REPRESENTATION THEORY An Electronic Journal of the American Mathematical Society Volume 00, 1997

FUNCTIONS ON THE MODEL ORBIT IN E_{8}

JEFFREY ADAMS, JING-SONG HUANG, AND DAVID A. VOGAN, JR.

1. Introduction

Perhaps the most intriguing and difficult problem in the representation theory of reductive Lie groups is this: how can one attach unitary representations to nilpotent coadjoint orbits? (More extensive discussions of this problem may be found in [Vunip] and [Vorange].) One of the difficulties about the problem is knowing when you have solved it; that is, knowing when a particular unitary representation may reasonably be regarded as “attached” to a particular orbit. In this direction some help is available from the general philosophy of the orbit method. We will not try to describe that philosophy in any generality here, concentrating instead on a very special part of it. What we will see is that the orbit method can sometimes predict how the representation “attached” to an orbit should restrict to a maximal compact subgroup. This prediction can be used in connection with the difficulty above: given an orbit and a unitary representation, one can try to compare the restriction of the representation to a maximal compact subgroup with the prediction of the orbit method. Agreement is evidence that the representation might be attached to the orbit.

If we try to put this idea into practice, there are two computational difficulties. First, there is the problem of computing the restriction to a maximal compact subgroup of a representation. This problem is solved in principle by the Blattner Conjecture and the Kazhdan-Lusztig Conjecture (both proved); the calculation may be unpleasant but any particular case can be done. Second, we need to compute explicitly the prediction provided by the orbit method. This (as we shall explain) is a problem in the representation theory of algebraic groups. It has not been solved in general. Even in the cases when it has been solved, the solution is difficult to compute with.

This brings us at last to the subject of the paper. We look first at a complex algebraic group G of type E_{8}. There is a unique nilpotent coadjoint orbit X_{ℝ} for G of (real) dimension 256. (Because G is complex, every coadjoint orbit is complex; so X_{ℝ} is actually a complex manifold of dimension 128. Some of what we say applies equally well to real reductive groups, and we are overlooking the complex structure here in order to maintain that compatibility.) We are concerned with the problem of attaching a unitary representation π(X_{ℝ}) to this orbit. A natural candidate for π(X_{ℝ}) has been proposed (implicitly) by McGovern in [McGmodel], Theorem 2.1: it is the irreducible spherical representation of G with parameter one fourth the

\begin{matrix}c \\ ◯ \end{matrix}1997 American Mathematical Society

1991 Mathematics Subject Classification. Primary 20G15, 22E46.

The first author was supported in part by NSF grant DMS-94-01074. The second author was partially supported by RGC-CERG grant number HKUST588/94P and HKUST713/96P. The third author was supported in part by NSF grant DMS-94-02994.

sum of the positive roots. (We do not know whether π(X_{ℝ}) is unitary.) McGovern calculated the restriction of π(X_{ℝ}) to a maximal compact subgroup K of G. (The group K is the compact form of E_{8}.) It is the sum of all irreducible representations of K, each occurring with multiplicity one.

In order to test whether π(X_{ℝ}) is really a good candidate for attachment to X_{ℝ}, one must also compute the orbit method’s predicted K-multiplicities. This McGovern was unable to do. (He carried out analogous calculations for almost all other complex simple groups.) To explain the problem, let us write K_{ℂ} for the complexification of the compact group K; this is an algebraic group. (Of course in our case it is isomorphic to G, but we maintain the notational distinction to allow us to discuss a more general situation.) Recall that algebraic representations of K_{ℂ} may be identified with locally finite representations of K. Attached to X_{ℝ} there is an algebraic homogeneous space X_{θ} for K_{ℂ}, whose complex dimension is half the real dimension of X_{ℝ}. (In our case X_{θ} may be identified with X_{ℝ}, regarded now as an algebraic homogeneous space for G.) As an algebraic homogeneous space, X_{θ} has an algebra of regular functions R(X_{θ}), which carries an algebraic representation of K_{ℂ}. The prediction of the orbit method is that this algebraic representation of K_{ℂ}, when regarded as a locally finite representation of K, is equivalent to the Kfinite part of π(X_{ℝ}). That is, an irreducible representation τ of K should have the same multiplicity in π(X_{ℝ}) as in the ring of regular functions on X_{θ}. (We will formulate a more general version of this conjecture in section 2.) Our main result is that this prediction agrees with McGovern’s calculation of the K-multiplicities in π(X_{ℝ}). Here is a statement.

Theorem 1.1. Suppose G is a complex algebraic group of type E_{8} and X is the nilpotent orbit of complex dimension 128. Then the algebra R(X) of regular functions on X contains every irreducible (algebraic) representation of G with multiplicity one.

We will complete the proof in section 6.

The statement of the theorem explains McGovern’s terminology “model orbit” for X. According to the terminology introduced in [BGG], a model representation of G is one containing every irreducible representation of G exactly once. (The term is deliberately vague about the category of representations in question.) Theorem 1.1 says that R(X) is a model (algebraic) representation of G.

The problem of calculating multiplicities in algebras of regular functions on nilpotent orbits has a long history. Perhaps the best results now available are those of McGovern in [McGmult]; here one can find an explicit description of the multiplicities as an alternating sum of certain partition functions. To pass from such a formula to Theorem 1.1 is a matter of finitely many calculations: too many to do by hand (which is why McGovern did not prove Theorem 1.1 in [McGmodel]) but probably not too many for a computer. Our contribution is a series of artful dodges making possible a proof of Theorem 1.1 by hand.

FUNCTIONS ON THE MODEL ORBIT IN E_{8} 3

Suppose next that G_{ℝ} is a connected and simply connected split real form of G. It turns out that there is a unique real form Y_{ℝ} of the orbit X; this is a coadjoint orbit for G_{ℝ} of real dimension 128. As we will explain in section 8 (Conjecture 8.14), there is a natural candidate for a unitary representation π(Y_{ℝ}) attached to this orbit. This representation is not nearly so well understood as McGovern’s π(X_{ℝ}), however. It is not known to be unitary, and we do not know its restriction to a maximal compact subgroup K_{ℝ} = Spin(16) of G_{ℝ}. Nevertheless, it makes sense to try to compute the prediction of the orbit method for this restriction, and this we do in section 8 (Theorem 8.10). Here we will describe the result only qualitatively. An obvious guess is that π(Y_{ℝ}) should be a model representation for K_{ℝ}: the sum of all the irreducible representations, each with multiplicity one. In fact it is much smaller than this. The irreducible representations of Spin(16) are parametrized by the cone of D_{8}-dominant weights inside an eight-dimensional lattice. The orbit method predicts roughly that only those representations whose highest weights are dominant for E_{8} should actually appear. (The predicted multiplicity is one.)

2. The multiplicity conjecture

In this section we will recall from [Vunip] some conjectures and theorems about K-multiplicities in representations attached to nilpotent orbits. In order to understand what we have proved for complex groups, the discussion before Theorem 1.1 is sufficient; but for the real E_{8}, and in order to understand what generalizations should be pursued, a larger context is useful. We begin with a real reductive Lie group G in Harish-Chandra’s class ([HC], section 3). We write 𝔤_{0} for the real Lie algebra of G and 𝔤 for its complexification; analogous notation will be used for other groups. Choose a Cartan involution θ of G, so that the group K of fixed points is a maximal compact subgroup of G. Define

𝔭_{0} = −1 eigenspace of θ on 𝔤_{0}.  (2.1)(a)

We will make use of

K_{ℂ} = complexification of K;  (2.1)(b)

this is a complex algebraic group.

The “coadjoint orbits” of the orbit method consist of linear functionals on Lie algebras. We are interested first of all in

i𝔤_{0}^{∗} = imaginary-valued ℝ-linear functionals on 𝔤_{0}.  (2.1)(c)

A coadjoint orbit is by definition an orbit of G on i𝔤_{0}^{∗}. An orbit is called nilpotent (or ℝ-nilpotent) if its closure is a cone. Formally,

N_{ℝ}^{∗} = {λ ∈ i𝔤_{0}^{∗} | for all t > 0, tλ ∈ G · λ}.  (2.1)(d)

We will also be concerned with the action of K_{ℂ} on (𝔤/𝔨)^{∗}. The corresponding nilpotent cone is

N_{θ}^{∗} = {λ ∈ (𝔤/𝔨)^{∗} | for all t ∈ ℂ^{×}, tλ ∈ K_{ℂ} · λ}.  (2.1)(e)

Elements of N_{θ}^{∗} may be called θ-nilpotent to distinguish them from elements of N_{ℝ}^{∗}.

We will be concerned first of all with the orbits of G on N_{ℝ}^{∗}. Now the orbit method seeks to attach unitary representations not to arbitrary coadjoint orbits, but only to those satisfying a certain additional condition. In the original work of Kirillov and Kostant, this additional condition, called integrality, was formulated in a straightforward way. All nilpotent coadjoint orbits are integral. But Duflo (see [Duflo]) later found that a treatment of more general groups was possible only if integrality was replaced by the more subtle requirement of admissibility. Not every nilpotent coadjoint orbit is admissible, so we need to understand this condition.

Definition 2.2 (see [Duflo] or [Vunip], Definition 7.2). Suppose G is a real Lie group, and λ ∈ i𝔤_{0}^{∗}. Write G(λ) for the isotropy group of the coadjoint action at λ, so that the coadjoint orbit X = G·λ may be identified with G/G(λ). Recall that the tangent space 𝔤_{0}/𝔤(λ)_{0} to X at λ carries a natural G(λ)-invariant imaginary-valued non-degenerate symplectic form ω_{λ}, defined by

ω_{λ}(A + 𝔤(λ)_{0}, B + 𝔤(λ)_{0}) = λ([A, B]) (A, B ∈ 𝔤_{0}).

The isotropy action therefore gives a natural group homomorphism

j(λ): G(λ) → Sp(ω_{λ}).

The symplectic group has a natural two-fold covering, the metaplectic group:

1 → {1, ϵ} → Mp(ω_{λ}) →^{p} Sp(ω_{λ}) → 1.

This covering may be pulled back via the homomorphism j(λ) to define the metaplectic double cover:

1 → {1, ϵ} → G̃(λ) →^{p(λ)} G(λ) → 1.

Explicitly,

G̃(λ) = {(g, m) ∈ G(λ) × Mp(ω_{λ}) | j(λ)(g) = p(m)}.

A representation χ of G̃(λ) is called genuine if χ(ϵ) = −I. It is called admissible if it is genuine, and in addition the differential of χ is a multiple of λ. Explicitly,

χ(exp A) = exp(λ(A)) · I (A ∈ 𝔤(λ)_{0}).

An admissible orbit datum is a pair (λ, χ) with λ ∈ i𝔤_{0}^{∗} and χ an irreducible unitary admissible representation of G̃(λ). The element λ (or the coadjoint orbit G · λ) is called admissible if admissible orbit data (λ, χ) exist. When we need to distinguish these definitions from parallel ones involving N_{θ}^{∗}, we will say ℝ-admissible.

The notion of admissibility is a bit involved, but fortunately it simplifies for nilpotent orbits.

Lemma 2.3 ([Vunip], Theorem 5.7 and Observation 7.4). Suppose G is a real reductive Lie group. Then an element λ ∈ i𝔤_{0}^{∗} is nilpotent (cf. (2.1)(d)) if and only if the restriction of λ to 𝔤(λ) is zero.

Suppose henceforth that λ is nilpotent. A representation χ of G̃(λ) is admissible if and only if χ(ϵ) = −I, and χ is trivial on the identity component of G̃(λ). Consequently λ is admissible if and only if the preimage under the metaplectic covering map p(λ) of the identity component G(λ)_{0} is disconnected.

In the case of complex groups, matters are even simpler.

Lemma 2.4. Suppose G is a complex Lie group and λ ∈ i𝔤_{0}^{∗}. Then the metaplectic double cover of G(λ) is trivial:

G̃(λ) ≃ {1, ϵ} × G(λ).

Suppose in addition that G is reductive and λ is nilpotent. Then λ is admissible, and the admissible representations of G̃(λ) are in one-to-one correspondence with irreducible representations of the group G(λ)/G(λ)_{0} of connected components of G(λ).

Here is a weak formulation of the orbit philosophy for nilpotent orbits.

Conjecture 2.5. Suppose G is a real reductive Lie group, and (λ, χ) is a nilpotent admissible orbit datum (cf. (2.1), Definition 2.2, and Lemma 2.3). Assume in addition that the boundary of the orbit closure \overline{G · λ} has codimension at least four. (This means that if G · λ′ is contained in the closure of G · λ, then dim G · λ′ ≤ dim G·λ−4.) Then there is attached to λ and χ an irreducible unitary representation π(λ, χ) of G.

Some motivation for the codimension condition may be found in [Vunip], Theorem 4.6. When the condition fails, there may still be unitary representations attached to the orbit, but it is not so clear how to parametrize them.

As mentioned in the introduction, one of the difficulties with Conjecture 2.5 is the interpretation of the word “attached.” Our main goal in this section is to formulate the multiplicity conjecture describing π(λ, χ) restricted to a maximal compact subgroup (in terms of λ and χ).

A second difficulty with Conjecture 2.5 is that it is false. There is a counterexample for the double cover of SL(3, ℝ), found by Torasso in [Tor], and considered further in [Vunip], Example 12.4. Because our concern here is essentially with evaluating candidates for π(λ, χ), we will not worry about the possibility that no candidates exist.

In order to formulate the multiplicity conjecture, we need first to recall the Sekiguchi correspondence. Here is some representation-theoretic motivation for it. We want to give a conjecture for the restriction to K of π(λ, χ). According to the general philosophy of the orbit method, this restriction should be obtained by applying geometric quantization to the action of K on the orbit X_{ℝ} = G · λ (with χ defining something like a bundle on X_{ℝ}). Geometric quantization requires a K-invariant polarization of the symplectic structure. In the present case this polarization will be a K-invariant complex structure on X_{ℝ}. The representation of K will then be on a space of holomorphic sections of the bundle related to χ.

It is convenient to begin with an analogue of Definition 2.2.

Definition 2.6 (see [Vunip], Definition 7.13). In the setting of (2.1), suppose λ ∈ N_{θ}^{∗}. Write K_{ℂ}(λ) for the isotropy group of the coadjoint action at λ, so that the orbit X = K_{ℂ} · λ may be identified with K_{ℂ}/K_{ℂ}(λ). Define γ(λ) to be the character by which K_{ℂ}(λ) acts on top degree differential forms at λ:

γ(λ): K_{ℂ}(λ) → ℂ^{×}, γ(λ)(k) = det(Ad(k)|_{(𝔨/𝔨(λ))^{∗}}.

A representation χ of K_{ℂ}(λ) is called admissible if the differential of χ is a multiple of \frac{1}{2}dγ(λ). Explicitly,

χ(exp A) = γ(λ)(exp(A/2)) · I (A ∈ 𝔨(λ)_{0}).

An admissible orbit datum is a pair (λ, χ) with λ ∈ N_{θ}^{∗} and χ an irreducible admissible representation of K_{ℂ}(λ). If (χ, V_{χ}) is an irreducible admissible representation, then we can define a K_{ℂ}-equivariant algebraic vector bundle

V_{χ} = K_{ℂ} ×_{K_{ℂ}(λ)} V_{χ}

over X ≃ K_{ℂ}/K_{ℂ}(λ). The element λ (or the orbit K_{ℂ} · λ) is called admissible if admissible orbit data (λ, χ) exist. When we need to distinguish these notions from those of Definition 2.2, we will say θ-admissible.

Theorem 2.7 ([Seki], [Vergne]). Suppose G is a real reductive Lie group in Harish-Chandra’s class; use the notation of (2.1). Then there is a natural bijection from G orbits on N_{ℝ}^{∗} to K_{ℂ} orbits on N_{θ}^{∗}. Suppose this bijection sends the G orbit X_{ℝ} = G · λ_{ℝ} to the K_{ℂ} orbit X_{θ} = K_{ℂ} · λ_{θ}. Then

1. There is a K-equivariant diffeomorphism X_{ℝ} ≃ X_{θ}.

2. For some λ_{ℝ} ∈ X_{ℝ}, the subgroup K(λ_{ℝ}) = G(λ_{ℝ}) ∩ K is a maximal compact subgroup of G(λ_{ℝ}).

3. For some λ_{θ} ∈ X_{θ}, the subgroup K(λ_{θ}) = K_{ℂ}(λ_{θ}) ∩ G is a maximal compact subgroup of K_{ℂ}(λ_{θ}).

4. For λ_{ℝ} and λ_{θ} as in 2 and 3 above, there is an isomorphism K(λ_{ℝ}) ≃ K(λ_{θ}), uniquely defined up to inner automorphisms.

