} + … + α_{k}, α_{1} + … + α_{k} + 2ρ_{0})_{Φ} ≤ k/m , (4.16)

and the equality holds if and only if 𝔞_{S} is a 𝔟_{0}-stable commutative Lie subalgebra of 𝔤_{1}. Moreover, every commutative 𝔟_{0}-stable subalgebra is (uniquely) of this form;

2. A is a multiplicity-free 𝔤_{0}-module;

3. Let S_{1}, …, S_{q} be all k-element subsets of Δ_{1} such that the equality in Eq. (4.16) holds. Then the non-zero elements in ∧^{k}𝔞_{S_{1}}, …, ∧^{k}𝔞_{S_{q}} are the only highest weight vectors in A_{k}.

Proof. 1. The sum μ := α_{1} + … + α_{k} is a weight of 𝔤_{0}-module ∧^{k}𝔤_{1}. Therefore this part is a consequence of Theorem 4.8 and the following property of weights: If μ occurs in an irreducible 𝔤_{0}-module V_{λ} (⊂ ∧^{k} 𝔤_{1}), then (μ, μ + 2ρ_{0})_{Φ} ≤ (λ, λ + 2ρ_{0})_{Φ} and the equality holds if and only if λ = μ. Recall that (λ, λ + 2ρ_{0})_{Φ} is the eigenvalue of C_{0} on V_{λ}.

2. In view of Theorem 4.8 and Lemma 4.11, the mapping 𝔞 ↦ ∧^{dim 𝔞}𝔞 establishes a one-to-one correspondence between the 𝔟_{0}-stable commutative subalgebras of 𝔤_{1} and the 𝔟_{0}-stable lines in A. Since the corresponding highest weight is |Δ_{𝔞}|, the assertion follows from Prop. 4.14.

3. This follows from parts 1,2 and Lemma 4.11. □

5 Spin_{0}(𝔤_{1}) and the maximal eigenvalue of C_{0}

In this section, Θ is an involutory automorphism of 𝔤. In other words, m = 2 and 𝔤 = 𝔤_{0} ⊕𝔤_{1}. We also assume that 𝔤 is simple. Let ℓ = max_{k} ℓ_{k} be the maximal eigenvalue of C_{0} on ∧𝔤_{1}. We know that ℓ_{k} ≤ k/2, but the maximal ℓ_{k} may fail to be equal to k/2. That is, ℓ is not necessarily attained on “commutative subalgebras”. In this section, we give a precise formula for ℓ and show that for the Hermitian involutory automorphisms ℓ does correspond to commutative subalgebras of maximal dimension.

It is a well-known result of Kostant that there exists an isomorphism of 𝔤-modules ∧𝔤 ≃ 2^{dim 𝔱}(V_{ρ} ⊗ V_{ρ}). In [Pa00], we obtained a generalization of this to any ℤ_{2}-grading of 𝔤. Roughly speaking, the theory of Clifford algebras implies that there exists a 𝔤_{0}-module, denoted by Spin_{0}(𝔤_{1}), such that

∧𝔤_{1} ≃ 2^{dim 𝔱_{1}}·Spin_{0}(𝔤_{1})^{⊗2} , (5.1)

and we gave an explicit general description of Spin_{0}(𝔤_{1}). We retain the notation of Section 4 concerning the weights of 𝔱_{0}, adapted to the case m = 2. Recall that Δ_{0} ∪ Δ_{1} = \{γ̄ | γ ∈ Δ\} =: \overline{Δ}. For our present purpose, it is important that Δ_{0}^{+} and Δ^{+} to be compatibly chosen. This is being achieved by putting Δ_{0}^{+} = \overline{Δ^{+}} ∩ Δ_{0}.

