December 18, 2004

RELEVANT AND PETITE K-TYPES FOR SPLIT GROUPS

DAN BARBASCH

1. Introduction

  [sec:1]

This paper discusses the role of the relevant and petite K-types in the classification of the spherical unitary dual for split real groups.

Let G be the rational points of a connected linear reductive group over 𝔽 = ℝ or a p-adic field. Assume G is split, and denote a maximal abelian torus by 𝔸 ≅ (𝔽^{×})^{n}. Let K be a maximal compact subgroup as in section 2. A representation is called spherical if it has nontrivial K-fixed vectors. In both the real and p-adic case, any irreducible spherical representation is realized in a canonical way as a subquotient, denoted L(χ), of a principal series X(χ) for a character χ ∈ 𝔸̂. We will use subscript R for the real case, and F for the p-adic case when we want to distinguish between them.

Two such representations L(χ) and L(χ′) are equivalent if and only if there is an element w of the Weyl group W such that wχ = χ′. The module L(χ) admits a nondegenerate hermitian form if and only if there is w ∈ W such that wχ = \overline{χ}^{−1}. The hermitian form is obtained from a hermitian form on X(χ); an L(χ) is unitary if and only if the form on X(χ) is positive semidefinite. For each K-type (E, μ), there is a hermitian form A(μ, χ) which depends meromorphically on χ. Thus L(χ) is unitary if and only if A(μ, χ) is positive semidefinite for all μ.

In the p-adic case, the results in [BM1] and [BM2] reduce the problem to a finite one, namely to show that certain a(σ, χ) (obtained from the A(μ, χ)) are positive semidefinite for all σ ∈ Ŵ. These ideas are used in [BM3] to obtain the spherical unitary dual for split p-adic groups of classical type. The full unitary dual for GL(n) was obtained by Tadić, and for G_{2} by Muić.

Consider now the real case. Let M be the centralizer of 𝔸 in K. group. Then (E_{μ}^{∗})^{M} is a representation of the Weyl group, which we denote by σ. The operator A(μ, χ) induces, by Frobenius reciprocity, a hermitian form a(μ, χ) on (E_{μ}^{∗})^{M}. In sections 3 and 4 we define a set of K-types, which we call petite. They have the property that a(μ, χ) only depends on σ, and in fact coincides with the a(σ, χ) defined in the context of p-adic groups for σ ∈ Ŵ.

In [B5], for the classical cases, I introduced a subset of Ŵ called relevant, which has the following properties:

(a) A representation L_{F} (χ) is unitary if and only if a(σ, χ) is positive semidefinite for all σ relevant.

(b) For every relevant σ there is a petite representation E_{μ} of the real

group such that (E_{μ}^{∗})^{M} ≅ σ. We will call the K-types in part (b) relevant as well. Finding such a set of W -representations, implies the following result.

Theorem (1). [t:0] A representation L_{R}(χ) is unitary only if L_{F} (χ) is unitary.

For split p-adic F_{4}, the spherical unitary dual (in fact the full Iwahori spherical unitary dual) is obtained in [C]. For p-adic split type E_{6}, E_{7}, E_{8}, the spherical unitary dual is obtained in [BC]. In all these cases, a set satisfying part (a) is given. In this paper we show that these W -representations also satisfy part (b). As a consequence, theorem (1) holds in all cases.

It is natural to conjecture that the only if can be strengthened to if and only if, in the theorem. This is done for the classical groups in [B5].

The proof of the matchup between petite K-types and relevant W -types is different from the one in [B5]. Instead of identifying the groups M and W explicitly and using restrictions to Levi components, I use properties of fine K-types, and tensor products. To decompose tensor products in the exceptional cases, I used the packages GAP and LiE. The results for E_{6} − E_{8} contain matchups for a larger class of petite K-types than needed for theorem (1). The definitions and background are in sections 2 and 3. The results are in section 4, and the calculations are in section 5.

Let us first consider the problem of restricting a K-type to M. The standard restriction formulas are designed to deal with restricting to a subgroup H ⊂ K which shares a Cartan subgroup with K. This is not the case for M. One strategy would be to compute the restriction of the fundamental representations, and set up an induction by realizing K-types as composition factors of tensor products of fundamental representations. We can then exploit the fact that there are good formulas for decomposing tensor products.

It is certainly feasible to obtain the restrictions of fundamental representations to M. Combined with the induction sketched above, this gives an efficient way to restrict a K-type to M. However the restrictions of the fundamental representations to M are not very nice. On the other hand, the fine K-types (section 3) have very nice restrictions. Precisely, each fine K-type decomposes into the Weyl orbit of a single M-type, and every Mtype occurs in at least one fine K-type. So the general aim is to replace the fundamental representations by the fine K-types. This works very well in the case of GL(n, ℝ) and SO(n, n), and Sp(n, ℝ) where all the fundamental representations are fine. To accomplish this, we need to use the groups, K̃ and M̃, which come from covers of the rational points of connected reductive groups.

For SO(n+1, n), the fine K-types do not generate the Grothendieck group of representations of K. However we can establish a weaker form of part (b) sufficient for theorem (1), by decomposing a tensor product of fine K-types of a cover of SO(n + 1, n). It is also not difficult to restrict the fundamental representations of K to M (this is implicit in [B5]). But we need a more explicit knowledge of the structure of M. In any case, these ideas give an efficient method for restricting representations of K to M.

For the real group of type G_{2}, the fine K-types do not generate the Grothendieck group of representations of K. However we can use the representation of K on mathfraks, from the Cartan decomposition mathfrakg = mathfrakk + mathfraks. Its restriction to M is easy to ascertain. In this case the full unitary is contained in the work of [V1]. So the if and only if analogue of theorem (1) is known to hold for spherical representations.

In the case of F_{4}, the fine K-types do not generate the Grothendieck group of representations of K. We can again use the representation of mathfraks.

In the cases of E_{6}, E_{7}, and E_{8}, the Grothendieck group is not generated by the fine K-types either. In addition to using mathfraks, the calculations are simplified considerably in the cases of E_{6} and E_{8} by the fact that M̃ has a unique genuine representation. Thus a genuine representation of K̃ restricts to a multiple of the genuine M̃-type,and the multiplicity is determined from the dimension.

To establish part (b), we also need to determine the Weyl group representation on the M-fixed vectors. This is done by using theorem 3.3 together with sections 5.1-5.2.

I would like to thank David Vogan for generously sharing his ideas about the relation between K-types, Weyl group representations and signatures. They were the catalyst for this paper. Thanks are also due to P. Trapa for carefully reading the manuscript, and making valuable suggestions for improvements.

The paper is an outgrowth of a series of talks given at the conference in Dubrovnik in the summer of 2003. I would like to thank the organizers for giving me the opportunity of participating, and express my gratitude for their work to make the conference a success.

This research was supported by NSF grants DMS-0070561 and DMS0300172.

2. Notation and background

  [sec:2]

2.1. [sec:2.1] Let 𝔽 be a local field, either ℝ or a p-adic field. In the latter case, let R ⊃ P be the ring of integers and its maximal prime ideal respectively. Let G := G(𝔽) be the rational points of a connected reductive group, 𝔽 real or p-adic. Assume G is split, and fix a rational Borel subgroup B = AN. In the real case, fix a Cartan involution θ, and let K be the maximal compact group which is its fixed points. In the p-adic case, let K := G(R). A character χ ∈ Â is called unramified, if χ|_{A∩K} is trivial. Since G is split, A ≅ (𝔽^{×})^{n} where n is the rank of G. Thus an unramified character is of the form

χ = (χ_{1}, …, χ_{n})  (2.1.1)

where χ_{i}(t) = |t|^{ν_{i}}. Let L := (0) if 𝔽 = ℝ, L := (\frac{2iπ}{log q} ℤ) in the p-adic case. We can therefore identify χ with an element in (ℂ/L)^{n}. As is well known, any irreducible spherical admissible representation of G is the unique spherical subquotient of a standard induced module

X_{B}(χ) := Ind_{B}^{G}(χ ⊗ δ_{B}^{−1/2}),  (2.1.2)

where χ is an unramified character, and δ_{B} is the modulus function of B. Denote this subquotient by L(χ). Two such quotients L(χ) and L(χ′) are isomorphic if and only if there is an element w ∈ W := N_{K}(A)/C_{K}(A) such that wχ = χ′.

2.2. [sec:2.2] Suppose Imχ ≠ 0. Then Imχ defines a proper standard parabolic subgroup P = MU, where

Δ(M) = {α : (α, Imχ) = 0},  Δ(U) = {α : (α, Imχ) > 0}. (2.2.1)

Let L_{M}(χ) be the spherical irreducible module of M corresponding to χ. Then L_{M}(χ) = L_{M}(χ_{r}) ⊗ χ_{i} where χ_{r} is such that Imχ_{r} = 0 and χ_{i} is unitary.

Theorem. [t:2.2]

L(χ) = Ind_{M}^{G}[L_{M}(χ_{r}) ⊗ χ_{i}],

and L(χ) is unitary if and only if L_{M}(χ_{r}) is unitary.

Proof. For the real case, this is theorem 16.10 in [K]. For the p-adic case and I-spherical representations it follows from [BM1] and [BM2]. □

Because of this theorem, we assume that χ is real, i.e. Imχ = 0. In this situation, the lattice L plays no role, and we can treat the real and p-adic case in the same way.

2.3. [sec:2.3] Let (χ, α) ≥ 0 for all roots α ∈ Δ(𝔫, 𝔞). then X(χ) has a unique irreducible quotient, namely L(χ). We call such a χ dominant. If on the other hand (Reχ, α) ≤ 0 for all α ∈ Δ(𝔫, 𝔞), then X(ν) has a unique irreducible submodule which is again L(χ). We call such a χ antidominant.

Let B′ = AN′ be another Borel subgroup. Then there is an intertwining operator

A(B, B′, χ) : X_{B}(χ) ⟶ X_{B′}(χ)  (2.3.1)

given by

A(B, B′, χ)f(g) := ∫_{N′/N∩N′} f(gn′) dn′.  (2.3.2)

The integral is convergent for χ very dominant, and has a meromorphic extension to all χ. It has no poles if χ is dominant. Let w ∈ W, and write B′ := w(B). There is an isomorphism R_{w} : X_{B′}(χ) ⟶ X_{B}(wχ) given by

R_{w}f(g) := f(gw^{−1}).  (2.3.3)

We write

A(w, χ) := R_{w} ∘ A(B, B′, χ) : X_{B}(χ) ⟶ X_{B}(wχ).  (2.3.4)

Recall that G = KB, and let (E, μ) be a K-type. Then the intertwining operator (2.3.1) gives rise to a map

A(μ, w, χ) : Hom_{K}[E, X_{B}(χ)] ⟶ Hom_{K}[E, X_{B}(wχ)].  (2.3.5)

By Frobenius reciprocity, we get a map

a(μ, w, χ) : (E^{∗})^{M} ⟶ (E^{∗})^{M}.  (2.3.6)

We normalize the intertwining operator (2.3.1) so that a(triv, w, χ) = 1 by multiplying it with the appropriate meromorphic function in χ.