Notice that Theorem 2.7.1, which is due to Vergne, allows us to endow X_{ℝ} with a K-invariant complex structure. According to the discussion after Conjecture 2.5, the restriction to K of π(λ_{ℝ}, χ_{ℝ}) ought to be something like a space of holomorphic functions on X_{θ}. More precisely, it should be a space of sections of a bundle related to χ_{ℝ}. The following result of James Schwartz provides a construction of the bundle. An equivariant bundle on a homogeneous space is the same thing as a representation of the isotropy group, and that is what the theorem provides.

Theorem 2.8 ([Schwartz], [Vunip], Theorem 7.14). In the setting of Theorem 2.7, choose λ_{ℝ} and λ_{θ} as in parts 2 and 3. Then there is a natural bijection from admissible representations χ_{ℝ} of G̃(λ_{ℝ}) (Definition 2.2) to admissible representations χ_{θ} of K_{ℂ}(λ_{θ}) (Definition 2.6).

We can now formulate the multiplicity conjecture.

Conjecture 2.9. Suppose G is a real reductive Lie group, and (λ_{ℝ}, χ_{ℝ}) is a nilpotent ℝ-admissible orbit datum (cf. (2.1), Definition 2.2, and Lemma 2.3). Assume in addition that the boundary of the orbit closure \overline{G · λ_{ℝ}} has codimension at least four. Let π(λ_{ℝ}, χ_{ℝ}) be the unitary representation of G conjecturally attached to λ_{ℝ} and χ_{ℝ}. Now let (λ_{θ}, χ_{θ}) be the corresponding θ-admissible data (Theorems 2.7 and 2.8), and V_{χ_{θ}} the corresponding vector bundle over the K_{ℂ} orbit X_{θ} (Definition 2.6). Then the space of K-finite vectors of π(λ_{ℝ}, χ_{ℝ}) to K is isomorphic to the space of algebraic sections of V_{χ_{θ}}.

This conjecture may also be regarded as part of Conjecture 2.5, saying something about what is required of the conjectural unitary representations π(λ_{ℝ}, χ_{ℝ}). In fact it is precisely this requirement that is used in [Vunip] as evidence that Conjecture 2.5 is false: for a certain admissible orbit datum (λ, χ) for the double cover of SL(3, ℝ), no unitary representation satisfies the requirement of Conjecture 2.9.

We conclude this section by outlining the simplifications in Conjecture 2.9 when G is a complex reductive group. We use the notation of (2.1). The assumption that G is complex means that there is an Ad(G)-equivariant map (multiplication by i)

J: 𝔤_{0} → 𝔤_{0},  J^{2} = −I,  Jθ = −θJ.  (2.10)(a)

From these facts it follows that the action of J defines an Ad(K)-equivariant isomorphism

J: 𝔨_{0} → 𝔭_{0}.  (2.10)(b)

Now 𝔤_{0} is the sum of 𝔨_{0} and 𝔭_{0}, and 𝔭 (the complexification of 𝔭_{0} is the sum of 𝔭_{0} and i𝔭_{0}. We can therefore construct an Ad(K)-equivariant isomorphism

ψ: 𝔤_{0} → 𝔭,  ψ(Z) = \frac{1}{2}(Z − θZ) − \frac{i}{2}(JZ − θJZ).  (2.10)(c)

The map ψ identifies 𝔭_{0} ⊂ 𝔤_{0} with the real part of 𝔭, and sends 𝔨_{0} ⊂ 𝔤_{0} isomorphically onto the imaginary part of 𝔭. One calculates immediately that

ψ(JZ) = iψ(Z);  (2.10)(d)

that is, that ψ is complex-linear. By dualizing and so on one can easily construct from ψ another Ad(K)-equivariant isomorphism

ψ: i𝔤_{0}^{∗} → (𝔤/𝔨)^{∗},  ψ(JZ) = iψ(Z).  (2.10)(e)

Now the group G is naturally isomorphic to the complexification K_{ℂ} of K. Because the action of K on (𝔤/𝔨)^{∗} respects the complex structure, it extends naturally and uniquely to a holomorphic action of K_{ℂ}; we already used this in (2.1). The action of G on i𝔤_{0}^{∗} is holomorphic for the complex structure defined by J. It therefore follows from (2.1)(e) that the map ψ carries the action of G to the action of K_{ℂ}, using the isomorphism of G with K_{ℂ}. In particular,

ψ: N_{ℝ}^{∗} →^{≃} N_{θ}^{∗},  (2.10)(f)

a K-equivariant diffeomorphism carrying G orbits to K_{ℂ} orbits. By calculation in SL(2, ℂ), one sees that this diffeomorphism implements the Sekiguchi bijection of Theorem 2.7. We saw in Lemma 2.4 that the metaplectic covers were all trivial in this case, so that ℝ-admissible orbit data were identified with G-equivariant local systems on orbits. Similarly, the isomorphism (2.10)(f) guarantees that the K_{ℂ} orbits are all symplectic, so the characters γ(λ) of Definition 2.6 are all trivial, and θ-admissible orbit data are identified with K_{ℂ}-equivariant local systems on orbits. The identification of Theorem 2.8 is the obvious one given by (2.10)(f). Finally, since all the orbits have even complex dimension, the codimension conditions in Conjectures 2.5 and 2.9 are automatic. Here is how the conjectures finally look for complex groups.

Conjecture 2.11. Suppose G is a complex reductive Lie group, and X = G · λ ≃ G/G(λ) is a nilpotent coadjoint orbit. Suppose we are given an irreducible Gequivariant local system on X; equivalently, an irreducible representation χ, V_{χ} of the finite group G(λ)/G(λ)_{0}, or an indecomposable G-equivariant holomorphic vector V_{χ} on X with a flat connection. Then there is attached to χ an irreducible unitary representation π(λ, χ) of G. The space of K-finite vectors of π(λ, χ) is isomorphic to the space of algebraic sections of the bundle V_{χ}.

We are not aware of counterexamples to Conjecture 2.11.

3. McGovern’s method for computing R(X).

In this section we will recall from [McGmult] McGovern’s method for calculating rings of functions on nilpotent orbits. As an excuse for repeating these known results, we will formulate the ideas in the more general context of Conjecture 2.9, even though the critical vanishing theorems of McGovern have not been proven there. McGovern’s method appears to depend on working not with nilpotent elements of 𝔤^{∗}, but rather with nilpotent elements of 𝔤 itself. An invariant bilinear form on 𝔤 identifies these two pictures, so for calculations the distinction is not significant; but philosophically it is a serious weakness.

At any rate, we begin with the nilpotent cone

N_{θ} = {e ∈ 𝔭 | ad(e) is nilpotent }.  (3.1)(a)

(Recall from (2.1) that 𝔭_{0} is the −1 eigenspace of θ on 𝔤_{0}.) Elements of N_{θ} may be called θ-nilpotent. We fix an Ad(G)-invariant, θ-invariant bilinear form ⟨, ⟩ on 𝔤_{0}, positive definite on 𝔭_{0} and negative definite on 𝔨_{0}. This form provides an isomorphism 𝔤 ≃ 𝔤^{∗} that carries N_{θ} K_{ℂ}-equivariantly onto N_{θ}^{∗} (see for example [Vunip], Corollary 5.11). We could of course define N_{ℝ} in a parallel way, and find a G-equivariant isomorphism N_{ℝ} ≃ N_{ℝ}^{∗}; but we will have no particular need for this.

We fix now a θ-nilpotent element

e ∈ N_{θ} ⊂ 𝔭.  (3.1)(b)

According to a version of the Jacobson-Morozov theorem due to Kostant and Rallis, we can find another θ-nilpotent element f ∈ 𝔭 and a semisimple element h ∈ 𝔨 so that

[h, e] = 2e,  [h, f] = −2f,  [e, f] = h  (3.1)(c)

([KR], page 767). This means that there is a Lie algebra homomorphism

ϕ: 𝔰𝔩(2, ℂ) → 𝔤,  ϕ ( \begin{matrix}a & b \\ c & −a \end{matrix} ) = ah + be + cf.  (3.1)(d)

(After replacing e, h, and f by conjugates under Ad(K_{ℂ}), it is possible to arrange

\overline{h} = −h,  \overline{e} = f,  \overline{f} = e.

Here bar denotes the complex conjugation for the real form 𝔤_{0}. In this case the map ϕ carries the real form 𝔰𝔲(1, 1) into 𝔤_{0}. This idea is at the heart of the proof of Theorem 2.7, but we will have no explicit need of it.) The operator ad(h) has integer eigenvalues (by the representation theory of 𝔰𝔩(2)). Accordingly we may define

𝔤(m) = {z ∈ 𝔤 | [h, z] = mz}, 𝔨(m) = 𝔤(m) ∩ 𝔨, 𝔭(m) = 𝔤(m) ∩ 𝔭. (3.1)(e)

These spaces define gradings of 𝔤, 𝔨, and 𝔭. We are particularly interested in the parabolic subalgebra 𝔮 = 𝔩 + 𝔲 ⊂ 𝔨 defined by

𝔮 = ∑_{m≥0} 𝔨(m),  𝔲 = ∑_{m>0} 𝔨(m),  𝔩 = 𝔨(0).

Write Q = L_{ℂ}U for the corresponding parabolic subgroup of K_{ℂ}. Here we define

L_{ℂ} = {k ∈ K_{ℂ} | Ad(k)h = h},  (3.1)(f)

and let U be the connected subgroup with Lie algebra 𝔲. (When h satisfies the condition \overline{h} = −h, then L_{ℂ} is actually the complexification of the compact group

L = {k ∈ K | Ad(k)h = h},

but in general the subscript ℂ has no particular meaning.) Finally, we define

𝔳 = ∑_{m≥2} 𝔭(m).  (3.1)(g)

McGovern’s idea is to study the action of K_{ℂ} on K_{ℂ} · e by first studying Q · e, then “inducing” from Q to K_{ℂ}. Here are the structural results necessary for this program.

Proposition 3.2. Suppose we are in the setting (3.1); write K_{ℂ}(e) for the stabilizer of e in the adjoint action.

1. The subgroup L_{ℂ}(e) is equal to the stabilizer in K_{ℂ} of the image of ϕ. It is therefore a reductive subgroup of K_{ℂ}.

2. The subgroup U(e) is a connected unipotent algebraic group.

3. There is a Levi decomposition K_{ℂ}(e) = L_{ℂ}(e)U(e). In particular, this isotropy group is contained in Q, and

K_{ℂ}(e)/K_{ℂ}(e)_{0} ≃ L_{ℂ}(e)/L_{ℂ}(e)_{0}.

4. The orbit Q · e is a dense open subvariety of 𝔳.

Proof. Parts 1–3 are essentially proved in [BV], Proposition 2.4. (What appears there is the complex analogue, but the present result follows easily.) Part 4 is at least implicit in [KR]; here is the argument. Clearly 𝔳 is preserved by Q, and e ∈ 𝔳 by (3.1)(e) and (3.1)(g). The claim of part 4 is therefore equivalent to ad(𝔮)(e) = 𝔳. This in turn may be formulated as

ad(e): 𝔨(m) → 𝔭(m + 2) is surjective for m ≥ 0.

This last statement follows from the representation theory of 𝔰𝔩(2). □

Here is McGovern’s construction. Because Q acts algebraically on the smooth variety 𝔳, we can form the fiber product

Z = K_{ℂ} ×_{Q} 𝔳 = (K_{ℂ} × 𝔳)/ ∼  (3.3)(a)

Here ∼ is the equivalence relation

(x, z) ∼ (x′, z′) ⇔ x = x′q and z′ = Ad(q)z  (some q ∈ Q).  (3.3)(b)

The space Z is a smooth variety with an action of K_{ℂ}; it is the total space of a homogeneous vector bundle on the flag variety K_{ℂ}/Q. (A brief general discussion of this construction may be found in an appendix to [VDix].) It follows from (3.3)(b) that

(x, z) ∼ (x′, z′) ⇒ Ad(x)z = Ad(x′)z′.  (3.3)(c)

We therefore have an algebraic map

π: Z → N_{θ},  (x, z) ↦ Ad(x)z.  (3.3)(d)

(Notice that 𝔳 obviously consists of nilpotent elements in 𝔭, and therefore the entire image of π does as well.)

Theorem 3.4. Suppose we are in the setting of (3.1) and (3.3). Then the map π is proper and birational, with image equal to the closure of K_{ℂ} · e. Consequently the algebra R(Z) of regular functions on Z is naturally isomorphic to the normalization of R(\overline{K_{ℂ} · e}).

Assume in addition that the boundary of the orbit closure \overline{K_{ℂ} · e} has complex codimension at least two. Then R(Z) is naturally isomorphic to the algebra R(K_{ℂ}·e) of regular functions on the orbit.

Sketch of proof. That π is proper is a consequence of the properness of the flag variety K_{ℂ}/Q and the properness of the inclusion 𝔳 ↪ N_{θ} ([VDix], Proposition A.2). Consequently the image is closed. By Proposition 3.2.4, it contains K_{ℂ} · e as a dense subvariety. That π is birational (more precisely, bijective over K_{ℂ} · e) is equivalent (by [VDix], (A.3)) to the fact that K_{ℂ}(e) ⊂ Q, proved in Proposition 3.2.3. The remaining statements are proved exactly as in [McGmult], Theorem 3.1. □

Next, we want to understand how Theorem 3.4 might allow us to compute rings of functions. The answer requires a little notation. It is convenient to change the meaning of G for a moment here.

Definition 3.5. Suppose G ⊃ Q are algebraic groups, and (δ, E) is an algebraic representation of Q. (This means that each element of E belongs to a finitedimensional Q-invariant subspace E_{0}, and that the restriction δ_{0}: Q → GL(E_{0}) is a morphism of algebraic groups.) Form the fiber product E = G ×_{Q} E, a vector bundle over G/Q. We may identify E with the corresponding quasicoherent sheaf on G/Q. Here is a description of the sheaf. Suppose U is an open set in G/Q; identify U with an open set Ũ in G closed under right multiplication by Q. Then the space of sections of E over U is

E(U) = {f: Ũ → E | f is algebraic, and f(xh) = δ^{−1}(h)f(x)}.

We therefore have sheaf cohomology groups H^{p}(G/Q, E); these carry natural algebraic representations of G, which we say are cohomologically induced from Q to G. We write

(Ind_{Q}^{G})^{p}(E) = H^{p}(G/Q, E).

When p = 0 we may omit the superscript. In particular

Ind_{Q}^{G}(E) = {f: G → E | f is algebraic, and f(xh) = δ^{−1}(h)f(x)}.

Proposition 3.6. Suppose G ⊃ Q are algebraic groups, and that Z_{Q} is an algebraic variety with an action of Q. Define Z = G ×_{Q} Z_{Q}, an algebraic variety with an action of G. Then there is a natural isomorphism of representations of G

R(Z) ≃ Ind_{Q}^{G} R(Z_{Q}).

This is elementary; some additional discussion of it may be found in [VDix], Proposition A.9.

Corollary 3.7. In the setting of Theorem 3.4,

R(Z) ≃ Ind_{Q}^{K_{ℂ}}(R(𝔳)) = ∑_{k=0}^{∞} Ind_{Q}^{K_{ℂ}}(S^{k}(𝔳^{∗})).

In the presence of the codimension condition of Theorem 3.4, this result begins to look like a computation of the ring of regular functions on an orbit. Roughly speaking, we need two more ingredients: an understanding of the induction functor from Q to K_{ℂ}, and an understanding of S^{k}(𝔳^{∗}) as a representation of Q. The first is provided by the Bott-Borel-Weil theorem (Theorem 3.10 and Corollary 3.11 below). For simplicity we confine our discussion to the case of connected K_{ℂ}. As in Definition 3.5, it is convenient to formulate the general results with different notation; in the application G will be replaced by K_{ℂ}.

Definition 3.8. Suppose G is a connected reductive complex algebraic group, Q ⊂ G is a parabolic subgroup, and H ⊂ Q is a maximal torus. Write U for the unipotent radical of Q and L for the Levi subgroup containing H. Choose a system of positive roots Δ^{+}(𝔩, 𝔥) for 𝔥 in 𝔩, so that Δ^{+}(𝔤, 𝔥) = Δ^{+}(𝔩, 𝔥)∪Δ(𝔲, 𝔥) is a positive system for 𝔥 in 𝔤. Write X^{∗}(H) ⊃ Δ(𝔤, 𝔥) for the lattice of weights, or rational characters of H (morphisms from H to ℂ^{×}). Similarly, we write X_{∗}(H) ⊃ Δ̌(𝔤, 𝔥) for the dual lattice of coweights, or one-parameter subgroups (morphisms from ℂ^{×} to H). If we fix an identification of the Lie algebra of ℂ^{×} with ℂ, then

X^{∗}(H) ⊂ 𝔥^{∗}, X_{∗}(H) ⊂ 𝔥,

and these inclusions are compatible with the natural dualities

X^{∗}(H) × X_{∗}(H) → ℤ, 𝔥^{∗} × 𝔥 → ℂ.