Let W (resp. W_{0}) be the Weyl group of (𝔤, 𝔱) (resp. (𝔤_{0}, 𝔱_{0})). Since 𝔱_{0} contains regular elements of 𝔱, W_{0} can be identified with a subgroup of W. It was shown in [Pa00, Sect. 5 & 6] that there exists an intermediate subgroup W_{0} ⊂ W\overline{0} ⊂ W, which is responsible for the description of Spin(𝔤_{1}). Namely, W\overline{0} is a Weyl group in its own right, acting as reflection group on 𝔱_{0}; the root system for W\overline{0} is \overline{Δ}. Next, each coset in W\overline{0}/W_{0} contains a unique element of minimal length, the length function on W\overline{0} being defined by the set of positive roots \overline{Δ^{+}}. Let W^{0} ⊂ W\overline{0} be the set of such “minimal” elements. A more geometric description of it is:

W^{0} = \{w ∈ W\overline{0} | w(Δ_{0}^{+}) ⊂ \overline{Δ^{+}}\} . (5.2)

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

5.3 Theorem [Pa00, 5.2, 6.5] .

(i) For each w ∈ W^{0}, λ_{w} := w^{−1}ρ − ρ_{0} is dominant relative to Δ_{0}^{+}; (ii) Spin_{0}(𝔤_{1}) = ⊕_{w∈W^{0}}V_{λ_{w}}.

Remark. It is worth mentioning two possibilities. If Θ is inner, then W\overline{0} = W and both the construction and the proof become much simpler. At the other extreme, for some outer Θ, we have W_{0} = W\overline{0}. Then W^{0} = \{id\} and Spin_{0}(𝔤_{1}) appears to be irreducible. This occurs precisely if 𝔤_{0} is non-homologous to zero in 𝔤. One obtains here an exact analogue of Kostant’s formula for the adjoint representation. (Note that the adjoint representation, as isotropy representation, corresponds to the involution Θ permuting the summands of 𝔤 ⊕ 𝔤. Here the diagonal is non-homologous to zero in 𝔤 ⊕ 𝔤.) Although Spin_{0}(𝔤_{1}) can be highly reducible 𝔤_{0}-module, the formula for λ_{w} implies that C_{0} acts scalarly on Spin_{0}(𝔤_{1}), with eigenvalue (ρ, ρ)_{Φ} − (ρ_{0}, ρ_{0})_{Φ} (see [Pa00, 7.7]). Combining Theorem 5.3 with Eq. (5.1), we get

5.4 Proposition. The maximal eigenvalue of C_{0} on ∧𝔤_{1} is equal to

ℓ = max_{w∈W^{0}} 4(ρ, ρ − w^{−1}ρ_{0})_{Φ}.

Proof. We use the following easy properties of the function λ ↦^{f} (λ, λ + 2ρ_{0})_{Φ} ={the eigenvalue of C_{0} on V_{λ}}:

• The maximal eigenvalue of C_{0} on V_{λ_{1}} ⊗ V_{λ_{2}} is being attained on the Cartan (highest) component of this product;

• f is a concave function of λ. It follows that the maximal eigenvalue of C_{0} on (V_{λ_{1}} ⊕…⊕V_{λ_{n}})^{⊗2} is equal to max_{i} f(2λ_{i}). Applying this to the collection of highest weights λ_{w} (w ∈ W^{0}), one obtains the required formula. (The equality (w^{−1}ρ, w^{−1}ρ)_{ϕ} = (ρ, ρ)_{Φ} can be used, since w ∈ W\overline{0} ⊂ W.) □

Remarks. 1. By Theorem 4.8, dimension of any commutative Lie subalgebra of 𝔤_{1} is at most [2ℓ]. Examples show that this usually gives a better bound than that in Corollary 4.13.

2. It is tempting to suggest that the maximal eigenvalue always corresponds to w = id. Indeed, this is true if W^{0} = {id} or for all Hermitian involutions, see below. But for 𝔤 = G_{2} this is not the case. It might be interesting to describe the corresponding w for all involutory automorphisms.