The hermitian dual of X_{B}(χ) is X_{B}(χ^{h}), where χ^{h} := (\overline{χ})^{−1} (χ not necessarily rea). Indeed, if v ∈ X_{B}(χ) and w ∈ X_{B}(χ^{h}), then v\overline{w} transforms according to δ_{B} under the right action of B. Thus the integral of v\overline{w} over G/B makes sense, and the hermitian pairing is

(v, w) := ∫_{G/B} v(g)\overline{w(g)} dg.  (2.3.7)

Thus the hermitian dual of L(χ) is L(χ^{h}), and L(χ) is hermitian if and only if there is w ∈ W such that wχ = χ^{h}. Suppose this is the case and B′ := w(B). If χ is dominant, then χ^{h} is antidominant. So the image of A(w, χ) is exactly L(χ). The hermitian form is given by

⟨v_{1}, v_{2}⟩ := (v_{1}, A(w, χ)(v_{2})),  (2.3.8)

where ( , ) is the pairing in (2.3.7). Fix a positive definite K-invariant hermitian form for each K-type (E, μ). Then as in (2.3.3)-(2.3.4), we get hermitian symmetric maps a(μ, χ). Then L(χ) is unitary if and only if a(μ, χ) is positive semidefinite for all μ.

3. Fine K-types

  [sec:3]

3.1. [sec:3.1] A reference for the results in this section is chapter 4 in [V3].

Let (μ_{a}, V_{a}) and (μ_{b}, V_{b}) be representations of K. Then Hom_{M}[V_{a}, V_{b}] is endowed with a representation of N_{K}(M) via

n · f(v) := μ_{a}(n)f(μ_{b}(n^{−1})v).

Under this action, M ⊂ N_{K}(M) acts trivially, so we get a representation of W. Note that

Hom_{M}[V_{a}, V_{b}] ≅ Hom_{M}[V_{a} ⊗ V_{b}^{∗}, T riv].

In particular, if V_{a} = E_{μ}, and V_{b} = T riv, we get the more familiar action of W on (E_{μ}^{∗})^{M}.

3.2. [sec:3.2] Let α be a simple root and P_{α} = M_{α}N be the standard parabolic subgroup so that M_{α} is θ-stable, and the Lie algebra of M_{α} is isomorphic to the sl(2, ℝ) generated by the root vectors E_{±α}.

The map ψ_{α} : sl(2, ℝ) ⟶ 𝔤 determined by

ψ_{α} [\begin{matrix}0 & 1 \\ 0 & 0 \end{matrix}] = E_{α}, ψ_{α} [\begin{matrix}0 & 1 \\ 0 & 0 \end{matrix}] = E_{−α},

gives a map

Ψ_{α} : SL(2, ℝ) ⟶ G  (3.2.1)

with image G_{α} ⊂ M_{α}, a connected group with Lie algebra isomorphic to sl(2, ℝ). We assume that θE_{α} = −E_{−α}. Let D_{α} = \sqrt{−1}(E_{α} − E_{−α}), and s_{α} = e^{\sqrt{−1}πD_{α}/2}. Then s_{α}^{2} = m_{α} is in M ∩ M_{α}, while s_{α} itself gives rise to the corresponding nontrivial element in the Weyl group.

Definition. [d:3.2] A K-type is called fine (Vogan), if μ(iD_{α}) = 0, ±1. More generally, a K-type is said to have level less than or equal to r, if

|μ(iD_{α})| ≤ r.

The fine K-types are the lowest K-types of principal series. For every δ, there is a fine K-type μ_{δ} whose restriction to M contains δ. The restriction of every fine K-type to M is the sum over the W -orbit of a single δ ∈ M̂.

Fix a representative δ for each W -orbit, and a fine K-type μ_{δ}. In the case of a linear group, M is abelian, so M̂ is formed of characters.

Then

μ_{δ} ⊗ μ_{δ′}^{∗} is formed of level ≤ 2 K-types only.

3.3. [sec:3.3] We will use the ideas in section 3.2 to determine the Weyl group representation on μ_{δ} ⊗ μ_{δ′}^{∗}. The following properties hold.

• ^{∨}Δ^{δ} := {α̌ | δ(m_{α}) = 1} is a roots system.

• The Weyl group generated by the roots in ^{∨}Δ^{δ} is a normal subgroup of W_{δ}.

• The quotient R_{δ} := W_{δ}/W_{δ}^{0} is is a product of ℤ_{2}’s.

• R̂_{δ} acts simply transitively on the fine K-types containing δ.

Since R_{δ} is a quotient of W_{δ}, we can inflate any τ ∈ R̂_{δ} to a representation of W_{δ}, trivial on W_{δ}^{0}. Having fixed a μ_{δ}, associate it to the trivial representation of W_{δ}, and set up a 1-1 correspondence

{τ ∈ Ŵ_{δ} | τ|_{W_{δ}^{0}} = triv} ⟷ {μ_{δ,τ} }, triv ⟷ μ_{δ}.

Theorem. [t:3.3] As a W-module,

Hom_{M}[μ_{δ,1}, μ_{δ,τ}] ≅ Ind_{W_{δ}}^{W}[τ].

Proof. Recall that a fine K-type has the property that its restriction to M is multiplicity free, and forms a single orbit under the action of M. The proof follows from the properties listed before the theorem, and standard properties of representations of finite groups. □

4. Relevant and Petite K-types

  [sec:4]

4.1. [sec:4.1] Recall the notation from section 3.2. Since the square of any element in M is in the center and M normalizes the root vectors, Ad m(D_{α}) = ±D_{α}. Let (E, μ) be a K-type, and grade E^{∗} = ⊕E_{i}^{∗} according to the absolute values of the eigenvalues of D_{α} (which are integers). Then M preserves this grading, and

(E^{∗})^{M} = ⨁_{i even}(E_{i}^{∗})^{M}.

Let A_{α} be the maps (2.3.4) for M_{α}. Recall that for SL(2, ℝ), K-types are parametrized by integers, and only K-types parametrized by even integers occur in the spherical principal series.

Proposition. [p:4.1] On (E_{2m}^{∗} + E_{−2m}^{∗})^{M},

a(2m, s_{α}, ν) = \begin{cases}Id & if m = 0, \\ ∏_{0<j≤m} \frac{2j−1−<ν,α̌>}{2j−1+<ν,α̌>} Id & if m ≠ 0. \end{cases}

Proof. The formula is well known for SL(2, ℝ). The general assertion follows from this and the listed properties of intertwining operators. □

Let w ∈ W. Then any reduced decomposition w = s_{1} ⋯ s_{k} gives rise to a factoring

A(w, χ) = ∏ A(s_{i}, s_{i+1} · . . . · s_{k}χ).  (4.1.1)

The resulting operator A(w, χ) does not depend on the particular reduced decomposition of w, and each A(s_{i}, s_{i+1} · · · · · s_{k}χ) is induced from the corresponding A_{α_{i}}.

Corollary. [c:4.1] In particular, A(w, χ) is an isomorphism unless ⟨ν, α̌⟩ ∈ ℕ for some root α.

Proof. This follows from the properties of the intertwining operator listed above. □

Definition. [d:4.1] A K-type μ is called petite if each A_{α_{i}} satisfies

a(μ, s_{i}, ν) = \begin{cases}Id & on the +1 eigenspace of s_{α_{i}}, \\ \frac{1−<ν,α̌>}{1+<ν,α̌>} Id & on the -1 eigenspace of s_{α_{i}}. \end{cases}

In other words, the restriction of μ to K_{α} := M_{α} ∩ K consists of K-types with 2m = 0, ±2 only. The hermitian symmetric matrix a(μ, χ) depends only on the W -structure of (E_{μ}^{∗})^{M}.

Theorem. [t:4.1] Level ≤ 3 K-types are petite.

Proof. This follows from the above discussion. □

4.2. Affine Hecke algebra. [sec:4.2] We now show that the formulas in the previous section coincide with corresponding ones in the p-adic case. Recall from [BM3] that the induced module is X(χ) := ℍ ⊗_{𝔸} ℂ_{χ} where ℍ = ℂ[W] ⊗ 𝔸 is the graded affine Hecke algebra. The abelian subalgebra 𝔸 is S(𝔞) (𝔞 = Lie(A)_{c}), and ℂ[W] is generated by {t_{α}}_{α simple} satisfying t_{α}^{2} = 1. They are subject to the relations

ωt_{α} = s_{α}(ω)t_{α} + c_{α} < ω, α >,  ω ∈ 𝔞.  (4.2.1)

Because the group is assumed to be split, c_{α} = 1. The intertwining operator A(w, ν) is a product of operators A_{α_{i}} according to a reduced decomposition of w = s_{α_{1}} · · · · · s_{α_{k}}. If α is a simple root, let

r_{α} := (t_{α}α̌ − c_{α})\frac{1}{α̌ − c_{α}} ,  A_{α} : x ⊗ 11_{ν} ↦ xr_{α} ⊗ 11_{s_{α}ν}.  (4.2.2)

The r_{α} are being multiplied on the right, so we can replace α̌ with ⟨s_{α}χ, α̌⟩ in the formulas. The A(w, χ) have the same properties as their counterparts in the real case.

Furthermore,

ℂ[W] = ∑_{σ∈Ŵ} V_{σ} ⊗ V_{σ}^{∗}.

Since r_{α} acts as multiplication on the right, it gives rise to an operator

r_{σ}(s_{α}, ν) : V_{σ}^{∗} ⟶ V_{σ}^{∗}.

Theorem. [t:4.2] Let (E, μ) be a petite K-type such that V_{σ}^{∗} ≅ (E_{μ}^{∗})^{M}. Then a(μ, w, χ) on (E_{μ}^{∗})^{M} coincide with r_{σ}(w, χ) on V_{σ}^{∗}.

Proof. The assertion is clear from definition (4.1) and formula (4.2.2). □

The algebra ℍ has a ∗, coming from the corresponding p-adic group, given by

ω^{∗} = −\overline{ω} + ∑_{α∈Δ(𝔫)}(ω, α̌)t_{α}.  (4.2.3)

So it makes sense to talk about its unitary dual.

We remark that in the work of Lusztig and Kazhdan-Lusztig, the Hecke algebra is defined in terms of the dual root system so that there is no discrepancy between α and α̌ in the formulas. For p-adic groups the determination of the Iwahori spherical unitary dual is equivalent to the determination of the unitary dual of ℍ. As explained in [BM1] and [BM2], an infinitesimal character is given by the conjugacy class of a semisimple element s ∈ Ǧ. In turn this element has a decomposition s = s_{e}s_{h}, where s_{e} is elliptic and s_{h} is hyperbolic. The Iwahori spherical dual consisting of representations with infinitesimal character s is in 1-1 correspondence with the unitary dual of an affine graded Hecke algebra ℍ(s_{e}) with real infinitesimal character. A representation is called spherical if it contains the trivial W-type. The classification of spherical (and hermitian) representations with real infinitesimal character is the same as for the real and p-adic cases.