If we identify coweights with elements of 𝔥 as above, then the coroot α̌ corresponding to a root α is the element often called h_{α}, part of a Chevalley basis of the Lie algebra. That is, we can choose root vectors e_{α} for α and f_{α} for −α so that

[h_{α}, e_{α}] = 2e_{α}, [h_{α}, f_{α}] = −2f_{α}, [e_{α}, f_{α}] = h_{α}.

The statement that e_{α} and f_{α} are root vectors may be expressed as

[h, e_{α}] = α(h)e_{α}, [h, f_{α}] = −α(h)f_{α} (h ∈ 𝔥).

A weight λ ∈ X^{∗}(H) is called G-dominant (or Δ^{+}(𝔤, 𝔥)-dominant) if for every positive root α ∈ Δ^{+}(𝔤, 𝔥), λ(α̌) ≥ 0.

In order to state the Bott-Borel-Weil theorem, we need a little notation related to the Weyl group.

Definition 3.9. In the setting of Definition 3.8, define

ρ = \frac{1}{2} ∑_{α∈Δ^{+}(𝔤,𝔥)} α.

For w ∈ W (𝔤, 𝔥), define

Δ^{+}(w) = {α ∈ Δ^{+}(𝔤, 𝔥) | w^{−1}α ∉ Δ^{+}}.

Then the length of w (the length of a minimal expression of w as a product of simple roots) is l(w) = |Δ^{+}(w)| ([Hump], Lemma 10.3A). The sign of w (the determinant of the action of w on 𝔥) is sgn(w) = (−1)^{l(w)}. Finally, recall that

ρ − w · ρ = ∑_{α∈Δ^{+}(w)} α ∈ X^{∗}(H).

Theorem 3.10 ([Bott], page 228, or [Kost], Theorem 6.4). Suppose we are in the setting of Definition 3.8; use also the notation of Definition 3.9. Suppose λ ∈ X^{∗}(H) is G-dominant, and V_{λ} is the irreducible algebraic representation of G of highest weight λ. Similarly, suppose μ ∈ X^{∗}(H) is L-dominant, and E_{μ} is the irreducible algebraic representation of L of highest weight μ. Then V_{λ}^{∗} appears in (Ind_{Q}^{G})^{p}(E_{μ}^{∗}) if and only if μ + ρ = w(λ + ρ), for some w ∈ W (𝔤, 𝔥) with l(w) = p. In this case V_{λ}^{∗} appears with multiplicity one.

To apply this result in Corollary 3.7, a slight reformulation is helpful.

Corollary 3.11. Suppose we are in the setting of Definitions 3.8 and 3.9. Suppose λ ∈ X^{∗}(H) is G-dominant, and V_{λ} is the irreducible algebraic representation of G of highest weight λ. Suppose E is a finite-dimensional algebraic representation of Q. Then the multiplicity of V_{λ}^{∗} in the virtual representation ∑_{p}(−1)^{p}(Ind_{Q}^{G})^{p}(E^{∗}) is equal to the sum over w of

sgn(w) · (multiplicity of E_{w(λ+ρ)−ρ} in E).

Here the sum runs over w such that w(λ + ρ) is L-dominant; E_{μ} is the irreducible representation of L of highest weight μ; and the multiplicity is as representations of L.

Proof. We proceed by induction on the length of E as a representation of L. If E is an irreducible representation of L, then the corollary is just a weakened restatement of Theorem 3.10. If not, then there is an exact sequence of Q-modules

0 → E_{1} → E → E_{2} → 0,

with E_{i} non-zero. Consequently E = E_{1} +E_{2} as representations of L. By inductive hypothesis, the multiplicity formula we want is true for E_{1} and E_{2}. The long exact sequence in sheaf cohomology on G/Q gives a long exact sequence of G-modules

⋯ → (Ind_{Q}^{G})^{p}(E_{2}^{∗}) → (Ind_{Q}^{G})^{p}(E^{∗}) → (Ind_{Q}^{G})^{p}(E_{1}^{∗}) → (Ind_{Q}^{G})^{p+1}(E_{2}^{∗}) → ⋯

By the Euler-Poincaré principle, it follows that

∑_{p}(−1)^{p}(Ind_{Q}^{G})^{p}(E^{∗}) = ∑_{p}(−1)^{p}(Ind_{Q}^{G})^{p}(E_{1}^{∗}) + ∑_{p}(−1)^{p}(Ind_{Q}^{G})^{p}(E_{2}^{∗})

as virtual representations of G. The multiplicity formula for E is therefore the sum of the formulas for E_{1} and E_{2}. □

Even though the virtual multiplicity formula in Corollary 3.11 refers only to the structure of E as an L-module, the Q-module structure is very important. It will control such questions as the vanishing of (Ind_{Q}^{G})^{p}(E^{∗}) for p > 0.

Corollary 3.7 asks us to compute Ind^{0}, and Corollary 3.11 computes instead an Euler characteristic. To use them together, we would like a vanishing theorem. Here is a special case.

Theorem 3.12 ([McGmult], page 212 and [Hin], Lemma 2.3). Suppose we are in the setting of (3.1) and (3.3), and suppose in addition that G is complex. Then

(Ind_{Q}^{K_{ℂ}})^{p}(S^{k}(𝔳^{∗})) = 0, (p > 0).

McGovern proves only that the alternating sum of the higher induced representations is zero; the conclusion that they vanish is due to Hinich (and apparently independently to Panyushev).

It is natural to hope that the vanishing result in Theorem 3.12 extends to the case of real groups; so we make

Conjecture 3.13. Suppose we are in the setting of (3.1) and (3.3). Then

(Ind_{Q}^{K_{ℂ}})^{p}(S^{k}(𝔳^{∗})) = 0, (p > 0).

We can make a few remarks about what makes this conjecture difficult. The argument used by McGovern in the complex case, as refined by Hinich, comes down to Lemma 2.3 of [Hin]. The idea is to find an approximate identification of the sheaf of functions (on a nilpotent orbit) with the sheaf of top degree differential forms. Such an identification exists because a coadjoint orbit is symplectic; the top power of the symplectic form trivializes the sheaf of differential forms. An orbit of K_{ℂ} on N_{θ}^{∗} need not be symplectic, and the sheaf of differential forms need not be trivial. In fact it is exactly the sheaf of differential forms that enters the definition of admissible orbit datum (Definition 2.6). We will see in (8.5)(b) that this sheaf is non-trivial in the case of the model orbit for E_{8}(ℝ) considered in sections 7 and 8. Consequently Lemma 2.3 of [Hin] cannot be directly applied to prove Conjecture 3.13. In fact the vanishing theorem of Grauert and Riemenschneider (which is at the heart of Hinich’s argument) seems not to offer any help toward Conjecture 3.13.

Combining this vanishing conjecture with Corollary 3.11 leads immediately to the following more explicit version of Corollary 3.7.

Corollary 3.14. Suppose we are in the setting of (3.1) and (3.3), and suppose that Conjecture 3.13 holds (for example if G is complex). Fix a maximal torus H_{ℂ} ⊂ L_{ℂ} and a system of positive roots Δ^{+}(𝔩, 𝔥). Extend this to a system of positive roots for H_{ℂ} in K_{ℂ} as in Definition 3.8, and write ρ_{c} for half the sum of the positive roots. Fix a K_{ℂ}-dominant weight λ ∈ X^{∗}(H_{ℂ}), and write V_{λ} for the corresponding irreducible representation of K_{ℂ}. Then the multiplicity of V_{λ}^{∗} in the ring of regular functions on the normalization of \overline{K_{ℂ} · e} is equal to the sum over w of

sgn(w) · (multiplicity of E_{w(λ+ρ_{c})−ρ_{c}} in S(𝔳)).

Here the sum is over w ∈ W (𝔨, 𝔥) such that w(λ + ρ_{c}) is L_{ℂ}-dominant.

Conjecture 2.9 suggests that one ought to study not the ring of functions on K_{ℂ} · e, but rather certain spaces of sections of line bundles. McGovern’s approach applies to these as well ([McGmult], §4). Here is the idea.

Proposition 3.15. In the setting of (3.1) and (3.3), suppose χ is an algebraic representation of K_{ℂ}(e) = Q(e); write V_{χ} for the corresponding vector bundle on K_{ℂ} · e (compare Definition 2.6). Assume that there is an algebraic representation δ of Q such that δ|_{Q(e)} ≃ χ. Then there is a natural inclusion

∑_{k=0}^{∞} Ind_{Q}^{K_{ℂ}}(S^{k}(𝔳^{∗}) ⊗ V_{δ}) ↪ sections of V_{χ} over K_{ℂ} · e.

The proof is parallel to that of Corollary 3.7; we omit the argument (and the precise formulation of an analogue of Corollary 3.14.) Of course the more difficult and interesting questions are left unresolved. (For which χ do representations δ exist? Is it possible to choose δ so that the inclusions of Proposition 3.15 are isomorphisms? So that the higher cohomologically induced representations vanish?)

4. The model orbit in E_{8}(ℂ).

In this section we will give an explicit description of the orbit we wish to study. In this section G will always be a complex algebraic group of type E_{8}. Instead of looking at the action of K_{ℂ} on 𝔭, we will look at the action of G on 𝔤_{0}; this is equivalent by (2.10). Otherwise our notation is more or less consistent with that of (3.1). We fix a maximal torus H of G, with Lie algebra 𝔥_{0}. (Recall from the beginning of section 2 that the subscript zero means we have not complexified the Lie algebra. Of course it is a complex Lie algebra, because of the complex structure on G.) There is a standard identification (as in [Hump], page 65, or [Bour], chapter IV, §4.10)

𝔥_{0}^{∗} ≃ ℂ^{8}  (4.1)(a)

with the property that the root system of 𝔥_{0} in 𝔤_{0} is

Δ(𝔤_{0}, 𝔥_{0}) = \{±e_{i} ± e_{j} | 1 ≤ i ≠ j ≤ 8\} ∪ \{\frac{1}{2}(ϵ_{1}, ⋯, ϵ_{8}) | ϵ_{i} = ±1, ∏_{i} ϵ_{i} = 1\}.

(4.1)(b)

Here {e_{i}} is the standard basis of ℂ^{8}. The first collection of roots described therefore has 4\binom{8}{2} = 112 elements, and the second has 2^{8}/2 = 128. Altogether there are 112 + 128 = 240 roots, so that G has complex dimension 8 + 240 = 248. For the calculations we wish to do, these coordinates are slightly inconvenient. The reason is this. By inspection of the Dynkin diagram, it is clear that 𝔤_{0} has a Levi subalgebra of type A_{7}; that is, isomorphic to 𝔤𝔩(8, ℂ). We will make extensive use of this subalgebra, and so we will need to identify its roots in our chosen coordinates. Of course the root system of 𝔤𝔩(8, ℂ) has a standard presentation, and it is natural to look for this standard form inside the root system for E_{8}. It is there: the 56 roots e_{i} − e_{j}. These roots do provide a subalgebra of 𝔤_{0} isomorphic to 𝔤𝔩(8, ℂ), but it is not the Levi subalgebra we want. (Here is one way to see this. Any root in the ℝ-span of the roots of a Levi subalgebra must actually be a root for that Levi subalgebra. The root \frac{1}{2}(1, 1, 1, 1, −1, −1, −1, −1) belongs to the ℝ-span of the e_{i} − e_{j}, but is not one of them.)

We therefore modify the isomorphism (4.1)(a) by composing it with the automorphism of ℂ^{8} that changes the sign of the first coordinate. In this new identification

𝔥_{0}^{∗} ≃ ℂ^{8},  (4.2)(a)

the root system of 𝔥_{0} in 𝔤_{0} is

Δ(𝔤_{0}, 𝔥_{0}) = \{±e_{i} ± e_{j} | 1 ≤ i ≠ j ≤ 8\} ∪ \{\frac{1}{2}(ϵ_{1}, ⋯, ϵ_{8}) | ϵ_{i} = ±1, ∏_{i} ϵ_{i} = −1\}.

(4.2)(b)

The standard bilinear form on ℂ^{8} is invariant by the Weyl group W = W (𝔤_{0}, 𝔥_{0}). It therefore provides an identification 𝔥_{0} ≃ ℂ^{8}. Using this identification, we define

h = (1, 1, 1, 1, 1, 1, 1, 1) ∈ 𝔥_{0}.  (4.2)(c)

Write L for the subgroup of G fixing h in the adjoint action. Then L is a Levi subgroup of G, with root system

Δ(𝔩_{0}, 𝔥_{0}) = {α ∈ Δ(𝔤_{0}, 𝔥_{0}) | α(H) = 0} = {e_{i} − e_{j} | 1 ≤ i ≠ j ≤ 8}. (4.2)(d)

Each root α gives rise to an algebraic homomorphism

ϕ_{α}: SL(2, ℂ) → G,  (4.3)(a)

with the property that ϕ_{α} carries diagonal matrices into H and upper triangular unipotent matrices into the root subgroup for α. These requirements characterize ϕ_{α} up to conjugation by a diagonal element in SL(2, ℂ). In particular, the coroot

h_{α} = dϕ_{α} ( \begin{matrix}1 & 0 \\ 0 & −1 \end{matrix} )  (4.3)(b)

is uniquely defined. In our coordinate system, the identification of both 𝔥_{0}^{∗} and 𝔥_{0} with ℂ^{8} makes α = h_{α}. We are interested in the four orthogonal roots

α_{1} = e_{1} + e_{2}, α_{2} = e_{3} + e_{4}, α_{3} = e_{5} + e_{6}, α_{4} = e_{7} + e_{8}.  (4.3)(c)

The corresponding root SL(2) subgroups commute with each other, and so define a homomorphism of SL(2, ℂ)^{4} into G. We are interested in the restriction of this homomorphism to the diagonal:

ϕ: SL(2, ℂ) → G,  ϕ(x) = ϕ_{α_{1}}(x)ϕ_{α_{2}}(x)ϕ_{α_{3}}(x)ϕ_{α_{4}}(x)  (4.3)(d)

By the remarks after (4.3)(b),

dϕ ( \begin{matrix}1 & 0 \\ 0 & −1 \end{matrix} ) = h_{α_{1}} + h_{α_{2}} + h_{α_{3}} + h_{α_{4}} = h,  (4.3)(e)

the element considered in (4.2)(c). We can now write down the nilpotent element we want to study:

e = dϕ ( \begin{matrix}0 & 1 \\ 0 & 0 \end{matrix} ) ,  (4.3)(f)

the sum of root vectors for the four orthogonal roots α_{i}.

In order to calculate the Dynkin diagram of the nilpotent element e, we need to choose a system of positive roots making h dominant. An obvious choice is

Δ^{+}(𝔤_{0}, 𝔥_{0}) = {e_{i} − e_{j} | i < j} ∪ {α | α(h) > 0}.  (4.4)(a)

It is easy to determine the simple roots for this positive system: they are

Π(𝔤_{0}, 𝔥_{0}) = \{e_{i} − e_{i+1} | 1 ≤ i ≤ 7\} ∪ \{\frac{1}{2}(−1, −1, −1, 1, 1, 1, 1, 1)\}.  (4.4)(b)

We label these simple roots in the order listed as {β_{1}, β_{3}, β_{4}, β_{5}, β_{6}, β_{7}, β_{8}, β_{2}}. (This is the numbering used in [Bour] and [Hump].) Then the Dynkin diagram of G is

\begin{aligned}& & β_{2} \\ & & | \\ β_{1} & β_{3} & β_{4} & β_{5} & β_{6} & β_{7} & β_{8} \end{aligned}  (4.4)(c)

To get the weighted Dynkin diagram of the nilpotent element e, we replace each vertex β by the non-negative integer β(h). These integers may be calculated immediately from (4.2)(c) and (4.4)(c); the result is

\begin{aligned}& & 1 \\ & & | \\ 0 & 0 & 0 & 0 & 0 & 0 & 0 \end{aligned}  (4.4)(d)

Proposition 4.5. In the setting of (4.2)–(4.4), let X = G · e be the adjoint orbit of the element e. Then X is the unique nilpotent orbit of complex dimension 128 in 𝔤_{0}, and its weighted Dynkin diagram is given by (4.4)(d). The stabilizer L(e) of e in L is isomorphic to Sp(8, ℂ). In particular, L(e) and G(e) are connected.