5.5 Example. Let 𝔤 = E_{6} and 𝔤_{0} = F_{4}. In this example, the notation α_{i}, φ_{i} refers to simple roots and fundamental weights of F_{4}. For instance, φ_{4} is the highest root and ρ_{0} = φ_{1} + … + φ_{4}. Here 𝔤_{1} = V_{φ_{1}} and W^{0} = {id}. Therefore ℓ = 4(ρ, ρ−ρ_{0})_{Φ}. It is convenient to perform all computations in terms of weights and roots of F_{4}. We have ρ−ρ_{0} = φ_{1}+φ_{2} = 5α_{1}+9α_{2}+ 6α_{3}+3α_{4} and ρ = 2φ_{1}+2φ_{2}+φ_{3}+φ_{4}, whence 4(ρ, ρ−ρ_{0})_{Φ} = 46(α_{4}, α_{4})_{Φ}. It is easily seen that any long root for F_{4} is also a long root for E_{6}. Therefore (α_{4}, α_{4})_{Φ} = 1/h^{∗}(E_{6}) = 1/12 and ℓ = 23/6. This implies that dimension of any commutative subalgebra of 𝔤_{1} is at most [23/3] = 7, whereas Corollary 4.13 provides the bound 12. However, it is not hard to calculate that ℓ_{2} = 1, ℓ_{3} = 3/2, and ℓ_{4} = 23/12 < 2. Thus, maximal commutative subalgebras are of dimension 3.

In the remainder of the section, we assume that Θ is a Hermitian involution, i.e., 𝔤_{0} is not semisimple. Then Θ is necessarily inner, the centre of 𝔤_{0} is 1-dimensional, and 𝔤_{1} is the sum of two irreducible dual 𝔤_{0}-modules. Since now 𝔱 = 𝔱_{0}, we have Δ = \overline{Δ} and Δ = Δ_{0} ⊔ Δ_{1}. Given α ∈ Π and μ in the root lattice, let ht_{α}μ denote the coefficient of α in the expression of μ through the simple roots. There is a one-to-one correspondence between the conjugacy classes of Hermitian involutions and the simple roots α such that ht_{α}Λ = 1. If α = α_{Θ} ∈ Π corresponds to a Hermitian Θ, then the sets Δ_{0}, Δ_{1} are the following. We have ht_{α}(Δ) = {−1, 0, 1}. Put Δ(i) = (ht_{α})^{−1}(i). Then Δ_{0} = Δ(0) and Δ_{1} = Δ(1) ∪ Δ(−1). The compatibility of Δ_{0}^{+} and Δ^{+} required for Theorem 5.3 means that Δ^{+} = Δ_{0}^{+} ∪ Δ(1). Note that α, Λ ∈ Δ(1). Let φ be the fundamental weight corresponding to α. Set 2ρ_{1} = |Δ(1)|, so that ρ = ρ_{0} + ρ_{1}.

5.6 Lemma. 1. 2ρ_{1} = h^{∗}(𝔤)φ; 2. ht_{α}2ρ = #Δ(1).

Proof. 1. This is easily checked case-by-case. However a conceptual proof is available. Since |Δ(1)| is the sum of all weights of a 𝔤_{0}-module, it must be orthogonal to all simple roots of 𝔤_{0}, t.e., to any β ∈ Π \ {α}. Hence 2ρ_{1} is proportional to φ, say 2ρ_{1} = xφ. Then x = (2ρ_{1}, α^{∨})_{Φ}. It is easily seen that the longest element in W_{0} takes ρ_{1} to itself and α to Λ. Therefore

x = (2ρ_{1}, Λ^{∨})_{Φ} = ∑_{γ∈Δ(1)}(γ, Λ^{∨})_{Φ} = #{γ ∈ Δ(1) | (γ, Λ) ≠ 0, γ ≠ Λ} + 2 .

Obviously, the set R of all positive roots γ such that (γ, Λ) ≠ 0 and γ ≠ Λ breaks into pairs {γ_{1}^{(i)}, γ_{2}^{(i)}}_{i∈I}, where γ_{1}^{(i)} + γ_{2}^{(i)} = Λ. (This is a manifestation of the fact that the nilpotent radical of the centralizer of a highest root vector is a Heisenberg Lie algebra.)