4.3. Relevant K-types. [sec:4.3] A set of K-types is called relevant, if the (E^{∗})^{M} form a relevant set, i.e. they satisfy part (a) of the introduction. We list them for the simple root systems. For the classical groups they are already in [B5].

The representations of W (A_{n−1}) = S_{n} are parametrized by partitions of n, written as (a_{1}, …, a_{k}), a_{i} ≤ a_{i+1}. The representations of W (B_{n}) ≅ W (C_{n}) are parametrized as in [L] by pairs of partitions

(a_{1}, …, a_{k}) × (b_{1}, …, b_{l}),
a_{i} ≤ a_{i+1}, b_{j} ≤ b_{j+1}, ∑ a_{i} + ∑ b_{j} = n.  (4.3.1)

Precisely, the representation σ parametrized by (4.3.1) is as follows. Let k = ∑ a_{i}, l = ∑ b_{j}. Recall that W ≅ S_{n} ⋊ ℤ_{2}^{n}. Let χ be the character of ℤ_{2}^{n} which is trivial on the first k ℤ_{2}’s, sign on the rest. Its centralizer in S_{n} is S_{k} × S_{l}. Let σ_{1} and σ_{2} be the representations of S_{k}, S_{l} corresponding to the partitions (a) and (b). Then σ, is

Ind_{(S_{k}×S_{l})×ℤ_{2}^{n}}^{W} [(σ_{1} ⊗ σ_{2}) ⊗ χ].

For W (D_{n}), the representations are parametrized as in (4.3.1) except that (a) × (b) and (b) × (a) parametrize the same representation, and when (a) = (b), there are two of them (a) × (a)_{I, II}. This is because the restriction of (a) × (b) to W (D_{n}) is irreducible when (a) ≠ (b) and equal to the restriction of (b) × (a), while the restriction of (a) × (a) consists of two nonisomorphic irreducible representations labelled (a) × (a)_{I, II}. These are usually easy to deal with.

4.4. Orthogonal groups. [sec:4.4] Because we are dealing with the spherical case, we can use the orthogonal group instead of its connected component. We follow Weyl’s conventions to parametrize the representations of O(n). Embed O(a) ⊂ U(a) in the standard way. An irreducible representation of O(n) is parametrized by

(a_{1}, …, a_{k}, 0, …, 0; ϵ),  a_{i} ≥ a_{i+1}, ϵ = ±1.  (4.4.1)

The ϵ is (sometimes) abbreviated as ±. The parameter in (4.4.1) is the irreducible representation generated by the highest weight vector of the irreducible representation of U(a) with highest weight

(a_{1}, …, a_{k}, \underbrace{1, …, 1}_{n−(1−ϵ)k}, 0, …, 0).  (4.4.2)

For O(n, n) we have K = O(n) × O(n). The fine K-types (in the sense of [V4]), and their restrictions to M are

\begin{aligned}K − type & M − type \\ (\underbrace{1, …, 1}_{k}, 0, …, 0) ⊗ (0, …, 0) \\ & \binom{n}{k} characters δ_{k} \\ (0, …, 0) ⊗ (\underbrace{1, …, 1}_{k}, 0, …, 0), \end{aligned}  (4.4.3)

and we suppress the ±. The meaning of (4.4.3) is that both representations of K in the left column restrict to an orbit of characters of M. The size of the orbit is \binom{n}{k}, and we fix a representative denoted δ_{k}.

The K-types

(0, …, 0; +) ⊗ (\underbrace{2, …, 2}_{r}, 0, …, 0; +)  (4.4.4)
(\underbrace{1, …, 1}_{k}, 0, …, 0; ϵ) ⊗ (\underbrace{1, …, 1}_{k}, 0, …, 0; ϵ)  ϵ = ±  (4.4.5)

form a relevant set. The representation of W on V^{M} is

(r, n − r) × (0)  ⟷ (4.4.4)  (4.4.6)
(n − k) ⊗ (k), k ≤ [n/2]  ⟷ (4.4.5) with +,  (4.4.7)
(n − k) ⊗ (k), k > [n/2]  ⟷ (4.4.5) with − .  (4.4.8)

For O(n + 1, n), we have K = O(n + 1) × O(n). The fine K-types ([V4]), and their restriction to M are

\begin{matrix}K − type & M − type \\ (0, …, 0) ⊗ (\underbrace{1, …, 1}_{k}, 0, …, 0) & \binom{n}{k} characters δ_{k} \end{matrix}  (4.4.9)

The K-types

(0, …, 0; +) ⊗ (\underbrace{2, …, 2}_{r}, 0, …, 0; +)  (4.4.10)
(\underbrace{1, …, 1}_{k}, 0, …, 0; +) ⊗ (\underbrace{1, …, 1}_{k}, 0, …, 0; +)  for k ≤ [n/2] (4.4.11)
(\underbrace{1, …, 1}_{n+1−k}, 0, …, 0; −) ⊗ (\underbrace{1, …, 1}_{n−k}, 0, …, 0; −)  for k > [n/2] (4.4.12)

form a relevant set, and the corresponding representations of W (B_{n}) on V^{M} are

(r, n − r) × (0), r ≤ n/2  ⟷ (4.4.11)  (4.4.13)
(n − k) ⊗ (k), k ≤ [n/2]  ⟷ (4.4.12) with +,  (4.4.14)
(n − k) ⊗ (k), k > [n/2]  ⟷ (4.4.12) with − .  (4.4.15)

Theorem ([B5]). [t:4.4] A spherical irreducible representation for ℍ is unitary if and only if the hermitian form is positive definite on the W-types

(r, n − r) × (0), (n − k) × (k).

Corollary ([B5]). [c:4.4] A spherical representation L_{R}(χ) for a real split orthogonal group is unitary only if the corresponding L_{F} (χ) for ℍ is unitary.

In other words, the set of spherical unitary parameters for the real split group is contained in the set of spherical unitary parameters for ℍ. In fact in [B5] the stronger result is proved that the unitary duals of G(ℝ) and ℍ coincide.

4.5. Symplectic groups. [sec:4.5] The maximal compact subgroup of Sp(n, ℝ) is U(n). Its irreducible representations are parametrized by

(a_{1}, …, a_{n}),  a_{i} ≥ a_{i+1}, a_{i} ∈ ℤ.  (4.5.1)

The fine K-types and their restrictions to M are

\begin{aligned}K − type & M − type \\ μ_{+}(k) := (\underbrace{1, …, 1}_{k}, 0, …, 0) \\ & \binom{n}{k} characters δ_{k} \\ μ_{−}(k) := (0, …, 0, \underbrace{−1, …, −1}_{k}). \end{aligned}  (4.5.2)

The K-types

(\underbrace{2, …, 2}_{k}, 0, …, 0) ⟷ (n − k) × (k),  (4.5.3)
(0, …, 0, \underbrace{−2, …, −2}_{k}) ⟷ (k) × (n − k),  (4.5.4)
(\underbrace{1, …, 1}_{r}, 0, …, 0, \underbrace{−1, …, −1}_{r}) ⟷ (r, n − r),  (4.5.5)

form a relevant set.

Theorem ([B5]). [t:4.5] A spherical representation for ℍ is unitary if and only if the hermitian form is positive definite on the W-types

(r, n − r) × (0), (n − k) × (k).

Corollary. [c:4.5] A spherical representation L_{R}(χ) for a real split orthogonal group is unitary only if the corresponding L_{F} (χ) for ℍ is unitary.

In fact in [B5] the stronger result is proved that the unitary duals of G(ℝ) and ℍ coincide.

4.6. General linear group. [sec:4.6] The maximal compact subgroup of GL(n, ℝ) is O(n). The fine K-types and their restrictions to M are

\begin{matrix}K − type & M − type \\ (\underbrace{1, …, 1}_{k}, 0, …, 0) & \binom{n}{k} characters δ_{k} \end{matrix}  (4.6.1)

According to [V2], a set of relevant K-types is given by

(\underbrace{2, …, 2}_{k}, 0, …, 0).  (4.6.2)

The corresponding representation of S_{n} on V^{M} is, according to [B5],

(\underbrace{2, …, 2}_{k}, 0, …, 0) ⟷ (n − k, k)  (4.6.3)

In the case n even, we have

(2, …, 2, ±2) ⟷ (n/2, n/2)_{I,II}.  (4.6.4)

The theorems and corollaries analogous to the ones in sections 4.4 and 4.5 follow from [V2] and [T]. For a construction of the relevant K-types for SL(n) starting from the Weyl group module on E^{M}, the interested reader may consult [P].

4.7. G2. [sec:4.6.1] We refer to [V1] for more details about the structure of the split group of type G_{2}. The split real form which consists of the real points of the complex simply connected linear group of type G_{2} has

K := [SU(2)_{s} × SU(2)_{l}]/{±Id}.  (4.7.1)

It is more convenient to work with the (nonlinear) double cover for which

K̃ = SU(2)_{s} × SU(2)_{l}.  (4.7.2)

The Borel subgroup is B̃ = M̃AN where M̃ is the group of order 8 which contains {±Id} in (4.7.1) as a central sugroup, with quotient ℤ_{2} × ℤ_{2}. Let T be the maximal compact subgroup. We identify the Lie algebra 𝔱 with {(a_{1}, a_{2}, a_{3}) | a_{1} + a_{2} + a_{3} = 0}, and use the standard positive system of roots

\begin{matrix}\{ & (1, −1, 0), (−1, 2, −1), (0, 1, −1), (1, 0, −1), \\ (1, 1, −2), (2, −1, −1)\}. \end{matrix}  (4.7.3)

A K-type is determined by its highest weight which we denote

(p | q) ⟷ (q, \frac{p − q}{2} , \frac{−p − q}{2} ),  p, q ∈ ℕ.  (4.7.4)

The positive roots in 𝔨 are (0, 1, −1) and (2, −1, −1). The fine K-types are

K̃ − type  M̃ − type
(0 | 0)  δ_{0}, the trivial character,
(1 | 0)  δ_{2}, a representation of dimension 2,
(2 | 0)  δ_{3}, three characters.

(4.7.5)

The one dimensional representations all factor to M, while the two dimensional one is a genuine representation of M̃. The Weyl group is generated by s_{s}, the reflection about the simple short root, and s_{l}, the reflection about the simple long root. The representations of W are parametrized as in [L],

\begin{matrix}1_{1} & trivial \\ 1_{2} & s_{s} = 1, s_{l} = −1, \\ 1_{3} & s_{s} = −1, s_{l} = +1, \\ 1_{4} & s_{s} = −1, s_{l} = −1, \\ 2_{1} & reflection, \\ 2_{2} & 2_{1} ⊗ 1_{2}. \end{matrix}  (4.7.6)

Theorem ([C]). [t:g2] A spherical representation of ℍ is unitary if and only if the r_{σ} are positive definite on the W -types

1_{1}, 2_{1}, 2_{2}.

Proposition. The following list consists of petite K-types.

\begin{matrix}K − type & W − type \\ (0 | 0) & 1_{1}, \\ (3 | 1) & 2_{1}, \\ (4 | 0) & 2_{2}. \end{matrix}

Thus the W-types in theorem 4.7 form a relevant set. More detailed results along these lines can be found in [C].