Proof. Because e belongs to the 2-eigenspace of ad h, it is certainly nilpotent. We have already computed the Dynkin diagram of e. Once the diagram is known, that X is the unique nilpotent orbit of dimension 128 may be read off from published tables (for example [Coll], page 132; [Cart], 13.1; or [Dynk], Table 20). (We will see in (4.6) how to calculate that X has dimension 128.) Here we will only outline the calculation of L(e). We will see in Lemma 5.2 that the representation of L on 𝔤_{0}(2) is ∧^{2}(τ), the second exterior power of the standard representation. The element e has an open L orbit in this representation (see for example [Coll], page 42); so we want to calculate the stabilizer in L of a generic element of ∧^{2}(τ). In GL(8) this stabilizer is Sp(8) more or less by definition (since ∧^{2}(V) may be identified with skew-symmetric bilinear forms on V^{∗}). The group L differs slightly from GL(8) (see the remarks at (5.1)) but nevertheless L(e) is still isomorphic to Sp(8). We will carry out the analogous calculation in the real case in detail in section 8. Now the connectedness of G(e) follows from Proposition 3.2.3. □

We will study the group G using a parabolic subgroup with Levi factor L. The operator ad(h) has integer eigenvalues on 𝔤_{0}, and we define

𝔤_{0}(m) = {z ∈ 𝔤_{0} | [h, z] = mz}.  (4.6)(a)

Each of these spaces is a representation of the group L, whose Lie algebra is precisely 𝔤_{0}(0). For m ≠ 0, 𝔤_{0}(m) is a sum of root spaces, for the roots α such that α(h) = m. These sets of roots are easily calculated. First,

Δ(𝔤_{0}(1), 𝔥_{0}) = \{\frac{1}{2}(ϵ_{1}, …, ϵ_{8}) | ϵ_{i} = ±1, ϵ_{i} = 1 for exactly 5 values of i\}. (4.6)(b)

This set has \binom{8}{3} = 56 elements, which is therefore the complex dimension of 𝔤_{0}(1). It is a general fact about nilpotent elements (see [Coll], Lemma 4.1.3, or [Cart]) that dim 𝔤_{0}(e) = dim 𝔤_{0}(0) + dim 𝔤_{0}(1), and therefore that dim G · e = dim G − dim 𝔤_{0}(0) − dim 𝔤_{0}(1). In our case, dim X = 248 − 64 − 56 = 128, as we had already claimed in Proposition 4.5. Next,

Δ(𝔤_{0}(2), 𝔥_{0}) = {e_{i} + e_{j} | i ≠ j}.  (4.6)(c)

This set has cardinality \binom{8}{2} = 28. Finally,

Δ(𝔤_{0}(3), 𝔥_{0}) = \{\frac{1}{2}(ϵ_{1}, …, ϵ_{8}) | ϵ_{i} = ±1, ϵ_{i} = 1 for exactly 7 values of i\}. (4.6)(d)

This set has cardinality \binom{8}{1} = 8. Now define

𝔲_{0} = ∑_{m≥1} 𝔤_{0}(m),  𝔮 = ∑_{m≥0} 𝔤_{0}(m) = 𝔩_{0} + 𝔲_{0}.  (4.6)(e)

Then 𝔮_{0} is a parabolic subalgebra of 𝔤_{0}, corresponding to a parabolic subgroup with Levi decomposition Q = LU. We will also need the subalgebra

𝔳_{0} = ∑_{m≥2} 𝔤_{0}(m) = 𝔤_{0}(2) + 𝔤_{0}(3).  (4.6)(f)

According to Theorem 3.4, the vector bundle

Z = G ×_{Q} 𝔳_{0}  (4.6)(g)

provides a resolution of singularities for the orbit closure \overline{X} ⊂ 𝔤_{0}. Applying Mc-Govern’s multiplicity formula (Corollary 3.14) to this case, we find

Theorem 4.7. Suppose we are in the setting of (4.2)–(4.4) and (4.6). Fix a Gdominant weight λ ∈ X^{∗}(H), and write V_{λ} for the corresponding irreducible representation of G. Then the multiplicity of V_{λ}^{∗} in the ring of regular functions on G · e is equal to the sum over w of

sgn(w) · (multiplicity of E_{w(λ+ρ)−ρ} in S(𝔳_{0})).

Here ρ is half the sum of the positive roots, and the sum is over w ∈ W (𝔤, 𝔥) such that w(λ + ρ) is L-dominant.

In order to make this formula more explicit, we need to calculate S(𝔳_{0}) as a representation of L. This we will do in the following section.

5. Decomposition of S(𝔳_{0}).

In this section we will work in the setting of (4.2)–(4.4) and (4.6), and calculate S(𝔳_{0}) as a representation of L. Because 𝔩_{0} ≃ 𝔤𝔩(8), this is entirely a classical invariant theory problem, and it can be solved by standard methods. We fix an isomorphism

j: 𝔤𝔩(8) → 𝔩_{0}  (5.1)(a)

carrying diagonal matrices to 𝔥_{0} ≃ ℂ^{8} in the obvious way. It is important to remember that the group L is not precisely isomorphic to GL(8). (One can construct L as a two-fold cover of GL(8) modulo a two-element central subgroup; details appear at (8.1) below.) The set of dominant weights for 𝔤𝔩(8) is

{λ = (λ_{1}, …, λ_{8}) ∈ ℂ^{8} | λ_{i} − λ_{j} is a non-negative integer for i < j}. (5.1)(b)

We write (τ, ℂ_{τ}^{8}) for the standard (tautological) representation of 𝔩_{0}; its weights are

Δ(τ, 𝔥_{0}) = {e_{i} | i = 1, …, 8}.  (5.1)(c)

The highest weight is (1, 0, …, 0). Similarly, the dual representation τ^{∗} has weights

Δ(τ^{∗}, 𝔥_{0}) = {−e_{i} | i = 1, …, 8}.  (5.1)(d)

The one-dimensional determinant character (det, ℂ_{det}) has weight

Δ(det, 𝔥_{0}) = {(1, …, 1)}.  (5.1)(e)

(We write det to suggest the group representation; on the Lie algebra the definition is det(x) = trace of x.) Even though we cannot extract the square root of the determinant on GL(8), there is a well-defined Lie algebra representation (det^{1/2}, ℂ_{det}_{1/2}), with weight

Δ(det^{1/2}, 𝔥_{0}) = {(1/2, …, 1/2)}.  (5.1)(f)

With these calculations in hand, we can easily determine the weights of other simple representations. For example, the second exterior power of the standard representation has weights

Δ(∧^{2}(τ), 𝔥_{0}) = {e_{i} + e_{j} | 1 ≤ i ≠ j ≤ 8}.  (5.1)(g)

The highest weight is (1, 1, 0 …, 0). Similarly, τ^{∗} twisted by det^{1/2} has weights

Δ(τ^{∗} ⊗ det^{1/2}, 𝔥_{0}) = \{\frac{1}{2}(ϵ_{1}, …, ϵ_{8}) | ϵ_{i} = ±1, ϵ_{i} = 1 for exactly 7 values of i\}.

(5.1)(h)

Now an algebraic representation of a reductive group is determined up to equivalence by its set of weights with multiplicities. Comparing (5.1) with (4.6), we deduce

Lemma 5.2. Suppose we are in the setting of (4.6). In terms of the isomorphism (5.1)(a), the representation of 𝔩_{0} on 𝔳_{0} is ∧^{2}(τ) ⊕ τ^{∗} ⊗ det^{1/2}. Consequently

S(𝔳_{0}) ≃ S(∧^{2}ℂ_{τ}^{8}) ⊗ (∑_{k=0}^{∞} S^{k}(ℂ_{τ^{∗}}^{8}) ⊗ det^{k/2}) .

In light of Lemma 5.2, we are left with two tasks: decomposing the symmetric algebra of ∧^{2}(τ) (as a representation of 𝔤𝔩(8)), and decomposing the tensor product of an arbitrary representation with the symmetric algebra of τ^{∗}. (The twist by the one-dimensional character det^{k/2} is easy to handle.) Here are the results.

Proposition 5.3 ([Sch], page 63, Behauptung c)). In the setting (5.1), the symmetric algebra S(∧^{2}τ) is the direct sum of the representations of GL(8) of highest weights (a, a, b, b, c, c, d, d), with a ≥ b ≥ c ≥ d ≥ 0 integers.

Proof. Consider the Hermitian symmetric space SO^{∗}(2n)/U(n) (see for example [Helg], page 445, or [Knapp], page 6). The holomorphic tangent space 𝔭_{+} at the identity coset is isomorphic to the second exterior power ∧^{2}(τ_{n}) of the standard representation of U(n). (This may be seen by explicit calculation with matrices, or by inspection of the root systems of SO(2n) and U(n).) The content of the result of Schmid cited in the statement is a decomposition of S(𝔭_{+}) as a representation of K for any Hermitian symmetric space G/K. In the case of SO^{∗}(4m)/U(2m), Schmid’s result is that the representations of U(2m) on S(∧^{2}τ_{2m}) are those of highest weights (a_{1}, a_{1}, …, a_{m}, a_{m}), with (a_{i}) a decreasing sequence of non-negative integers. Taking m = 4 gives the proposition. (We use also the equivalence of categories between finite-dimensional representations of U(n) and algebraic representations of GL(n, ℂ).) □

Proposition 5.4 (Pieri’s formula; see for example [Mac], Chapter I, (5.12) (page 73)). Suppose λ = (λ_{1}, …, λ_{n}) ∈ ℝ^{n} is a dominant weight for 𝔤 = 𝔤𝔩(n, ℂ); that is, that λ_{i} − λ_{j} is a non-negative integer whenever i < j. Let V_{λ} be the irreducible representation of 𝔤 of highest weight λ, and let τ be the n-dimensional standard representation. Then the tensor product V_{λ} ⊗ S^{k}(τ_{n}) is the direct sum of the representations V_{μ}, where μ ∈ ℝ^{n} satisfies

μ_{1} ≥ λ_{1} ≥ μ_{2} ≥ λ_{2} ≥ ⋯ ≥ μ_{n} ≥ λ_{n},  μ − λ ∈ ℤ^{n},  ∑_{i} μ_{i} − λ_{i} = k.

Each of these constituents V_{μ} appears with multiplicity one.

The formula stated in Macdonald’s book is about multiplying Schur functions. Because Schur functions are characters for GL(n) ([Mac], page 163), it amounts to a statement about the tensor product of two representations. To make this translation, one needs also the formula (3.9) on page 42 of [Mac], and the discussion on the top of page 5 there. What emerges when the dust has settled is Proposition 5.4 in the case that λ_{n} is a non-negative integer. The general case follows at once by tensoring with powers of the determinant character.

(We are grateful to Chris Woodward for showing us a beautiful geometric proof of Proposition 5.4. He later found his proof in a paper of Brion, who in turn refers to Macdonald.)

The setting of Lemma 5.2 actually involves the symmetric algebra of the dual of the standard representation. It is a simple matter to take duals of everything in Proposition 5.4; the result is

Proposition 5.5 (Pieri’s formula). Suppose λ = (λ_{1}, …, λ_{n}) ∈ ℝ^{n} is a dominant weight for 𝔤 = 𝔤𝔩(n, ℂ). Let V_{λ} be the irreducible representation of 𝔤 of highest weight λ, and let τ be the n-dimensional standard representation. Then the tensor product V_{λ} ⊗ S^{k}(τ_{n}^{∗}) is the direct sum of the representations V_{μ}, where μ ∈ ℝ^{n} satisfies

λ_{1} ≥ μ_{1} ≥ λ_{2} ≥ μ_{2} ≥ ⋯ ≥ λ_{n} ≥ μ_{n},  μ − λ ∈ ℤ^{n},  ∑_{i} λ_{i} − μ_{i} = k.

Each of these constituents V_{μ} appears with multiplicity one.

The decomposition of S(𝔳_{0}) is now straightforward. Here is the result.

Proposition 5.6. Suppose we are in the setting of (4.6); use the notation of (5.1). Suppose λ = (λ_{1}, …, λ_{8}) ∈ ℂ^{8} is a dominant weight for 𝔩_{0} ≃ 𝔤𝔩(8); write E_{λ} for the irreducible representation of highest weight λ. Then the multiplicity of E_{λ} in S(𝔳_{0}) is equal to 1 if

λ_{i} − λ_{i+1} is a non-negative integer for i = 1, …, 7, and
\frac{1}{2}(−λ_{1} + λ_{2} − λ_{3} + λ_{4} − λ_{5} + λ_{6} + λ_{7} + λ_{8}) is a non-negative integer.

Otherwise the multiplicity is zero. In terms of the eight simple roots β_{i} of (4.4)(c) and the corresponding coroots h_{i}, the conditions for multiplicity one may be expressed as

λ(h_{i}) ∈ ℕ  (i = 1, 3, 4, 5, 6, 7, 8),  λ(h_{2} + h_{3} + h_{4} + h_{5}) ∈ ℕ.

Proof. We use Lemma 5.2 to describe S(𝔳_{0}). According to Proposition 5.3, the highest weights of S(∧^{2}τ) are those of the form (a, a, b, b, c, c, d, d), with (a, b, c, d) a decreasing sequence of non-negative integers. According to Proposition 5.5, the highest weights of S(∧^{2}τ) ⊗ S(τ^{∗}) are those μ ∈ ℤ^{8} satisfying

a ≥ μ_{1} ≥ a ≥ μ_{2} ≥ b ≥ ⋯ ≥ d ≥ μ_{8}.  (5.7)(a)

These conditions force μ_{1} = a, μ_{3} = b, μ_{5} = c, and μ_{7} = d. This shows that μ determines (a, b, c, d); and any dominant μ can occur, as long as μ_{7} ≥ 0. The integer k of Proposition 5.5 can also be read off from μ: it is

k = μ_{1} − μ_{2} + μ_{3} − μ_{4} + μ_{5} − μ_{6} + μ_{7} − μ_{8}.  (5.7)(b)

According to Lemma 5.2, the highest weights λ that we want are obtained by twisting these μ by the weight of det^{k/2}. That is,

λ = μ + \frac{k}{2} (1, 1, 1, 1, 1, 1, 1, 1).  (5.7)(c)

Comparing (5.7)(b) and (5.7)(c), we see that

k = λ_{1} − λ_{2} + λ_{3} − λ_{4} + λ_{5} − λ_{6} + λ_{7} − λ_{8}.  (5.7)(d)

We can interpret (5.7)(c) and (5.7)(d) as formulas for μ in terms of λ. In particular, they give

μ_{7} = λ_{7} − \frac{1}{2}(λ_{1} − λ_{2} + λ_{3} − λ_{4} + λ_{5} − λ_{6} + λ_{7} − λ_{8}),

or

μ_{7} = \frac{1}{2}(−λ_{1} + λ_{2} − λ_{3} + λ_{4} − λ_{5} + λ_{6} + λ_{7} + λ_{8})  (5.7)(e)

We know that μ occurs in S(∧^{2}τ)⊗S(τ^{∗}) if and only if μ_{7} is a non-negative integer; so it follows that λ appears in S(𝔳_{0}) if and only if the expression in (5.7)(e) is a nonnegative integer. This is the first formula in the proposition. The second follows from the formulas (4.4)(b) for the simple positive roots. □

Corollary 5.8. Suppose we are in the setting of (4.2)–(4.4) and (4.6). Let α be the positive root

α = β_{2} + β_{3} + β_{4} + β_{5} = \frac{1}{2}(−1, 1, −1, 1, −1, 1, 1, 1).

Write h_{α} for the corresponding coroot. Define a subset S ⊂ W of the Weyl group by

S = {w ∈ W | w^{−1}(β_{i}) > 0 (i = 1, 3, 4, …, 8), w^{−1}(α) > 0}.

Fix a G-dominant weight λ ∈ X^{∗}(H), and write V_{λ} for the corresponding irreducible representation of G. Define a subset S(λ) ⊂ W by

S(λ) = {w ∈ S | (λ + ρ)(w^{−1}(h_{α})) ≥ 4}.

Then the multiplicity of V_{λ}^{∗} in the ring of regular functions on G · e is equal to

∑_{w∈S(λ)} sgn(w).

Proof. We combine the formula of Theorem 4.7 with Proposition 5.6. Define S(λ) to be the set of w ∈ W for which w(λ + ρ) − ρ is the highest weight (for 𝔩_{0}) of a representation in S(𝔳_{0}). (We will see that this is the set defined in the Corollary.) Theorem 4.7 says that we must sum sgn w times a multiplicity over S(λ). Proposition 4.6 says that all the multiplicities are one, and that these w are characterized by the conditions

[w(λ + ρ) − ρ](h_{i}) ≥ 0 (i = 1, 3, 4, …, 8),

and

[w(λ + ρ) − ρ](h_{α}) ≥ 0.

Now ρ(h_{i}) = 1 (see for example [Hump], Lemma 13.3A). These two conditions may therefore be written as

w(λ + ρ)(h_{i}) ≥ 1 (i = 1, 3, 4, …, 8),

and

w(λ + ρ)(h_{α}) ≥ 4.

Because λ + ρ is dominant, integral, and regular, the first condition is equivalent to w^{−1}(β_{i}) being a positive root. The second implies that w^{−1}(α) is a positive root. The two together may be written as

w^{−1}(β_{i}) > 0 (i = 1, 3, 4, …, 8), w^{−1}(α) > 0

and

(λ + ρ)(w^{−1}(h_{α})) ≥ 4.