Since ht_{α}(γ_{1}^{(i)} + γ_{2}^{(i)}) = 1, exactly one element in each pair belongs to Δ(1). Hence x = #I +2 = (#R)/2+2. Remembering Lemma 1.3, we see that x = (dim O_{min} −2)/2+ 2 = h^{∗}(𝔤).

2. This is obvious, since ht_{α}μ = 1 if and only if μ ∈ Δ(1). □

5.7 Theorem.

1. If Θ is Hermitian, then the maximal eigenvalue of C_{0} on ∧𝔤_{1} is attained with w = id ∈ W^{0} (in the formula of Prop. 5.4).

2. This maximal eigenvalue is equal to dim 𝔤_{1}/4.

Proof. 1. The difference between the value corresponding to w = id and the arbitrary w is equal to 4(w^{−1}ρ−ρ, ρ_{0})_{Φ} = 2 ∑_{γ∈Δ_{0}^{+}}(w^{−1}ρ−ρ, γ)_{Φ}. Let us show that every summand on the right hand side is non-negative. It is enough to check this for the set Π_{0} of all simple roots in Δ_{0}^{+}. But the Hermitian case can be characterized by the property that Π_{0} ⊂ Π. Indeed, Π_{0} = Π \ {α}. Thus, if γ ∈ Π_{0} then

(w^{−1}ρ − ρ, γ^{∨})_{Φ} = (ρ, (wγ)^{∨})_{Φ} − (ρ, γ^{∨})_{Φ} = (ρ, (wγ)^{∨})_{Φ} − 1 .

By Eq. (5.2), we have wγ is positive. Whence (ρ, (wγ)^{∨})_{Φ} ≥ 1.

2. By part 1, the maximal eigenvalue is equal to 4(ρ, ρ) − 4(ρ, ρ_{0}) = (2ρ, 2ρ_{1}). By Lemma 5.6, the latter is equal to

#Δ(1)·h^{∗}(𝔤)·(α, φ)_{Φ} = \frac{dim 𝔤_{1}}{2} ·h^{∗}(𝔤)·\frac{(α, α)_{Φ}}{2} .

Since α is long, we conclude by the definition of the dual Coxeter number. □

Remarks. 1. Since 𝔤(1) = ⊕_{α∈Δ(1)}𝔤^{α} is a commutative Lie subalgebra and dim 𝔤(1) = dim 𝔤_{1}/2 =: a, we have 𝔤(1) is maximal commutative by Corollary 4.13. This subalgebra generates a submodule of A_{a} ⊂ ∧^{a}𝔤_{1}, where, by Theorem 4.8, C_{0} takes value dim 𝔤_{1}/4. So, the maximal eigenvalue of C_{0} is attained on commutative subalgebras. Example 5.5 shows that this is not a general phenomenon, see also a discussion for the adjoint representation in [Ko98, 1.1].

2. Because ∧^{i}𝔤_{1} ≃ ∧^{2a−i}𝔤_{1} and 𝔤_{1} contains a-dimensional commutative subalgebras, the second claim of Theorem 5.7 follows from Theorem 4.8 as well.

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[Su98] Suter, R.: Coxeter and dual Coxeter numbers, Comm. Alg. 26(1998), 147–153.

[Vi76] Vinberg, E.B.: The Weyl group of a graded Lie algebra, Izv. AN SSSR. Ser. Matem. 40(1976), � 3, 488–526 (Russian). English translation: Math. USSR-Izv. 10(1976), 463–495.

[VO88] Vinberg, E.B.; Onishchik, A.L.: “Seminar on Lie groups and algebraic groups”, Moskva: “Nauka” 1988 (Russian). English translation: Onishchik, A.L.; Vinberg, E.B.: Lie groups and algebraic groups. Berlin Heidelberg New York: Springer 1990.

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