4.8. F4. [sec:4.7] The split real form which is the rational points of the complex linear group of type F_{4} has maximal compact subgroup

K := [Sp(1) × Sp(3)]/{±Id}.  (4.8.1)

It is more convenient to use the double cover which has

K̃ = [Sp(1) × Sp(3)]  (4.8.2)

The Borel subgroup is B̃ = M̃AN, where M̃ is a finite nonabelian group of order 2^{5}. It has a central subgroup of order 2 (the ±Id in (4.8.1)) such that the quotient is ℤ_{2}^{4}.

Let T be a maximal compact Cartan subgroup. We use the standard positive system and roots,

{2ϵ_{1}, 2ϵ_{k}, ϵ_{k} ± ϵ_{ℓ}}_{2≤k≤ℓ≤4}.  (4.8.3)

The highest weight of a K̃-type will be denoted

(a_{1} | a_{2}, a_{3}, a_{4}, a_{5}),  a_{i} ∈ ℕ, a_{2} ≥ a_{3} ≥ a_{4} ≥ 0.  (4.8.4)

The fine K-types (in the sense of [V4]), and their restrictions to M̃ are

K̃ − type  M̃ − type W_{δ}
(0 | 0, 0, 0)  δ_{0}, one character F_{4},
(1 | 0, 0, 0)  δ_{2}, a representation of dimension 2 F_{4}
(2 | 0, 0, 0)  δ_{3}, three characters C_{4},
(0 | 1, 0, 0) δ_{6}, three representations of dimension 2 B_{4},
(1 | 1, 0, 0)  δ_{12}, twelve characters B_{3}A_{1}

(4.8.5)

The 1-dimensional representations all factor to M. The remaining ones are genuine for M̃.

The representations of the Weyl group of type F_{4} are parametrized as in [L].

Theorem ([C]). [t:4.7] A spherical representation of ℍ of type F_{4} is unitary if and only if the r_{σ} are positive semidefinite on the W-types

1_{1}, 2_{3}, 8_{1}, 4_{2}, 9_{1}.

The next result was obtained joint with D. Vogan.

Proposition. [p:4.7] The following list consists of petite K-types.

\begin{matrix}K − type & W − type \\ (0 | 0, 0, 0) & 1_{1}, \\ (0 | 1, 1, 0) & 2_{1}, \\ (4 | 0, 0, 0) & 2_{3}, \\ (1 | 2, 1, 0) & 8_{1}, \\ (1 | 1, 1, 1) & 4_{2}, \\ (2 | 2, 0, 0) & 9_{1}. \end{matrix}

Thus the W -types in theorem 4.8 form a relevant set.

Corollary. [c:4.7] A spherical representation for the real split group G of type F_{4} is unitary only if the corresponding L_{F} (χ) for ℍ is unitary.

The results in [C] provide an explicit list of the unitary dual of ℍ. The unitarity of the unipotent representations and the relevant irreducibility results for the real case are not (yet) available.

4.9. E6. [sec:4.8]

The maximal compact subgroup of the rational points of the simply connected complex group of type E_{6} is

K = Sp(4)/{±Id}.  (4.9.1)

Again it is more convenient to use the double cover which has

K̃ = Sp(4).  (4.9.2)

We use the same conventions as in section 4.8 for the positive roots and highest weights of representations of K̃. The finite group M̃ has size 2^{7}, and its quotient by the central group {±Id} is a ℤ_{2}^{6}. The fine K-types and their restrictions to M̃ are

K̃ − type  M̃ − type  W_{δ}^{0}
(0, 0, 0, 0)  δ_{1}, the trivial character,  E_{6},
(1, 0, 0, 0) δ_{8}, a representation of dimension 8,  E_{6},
(1, 1, 0, 0)  δ_{27}, twenty seven characters,  D_{5},
(2, 0, 0, 0)  δ_{36}, thirty six characters,  A_{5}A_{1}.

(4.9.3)

The second representation is genuine for M̃, the others all factor to M. The W -types are parametrized as in [L].

Theorem ([BC]). [t:4.8] A spherical representation of ℍ of type E_{6} is unitary if and only if the r_{σ} are positive semidefinite on the W-types

1_{p}, 6_{p}, 20_{p}, 30_{p}, 15_{q}.

We denote by ω_{i} the fundamental weights of sp(4). In coordinates they are ω_{i} = ∑_{j≤i} ϵ_{j}.

Proposition. [p:4.8] The following list consists of petite K-types.

\begin{matrix}K − type & W − type \\ (0) = (0, 0, 0, 0) & 1_{p}, \\ ω_{4} = (1, 1, 1, 1) & 6_{p} \\ 2ω_{2} = (2, 2, 0, 0) & 20_{p}, \\ 4ω_{1} = (4, 0, 0, 0) & 15_{q}, \\ 2ω_{1} + ω_{4} = (3, 1, 1, 1) & 30_{p}, \\ ω_{1} + ω_{2} + ω_{3} = (3, 2, 1, 0) & 64_{p}, \\ 3ω_{1} + ω_{3} = (4, 1, 1, 0) & 60_{p}, \\ 2ω_{3} = (2, 2, 2, 0) & 15_{p}, \\ 2ω_{1} + 2ω_{2} = (4, 2, 0, 0) & 81_{p}, \\ 3ω_{2} = (3, 3, 0, 0) & 24_{p}, \\ 6ω_{1} = (6, 0, 0, 0) & 24′_{p}. \end{matrix}

Corollary. [c:4.8] A spherical representation L_{R}(χ) for the real group G of type E_{6} is unitary only if the corresponding L_{F} (χ) for ℍ is unitary.

Thus the W-types in theorem 4.9 form a relevant set.

4.10. E7. [sec:4.9]

The maximal compact subgroup of the split real form of the simply connected group of type E_{7} is

K = SU(8)/{±Id}.  (4.10.1)

We work with the double cover for which

K̃ = SU(8).  (4.10.2)

The finite group M̃ has size 2^{8}, and its quotient by the center in (4.10.1) is a ℤ_{2}^{7}. The fine K-types are

K̃ − type  M̃ − type  W_{δ}^{0}
(0)  δ_{1}, trivial representation,  E_{7}
ω_{1}  δ_{8}, eight dimensional representation,  E_{7}
ω_{7}  δ_{8}^{∗}, eight dimensional representation,  E_{7}
ω_{2}, ω_{6}  δ_{28}, twenty eight characters,  E_{6}
2ω_{1}, 2ω_{7}  δ_{36}, thirty six characters,  A_{7}
ω_{1} + ω_{7}  δ_{63}, sixty three characters,  D_{6}A_{1}

(4.10.3)

In (4.10.3), the ω refer to the fundamental weights of K̃, usual labelling.

The second and third are genuine representations of M̃, the others are single orbits under the action of W.

The Weyl group representations are parametrized as in [L].

Theorem ([BC]). [t:4.9] A spherical representation of ℍ of type E_{7} is unitary if and only if r_{σ} is positive semidefinite for

1_{a}, 7′_{a}, 27_{a}, 56′_{a}, 21′_{b}, 35_{b}, 105_{b}.

Proposition. [p:4.9] The following list gives the petite K-types with Mspherical vectors, and the corresponding Weyl group representations:

\begin{matrix}K − type & W − type \\ (0) & 1_{a}, \\ ω_{4} & 7′_{a} \\ 2ω_{2}, 2ω_{6} & 21′_{b}, \\ ω_{2} + ω_{6} & 27_{a}, \\ 2ω_{1} + 2ω_{7} & 35_{b}, \end{matrix}

\begin{matrix}K − type & W − type \\ 4ω_{1}, 4ω_{7} & 15′_{a}, \\ ω_{2} + ω_{3} + ω_{7}, ω_{1} + ω_{5} + ω_{6} & 105′_{a}, \\ ω_{1} + ω_{4} + ω_{7} & 56′_{a}, \\ 2ω_{1} + ω_{3} + ω_{7}, ω_{1} + ω_{5} + 2ω_{7} & 189′_{b}, \\ ω_{1} + ω_{3} + ω_{6}, ω_{2} + ω_{5} + ω_{7} & 168_{a}, \\ 3ω_{1} + ω_{5}, ω_{3} + 3ω_{7} & 105_{b}, \\ ω_{3} + ω_{5} & 21_{a}, \\ ω_{1} + ω_{2} + ω_{5}, ω_{3} + ω_{6} + ω_{7} & 120_{a}, \\ ω_{1} + ω_{2} + ω_{6} + ω_{7}, & 168_{a} + 210_{a}, \\ 3ω_{1} + 3ω_{7}, & 84_{a} + 105_{c}, \\ ω_{1} + 2ω_{2} + ω_{7}, ω_{1} + ω_{6} + ω_{7}, & 189′_{c} \\ 3ω_{1} + ω_{2} + ω_{7}, ω_{1} + ω_{6} + 3ω_{7}, & 216′_{a}, \\ 3ω_{1} + ω_{6} + ω_{7}, ω_{1} + ω_{2} + 3ω_{7}, & 280_{b}, \\ 5ω_{1} + ω_{7}, ω_{1} + 5ω_{7}, & 84′_{a}. \end{matrix}

Corollary. [c:4.9] A spherical representation L_{R}(χ) for the real group G of type E_{7} is unitary only if the corresponding L_{F} (χ) for ℍ is unitary.

Thus the W-types in theorem 4.10 form a relevant set.

4.11. E8. [sec:3a.10] The maximal compact subgroup of the split real form of the simply connected complex group of type E_{8} is

Spin(16)/{Id, ω},  (4.11.1)

for ω the appropriate element of order two in the center (the quotient is not SO(16)). As before we work with the double cover which has

K̃ = Spin(16).  (4.11.2)

The group M̃ has size 2^{9}, and its quotient by the center in (4.11.1) is ℤ_{2}^{8}. The fine K-types are

K̃ − type  M̃ − type
(0)  δ_{1}, trivial representation,
ω_{1}  δ_{16}, sixteen dimensional representation,
ω_{2}  δ_{120}, one hundred and twenty characters,
2ω_{1}  δ_{135}, one hundred thirty five characters,

(4.11.3)

Only the second representation is genuine.

The representations of the Weyl group are parametrized as in [L].

Theorem. [t:3a.10] A spherical representation of ℍ of type E_{8} is unitary if and only if r_{σ} is positive semidefinite for

1_{x}, 8_{z}, 35_{x}, 50_{x}, 84_{x}, 112_{z}, 400_{z}, 300_{x}, 210_{x}.