The first condition is just the definition of S in the Corollary, and the addition of the second gives S(λ) as in the Corollary. □

What remains is to calculate S and to understand the possibilities for S(λ). This we will do in the next section.

6. Calculations in the Weyl group of E_{8}.

In this section we will calculate the subsets of the Weyl group of E_{8} described in Corollary 5.8. We begin with some general discussion of the nature of the subsets. The conditions w^{−1}(β_{i}) > 0 for i ∈ I are very familiar. They define natural coset representatives for the subgroup W_{I} of W generated by simple reflections in the roots β_{i} (for i ∈ I). In our case, I consists of all the simple roots except β_{2}. The other seven simple roots span a system of type A_{7}, so W_{I} is S_{8}, the symmetric group on 8 elements. The number of Weyl group elements satisfying w^{−1}(β_{i}) > 0 for i = 1, 3, 4, …, 8 is therefore equal to

|W (E_{8})|/8! = 17280.

This is a discouragingly large number for hand calculations. It is not inaccessible to a computer, however. Our original determination of S essentially listed these elements (by computer) and calculated w^{−1}(α) for each of them. We discovered in this way that S had only seventeen elements. Once we knew that S was small, we were able to prove that without computer assistance.

First we write the definition of S (Corollary 5.8) in the language of Definition 3.9. This is

S = {w ∈ W | Δ^{+}(w) does not contain α or any β_{i}, i = 1, 3, 4, …, 8}. (6.1)

(The equivalence with the definition in Corollary 5.8 is immediate.) We propose to find all the elements w satisfying (6.1) by induction on the length l(w). Longer elements of W are produced from shorter ones by multiplication by simple reflections; so we need to see what effect that multiplication has on Δ^{+}(w).

Lemma 6.2. In the setting of Definition 3.9, suppose w ∈ W and β is a simple root. Then exactly one of the following possibilities holds:

1. l(ws_{β}) = l(w) + 1, wβ > 0, and Δ^{+}(ws_{β}) = Δ^{+}(w) ∪ {wβ}.

2. l(ws_{β}) = l(w) − 1, wβ < 0, and Δ^{+}(ws_{β}) = Δ^{+}(w) − {−wβ}.

Similarly, exactly one of the following possibilities holds:

1. l(s_{β}w) = l(w) + 1, w^{−1}β > 0, and Δ^{+}(s_{β}w) = s_{β}Δ^{+}(w) ∪ {β}.

2. l(s_{β}w) = l(w) − 1, w^{−1}β < 0, and Δ^{+}(s_{β}w) = s_{β}Δ^{+}(w) − {−β}.

This is elementary and well-known (compare [Hump], proof of Lemma 10.3A).

Corollary 6.3. In the setting of (6.1), suppose w ∈ S has length m. Let β be a simple root such that

wβ > 0, wβ ∉ {α, β_{1}, β_{3}, β_{4}, …, β_{8}}.

Then ws_{β} ∈ S has length m + 1. Every element of S of length m + 1 arises by this construction.

Using this corollary, we can construct a list of elements of S. The results appear in the following table. The first column gives a reduced expression w = s_{i_{1}} ⋯ s_{i_{m}}. Here s_{i} = s_{β_{i}}, and as usual the roots are labelled as in (4.4)(c). The next eight columns show the roots wβ_{i}. In order to keep the table small, we have written here −2456 instead of −(β_{2} +β_{4} +β_{5} +β_{6}). The elements are grouped according to their lengths.

\begin{matrix}w & wβ_{1} & wβ_{2} & wβ_{3} & wβ_{4} & wβ_{5} & wβ_{6} & wβ_{7} & wβ_{8} \\ 1 & 1 & 2 & 3 & 4 & 5 & 6 & 7 & 8 \\ s_{2} & 1 & −2 & 3 & 24 & 5 & 6 & 7 & 8 \\ s_{2}s_{4} & 1 & 4 & 234 & −24 & 245 & 6 & 7 & 8 \\ s_{2}s_{4}s_{3} & 1234 & 4 & −234 & 3 & 245 & 6 & 7 & 8 \\ s_{2}s_{4}s_{5} & 1 & 4 & 234 & 5 & −245 & 2456 & 7 & 8 \\ s_{2}s_{4}s_{3}s_{1} & −1234 & 4 & 1 & 3 & 245 & 6 & 7 & 8 \\ s_{2}s_{4}s_{3}s_{5} & 1234 & 4 & −234 & 2345 & −245 & 2456 & 7 & 8 \\ s_{2}s_{4}s_{5}s_{6} & 1 & 4 & 234 & 5 & 6 & −2456 & 24567 & 8 \\ s_{2}s_{4}s_{3}s_{1}s_{5} & −1234 & 4 & 1 & 2345 & −245 & 2456 & 7 & 8 \\ s_{2}s_{4}s_{3}s_{5}s_{6} & 1234 & 4 & −234 & 2345 & 6 & −2456 & 24567 & 8 \\ s_{2}s_{4}s_{5}s_{6}s_{7} & 1 & 4 & 234 & 5 & 6 & 7 & −24567 & 245678 \\ s_{2}s_{4}s_{3}s_{1}s_{5}s_{6} & −1234 & 4 & 1 & 2345 & 6 & −2456 & 24567 & 8 \\ s_{2}s_{4}s_{3}s_{5}s_{6}s_{7} & 1234 & 4 & −234 & 2345 & 6 & 7 & −24567 & 245678 \\ s_{2}s_{4}s_{5}s_{6}s_{7}s_{8} & 1 & 4 & 234 & 5 & 6 & 7 & 8 & −245678 \\ s_{2}s_{4}s_{3}s_{1}s_{5}s_{6}s_{7} & −1234 & 4 & 1 & 2345 & 6 & 7 & −24567 & 245678 \\ s_{2}s_{4}s_{3}s_{5}s_{6}s_{7}s_{8} & 1234 & 4 & −234 & 2345 & 6 & 7 & 8 & −245678 \\ s_{2}s_{4}s_{3}s_{1}s_{5}s_{6}s_{7}s_{8} & −1234 & 4 & 1 & 2345 & 6 & 7 & 8 & −245678 \end{matrix}

Table 6.4. Elements of S.

We now explain more carefully how Table 6.4 was constructed. Suppose we have constructed the lines for elements of S of length m; we want to find the lines for elements of length m + 1. According to Corollary 6.3, each new element will be of the form ws_{β}, where w ∈ S has length m, β is a simple root, and wβ is a positive root not in {α, β_{1}, β_{3}, β_{4}, …, β_{8}}. Our table through length m contains all of this information. Suppose for example that m = 5, so that we are trying to construct elements of S of length 6. Our table lists three elements s_{2}s_{4}s_{3}s_{1}s_{5}, s_{2}s_{4}s_{3}s_{5}s_{6}, and s_{2}s_{4}s_{5}s_{6}s_{7} of length 5; let us call them x, y, and z. Looking at the table for xβ_{i}, we see that xβ_{i} is positive only for i = 2, 3, 4, 6, 7, and 8. Of these possibilities, 2, 3, 7, and 8 are excluded because xβ_{i} is one of the forbidden simple roots β_{1}, β_{3}, …, β_{8}. Similarly 4 is excluded because xβ_{4} = β_{2} + β_{3} + β_{4} + β_{5} = α. So the only candidate for an element of length 6 coming from x is xs_{6} = s_{2}s_{4}s_{3}s_{1}s_{5}s_{6}, and this is the first on the list in Table 6.4. In the same way we see that y gives rise to ys_{1} and ys_{7}, and z to zs_{3} and zs_{8}. Table 6.4 shows ys_{7} and zs_{8}, but seems to omit ys_{1} and zs_{3}. The reason is that ys_{1} = xs_{6}, and zs_{3} = ys_{7}. There are many ways to detect these equalities. For us the easiest is in the rest of the table: two Weyl group elements agree exactly when they have the same effect on all simple roots. So we need to know how to calculate these remaining columns of the table; that is, how to calculate ws_{β}(β′) given knowledge of all wβ′. (Here β and β′ are simple roots.)

Because all the simple roots for E_{8} have the same length, we have

s_{β}(β′) = \{ \begin{matrix}β′, & if β is not adjacent to β′ \\ β + β′, & if β is adjacent to β′ \\ −β, & if β is equal to β′. \end{matrix}  (6.5)

Therefore each entry ws_{β}(β_{i}) in the row for ws_{β} is either equal to the corresponding entry wβ_{i} for w (if β is not adjacent to β_{i}), or it is the sum of the entries wβ_{i} and wβ (if β is adjacent to β_{i}), or it is the negative of the wβ (if β = β_{i}). In the example considered above, the entry ys_{7}(β_{6}) is computed as the sum of yβ_{6} and yβ_{7}; that is, as −2456 + 24567 = 7.

We leave to the reader the rest of the task of verifying Table 6.4, which can be done more or less by visual inspection. Next we want to understand the subsets S(λ) ⊂ S defined in Corollary 5.8. Their definition involves the positive root w^{−1}α. This is easy to calculate in our inductive calculation of S, since (ws_{β})^{−1}(α) = s_{β}(w^{−1}α). At each stage we will express w^{−1}α in terms of simple roots, and then (6.5) tells us how to apply s_{β} to this expression. Here is the result. The last four columns of the table are computed in the same way; they will be needed in section 7. We use the same shorthand as in Table 6.4 for roots.

\begin{matrix}w & l(w) & w^{−1}α & w^{−1}β_{1} & w^{−1}β_{4} & w^{−1}β_{6} & w^{−1}β_{8} \\ 1 & 0 & 2345 & 1 & 4 & 6 & 8 \\ s_{2} & 1 & 345 & 1 & 24 & 6 & 8 \\ s_{2}s_{4} & 2 & 345 & 1 & 2 & 6 & 8 \\ s_{2}s_{4}s_{3} & 3 & 45 & 13 & 2 & 6 & 8 \\ s_{2}s_{4}s_{5} & 3 & 34 & 1 & 2 & 56 & 8 \\ s_{2}s_{4}s_{3}s_{1} & 4 & 45 & 3 & 2 & 6 & 8 \\ s_{2}s_{4}s_{3}s_{5} & 4 & 4 & 13 & 2 & 56 & 8 \\ s_{2}s_{4}s_{5}s_{6} & 4 & 34 & 1 & 2 & 5 & 8 \\ s_{2}s_{4}s_{3}s_{1}s_{5} & 5 & 4 & 3 & 2 & 56 & 8 \\ s_{2}s_{4}s_{3}s_{5}s_{6} & 5 & 4 & 13 & 2 & 5 & 8 \\ s_{2}s_{4}s_{5}s_{6}s_{7} & 5 & 34 & 1 & 2 & 5 & 78 \\ s_{2}s_{4}s_{3}s_{1}s_{5}s_{6} & 6 & 4 & 3 & 2 & 5 & 8 \\ s_{2}s_{4}s_{3}s_{5}s_{6}s_{7} & 6 & 4 & 13 & 2 & 5 & 78 \\ s_{2}s_{4}s_{5}s_{6}s_{7}s_{8} & 6 & 34 & 1 & 2 & 5 & 7 \\ s_{2}s_{4}s_{3}s_{1}s_{5}s_{6}s_{7} & 7 & 4 & 3 & 2 & 5 & 78 \\ s_{2}s_{4}s_{3}s_{5}s_{6}s_{7}s_{8} & 7 & 4 & 13 & 2 & 5 & 7 \\ s_{2}s_{4}s_{3}s_{1}s_{5}s_{6}s_{7}s_{8} & 8 & 4 & 3 & 2 & 5 & 7 \end{matrix}

Table 6.6. Data for S(λ).

Inspection of this table establishes

Lemma 6.7. The non-identity elements of S may be partitioned into pairs (u, v) in such a way that

u^{−1}(α) = v^{−1}(α), sgn(u) = − sgn(v).

Every set S(λ) (Corollary 5.8) consists of the identity element and some of these pairs. Consequently

∑_{w∈S(λ)} sgn(w) = 1.

Proof. The partition into pairs is clear from Table 6.6; we can even arrange v = us_{β} for a simple root β, and then the partition is unique. The statement about S(λ) is clear from Corollary 5.8, and the sum formula follows. □

Proof of Theorem 1.1. We use the notation (4.2)–(4.4) and (4.6). Every irreducible (algebraic) representation of G is of the form V_{λ} for a dominant weight λ ∈ X^{∗}(H). The multiplicity of V_{λ} in the ring of functions on X = G·e is computed by Corollary 5.8 as a sum over S(λ); and this sum is computed by Lemma 6.7 to be 1. □

7. The model orbit in E_{8}(ℝ).

In this section we consider real group analogues of Theorem 1.1. In the notation of section 2, we therefore want to take G to be a real form of E_{8}, and X_{θ} to be an orbit of K_{ℂ} on N_{θ}^{∗}, whose complexification is the model orbit in E_{8}. Real forms of complex orbits have been tabulated in [Djok], and those tables are reproduced in chapter 9 of [Coll]. What one finds from those tables is that the model orbit has a real form only in the split real form of E_{8} (sometimes denoted E_{8(8)}); and in that case there is only one real form. In this section we will therefore take for G a simply connected split real form of E_{8}. The maximal compact subgroup K is then isomorphic to Spin(16), the compact simply connected double cover of SO(16). In accordance with (4.2), we arrange this isomorphism so that the representation of K on 𝔭 is the negative half spin representation of dimension 2^{8−1} = 128. A little more precisely, we let H be a compact Cartan subgroup of K, and choose an isomorphism

𝔥^{∗} ≃ ℂ^{8}  (7.1)(a)

in such a way that

Δ(𝔨, 𝔥) = {(±e_{i} ± e_{j}) | 1 ≤ i ≠ j ≤ 8}  (7.1)(b)

and

Δ(𝔭, 𝔥) = \{\frac{1}{2}(ϵ_{1}, ⋯, ϵ_{8}) | ϵ_{i} = ±1, ∏_{i} ϵ_{i} = −1\}.  (7.1)(c)

It will be convenient to notice that

Δ(𝔨, 𝔥) = {α ∈ Δ(𝔤, 𝔥) | α(1, 1, 1, 1, 1, 1, 1, 1) ∈ 2ℤ}.  (7.1)(b′)

We now choose e, h, and f as in (3.1), in such a way that e ∈ N_{θ}^{∗} belongs to the model orbit; equivalently, so that

dim K_{ℂ} · e = 64.  (7.1)(d)

After conjugation by K_{ℂ}, we may assume that h ∈ 𝔥 is dominant with respect to a standard positive root system for 𝔨. The tables of Djoković allow us to compute h (see [Djok], entry 6 of Table XIV, or [Coll], entry 6 on page 161). The result (determined up to the Weyl group of Spin(16)) is

h = (1, 1, 1, 1, 2, 0, 0, 0).  (7.1)(e)

Just as in section 4, we will be interested in the parabolic subalgebra defined by h, and it will again be convenient to have coordinates in which h takes the simpler form of (4.2)(c). For this purpose we will twist the identifications in (7.1)(a) by an element of the Weyl group. Specifically, consider the two roots

ϕ = \frac{1}{2}(1, −1, −1, 1, 1, −1, −1, −1), ψ = \frac{1}{2}(−1, 1, 1, −1, 1, −1, −1, −1).

These roots are orthogonal, so the product w = s_{ϕ}s_{ψ} is an element of order two in the Weyl group. We compute

w(1, 1, 1, 1, 2, 0, 0, 0) = (1, 1, 1, 1, 1, 1, 1, 1),

which is equivalent to

w(1, 1, 1, 1, 1, 1, 1, 1) = (1, 1, 1, 1, 2, 0, 0, 0).  (7.2)(a)

Our identification of 𝔥^{∗} with ℂ^{8} is that of (7.1)(a) twisted by w. The description of the compact roots in (7.1)(b′) translates to

Δ(𝔨, 𝔥) = {α ∈ Δ(𝔤, 𝔥) | α(1, 1, 1, 1, 2, 0, 0, 0) ∈ 2ℤ}.  (7.2)(b′)

From this one calculates easily

Δ(𝔨, 𝔥) = A ∪ B,
A = {±e_{i} ± e_{j} | 1 ≤ i, j ≤ 4 or 5 ≤ i, j ≤ 8}  (7.2)(b)

B = \{\frac{1}{2}(ϵ_{1}, …, ϵ_{8}) | ϵ_{i} = ±1, ϵ_{1} ⋯ ϵ_{4} = −1, ϵ_{5} ⋯ ϵ_{8} = 1\}.

The remaining roots are

Δ(𝔭, 𝔥) = C ∪ D,
C = {±e_{i} ± e_{j} | 1 ≤ i ≤ 4, 5 ≤ j ≤ 8},  (7.2)(c)

D = \{\frac{1}{2}(ϵ_{1}, …, ϵ_{8}) | ϵ_{i} = ±1, ϵ_{1} ⋯ ϵ_{4} = 1, ϵ_{5} ⋯ ϵ_{8} = −1\}.