Proposition. [p:3a.10] The following list gives petite K-types and the corresponding Weyl group representations on (V^{∗})^{M} :

\begin{matrix}K − type & W-type on (V^{∗})^{M} \\ (0) & 1_{x}, \\ ω_{8} & 8_{z} \\ ω_{4} & 35_{x}, \\ 2ω_{2} & 84_{x}, \\ ω_{2} + ω_{8} & 112_{z}, \\ 4ω_{1} & 50_{x}, \\ 3ω_{1} + ω_{7} & 400_{z}, \\ ω_{3} + ω_{7} & 160_{z}, \\ ω_{6} & 28_{x}, \\ ω_{1} + ω_{5} & 210_{x}, \\ ω_{1} + ω_{2} + ω_{7} & 560_{x}, \\ ω_{2} + ω_{4} & 567_{x}, \\ 2ω_{3} & 300_{x}, \\ 2ω_{1} + ω_{4} & 700_{x}, \\ 3ω_{1} + ω_{3} & 1050_{x}, \\ ω_{1} + ω_{2} + ω_{3} & 1344_{x}, \\ 3ω_{2} & 525_{x}, \\ 2ω_{1} + 2ω_{2} & 972_{x}, \\ 4ω_{1} + ω_{2} & 700_{xx}, \\ 6ω_{1} & 168_{y}. \end{matrix}

Corollary. [c:3a] A spherical representation L_{R}(χ) for the real group G of type E_{8} is unitary only if the corresponding L_{F} (χ) for ℍ is unitary.

5. Some proofs

  [sec:5]

5.1. [sec:5.1] Let K̃ be a compact group, M̃ ⊂ Ñ finite subgroups such that M̃ is normal in Ñ. Denote by W the quotient Ñ/M̃. In the applications, Ñ is the normalizer of 𝔞 in K̃, which is simply connected.

Let V_{a}, V_{b} be representations of K̃. In the nicer cases, the restrictions of V_{a}, V_{b} to M̃ are multiples of the same representation V_{δ}, which in turn extends to a representation of K̃. This is the case for E_{8}. The standard 16-dimensional representation is genuine, and restricts to a single M̃-type denoted δ_{16}. We already defined the action of W in the next proposition in section 3.2.

Proposition. [p:4] There is a natural action of W on Hom_{M}[V_{a}, V_{b}].

Proof. The action on f ∈ Hom[V_{a}, V_{b}] is given by

(n · f)(v) = μ_{a}(n)f(μ_{b}(n^{−1})v).  (5.1.1)

This is clearly an action of Ñ. The fact that the action does not depend on the right M̃-coset is straightforward. □

This action is compatible with the canonical isomorphism

HomM̃[V_{a}, V_{b}] ≅ [V_{a}^{∗} ⊗ V_{b}]M̃,  (5.1.2)

and is also compatible with the isomorphism

HomM̃[V_{a}, V_{b}]^{∗} ≅ HomM̃[V_{b}^{∗}, V_{a}^{∗}].  (5.1.3)

This action is a generalization of the usual one on V M̃.

5.2. [sec:5.2] Now consider two genuine modules V_{a}, V_{b}. Then there is a homomorphism

HomM̃[V_{a}, V_{δ}] ⊗ HomM̃[V_{δ}, V_{a}] ⟶ HomM̃[V_{a}, V_{b}].  (5.2.1)

This is compatible with the action of W. When V_{a}, V_{b} are multiples of a single orbit of V_{δ}, this is an isomorphism.

5.3. Orthogonal groups. [sec:5.2.1] In the case of SO(n + 1, n), the Weyl group is W (B_{n}). The centralizer W_{δ}^{0} = W_{δ} corresponding to (0) ⊗ (\underbrace{1, …, 1}_{k}, 0, …, 0) is isomorphic to W (C_{k} × C_{n−k}). Then

Ind_{W (C_{k}×C_{n−k})}^{W (C_{n}}[triv] = ∑(n − k + ℓ, k − ℓ) × (0).  (5.3.1)

The corresponding tensor product is

∑(0) ⊗ (\underbrace{2, …, 2}_{a}, 1, …, 1, \underbrace{0, …, 0}_{b}).  (5.3.2)

These K-types are automatically level ≤ 2. The formulas imply that these W -types are realized in the M-spherical vectors of petite K-types. This is sufficient for the purpose of corollary 4.4. For the more precise results in (4.4.13-4.4.15) one needs to show first that the factors occuring in (5.3.2) with 1’s do not have any M fixed vectors. This can be done by showing that such factors occur in tensor products of distinct fine K-types which cannot have any M-spherical vectors. To sort out the matchup of the remaining K-types with the W-representations, one needs to compare the formulas (5.3.1) and (5.4.2) for various values of k.

Tensor products of fine K-types of SO(n + 1, n) are not sufficient to realize the remaining relevant Weyl group representations in the M-spherical vectors of petite K-types. To achieve this, we observe that Spin(n + 1, n), the rational points of Spin(2n + 1, ℂ) has another pair of fine K-types

spin ⊗ spin^{±}.  (5.3.3)

In this case W_{δ}^{0} = S_{n}, while W_{δ} = S_{n} × ℤ_{2}. Then

\begin{matrix}Ind_{S_{n}×ℤ_{2}}[triv ⊗ triv] = ∑(n − k) × (k),  k even, \\ Ind_{S_{n}×ℤ_{2}}[triv ⊗ sgn] = ∑(n − k) × (k),  k odd. \end{matrix}  (5.3.4)

We combine these formulas with

spin ⊗ spin = ∑(\underbrace{1, …, 1}_{k}, 0, …, 0), for Spin(2n+1),

spin^{+} ⊗ spin^{−} = ∑_{k even}(\underbrace{1, …, 1}_{n−k−1}, 0, …, 0),

spin^{+} ⊗ spin^{+} = (1, …, 1) + ∑_{k>0 even}(\underbrace{1, …, 1}_{n−k}, 0, …, 0),

spin^{−} ⊗ spin^{−} = (1, … 1, −1) + ∑_{k>0 even}(\underbrace{1, …, 1}_{n−k}, 0, …, 0).

(5.3.5)

This proves that the sum of petite representations in formulas (4.4.9) contains the sum of all the relevant W-representations. This is sufficient for corollary 4.4.

In the case O(n, n), we do not need to use the covers. We can also derive the matchup of the petite K-types with the relevant W-representations in (4.4.6-4.4.8) in a manner similar to the symplectic group case, where we give a few more details.

5.4. Symplectic groups. [sec:5.2.2] The basic tool is the tensor product formulas

\begin{matrix}μ_{+}(k) ⊗ μ_{−}(k) = ∑(\underbrace{1, …, 1}_{a}, \underbrace{0, …, 0}_{b}, −1, …, −1), \\ μ_{+}(k) ⊗ μ_{+}(k) = ∑(\underbrace{2, …, 2}_{a}, \underbrace{1, …, 1}_{b}, 0, …, 0),  2k ≤ n \end{matrix} (5.4.1)

with 2a + b = 2k. These K-types are automatically level ≤ 2, so petite. The stabilizer W_{δ}^{0} for μ_{±}(k) is W (D_{k}) × W (C_{n−k}), while W_{δ} = W (C_{k}) × W (C_{n−k}). Then

Ind_{W (D_{k})}^{W (C_{k})}[triv] = (k) × (0) + (0) × (k).  (5.4.2)

The induced modules corresponding to (5.4.1) are

\begin{matrix}Ind_{W (C_{k})×W (C_{n−k})}^{W (C_{n})}[(k) × (0) ⊗ (n − k) × (0)] = ∑(n − k + ℓ, k − ℓ) × (0), \\ Ind_{W (C_{k})×W (C_{n−k})}^{W (C_{n})}[(0) × (k) ⊗ (n − k) × (0)] = (n − k) × (k). \end{matrix}

(5.4.3)

These results imply that any relevant Weyl group representation is realized in (E_{μ}^{+})^{M} for some petite K-type. To get the more precise result of (4.5.3-4.5.4), one needs to decompose μ_{+}(r) ⊗ μ_{+}(s) for all r, s. For r ≠ s, none of the factors have M-fixed vectors. Then one makes a comparison of the remaining representations occuring in the tensor products for r = s with the induced representations of the corresponding Weyl groups.

5.5. G2. [sec:5.2.3] The stabilizer of δ_{3} is the subgroup W (A_{1,s} × A_{1,l}). The induced module

Ind_{W (A_{1,s}×A_{1,l}}^{W (G2)}[triv] = 1_{1} + 2_{2}  (5.5.1)

corresponds to the tensor product

(2 | 0) ⊗ (2 | 0) = (4 | 0) + (2 | 0) + (0 | 0).  (5.5.2)

On the other hand,

(1 | 0) ⊗ (1 | 0) = (2 | 0) + (0 | 0).  (5.5.3)

Since the M-fixed vectors in the tensor product have dimension 1, it follows that (2 | 0) has no M-fixed vectors. Combining this with the fact that the representation of W on the M-fixed vectors of 𝔰 = (3 | 1) is the reflection representation, we conclude

\begin{matrix}(0 | 0) ⟷ & 1_{1}, \\ (4 | 0) ⟷ & 2_{2}, \\ (3 | 1) ⟷ & 2_{1}. \end{matrix}  (5.5.4)

22 DAN BARBASCH

5.6. F4. [sec:5.3]

We first compute some tensor products of the δ’s. In the process we get the restrictions of the K̃-types in theorem 4.8 to M̃.

• δ_{2} ⊗ δ_{2} = δ_{0} + δ_{3} because it equals (1 | 0, 0, 0) ⊗ (1 | 0, 0, 0) = (2 | 0, 0, 0) + (0 | 0, 0, 0).

• δ_{2} ⊗ δ_{3} equals (1 | 0, 0, 0) ⊗ (2 | 0, 0, 0) = (3 | 0, 0, 0) + (1 | 0, 0, 0) = 2δ_{2} + δ_{2}, because there is no room for a δ_{6}. So we also get (3 | 0, 0, 0) = 2δ_{2}.

• δ_{2} ⊗ δ_{6} equals (1 | 0, 0, 0) ⊗ (0 | 1, 0, 0) = (1 | 1, 0, 0) = δ_{12}.

• δ_{2} ⊗ δ_{12} = δ_{2} ⊗ δ_{2} ⊗ δ_{6} equals

(δ_{1} + δ_{3}) ⊗ δ_{6} = δ_{6} + δ_{3} ⊗ δ_{6}.

It also equals (1 | 0, 0, 0) ⊗ (1 | 1, 0, 0) = (2 | 1, 0, 0) + (0 | 1, 0, 0) = δ_{3} ⊗ δ_{6} + δ_{6} = 4δ_{6}.

Then δ_{3} ⊗ δ_{6} = aδ_{2} + bδ_{6}. But

δ_{2} ⊗ δ_{3} ⊗ δ_{6} = 3δ_{2} ⊗ δ_{6} = 3δ_{12},

so a = 0. So also (2 | 1, 0, 0) = δ_{3} ⊗ δ_{6} = 3δ_{6}.