Roughly speaking, this says that 𝔨 is built from a subalgebra 𝔰𝔬(8) × 𝔰𝔬(8) (the roots in A) by adjoining a copy of the representation spin^{−} ⊗ spin^{+}. Similarly, 𝔭_{0} is the sum of the two representations ℝ^{8} ⊗ ℝ^{8} and spin^{+} ⊗ spin^{−}. (Our passage from the standard realization of 𝔰𝔬(16) in (7.1) is an incarnation of the “triality” linking the three eight-dimensional representations ℝ^{8}, spin^{+}, and spin^{−} of 𝔰𝔬(8).) Now (7.2)(a) ensures that in our new coordinates the element h is

h = (1, 1, 1, 1, 1, 1, 1, 1).  (7.2)(d)

By analogy with (4.6), we find

Δ(𝔭(2), 𝔥) = {e_{i} + e_{j} | 1 ≤ i ≤ 4, 5 ≤ j ≤ 8},  (7.2)(e)
Δ(𝔭(3), 𝔥) = \{\frac{1}{2}(1, …, 1, ϵ_{5}, …, ϵ_{8}) | ϵ_{i} = 1 for exactly 3 values of i\}. (7.2)(f)

We will need a set of positive roots making h dominant; but the one chosen in (4.4) is not so convenient here. We modify it by permuting the coordinates, sending

(1, 2, 3, 4, 5, 6, 7, 8) → (1, 5, 2, 6, 3, 7, 4, 8).

The corresponding simple roots are

\{(e_{1} − e_{5}, e_{5} − e_{2}, e_{2} − e_{6}, e_{6} − e_{3}, e_{3} − e_{7}, e_{7} − e_{4}, e_{4} − e_{8}, \frac{1}{2}(−1, −1, 1, 1, −1, 1, 1, 1)\}.

(7.2)(g)

Just as in (4.4), we label these roots as {β_{1}, β_{3}, β_{4}, β_{5}, β_{6}, β_{7}, β_{8}, β_{2}}. One interesting property (following from (7.2)(b) and (7.2)(c)) is that all of these simple roots are noncompact.

We want to calculate the various objects introduced in (3.1)–(3.3). We imitate the ideas in section 5. First, we fix an isomorphism

j: 𝔤𝔩(4)_{L} × 𝔤𝔩(4)_{R} → 𝔩  (7.2)(h)

carrying pairs of diagonal matrices to 𝔥 ≃ ℂ^{8} in the obvious way. (The labels L and R, standing for “left” and “right,” are intended simply to distinguish the two factors of 𝔩.)

Lemma 7.3. Suppose we are in the setting of (7.2). In terms of the isomorphism (7.2)(g), the representation of 𝔩 on 𝔭(2) is (τ_{4,L}) ⊗ (τ_{4,R}), and that on 𝔭(3) is (det_{L}^{1/2}) ⊗ (τ_{4,R}^{∗} ⊗ det_{R}^{1/2}). In each case the factors enclosed in parentheses are representations of 𝔤𝔩(4). Consequently

S(𝔳) ≃ [S(τ_{4,L} ⊗ τ_{4,R})] ⊗ ∑_{k=0}^{∞}(det_{L}^{k/2}) ⊗ (S^{k}(τ_{4,R}^{∗}) ⊗ det_{R}^{k/2})

This is clear from (7.2). Of course it is precisely analogous to Lemma 5.2. In order to proceed, we need an analogue of Proposition 5.3 to calculate the term in square brackets. This is a standard fact in invariant theory, but we will again deduce it from Schmid’s result.

Proposition 7.4 ([Sch], page 63, Behauptung c)). In the setting (7.2), the symmetric algebra [S(τ_{n,L} ⊗ τ_{n,R})] is the direct sum of the representations of GL(n)×GL(n) of highest weights [(a_{1}, …, a_{n}), (a_{1}, …, a_{n})], with a_{1} ≥ ⋯ ≥ a_{n} ≥ 0 integers.

Proof. Consider the Hermitian symmetric space U(n, n)/U(n) × U(n) (see for example [Helg], page 444, or [Knapp], page 6). The holomorphic tangent space 𝔭_{+} at the identity coset is isomorphic to the outer tensor product τ_{n} ⊗ τ_{n}^{∗} of the standard representations of U(n). (This may be seen by explicit calculation with matrices, or by inspection of the root systems.) If we now twist the identification of the second factor of K by the outer automorphism inverse transpose of U(n), we get instead τ_{n} ⊗ τ_{n}. In these twisted coordinates, Schmid’s result becomes precisely what is stated here. (Just as in Proposition 5.3, we use the equivalence between representations of U(n) and GL(n, ℂ).) □

Proposition 7.5. Suppose we are in the setting (7.2). Suppose λ = (λ_{1}, …, λ_{8}) ∈ ℂ^{8} is a dominant weight for 𝔩 ≃ 𝔤𝔩(4) × 𝔤𝔩(4); write E_{λ} for the irreducible representation of highest weight λ. Then the multiplicity of E_{λ} in S(𝔳) is equal to 1 if

λ_{1} ≥ λ_{5} ≥ λ_{2} ≥ λ_{6} ≥ λ_{3} ≥ λ_{7} ≥ λ_{4} ≥ λ_{8},

with all differences integers, and

\frac{1}{2}(−λ_{1} − λ_{2} − λ_{3} + λ_{4} + λ_{5} + λ_{6} + λ_{7} + λ_{8})

is a non-negative integer. Otherwise the multiplicity is zero. In terms of the eight simple roots β_{i} of (7.2)(g) and the corresponding coroots h_{i}, the conditions for multiplicity one may be written as

λ(h_{i}) ∈ ℕ  (i = 1, 3, 4, 5, 6, 7, 8),  λ(h_{2} + h_{3} + h_{4} + h_{5}) ∈ ℕ.

Proof. The argument is exactly like the one for Proposition 5.6, using Lemma 7.3 and Proposition 7.4 in place of Lemma 5.2 and Proposition 5.3. We leave the details to the reader. □

We are going to use Corollary 3.14 to translate this analogue of Proposition 5.6 into a multiplicity formula analogous to Corollary 5.8. To do that, we need to calculate the weight ρ_{c}, half the sum of the positive compact roots. Because all of the simple roots β_{i} are noncompact, the compact roots are precisely the sums of even numbers of simple roots. If β_{i} and β_{j} are adjacent in the Dynkin diagram, it follows that β_{i} + β_{j} is a simple compact root. This accounts for seven of the eight simple compact roots. The eighth is β_{2} + β_{3} + β_{4} + β_{5}. Computing inner products of these roots, we find that the Dynkin diagram of K is

\begin{aligned}& & & & & β_{78} \\ & & & & & | \\ β_{2345} & β_{67} & β_{45} & β_{13} & β_{24} & β_{56} & β_{34} \end{aligned} (7.6)(a)

Here we have written β_{ij} for β_{i} + β_{j}. Now the value of ρ_{c} on a coroot for a compact simple root is one. This provides eight equations for the eight values ρ_{c}(h_{i}), which may be solved to give

ρ_{c}(h_{i}) = \{ \begin{matrix}1, & if i = 1, 4, 6, or 8 \\ 0, & if i = 2, 3, 5, or 7. \end{matrix}  (7.6)(b)

Corollary 7.7. Suppose we are in the setting (7.2). Write α = β_{2} + β_{3} + β_{4} + β_{5}, and h_{α} for the corresponding coroot. Fix a K-dominant weight μ ∈ X^{∗}(H), and write V_{μ} for the corresponding irreducible representation of K. Define a subset S_{0}(μ) ⊂ W_{K} by

S_{0}(μ) = {x ∈ W_{K} |x(μ + ρ_{c})(h_{α}) ∈ ℕ + 1, x(μ + ρ_{c})(h_{i}) ∈ ℕ (i = 3, 5, 7),
x(μ + ρ_{c})(h_{j}) ∈ ℕ + 1 (j = 1, 4, 6, 8)}.

Assume that Conjecture 3.13 holds. Then the multiplicity of V_{μ}^{∗} in the ring of regular functions on the normalization of \overline{K_{ℂ} · e} is

∑_{x∈S_{0}(μ)} sgn(x).

Proof. The argument is identical to the one for Corollary 5.8, with the formula in (7.6) for ρ_{c} replacing the fact that ρ(h_{i}) = 1. □

The conditions defining S_{0}(μ) involve mostly noncompact roots, so it is not so clear how to control S_{0}(μ) as a subset of the Weyl group of K. On the other hand, the conditions are very close to those defining the set S of Corollary 5.8. In order to continue, we need some elementary information about how W_{K} is included in the full Weyl group. For this we temporarily weaken the hypotheses of (7.2).

Proposition 7.8. Suppose Δ ⊂ 𝔥^{∗} is a root system, and Δ_{c} ⊂ Δ is a subroot system. Write W_{K} ⊂ W for the corresponding Weyl groups. Fix a positive system Δ^{+} ⊂ Δ, and write Δ_{c}^{+} for the corresponding positive system in Δ_{c}. Define

W^{1} = {t ∈ W | tΔ^{+} ⊃ Δ_{c}^{+}}.

In the notation of Definition 3.9, this amounts to

W^{1} = {t ∈ W | Δ^{+}(t) ∩ Δ_{c}^{+} = ∅}.

Then W^{1} is a set of coset representatives for W_{K} in W. More precisely, every element w of W has a unique decomposition

w = xt, x ∈ W_{K}, t ∈ W^{1}.

The element x is characterized by the property that

Δ^{+}(w) ∩ Δ_{c} = Δ_{c}^{+}(x)

(notation as in Definition 3.9). That is, if α is any root in Δ_{K}, then w^{−1}α is positive if and only if x^{−1}α is positive.

The special case when Δ_{c} is a Levi subsystem appears in [Kost], Proposition 5.13. The proof in general is identical.

We turn now to the computation of the set S_{0}(μ) in Corollary 7.7.

Lemma 7.9. Fix a K-dominant weight μ, and let λ be the unique G-dominant weight conjugate by W to μ + ρ_{c}. Then the set S_{0}(μ) (Corollary 7.7) is non-empty only if λ is integral for G. Assume therefore that λ is integral. For each x ∈ S_{0}(μ), let w be the unique shortest element of W such that

x(μ + ρ_{c}) = wλ.  (7.9)

Then w ∈ S (Corollary 5.8). The element x may be computed from w by Proposition 7.8, as the first term in the factorization w = xt. Furthermore μ+ρ_{c} = tλ. Finally, we have

S_{0}(μ) = {x′ ∈ W_{K} | w′ = x′t ∈ S}.

(Here the factorization of w′ is the one in Proposition 7.8.)

Proof. We first record the condition on w for it to be the shortest element solving the equation (7.9). This equation puts w in a certain coset of the stabilizer of λ, which is generated by the reflections fixing λ. By Lemma 6.2, the condition is therefore

λ(h_{i}) = 0 ⇒ wβ_{i} > 0.  (7.10)(a)

It will be convenient to rewrite this condition as

wβ_{i} < 0 ⇒ λ(h_{i}) > 0.  (7.10)(a′)

Suppose S_{0}(μ) contains an element x. The conditions in Corollary 7.7 imply that x(μ + ρ_{c}) takes integer values on all the h_{i} except perhaps for i = 2; and an integer value on h_{α} = h_{2} + h_{3} + h_{4} + h_{5}. From this it follows that the value on h_{2} is an integer as well, so x(μ + ρ_{c}) is integral. Because the Weyl group preserves integrality, λ must be integral as well.

Next, the condition for x to belong to S_{0}(μ) may be written using (7.9) as

wλ(h_{α}) ∈ ℕ + 1, wλ(h_{i}) ∈ ℕ (i = 3, 5, 7), wλ(h_{j}) ∈ ℕ + 1 (j = 1, 4, 6, 8).

(7.10)(b)

The first condition implies that λ(w^{−1}h_{α}) > 0. Because λ is dominant, we conclude that w^{−1}α is a positive root. Similarly w^{−1}β_{j} > 0 (j = 1, 4, 6, 8). To prove that w ∈ S, we only need to show that w^{−1}β_{i} > 0 (i = 3, 5, 7). Suppose not. We conclude from (7.10)(b) that β_{i} must be orthogonal to wλ, and therefore that s_{i}wλ = wλ. We may replace w by w′ = s_{i}w without affecting wλ, but Lemma 6.2 says that w′ is shorter than w. This contradicts the choice of w, completing the proof that w ∈ S.

Because μ+ρ_{c} is dominant and regular for K, (7.9) guarantees that Δ^{+}(w)∩Δ_{c} = Δ_{c}^{+}(x). Proposition 7.8 therefore implies that x is the first term in the factorization of w, and the formula for μ + ρ_{c} follows immediately.

The description of S_{0}(μ) appears at first glance to be immediate, but there are two non-trivial points to check. First, S_{0}(μ) could have an element x′ for which the corresponding w′ = x′t′ had a different factor t′, still satisfying tλ = t′λ. Second, an element w′ = x′t in S might fail to satisfy the conditions (7.10)(a) and (b). Both of these possibilities are best addressed by explicit calculation, so we postpone the end of the proof until we have some explicit information. □

Corollary 7.7 and Lemma 7.9 allow us to compute K multiplicities in the ring of functions on \overline{K_{ℂ} · e} (always assuming Conjecture 3.13). In order to make the result explicit, we need to compute the factorization of Proposition 7.8 for the elements of S. The result is in Table 7.11. The first column lists the elements of S (as for example in Table 6.4). The second lists the elements x of W_{K}, written as products of simple reflections for the positive system shown in (7.6)(a). (Here for example s_{34} denotes reflection in the root β_{34} = β_{3} + β_{4}.) The third column lists the coset representatives t ∈ W^{1}. The fourth column is the set Δ^{+}(w) of positive roots that change sign under w^{−1} (Definition 3.9). These are written using the shorthand of Table 6.4: 2456 denotes β_{2} + β_{4} + β_{5} + β_{6}. Finally, the compact roots (roots of H in 𝔨) are underlined.

Here is how Table 7.11 was computed. The first column comes from Table 6.4. The fourth column may be computed by induction on the length of w, using Lemma 6.2 and the information in Table 6.4. Because every simple root is noncompact, the compact roots are exactly those involving an even number of simple roots. (More precisely, a root ∑ m_{i}β_{i} is compact if and only if ∑ m_{i} is even.) The second column is computed as follows. According to Proposition 7.8, the element x ∈ W_{K} is characterized by the property that Δ_{c}^{+}(x) consists of the compact roots in Δ^{+}(w). That is, Δ_{c}^{+}(x) consists of the underlined roots in the fourth column. Lemma 6.2 may be used in reverse to compute x from Δ_{c}^{+}(x); this gives the second column.

Finally, the element t is x^{−1}w. This is computed from the first two columns using facts like s_{24} = s_{2}s_{4}s_{2}.

w  x  t  Δ^{+}(w)
1  1  1  ∅
s_{2}  1  s_{2}  2
s_{2}s_{4}  s_{24}  s_{2}  2, \underline{24}
s_{2}s_{4}s_{3}  s_{24}  s_{2}s_{3}  2, \underline{24}, 234
s_{2}s_{4}s_{5}  s_{24}  s_{2}s_{5}  2, \underline{24}, 245
s_{2}s_{4}s_{3}s_{1}  s_{24}s_{13}  s_{2}s_{3}  2, \underline{24}, 234, \underline{1234}
s_{2}s_{4}s_{3}s_{5}  s_{24}  s_{2}s_{3}s_{5}  2, \underline{24}, 234, 245
s_{2}s_{4}s_{5}s_{6}  s_{24}s_{56}  s_{2}s_{5}  2, \underline{24}, 245, \underline{2456}
s_{2}s_{4}s_{3}s_{1}s_{5}  s_{24}s_{13}  s_{2}s_{3}s_{5}  2, \underline{24}, 234, \underline{1234}, 245
s_{2}s_{4}s_{3}s_{5}s_{6}  s_{24}s_{56}  s_{2}s_{3}s_{5}  2, \underline{24}, 234, 245, \underline{2456}
s_{2}s_{4}s_{5}s_{6}s_{7}  s_{24}s_{56}  s_{2}s_{5}s_{7}  2, \underline{24}, 245, \underline{2456}, 24567
s_{2}s_{4}s_{3}s_{1}s_{5}s_{6}  s_{24}s_{13}s_{56}  s_{2}s_{3}s_{5}  2, \underline{24}, 234, \underline{1234}, 245, \underline{2456}
s_{2}s_{4}s_{3}s_{5}s_{6}s_{7}  s_{24}s_{56}  s_{2}s_{3}s_{5}s_{7}  2, \underline{24}, 234, 245, \underline{2456}, 24567
s_{2}s_{4}s_{5}s_{6}s_{7}s_{8}  s_{24}s_{56}s_{78}  s_{2}s_{5}s_{7}  2, \underline{24}, 245, \underline{2456}, 24567, \underline{245678}
s_{2}s_{4}s_{3}s_{1}s_{5}s_{6}s_{7}  s_{24}s_{13}s_{56} s_{2}s_{3}s_{5}s_{7} 2, \underline{24}, 234, \underline{1234}, 245, \underline{2456}, 24567
s_{2}s_{4}s_{3}s_{5}s_{6}s_{7}s_{8}  s_{24}s_{56}s_{78} s_{2}s_{3}s_{5}s_{7} 2, \underline{24}, 234, 245, \underline{2456}, 24567, \underline{245678}
s_{2}s_{4}s_{3}s_{1}s_{5}s_{6}s_{7}s_{8} s_{24}s_{56}s_{78}s_{13} s_{2}s_{3}s_{5}s_{7}  2, \underline{24}, 234, \underline{1234}, 245, \underline{2456},
24567, \underline{245678}

Table 7.11. Factorization of elements of S.