• δ_{6} ⊗ δ_{6} = 3δ_{1} + aδ_{3} + bδ_{12}. Because

δ_{3} ⊗ δ_{6} ⊗ δ_{6} = 3δ_{6} ⊗ δ_{6},

a = 3 and b = 2. Since also (0 | 1, 0, 0) ⊗ (0 | 1, 0, 0) = (0 | 2, 0, 0) + (0 | 1, 1, 0) + (0 | 0, 0, 0), we conclude

\begin{matrix}(0 | 2, 0, 0) = 3δ_{3} + δ_{12}, \\ (0 | 1, 1, 0) = 2δ_{1} + δ_{12}. \end{matrix}

We now compute the induced modules

\begin{matrix}Ind_{W (B_{4})}^{W (F_{4})}[triv] = 1_{1} + 2_{1}, \\ Ind_{W (C_{4})}^{W (F_{4})}[triv] = 1_{1} + 2_{3}, \\ Ind_{W (B_{3}A_{1})}^{W (F_{4})}[triv] = 1_{1} + 2_{1} + 9_{1}, \end{matrix}  (5.6.1)

corresponding to the tensor products

\begin{matrix}(0 | 1, 0, 0) ⊗ (0 | 1, 0, 0) = (0 | 2, 0, 0) + (0 | 1, 1, 0) + (0 | 0, 0, 0), \\ (2 | 0, 0, 0) ⊗ (4 | 0, 0, 0) = (2 | 0, 0, 0) + (0 | 0, 0, 0), \\ (1 | 1, 0, 0) ⊗ (1 | 1, 0, 0) = (2 | 2, 0, 0) + (2 | 1, 1, 0) + (2 | 0, 0, 0)+ \\ + (0 | 2, 0, 0) + (0 | 1, 1, 0) + (0 | 0, 0, 0). \end{matrix} (5.6.2)

Noting also that

(2 | 2, 0, 0) = (2 | 0, 0, 0) ⊗ (0 | 2, 0, 0) = δ_{3} ⊗ (3δ_{3} + δ_{12}),  (5.6.3)

we conclude that the M-fixed vectors of (2 | 0, 0, 0) are nine dimensional. Therefore

\begin{matrix}(0 | 1, 1, 0) ⟷ 2_{1}, \\ (4 | 0, 0, 0) ⟷ 2_{3}, \\ (2 | 2, 0, 0) ⟷ 9_{1}. \end{matrix}  (5.6.4)

Because (1 | 1, 1, 1) is the representation of K on 𝔰, the Weyl group representation on the fixed vectors is the reflection representation 4_{2}.

Now consider

Hom_{M}[(1 | 1, 0, 0), (0 | 1, 1, 0)] = Hom_{M}[δ_{12}, 2δ_{1} + δ_{12}].  (5.6.5)

It is therefore 12-dimensional. Because δ_{12} corresponds to B_{3}A_{1}, in (4.8.5), the W -module is induced from a character of W (B_{3}A_{1}). Thus the tensor product

(1 | 1, 0, 0) ⊗ (0 | 1, 1, 0) = (1 | 2, 1, 0) + (1 | 1, 1, 1) + (1 | 1, 0, 0) (5.6.6)

has the property that the W -representation on the M-fixed vectors is induced from this 1-dimensional character. It follows that the M-fixed vectors of (1 | 2, 1, 0) are 8-dimensional. To determine the character, we note that the restriction of 4_{2} to W (B_{3}A_{1}) is

4_{2} ⟶ [(2 × 1) ⊗ (2)] + [(3 × 0) ⊗ (1^{2})].  (5.6.7)

The induced module from the character which occurs, is

[(3 × 0) ⊗ (1^{2})] ⟶ 4_{2} + 8_{1}.  (5.6.8)

The proof of theorem 4.8 follows.

5.7. E6. [sec:4.8a] The proof will occupy several sections. We will exploit the fact that genuine representations are all multiples of δ_{8}.

The following induced modules,

\begin{matrix}Ind_{W (D5)}^{W (E6)}[triv] = 1_{p} + 6_{p} + 20_{p}, \\ Ind_{W (A5A1)}^{W (E6)}[triv] = 1_{p} + 15_{q} + 20_{p}, \end{matrix}  (5.7.1)

correspond to the tensor products

\begin{matrix}ω_{2} ⊗ ω_{2} = (2ω_{1}) + (ω_{1} + ω_{3}) + (2ω_{2}) + (ω_{4}) + (ω_{2}) + (0), \\ 2ω_{1} ⊗ 2ω_{1} = (4ω_{1}) + (2ω_{1} + ω_{2}) + (2ω_{1}) + (2ω_{2}) + (ω_{2}) + (0). \end{matrix} (5.7.2)

We know that ω_{4} ⟷ 6_{p} because this is the representation of K̃ on 𝔰 of the Cartan decomposition. We conclude that

\begin{matrix}(2ω_{1} + ω_{2}) + (2ω_{2}) ⟷ 20_{p}, \\ (ω_{1} + ω_{3}) + (2ω_{2}) ⟷ 20_{p}. \end{matrix}  (5.7.3)

But (2ω_{1}) ⊗ (ω_{2}) does not have any M-fixed vectors, so 2ω_{1} + ω_{2} which is a factor does not have any either. We conclude that

\begin{matrix}(0) ⟷ & 1_{p}, \\ (ω_{4}) ⟷ & 6_{p}, \\ (2ω_{2}) ⟷ & 20_{p}, \\ (4ω_{1}) ⟷ & 15_{q}. \end{matrix}  (5.7.4)

The rest of the representations considered so far do not have any M-fixed vectors. Using tensoring with ω_{1} we also find that

\begin{matrix}(ω_{1}) ⟷ & 1_{p}, \\ (ω_{3}) ⟷ & 6_{p}, \\ (ω_{1} + ω_{2}) ⟷ & 20_{p}, \\ (3ω_{1}) ⟷ & 15_{q}. \end{matrix}  (5.7.5)

In these matchups, the W representations are on Hom_{M}[ω_{1}, V].

5.8. [sec:4.8b] The tensor product

(ω_{3})⊗(ω_{3}) = (2ω_{1})+(ω_{1}+ω_{3})+(2ω_{2})+(ω_{2}+ω_{4})+(ω_{2})+(2ω_{3})+(0) (5.8.1)

corresponds to

6_{p} ⊗ 6_{p} = 1_{p} + 20_{p} + 15_{p}.  (5.8.2)

Since ω_{4} = 6δ_{1} + δ_{36}, (ω_{2}) ⊗ (ω_{4}) does not have any M-fixed vectors. Thus (ω_{2} + ω_{4}) doesn’t either. We conclude

(2ω_{3}) ⟷ 15_{p}.  (5.8.3)

5.9. [sec:4.8c] The representations (3ω_{1}) and ω_{3} are genuine, and

(3ω_{1}) ⊗ (ω_{3}) = (3ω_{1} + ω_{3}) + (2ω_{1} + ω_{2}) + (2ω_{1} + ω_{4}) + (ω_{1} + ω_{3}) (5.9.1)

corresponds to

6_{p} ⊗ 15_{q} = 30_{p} + 60_{p}.  (5.9.2)

Since

(ω_{1}) ⊗ (ω_{1} + ω_{4}) = (2ω_{1} + ω_{4}) + (ω_{1} + ω_{3}) + (ω_{2} + ω_{4}) + (ω_{4}), (5.9.3)

and the dimension of (ω_{1} + ω_{4}) is 8 · 36, we conclude that

\begin{matrix}(2ω_{1} + ω_{4}) ⟷ & 30_{p}, \\ (3ω_{1} + ω_{3}) ⟷ & 60_{p}. \end{matrix}  (5.9.4)

This completes the proof of corollary 4.9.

RELEVANT AND PETITE K-TYPES FOR SPLIT GROUPS 25

5.10. [sec:4.8d] The tensor product

\begin{matrix}(ω_{1} + ω_{2}) ⊗ (ω_{3}) = (ω_{1} + ω_{2} + ω_{3}) + (2ω_{1} + ω_{2}) + (2ω_{1} + ω_{4})+ \\ (2ω_{1}) + 2(ω_{1} + ω_{3}) + (2ω_{2}) + (ω_{2} + ω_{4}) + (ω_{2}) + (ω_{4}) \end{matrix} (5.10.1)

corresponds to

6_{p} ⊗ 20_{p} = 6_{p} + 30_{p} + 20_{p} + 64_{p}.  (5.10.2)

We conclude that

(ω_{1} + ω_{2} + ω_{3}) ⟷ 64_{p}.  (5.10.3)

5.11. [sec:4.8e] The tensor product

\begin{matrix}(3ω_{1}) ⊗ (ω_{1} + ω_{2}) = (4ω_{1} + ω_{2}) + (4ω_{1}) + (3ω_{1} + ω_{3})+ \\ (2ω_{1} + 2ω_{2}) + 2(2ω_{1} + ω_{2}) + (2ω_{1})+ \\ (ω_{1} + ω_{2} + ω_{3}) + (ω_{1} + ω_{3}) + (2ω_{2}) + (ω_{2}) \end{matrix}  (5.11.1)

corresponds to

15_{q} ⊗ 20_{p} = 15_{q} + 20_{p} + 60_{p} + 60_{s} + 64_{p} + 81_{p}.  (5.11.2)

We conclude that

(4ω_{1} + ω_{2}) + (2ω_{1} + 2ω_{2}) ⟷ 81_{p} + 60_{s}.  (5.11.3)

5.12. [sec:4.8f] The tensor product

\begin{matrix}(ω_{1} + ω_{2}) ⊗ (ω_{1} + ω_{2}) = \\ (4ω_{1}) + (3ω_{1} + ω_{3}) + (2ω_{1} + 2ω_{2}) + 3(2ω_{1} + ω_{2})+ \\ (2ω_{1} + ω_{4}) + 2(2ω_{1}) + 2(ω_{1} + ω_{2} + ω_{3}) + 3(ω_{1} + ω_{3})+ \\ (3ω_{2}) + 2(2ω_{2}) + (ω_{2} + ω_{4}) + (2ω_{2}) + (2ω_{3}) + (ω_{4}) + (0) \end{matrix}  (5.12.1)

corresponds to

20_{p}⊗20_{p} = 1_{p}+6_{p}+15_{q}+15_{p}+2·20_{p}+24_{p}+30_{p}+60_{p}+2·64_{p}+81_{p}. (5.12.2)

We conclude that

(2ω_{1} + 2ω_{2}) + (3ω_{2}) ⟷ 81_{p} + 24_{p}.  (5.12.3)

5.13. [sec:4.8g] The tensor product

\begin{matrix}(3ω_{1}) ⊗ (3ω_{1}) = (6ω_{1}) + (4ω_{1} + ω_{2}) + (4ω_{1})+ \\ (2ω_{1} + 2ω_{2}) + 2(2ω_{1} + ω_{2}) + (2ω_{1})+ \\ (3ω_{2}) + (2ω_{2}) + (ω_{1}) + (0) \end{matrix}  (5.13.1)

corresponds to

15_{q} ⊗ 15_{q} = 1_{p} + 15_{q} + 20_{p} + 24_{p} + 24′_{p} + 60_{s} + 81_{p}.  (5.13.2)

We conclude that

(6ω_{1})+(4ω_{1} +ω_{2})+(2ω_{1} +2ω_{2})+(3ω_{2}) ⟷ 24_{p} +81_{p} +60_{s} +24′_{p}. (5.13.3)

Combining this with (5.12.3), and (5.11.3), we get

\begin{matrix}(3ω_{2}) ↔ 24_{p}, \\ (4ω_{1} + ω_{2}) ↔ 60_{s}, \\ (2ω_{1} + 2ω_{2}) ↔ 81_{p}, \\ (6ω_{1}) ↔ 24′_{p}. \end{matrix}  (5.13.4)

5.14. E7. [sec:4.10] In this case, genuine representations restrict to combinations of δ_{8} and δ_{8}^{∗}, so the proof will be more involved. We first note that

(ω_{4}) ⟷ 7′_{a},  (5.14.1)

because it is the representation of K on 𝔰 in the Cartan decomposition. We record the following induced representations:

\begin{matrix}Ind_{A7}^{E7}[triv] = 1_{a} + 35_{b} + 21′_{b} + 15′_{a}, \\ Ind_{E6}^{E7}[triv] = 1_{a} + 27_{a} + 7′_{a} + 21′_{b}, \\ Ind_{D6}^{E7}[triv] = 1_{a} + 27_{a} + 35_{b}. \end{matrix}  (5.14.2)

For the first one, W_{δ} ≠ W_{δ}^{0}. It corresponds to the sum of

\begin{matrix}(2ω_{1}) ⊗ (2ω_{1}) = (4ω_{1}) + (2ω_{1} + ω_{2}) + (2ω_{2}), \\ and \\ (2ω_{1}) ⊗ (2ω_{7}) = (2ω_{1} + 2ω_{7}) + (ω_{1} + ω_{7}) + (0). \end{matrix}  (5.14.3)

The first line is 21′_{b} + 15_{a}, the second one 1_{a} + 35_{b}. Since

\begin{matrix}(ω_{1}) ⊗ (ω_{7}) = (ω_{1} + ω_{7}) + (0), \\ (ω_{1}) ⊗ (ω_{1}) = (2ω_{1}) + (ω_{2}), \\ (ω_{7}) ⊗ (ω_{7}) = (2ω_{7}) + (ω_{6}), \end{matrix}  (5.14.4)

we conclude that none of the representations on the RHS of these representations (except (0)) are M-spherical. We conclude that

\begin{matrix}(2ω_{1} + 2ω_{7}) ⟷ 35_{b}, \\ (4ω_{1}) + (2ω_{1} + ω_{2}) + (2ω_{2}) ⟷ 21′_{b} + 15_{a}. \end{matrix}  (5.14.5)

For the second equation, W_{δ} ≠ W_{δ}^{0} as well. It corresponds to the sum

\begin{matrix}(ω_{2}) ⊗ (ω_{2}) = (2ω_{2}) + (ω_{1} + ω_{3}) + (ω_{4}), \\ and \\ (ω_{2}) ⊗ (ω_{6}) = (ω_{2} + ω_{6}) + (ω_{1} + ω_{7}) + (0). \end{matrix}  (5.14.6)

The first line is 7′_{a} + 21′_{b}, the second one 1_{a} + 27_{a}. The same argument as before implies

\begin{matrix}(ω_{2} + ω_{6}) ⟷ 27_{a}, \\ (2ω_{2}) + (ω_{1} + ω_{3}) ⟷ 21′_{b}. \end{matrix}  (5.14.7)

But

(ω_{1}) ⊗ (ω_{3}) = (ω_{1} + ω_{3}) + (ω_{4}),  (5.14.8)

RELEVANT AND PETITE K-TYPES FOR SPLIT GROUPS 27

and the dimension of ω_{3} is 8 · 7, So ω_{1} + ω_{3} cannot have a 21-dimensional space of M-fixed vectors. It follows that

\begin{matrix}(2ω_{2}) ⟷ 21′_{b}, \\ (ω_{3}) ⟷ 21′_{b}  on δ_{7}^{∗}. \end{matrix}  (5.14.9)

The dimension of (ω_{1} + ω_{2}) is 8 · 21, and

(ω_{1} + ω_{2}) ⊗ (ω_{1}) = (2ω_{1} + ω_{2}) + (ω_{1} + ω_{3}) + (2ω_{2}).  (5.14.10)

It follows that neither (2ω_{1} + ω_{2}) nor (ω_{1} + ω_{3}) can be M-spherical. So

(4ω_{1}) ⟷ 15′_{a}.  (5.14.11)

For the third equation, W_{δ} = W_{δ}^{0}. It corresponds to

\begin{aligned}(ω_{1} + ω_{7}) & ⊗ (ω_{1} + ω_{7}) = (2ω_{1} + 2ω_{7}) + (2ω_{1} + ω_{6})+ \\ & + 2(ω_{1} + ω_{7}) + (ω_{2} + ω_{6}) + (ω_{2} + ω_{7}) + (0). \end{aligned}  (5.14.12)

We conclude that neither of

(2ω_{1} + ω_{6}), 2(ω_{1} + ω_{7}), (ω_{2} + ω_{7})  (5.14.13)

are M-spherical.

5.15. [sec:4.11] The following tensor products,

\begin{matrix}(ω_{1} + ω_{6}) ⊗ (ω_{1}) = (2ω_{1} + ω_{6}) + (ω_{1} + ω_{7}) + (ω_{2} + ω_{6}), \\ (ω_{1} + 2ω_{7}) ⊗ (ω_{1}) = (2ω_{1} + 2ω_{7}) + (ω_{1} + ω_{7}) + (ω_{2} + 2ω_{7}), \end{matrix} (5.15.1)

and the results in section 5.14 allow us to conclude that

\begin{matrix}(ω_{1}) ⟷ 1_{a}, \\ (ω_{3}) ⟷ 7′_{a} \\ (3ω_{1}) ⟷ 15′_{a} \\ (ω_{1} + ω_{2}) ⟷ 21′_{b}, \\ (ω_{1} + ω_{6}) ⟷ 27_{a}, \\ (ω_{1} + 2ω_{7}) ⟷ 35_{b}, \end{matrix}  (5.15.2)

on the δ_{7}-isotypic component. From

\begin{matrix}(ω_{1} + ω_{6}) ⊗ (ω_{1}) = (2ω_{1} + ω_{6}) + (ω_{1} + ω_{7}) + (ω_{2} + ω_{6}) \\ (3ω_{1}) ⊗ (ω_{7}) = (3ω_{1} + ω_{7}) + (2ω_{1}) \end{matrix}  (5.15.3)

we conclude that (3ω_{1} + ω_{7}) and (2ω_{1} + ω_{6}) are not M-spherical.

5.16. [sec:4.12] Now consider

7′_{a} ⊗ 7′_{a} = 1_{a} + 21_{a} + 27_{a},  (5.16.1)

and

\begin{matrix}(ω_{3}) ⊗ (ω_{3}) = (2ω_{3}) + (ω_{1} + ω_{5}) + (ω_{2} + ω_{4}) + (ω_{6}), \\ (ω_{3}) ⊗ (ω_{5}) = (ω_{3} + ω_{5}) + (ω_{1} + ω_{7}) + (ω_{2} + ω_{6}) + (0). \end{matrix}  (5.16.2)

It follows that

(ω_{3} + ω_{5}) ⟷ 21_{a},  (5.16.3)

and aside from the already listed cases, none of the representations occuring in (5.16.2) are M-spherical.

Next observe that

(ω_{5})⊗(ω_{1}+ω_{2}) = (ω_{1}+ω_{2}+ω_{5})+(2ω_{1}+ω_{6})+(ω_{1}+ω_{7})+(ω_{2}+ω_{6}) (5.16.4)

corresponds to

7′_{a} ⊗ 21′_{b} = 27_{a} + 120_{a}.  (5.16.5)

Because (2ω_{1} + ω_{6}) does not have M-fixed vectors, we conclude that

(ω_{1} + ω_{2} + ω_{5}) ⟷ 120_{a}.  (5.16.6)

The same argument using

(ω_{3})⊗(ω_{1}+ω_{2}) = (ω_{1}+ω_{2}+ω_{3})+(2ω_{1}+ω_{4})+(ω_{1}+ω_{5})+(ω_{2}+ω_{4}) (5.16.7)

shows that none of the representations in the RHS of (5.16.7) can be Mspherical.

5.17. [sec:4.13] Note that

\begin{aligned}(ω_{3}) & ⊗ (ω_{1} + ω_{6}) = (ω_{1} + ω_{3} + ω_{6}) + (2ω_{1})+ \\ & + (ω_{1} + ω_{2} + ω_{7}) + (ω_{2}) + (ω_{3} + ω_{7}) + (ω_{4} + ω_{6}) \\ (ω_{5}) & ⊗ (ω_{1} + ω_{6}) = (ω_{1} + ω_{5} + ω_{6}) + (ω_{1} + ω_{4} + ω_{7})+ \\ & + (ω_{1} + ω_{3}) + (ω_{4}) + (ω_{5} + ω_{7}) + (2ω_{6}). \end{aligned}  (5.17.1)

The representations in the first equation do not have M-fixed vectors, while the ones in the second equation correspond to

7′_{a} ⊗ 27_{a} = 7′_{a} + 21′_{b} + 56′_{a} + 105′_{a}.  (5.17.2)

Since

(ω_{7}) ⊗ (ω_{1} + ω_{4}) = (ω_{1} + ω_{4} + ω_{7}) + (ω_{1} + ω_{3}) + (ω_{4}),  (5.17.3)

and the dimension of (ω_{1} + ω_{4}) is 8 · 63, we conclude that the dimension of the M-fixed vectors in (ω_{1} + ω_{4} + ω_{7}) is 56. So it follows that

\begin{matrix}(ω_{1} + ω_{5} + ω_{6}) ⟷ 105′_{a}, \\ (ω_{1} + ω_{4} + ω_{7}) ⟷ 56′_{a}. \end{matrix}  (5.17.4)

We also conclude that

(ω_{1} + ω_{4}) ⟷ 7′_{a} + 56′_{a}.  (5.17.5)

5.18. [sec:4.14] Note that

\begin{matrix}(ω_{3}) ⊗ (ω_{1} + 2ω_{7}) = & (ω_{1} + ω_{3} + 2ω_{7}) + (ω_{1} + ω_{2} + ω_{7})+ \\ + (ω_{3} + ω_{7}) + (ω_{4} + 2ω_{7}), \\ (ω_{3}) ⊗ (ω_{1} + 2ω_{7}) = & (ω_{1} + ω_{3} + 2ω_{7}) + (ω_{1} + ω_{2} + ω_{7})+ \\ + (ω_{3} + ω_{7}) + (ω_{4} + 2ω_{7}). \end{matrix}  (5.18.1)

The first equation consists of representations without M-fixed vectors, the second one corresponds to

7′_{a} ⊗ 35_{b} = 56′_{a} + 189′_{b}.  (5.18.2)

It follows that

(ω_{1} + ω_{5} + 2ω_{7}) ↔ 189′_{b}.  (5.18.3)

5.19. [sec:4.15] Note that

\begin{matrix}(ω_{3}) ⊗ (3ω_{1}) = (3ω_{1} + ω_{3}) + (2ω_{1} + ω_{4}), \\ (ω_{5}) ⊗ (3ω_{1}) = (3ω_{1} + ω_{5}) + (2ω_{1} + ω_{6}). \end{matrix}  (5.19.1)

The first equation does not contain any representations with M-fixed vectors, while the second one corresponds to

7′_{a} ⊗ 15′_{a} = 105_{b}.  (5.19.2)

So

(3ω_{1} + ω_{5}) ⟷ 105_{b}.  (5.19.3)

The proof of corollary 4.10 is complete.