With this information in hand, we can complete the proof of Lemma 7.9. Recall that we have a dominant integral weight λ, assumed conjugate by W to some μ+ρ_{c}; say zλ = μ + ρ_{c}. If α and β are adjacent simple roots, then at least one of the three roots zα, zβ, and z(α + β) must be compact. A compact root has non-zero inner product with μ + ρ_{c}, so it follows that zλ cannot be orthogonal to both zα and zβ. Therefore λ can be orthogonal only to a collection of mutually orthogonal simple roots.

Suppose now that w′ = x′t′ ∈ S is factored as in Table 7.11. We want to understand when μ + ρ_{c} = t′λ, and x′ gives a solution to the equations (7.9) and (7.10). It will turn out that this happens only when λ is strictly positive on certain simple coroots, indicated in Table 7.12 below. The first requirement is specified in (7.10)(a′): whenever w′β_{i} < 0, λ(h_{i}) must be strictly positive. The simple roots with w′β_{i} negative are listed in Table 6.4; they appear in the third column of Table 7.12. Second, (7.10)(b) implies that λ must be strictly positive on w^{−1}h_{α} and on w^{−1}h_{j} for j = 1, 4, 6, 8. These elements are computed in Table 6.6: each is a sum of one, two, or (in the case of α) three or four simple coroots. Because λ is dominant and singular only on certain mutually orthogonal roots, this positivity is automatic on the sums of two or more simple coroots. The simple coroots appearing among w^{−1}h_{α} and w^{−1}h_{j} (for j = 1, 4, 6, 8) may be found in Table 6.6; they are listed in the fourth column of Table 7.12.

By inspection of Table 7.12, we find first of all that the element t is always a product of a set of orthogonal simple reflections from the set {s_{2}, s_{3}, s_{5}, s_{7}}. Write I(w) for the corresponding subset of {2, 3, 5, 7}, so that t = ∏_{i∈I(w)} s_{i}. We observe that all the elements of I(w) appear in the third and fourth columns of Table 7.12, and therefore that λ takes strictly positive values on the corresponding h_{i}. In the setting of (7.9)–(7.10), we therefore have

μ + ρ_{c} = tλ = λ − ∑_{i∈I(w)} m_{i}β_{i},

with m_{i} > 0. It follows from this equation that I(w), and therefore t, is determined uniquely by μ.

w  t  wβ_{i} < 0  w^{−1}h_{α},  wβ compact
w^{−1}h_{j} (j = 1, 4, 6, 8)
1  1  1, 4, 6, 8
s_{2}  s_{2}  2  1, 6, 8  4
s_{2}s_{4}  s_{2}  4  1, 2, 6, 8  4
s_{2}s_{4}s_{3}  s_{2}s_{3}  3  2, 6, 8  1
s_{2}s_{4}s_{5}  s_{2}s_{5}  5  1, 2, 8  6
s_{2}s_{4}s_{3}s_{1}  s_{2}s_{3}  1  2, 3, 6, 8  1
s_{2}s_{4}s_{3}s_{5}  s_{2}s_{3}s_{5}  3, 5  2, 4, 8  1, 4, 6
s_{2}s_{4}s_{5}s_{6}  s_{2}s_{5}  6  1, 2, 5, 8  6
s_{2}s_{4}s_{3}s_{1}s_{5}  s_{2}s_{3}s_{5}  1, 5  2, 3, 4, 8  1, 4, 6
s_{2}s_{4}s_{3}s_{5}s_{6}  s_{2}s_{3}s_{5}  3, 6  2, 4, 5, 8  1, 4, 6
s_{2}s_{4}s_{5}s_{6}s_{7}  s_{2}s_{5}s_{7}  7  1, 2, 5  8
s_{2}s_{4}s_{3}s_{1}s_{5}s_{6}  s_{2}s_{3}s_{5}  1, 6  2, 3, 4, 5, 8  1, 4, 6
s_{2}s_{4}s_{3}s_{5}s_{6}s_{7} s_{2}s_{3}s_{5}s_{7}  3, 7  2, 4, 5  1, 4, 8
s_{2}s_{4}s_{5}s_{6}s_{7}s_{8}  s_{2}s_{5}s_{7}  8  1, 2, 5, 7  8
s_{2}s_{4}s_{3}s_{1}s_{5}s_{6}s_{7} s_{2}s_{3}s_{5}s_{7}  1, 7  2, 3, 4, 5  1, 4, 8
s_{2}s_{4}s_{3}s_{5}s_{6}s_{7}s_{8} s_{2}s_{3}s_{5}s_{7}  3, 8  2, 4, 5, 7  1, 4, 8
s_{2}s_{4}s_{3}s_{1}s_{5}s_{6}s_{7}s_{8} s_{2}s_{3}s_{5}s_{7}  1, 8  2, 3, 4, 5, 7  1, 4, 8

Table 7.12. Roots on which λ must be positive to satisfy (7.9)–(7.10).

This resolves the first of the two problems mentioned at the end of the argument for Lemma 7.9: since t is determined by μ, S_{0}(μ) is clearly a subset of the set in Lemma 7.9. What remains is to show that every element of this set—that is, every w′ = x′t in S—actually contributes a solution to the equations (7.9)–(7.10). The nature of the difficulty is apparent for example when w = s_{2} and w′ = s_{2}s_{4}. We are given μ, and so we know that the corresponding λ solves the equations with w. From Table 7.12 we deduce that λ must be strictly positive on the roots β_{1}, β_{2}, β_{6}, and β_{8}. In order for w′ to give a solution as well (as Lemma 7.9 says it should) Table 7.12 says that we also need λ to be strictly positive on β_{4}. But we compute

λ(h_{4}) = w′λ(w′h_{4}) = x′(μ + ρ_{c})(−h_{24}).

Here in the last equality we use (7.9) and the calculation in Table 6.4. Since h_{24} is a compact coroot (because it involves an even number of simple coroots) the K-regular weight x′(μ + ρ_{c}) must take a non-zero value on it. We conclude that λ(h_{4}) ≠ 0, as we wished to show. More generally, we find that if λ and w satisfy (7.9)–(7.10), β is a simple root, and wβ is compact, then λ(h_{β}) is automatically positive. These roots are listed in the last column of Table 7.12. They are the ones for which wβ is a sum of an even number of simple roots, so they may be read off from Table 6.4.

We now find that for a fixed value of t, the same simple roots appear in the last three columns of Table 7.12 for all elements w with this fixed t. This completes the verification of the formula for S_{0}(μ), and finishes the omitted part of the proof of Lemma 7.9.

Theorem 7.13. Suppose we are in the setting (7.2). Fix a K-dominant weight μ ∈ X^{∗}(H), and write V_{μ} for the corresponding irreducible representation of K. Assume that Conjecture 3.13 holds. Then the multiplicity of V_{μ}^{∗} in the ring of regular functions on the normalization of \overline{K_{ℂ} · e} is equal to 1 if μ is dominant and integral for G (and the positive system (7.2)(g)) and zero otherwise.

Proof. The multiplicity is calculated by Corollary 7.7 as a sum of signs of certain Weyl group elements S_{0}(μ). This set is empty unless μ+ρ_{c} is integral for G (Lemma 7.9). Because ρ_{c} is integral for G (by (7.6)(b)), this is equivalent to the integrality of μ. Now assume μ is integral, and define λ to be the dominant weight conjugate to μ + ρ_{c}. Lemma 7.9 calculates S_{0}(μ) whenever it is non-empty; it consists of all elements x in W_{K} so that xt belongs to S, with t a certain fixed element of W^{1}. Table 7.11 shows that S_{0}(μ) (when it is non-empty) has 1, 2, or 4 elements; and that the sum of the signs of these elements is zero except in the first case. So the multiplicity is zero unless the elements t and w are both trivial; that is, unless μ+ρ_{c} is G-dominant. Table 7.12 shows that in this case λ = μ + ρ_{c} must actually have strictly positive inner product with the roots β_{1}, β_{4}, β_{6}, and β_{8}; so by (7.6)(b), μ is also G-dominant.

We have shown that the multiplicity is zero unless μ is integral and dominant. Conversely, if μ is integral and dominant, then x = 1 belongs to S_{0}(μ) (by inspection of (7.6) and Corollary 7.7), and the corresponding element w from Lemma 7.9 is 1. Therefore Lemma 7.9 shows that S_{0}(μ) = {1}, so the multiplicity is one. □

8. Representations of E_{8}(ℝ)

In this section we want to consider the representation-theoretic consequences of the calculations in the last section. As explained in section 2, this means that we want to consider not just functions on K_{ℂ} · e, but sections of certain bundles over the orbit. In order to describe the bundles, we need to understand more precisely the isotropy group K_{ℂ}(e). Recall that K is Spin(16). We will work mostly inside the Levi subgroup L_{ℂ} of K_{ℂ} with Lie algebra 𝔩 = 𝔤𝔩(4)_{L} × 𝔤𝔩(4)_{R} (see (7.2)(h)); recall from (3.1) that L_{ℂ} is the subgroup fixing h. We need to understand the group L_{ℂ}. The torus H_{ℂ} is a maximal torus of L_{ℂ}. We will first compute the lattice of rational characters X^{∗}(H) for this torus, and then use that to deduce the structure of L_{ℂ}. Recall that we have identified 𝔥^{∗} with ℂ^{8} in (7.1)(a). It will be convenient to write this as

𝔥^{∗} = {λ = (λ_{L}, λ_{R}) | λ_{L}, λ_{R} ∈ ℂ^{4}}.  (8.1)(a)

It will be convenient to understand at the same time structures related to the adjoint group

\overline{G} = G/Z(G);  (8.1)(b)

we write \overline{H}, \overline{K}, \overline{L}, and so on for the corresponding subgroups. The character lattice of \overline{H}_{ℂ} is just the root lattice of E_{8}. This is easily computed from (4.2) to be

X^{∗}(\overline{H}) = \{λ ∈ ℤ^{8} | ∑ λ_{i} ∈ 2ℤ\} ∪ \{λ ∈ (ℤ + \frac{1}{2})^{8} | ∑ λ_{i} ∈ 2ℤ + 1\}. (8.1)(c)

As an introduction to the calculations to come, we will use this description to describe the Levi subgroup of E_{8}(ℂ) appearing in (4.2). We will call this group F instead of L to avoid confusion with the Levi subgroup L of K. The Lie algebra 𝔣 is 𝔤𝔩(8, ℂ) by (5.1). If Λ is a lattice containing the root lattice of GL(8, ℂ) and satisfying appropriate integrality conditions, then there is a complex group F (Λ) locally isomorphic to GL(8), with weight lattice Λ. It is some of these groups F (Λ) that we wish to understand. Notice first that X^{∗}(\overline{H}) has index two in the lattice

Λ_{1/2}^{8} = ℤ^{8} ∪ (ℤ + \frac{1}{2})^{8}.  (8.2)(a)

It follows that the group F = F (X^{∗}(\overline{H})) is a quotient of F (Λ_{1/2}^{8}) by a two-element central subgroup. On the other hand, Λ_{1/2}^{8} contains ℤ^{8} as a sublattice of index 2. It follows that F (Λ_{1/2}^{8}) is a two-fold cover of F (ℤ^{8}). This last group is just GL(8, ℂ). The two-fold cover may be constructed by the “square root of the determinant” construction:

F (Λ_{1/2}^{8}) = {(g, z) | g ∈ GL(8, ℂ), z ∈ ℂ^{×}, det(g) = z^{2}}.  (8.2)(b)

The covering map from F (Λ_{1/2}^{8}) to GL(8) is projection on the first factor. Projection on the second factor defines a character of F (Λ_{1/2}^{8}), which is a square root of the determinant character of GL(8). The differential of this character is therefore half the differential of the determinant character, or (1/2, …, 1/2). Using these ideas, it is easy to verify that F (Λ_{1/2}^{8}) really has the weight lattice we want. Finally, we must divide F (Λ_{1/2}^{8}) by a two-element central subgroup on which the weights of X^{∗}(\overline{H}) act trivially. This is easy to compute. Define

ϵ = (−I_{8}, −1) ∈ F (Λ_{1/2}^{8}).  (8.2)(c)

A character λ ∈ ℤ^{8} sends ϵ to (−1)^{∑ λ_{i}}, which is one exactly when λ ∈ X^{∗}(\overline{H}). To calculate the action of λ ∈ (ℤ + \frac{1}{2})^{8}, we write it as a sum of (1/2, …, 1/2) and an element of ℤ^{8}. The first summand acts on ϵ by projection on the second factor (hence by -1); so altogether we find

λ(ϵ) = −(−1)^{∑(λ_{i}−1/2)} = −(−1)^{∑ λ_{i}}.  (8.2)(d)

This is one exactly when λ ∈ X^{∗}(\overline{H}). Consequently

F (X^{∗}(\overline{H})) = F (Λ_{1/2}^{8})/\{1, ϵ\}.  (8.2)(e)

This is the Levi subgroup considered in (4.2).

We return now to the Levi subgroup L_{ℂ} ⊂ K_{ℂ} considered in (7.2). Recall that L_{ℂ} is locally isomorphic to GL(4) × GL(4). Just as above, if Λ is an appropriately integral lattice containing the root lattice of GL(4) × GL(4), we will write L_{ℂ}(Λ) for a locally isomorphic group with weight lattice Λ. The image of L in the adjoint group is easily computed from (8.2). It is a subgroup of F, namely

\overline{L_{ℂ}} = L_{ℂ}(X^{∗}(\overline{H})) = \{(g, h, z) | g, h ∈ GL(4), z ∈ ℂ^{×}, det(gh) = z^{2}\}/\{1, ϵ\}.

(8.3)(a)

Here ϵ = (−I_{4}, −I_{4}, −1) is the element from (8.2)(c).

Because the maximal compact subgroup of a simply connected group is simply connected, the lattice X^{∗}(H) consists of the weights λ ∈ 𝔥^{∗} integral with respect to the compact roots. Because the roots all have length two in the standard inner product, these are the weights having integer inner product with all compact roots. The compact roots are listed in (7.2)(b). Integrality with respect to the roots A leads to the lattice

Λ_{1/2}^{4} = ℤ^{4} ∪ (ℤ + \frac{1}{2})^{4}.  (8.3)(b)

Imposing integrality with respect to the roots B gives

X^{∗}(H) = {(λ_{L}, λ_{R}) ∈ Λ_{1/2}^{4} × Λ_{1/2}^{4} | ∑ λ_{i} ≡ 2λ_{8} (mod 2ℤ)}  (8.3)(c)

The group with weight lattice Λ_{1/2}^{4} may be computed as a double cover of GL(4) just as in (8.2)(b); L_{ℂ}(Λ_{1/2}^{4} × Λ_{1/2}^{4}) is a product of two copies of this double cover. Because X^{∗}(H) is a sublattice of index 2 in the product lattice, the group L_{ℂ} is a quotient of the product group by a subgroup of order two. Explicitly,

L_{ℂ} = L_{ℂ}(X^{∗}(H)) = {(g_{L}, z_{L}, g_{R}, z_{R}) ∈ (GL(4) × ℂ^{×})^{2} | z_{x}^{2} = det(g_{x})}/{1, δ}.

(8.3)(d)

Here δ = (−I_{4}, 1, −I_{4}, −1). The verification that this group has the correct weight lattice is parallel to the argument for (8.2)(e), and we omit it. The two-to-one covering map from L_{ℂ} to \overline{L_{ℂ}} may be defined before the quotients as

(g_{L}, z_{L}, g_{R}, z_{R}) ↦ (g_{L}, g_{R}, z_{L}z_{R})  (8.3)(e)

Since this map carries δ to ϵ, it passes to the quotients.

The representation of L_{ℂ} on 𝔭(2) has been identified in Lemma 7.3. Identifying ℂ^{4} ⊗ ℂ^{4} with 4 by 4 matrices, we find that the class of (g_{L}, z_{L}, g_{R}, z_{R}) acts by

(g_{L}, z_{L}, g_{R}, z_{R}) · A = g_{L}A^{t}g_{R}.  (8.3)(f)

In this identification the matrix unit e_{ij} has weight (e_{i}, e_{j}) ∈ ℤ^{4} × ℤ^{4}. Using (4.3), we deduce that the element e may be taken to be the identity matrix I_{4}. Now (8.3)(f) and (8.3)(d) allow us to compute the stabilizer of e in L_{ℂ}: it is

L_{ℂ}(e) = {(g, z,^{t}g^{−1}, w) | z^{2} = det(g) = w^{−2}}/{1, δ}.  (8.3)(g)

Before passage to the quotient, this is easily seen to be a direct product

{(g, z,^{t}g^{−1}, z^{−1}) | z^{2} = det(g)} × {1, δ}.

Consequently L_{ℂ}(e) is isomorphic to the first factor. This in turn is the connected double cover of GL(4):

L_{ℂ}(e) ≃ {(g, z) | g ∈ GL(4), z ∈ ℂ^{×}, z^{2} = det(g)}.  (8.3)(h)

Now Proposition 3.2 implies that K_{ℂ}(e) is connected.

We turn next to the determination of the admissible orbit data. According to Definition 2.6, we must begin by computing the character γ by which K_{ℂ}(e) acts on the top exterior power of (𝔨/𝔨(e))^{∗}. This is an algebraic character, so the unipotent radical U(e) acts trivially. It is therefore enough to compute the action of L_{ℂ}(e) (Proposition 3.2). Using the representation theory of SL(2), one can reorganize this problem in various ways. Here is one that will suffice for our purposes.

Lemma 8.4. Suppose we are in the setting (3.1). Define τ(m) to be the character of L_{ℂ} given by the determinant of the adjoint action on 𝔨(m). That is, the weight of τ(m) (with respect to a Cartan subalgebra of 𝔨 containing h) is the sum of the compact roots α such that α(h) = m. Consider the character

τ = τ(1) ⊗ τ(3)^{2} ⊗ τ(5)^{2} ⊗ ⋯

of L_{ℂ}. Then the restriction of τ to L_{ℂ}(e) agrees with γ(e) (Definition 2.6) up to sign. In particular, τ and γ(e) agree on the identity component of L_{ℂ}(e).

This result refines Corollary 7.27 of [Vunip]. We omit the proof; the necessary ideas (taken from [Schwartz]) may be found in section 7 of [Vunip] (particularly (7.26)(a) and (7.24)(f)). In our case the weights τ(m) may be computed from (7.2)(b). We find

τ(1) = (5, 5, 5, 5, 2, 2, 2, 2), τ(3) = (1, 1, 1, 1, 2, 2, 2, 2),

and therefore

τ = τ(1) + 2τ(3) = (7, 7, 7, 7, 6, 6, 6, 6).  (8.5)(a)

The restriction to L_{ℂ}(e) of any weight (λ_{L}, λ_{R}) for L_{ℂ} is λ_{L} − λ_{R} by (8.3)(g). Lemma 8.4 now gives the character γ(e) of Definition 2.6 as

γ(e) = (1, 1, 1, 1),  (8.5)(b)

the ordinary determinant character of GL(4); of course this is using the identification (8.3)(h) of L_{ℂ}(e). This character has a unique square root χ(e), which is the unique admissible orbit datum for K_{ℂ} · e (Definition 2.6). Explicitly,

χ(e) = (1/2, 1/2, 1/2, 1/2).  (8.5)(c)

Conjecture 2.9 says that the K-types of a representation associated to this nilpotent orbit should be given by the sections of the line bundle V_{χ(e)} over K_{ℂ}·e. Proposition 3.15 suggests a way to compute these sections. We first need a character δ(e) of L_{ℂ} whose restriction to L_{ℂ}(e) is χ(e). A natural choice is

δ(e) = (1/2, 1/2, 1/2, 1/2, 0, 0, 0, 0).  (8.5)(d)

(Many other characters, such as (1/2 + m, 1/2 + m, 1/2 + m, 1/2 + m, m, m, m, m), also restrict to χ(e). Our choice is the smallest one possible, and the weight δ(e) is dominant for K. The first property of δ(e) should make it possible to prove a cohomology vanishing theorem in the setting of Proposition 3.15; and the second should make the inclusion of Proposition 3.15 more nearly an isomorphism. We will not prove any results in this direction, but an analogous argument appears in [Gra] (Proposition 1.4 and Theorem 1.3).) At any rate, we make Conjecture 8.6. Suppose G is the simply connected split real group of type E_{8}, K ≃ Spin(16) is a maximal compact subgroup, and e ∈ 𝔭 is a representative of the nilpotent K_{ℂ} orbit of complex dimension 64. In the notation of (3.1) and (3.3), let δ(e) be the character of Q (a parabolic subgroup of K_{ℂ}) be the character introduced in (8.5).

1. The inclusion

∑_{k=0}^{∞} Ind_{Q}^{K_{ℂ}}(S^{k}(𝔳^{∗}) ⊗ V_{δ(e)}) ↪ sections of V_{χ} over K_{ℂ} · e

of Proposition 3.15 is an isomorphism.

2. For all p > 0, the higher cohomology spaces (Ind_{Q}^{K_{ℂ}})^{p}(S^{k}(𝔳^{∗}) ⊗ V_{δ(e)}) are equal to zero.

We can now imitate the calculation of section 7 with the twist by δ(e). In the setting of Proposition 7.5, E_{λ} occurs in S(𝔳) ⊗ V_{δ(e)}^{∗} if and only if E_{λ} ⊗ V_{δ(e)} occurs in S(𝔳). The highest weight of E_{λ} ⊗ V_{δ(e)} is

(λ_{1} + 1/2, …, λ_{4} + 1/2, λ_{5}, …, λ_{8}).

Proposition 8.7. Suppose we are in the setting (7.2), and δ(e) is the character of (8.5)(d). Suppose λ = (λ_{1}, …, λ_{8}) ∈ ℂ^{8} is a dominant weight for 𝔩 ≃ 𝔤𝔩(4) × 𝔤𝔩(4); write E_{λ} for the irreducible representation of highest weight λ. Then the multiplicity of E_{λ} in S(𝔳) ⊗ V_{δ(e)}^{∗} is equal to 1 if

λ_{1} + 1/2 ≥ λ_{5} ≥ λ_{2} + 1/2 ≥ λ_{6} ≥ λ_{3} + 1/2 ≥ λ_{7} ≥ λ_{4} + 1/2 ≥ λ_{8},

with all differences integers, and

\frac{1}{2}(−λ_{1} − λ_{2} − λ_{3} + λ_{4} + λ_{5} + λ_{6} + λ_{7} + λ_{8}) − 1/2

is a non-negative integer. Otherwise the multiplicity is zero. In terms of the eight simple roots β_{i} of (7.2)(g) and the corresponding coroots h_{i}, the conditions for multiplicity one may be written as

λ(h_{i}) ∈ ℕ + 1/2  (i = 3, 5, 7),  λ(h_{j}) ∈ ℕ − 1/2  (j = 1, 4, 6, 8),
λ(h_{2} + h_{3} + h_{4} + h_{5}) ∈ ℕ + 1/2.

This is just a translation of Proposition 7.5.

Corollary 8.8. Suppose we are in the setting (7.2). Write α = β_{2} + β_{3} + β_{4} + β_{5}, and h_{α} for the corresponding coroot. Fix a K-dominant weight μ ∈ X^{∗}(H), and write V_{μ} for the corresponding irreducible representation of K. Define a subset S̃_{0}(μ) ⊂ W_{K} by

S̃_{0}(μ) = {x ∈ W_{K} | x(μ + ρ_{c})(h_{α}) ∈ ℕ + 3/2, x(μ + ρ_{c})(h_{i}) ∈ ℕ + 1/2 (i ≠ 2)}

Assume that Conjecture 8.6 holds. Then the multiplicity of V_{μ}^{∗} in the space of sections of V_{χ} over K_{ℂ} · e is

∑_{x∈S̃_{0}(μ)} sgn(x).

The proof is identical to that of Corollary 7.7.

The calculation of the sets S̃_{0}(μ) is actually easier than in section 7. The reason is that the conditions in Corollary 8.8 force μ + ρ_{c} to be regular. (The reason is that its value on each noncompact coroot must belong to ℤ + 1/2, and we already know that the values on compact coroots are non-zero.) Here is the main step.

Lemma 8.9. Fix a K-dominant weight μ, and let λ be the unique G-dominant weight conjugate by W to μ + ρ_{c}. Then the set S̃_{0}(μ) (Corollary 7.7) is nonempty only if λ(h_{i}) is regular for G. Assume therefore that λ is regular. For each x ∈ S̃_{0}(μ), let w be the unique element of W such that

x(μ + ρ_{c}) = wλ.  (8.9)

Then w ∈ S (Corollary 5.8). The element x may be computed from w by Proposition 7.8, as the first term in the factorization w = xt. Furthermore μ+ρ_{c} = tλ. Finally, we have

S̃_{0}(μ) = {x′ ∈ W_{K} | w′ = x′t ∈ S}.

(Here the factorization of w′ is the one in Proposition 7.8.)

The proof is a simplified version of that of Lemma 7.9, so we omit it.

Theorem 8.10. Suppose we are in the setting (7.2). Fix a K-dominant weight μ ∈ X^{∗}(H), and write V_{μ} for the corresponding irreducible representation of K. Assume that Conjecture 8.6 holds. Then the multiplicity of V_{μ}^{∗} in the space of sections of the line bundle V_{χ} over K_{ℂ} · e is equal to 1 if

(μ + ρ_{c})(h_{i}) ∈ ℕ + 1/2, (i = 1, 2, …, 8).

Otherwise the multiplicity is equal to zero.

The proof is a simplified version of that of Theorem 7.13. The main point is that in Table 7.11, the elements x appearing in factorizations xt (for fixed non-identity t) have signs summing to zero. We omit the details.

Recall that Conjecture 2.9 says that the K-types computed in Theorem 8.10 should be those of a unitary unipotent representation. It is not completely trivial to list the representations of Spin(16) given by Theorem 8.10, but there is certainly an obvious “lowest” one. (In fact it is very easy to see that it is lowest in the technical sense of [Green].) This is the one with highest weight μ_{0} defined by

(μ_{0} + ρ_{c})(h_{i}) = 1/2, (i = 1, …, 8).  (8.11)(a)

Using (7.6)(b) we deduce

μ_{0}(h_{i}) = \{ \begin{matrix}−1/2, & if i = 1, 4, 6, or 8 \\ 1/2, & if i = 2, 3, 5, or 7. \end{matrix}  (8.11)(b)

Using (7.6)(a), we can now compute the value of μ_{0} on the compact simple coroots h_{β}. The result is

μ_{0}(h_{β}) = \{ \begin{matrix}1, & if β = α_{2} + α_{3} + α_{4} + α_{5} \\ 0, & otherwise. \end{matrix}  (8.11)(c)

That is, V_{μ_{0}} is the fundamental representation of Spin(16) attached to the simple root at the unbranched end of the Dynkin diagram. This is the standard 16dimensional representation:

V_{μ_{0}} = ℂ^{16}.

We would like to identify the representation of G that (according to Conjecture 2.5) is attached to the real nilpotent orbit of dimension 128. According to Conjecture 2.9 and (8.11), it should have lowest K-type the 16-dimensional representation. We therefore look at the entire family of representations of G having this lowest Ktype. Fix a minimal parabolic subgroup P = MAN of G; here A is a vector group with Lie algebra 𝔞_{0} a maximal abelian subalgebra of 𝔭_{0}, and M is the centralizer of A in G. Because \overline{G} is a linear group split over ℝ, its Cartan subgroup \overline{M}\overline{A} is a product of 8 copies of ℝ^{×}. Consequently

\overline{M} ≃ (ℤ/2ℤ)^{8},  (8.12)(a)

an abelian group of order 256. The preimage M of \overline{M} in G is therefore a group of order 512. According to the Bernstein-Gelfand-Gelfand theory of fine representations of \overline{K} (see for example [Green], Theorem 4.3.16,)

(S^{2}(ℂ^{16}) + ∧^{2}ℂ^{16}) |\overline{M} ≃ regular representation of \overline{M}.  (8.12)(b)

Now the left side here is just the tensor product of the (self-dual) representation ℂ^{16} with itself. By Schur’s lemma, the fact that the trivial representation of M appears just once in the tensor product guarantees that

ℂ^{16} |_{M} ≃ irreducible representation δ of M.  (8.12)(c)

Because the sum of the squares of the dimensions of the irreducible representations of the finite group M must be 512, we see that the 256 one-dimensional representations of \overline{M} and the 16-dimensional representation δ of M together exhaust M̂. In fact the theory of fine representations extends to the non-linear group G (see [Alg], section 6); one finds that ℂ^{16} is a fine representation of K.

Proposition 8.13 ([Alg], Proposition 6.7). Suppose as above that P = MAN is a minimal parabolic subgroup of a split real group of type E_{8}, and that δ is the 16dimensional irreducible representation of M. Write K ≃ Spin(16) for a maximal compact subgroup of G, and \overline{K} for its image in the linear group \overline{G}.

1. If V is any irreducible representation of K not factoring to \overline{K}, then V |_{M} is a multiple of δ. In particular, the dimension of V is divisible by 16.

2. If ν ∈ Â ≃ 𝔞^{∗} is any character, define

I(δ ⊗ ν) = Ind_{P}^{G}(δ ⊗ ν ⊗ 1),

a principal series representation of G. The restriction of I(δ ⊗ ν) to K is the sum of all the irreducible representations of K not factoring to \overline{K}; each occurs with multiplicity equal to its dimension divided by 16.

3. The infinitesimal character of I(δ ⊗ν) is given in the Harish-Chandra parametrization by the weight ν ∈ 𝔞^{∗}.

4. The character of I(δ ⊗ ν) depends only on the Weyl group orbit of ν.

5. Define J(δ ⊗ ν) to be the unique irreducible subquotient of I(δ ⊗ ν) containing the K-type ℂ^{16} (which has multiplicity one). Then the representations J(δ ⊗ ν) exhaust the irreducible (𝔤, K)-modules containing the K-type ℂ^{16}. Two of these are equivalent if and only the weights in 𝔞^{∗} are conjugate by the Weyl group.

In addition to the results in [Alg], the main observation incorporated here is that the stabilizer in the Weyl group of δ is the entire Weyl group. This is clear from (8.12)(c), or from the fact that δ is the only representation of M of dimension 16.

Proposition 8.13 says that there is exactly one irreducible representation of G containing the K-type ℂ^{16} for each infinitesimal character. In order to specify the representation attached to our 128-dimensional orbit by Conjecture 2.5, we need only specify the infinitesimal character. Here is a precise statement.

Conjecture 8.14. Suppose G is the simply connected split real Lie group of type E_{8}, and λ ∈ i𝔤_{0}^{∗} is a representative of the unique nilpotent coadjoint orbit of dimension 128. With notation as in Proposition 8.13, let ρ ∈ 𝔞^{∗} be half the sum of the positive roots. Then the irreducible (𝔤, K)-module J(δ⊗(ρ/2)) is an irreducible unitary representation of G attached to the orbit G·λ. In particular, the restriction of J(δ⊗(ρ/2)) to K is described by Theorem 8.10: the K-types all have multiplicity one, and lie in the translate of the cone of dominant integral weights for E_{8} by the highest weight of ℂ^{16}.

The concrete content of the conjecture is in the assertion that J(δ ⊗ (ρ/2)) is unitary, and in the description of its K-types. It is not very difficult to show that the annihilator of J(δ⊗(ρ/2)) in U(𝔤) is the maximal primitive ideal of infinitesimal character ρ/2. The corresponding primitive quotient U(𝔤)/I is the Harish-Chandra module for complex E_{8} studied by McGovern in [McGmodel]. He shows that its Ktypes are a model representation for the compact form of E_{8}, in agreement with the prediction given by Theorem 1.1. According to the general philosophy of unipotent representations of real groups, having this kind of “unipotent annihilator” is a necessary condition for being a unipotent representation (see [Vorange], chapter 9). But we do not know how to prove either of the concrete assertions in the conjecture.

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Department of Mathematics, University of Maryland, College Park, Maryland 20742

E-mail address: jda@math.umd.edu

Department of Mathematics, Hong Kong University of Science and Technology, Clear Water Bay, Kowloon, Hong Kong

E-mail address: mahuang@uxmail.ust.hk

Department of Mathematics, Massachusetts Institute of Technology, Cambridge, Massachusetts 02139

E-mail address: dav@math.mit.edu