5.20. [sec:4.16] The tensor products decompose

\begin{aligned}(ω_{1} + ω_{2}) & ⊗ (ω_{1} + ω_{2}) = (2ω_{1} + 2ω_{2}) + (3ω_{1} + ω_{3}) + (2ω_{1} + ω_{4})+ \\ & + 2(ω_{1} + ω_{2} + ω_{3}) + (3ω_{2}) + (ω_{2} + ω_{4}) + (2ω_{3}), \\ (ω_{1} + ω_{2}) & ⊗ (ω_{6} + ω_{7}) = (ω_{1} + ω_{2} + ω_{6} + ω_{7}) + (2ω_{1} + ω_{6})+ \\ & + (2ω_{1} + 2ω_{7}) + 2(ω_{1} + ω_{7}) + (ω_{2} + ω_{6})+ \\ & + (ω_{2} + 2ω_{7}) + (0). \end{aligned} (5.20.1)

The first equation does not have any representations with M-fixed vectors, while the second one corresponds to

21′_{b} ⊗ 21′_{b} = 1_{a} + 27_{a} + 35_{b} + 168_{a} + 210_{a}.  (5.20.2)

Thus

(ω_{1} + ω_{2} + ω_{6} + ω_{7}) ⟷ 168_{a} + 210_{a}.  (5.20.3)

5.21. [sec:4.17] We decompose the tensor products

(ω_{1} + ω_{2}) ⊗ (ω_{1} + ω_{6}) = (3ω_{1} + ω_{7}) + (2ω_{1} + ω_{2} + ω_{6}) + (2ω_{1})+ 2(ω_{1} + ω_{2} + ω_{7}) + (ω_{1} + ω_{3} + ω_{6}) + (2ω_{2} + ω_{6}) + (ω_{2}) + (ω_{3} + ω_{7}),

(5.21.1)

(ω_{1} + ω_{2}) ⊗ (ω_{2} + ω_{7}) = (ω_{1} + 2ω_{2} + ω_{7}) + (2ω_{1} + ω_{3} + ω_{7}) + (2ω_{1} + ω_{2})+ 2(ω_{1} + ω_{3}) + (ω_{1} + ω_{4} + ω_{7}) + (ω_{1} + ω_{4} + ω_{7}) + (2ω_{2}) + (ω_{2} + ω_{3} + ω_{7}). Since also

21′_{b} ⊗ 27_{a} = 7′_{a} + 21′_{b} + 56′_{a} + 105′_{a} + 189′_{b} + 189′_{c},  (5.21.2)

and the first equation does not contain any representations with M-fixed vectors, we conclude

\begin{matrix}(2ω_{1} + ω_{3} + ω_{7}) ⟷ 189′_{b,c}, \\ (ω_{1} + 2ω_{2} + ω_{7}) ⟷ 189′_{b,c}. \end{matrix}  (5.21.3)

The ambiguity is resolved in section 5.25.

5.22. [sec:4.18] We decompose the tensor products

\begin{matrix}(3ω_{1}) ⊗ (3ω_{1}) = (6ω_{1}) + (4ω_{1} + ω_{2}) + (2ω_{1} + 2ω_{2}) + (3ω_{2}), \\ (3ω_{1}) ⊗ (3ω_{7}) = (3ω_{1} + 3ω_{7}) + (2ω_{1} + 2ω_{7}) + (ω_{1} + ω_{7}) + (0). \end{matrix} (5.22.1)

The first equation does not contain any representations with M-fixed vectors, the second one corresponds to

15′_{a} ⊗ 15′_{a} = 1_{a} + 35_{b} + 84_{a} + 105_{c}.  (5.22.2)

It follows that

(3ω_{1} + 3ω_{2}) ⟷ 84_{a} + 105_{c}.  (5.22.3)

5.23. [sec:4.20] We decompose the tensor products

\begin{matrix}(3ω_{1}) ⊗ (ω_{1} + ω_{6}) = (4ω_{1} + ω_{6}) + (3ω_{1} + ω_{7})+ \\ (2ω_{1} + ω_{2} + ω_{6}) + (ω_{1} + ω_{2} + ω_{7}), \\ (3ω_{1}) ⊗ (ω_{2} + ω_{7}) = (3ω_{1} + ω_{2} + ω_{7}) + (2ω_{1} + ω_{2})+ \\ (2ω_{1} + ω_{3} + ω_{7}) + (ω_{1} + ω_{3}). \end{matrix}  (5.23.1)

The first equation does not contain any M-spherical representations, while the second one corresponds to

15′_{a} ⊗ 27_{a} = 189′_{b} + 216′_{a}.  (5.23.2)

We conclude that

(3ω_{1} + ω_{2} + ω_{7}) ⟷ 216′_{a}.  (5.23.3)

5.24. [sec:4.21] We decompose the tensor products

\begin{matrix}(3ω_{1}) ⊗ (ω_{1} + ω_{2}) = (4ω_{1} + ω_{2}) + (3ω_{1} + ω_{3}) + (2ω_{1} + 2ω_{2})+ \\ (ω_{1} + ω_{2} + ω_{3}), \\ (3ω_{1}) ⊗ (ω_{6} + ω_{7}) = (3ω_{1} + ω_{6} + ω_{7}) + (2ω_{1} + ω_{6})+ \\ (2ω_{1} + 2ω_{7}) + (ω_{1} + ω_{7}). \end{matrix} (5.24.1)

The first equation does not contain any M-spherical representations, while the second on corresponds to

15′_{a} ⊗ 21′_{b} = 35_{b} + 280_{b}  (5.24.2)

We conclude that

(3ω_{1} + ω_{6} + ω_{7}) ⟷ 280_{b}.  (5.24.3)

5.25. [sec:4.22] We decompose the tensor products

\begin{matrix}(3ω_{1}) ⊗ (ω_{1} + 2ω_{7}) = (4ω_{1} + 2ω_{2}) + (3ω_{1} + ω_{7})+ \\ (2ω_{1} + ω_{2} + 2ω_{7}) + (2ω_{1}) + (ω_{1} + ω_{2} + ω_{7}) + (ω_{2}), \\ (3ω_{1}) ⊗ (2ω_{1} + ω_{7}) = (5ω_{1} + ω_{7}) + (4ω_{1})+ \\ (3ω_{1} + ω_{2} + ω_{7}) + (2ω_{1} + ω_{2}) + (ω_{1} + 2ω_{2} + ω_{7}) + (2ω_{2}). \end{matrix}  (5.25.1)

The first equation does not contain any M-spherical representations, while the second on corresponds to

15′_{a} ⊗ 35_{b} = 15′_{a} + 21′_{b} + 84′_{a} + 189′_{c} + 216′_{a}.  (5.25.2)

Combining this with section 5.21, we conclude that

\begin{matrix}(5ω_{1} + ω_{7}) ⟷ 84′_{a}, \\ (ω_{1} + 2ω_{2} + ω_{7}) ⟷ 189′_{c}, \\ (2ω_{1} + ω_{3} + ω_{7}) ⟷ 189′_{b} \end{matrix}  (5.25.3)

5.26. E8. [sec:5.3.1.1] Let (μ, V) be the fine K-type (ω_{2}). Then

\begin{matrix}Hom_{M}[V, V] ≅ Ind_{W (E7A1)}^{W (E8)}[triv] = 1_{x} + 35_{x} + 84_{x}, \\ (ω_{2}) ⊗ (ω_{2}) = (2ω_{2}) + (ω_{1} + ω_{3}) + (2ω_{1}) + (ω_{2}) + (ω_{4}) + (0). \end{matrix} (5.26.1)

This is because the stabilizer of δ_{120} in W (E8) is W (E7A1).

Similarly, if (μ, V) is the fine K-type (2ω_{1}),

\begin{matrix}Hom_{M}[V, V] ≅ Ind_{W (D8)}^{W (E8)}[triv] = 1_{x} + 84_{x} + 50_{x}, \\ (2ω_{1}) ⊗ (2ω_{1}) = (4ω_{1}) + (2ω_{1} + ω_{2}) + (2ω_{1}) + (2ω_{2}) + (ω_{2}) + (0). \end{matrix} (5.26.2)

Then (ω_{3}) has dimension 35 · 16, so δ_{16} occurs 35 times. The tensor product is

(ω_{1}) ⊗ (ω_{3}) = (ω_{1} + ω_{3}) + (ω_{2}) + (ω_{4}),  (5.26.3)

and thus the multiplicity of δ_{1} in (ω_{1} + ω_{3}) + (ω_{4}) is 35, and (5.26.1) implies that the representation on the M fixed vectors is 35_{x}. On the other hand, dim ω_{4} = 1820, so the multiplicity of δ_{1} in (ω_{4}) is nonzero. This is because 1820 is not divisible by 3, but the contribution of the other characters occuring must be divisible by 3. Using (5.26.1) again, we conclude that

(ω_{4}) ⟷ 35_{x}, (2ω_{2}) ⟷ 84_{x}.  (5.26.4)

We also conclude that the multiplicity of δ_{1} in (ω_{1} + ω_{3}) is zero, and

(ω_{3}) ⟷ 35_{x}  (5.26.5)

(on the δ_{16} isotypic component).

Consider (ω_{1} + ω_{2}) which has dimension 84 · 16, so δ_{16} occurs 84 times. Then

(ω_{1} + ω_{2}) ⊗ (ω_{1}) = (2ω_{1} + ω_{2}) + (2ω_{1}) + (ω_{1} + ω_{3}) + (2ω_{2}) + (ω_{2}). (5.26.6)

Thus only (2ω_{2}) contains δ_{1}. Combined with (5.26.2) we get

(4ω_{1}) ⟷ 50_{x}.  (5.26.7)

5.27. [sec:5.4] The representation ω_{7} equals 8δ_{16}. Also, ω_{8} = 8δ_{1} + δ_{120}, and the W-representation on (ω_{8})^{M} is 8_{x}, because this is the representation on 𝔰. Then

(ω_{1}) ⊗ (ω_{7}) = (ω_{1} + ω_{7}) + (ω_{8}).  (5.27.1)

So δ_{1} does not occur in (ω_{1} + ω_{7}), and it occurs 8 times in (ω_{8}). This implies that Hom_{M}[ω_{1}, ω_{7}] is 8_{x}. We combine this with

\begin{matrix}35_{x} ⊗ 8_{x} = 160_{z} + 112_{x} + 8_{x}, \\ (ω_{3}) ⊗ (ω_{7}) = (ω_{3} + ω_{7}) + (ω_{2} + ω_{8}) + (ω_{1} + ω_{7}) + (ω_{8}). \end{matrix}  (5.27.2)

We have to distribute the other two representations among (ω_{3} + ω_{7}) and (ω_{2} + ω_{8}). We have

(ω_{2}) ⊗ (ω_{8}) = (ω_{2} + ω_{8}) + (ω_{2}) + (ω_{8}),  (5.27.3)

and δ_{1} occurs 120 times. Thus (ω_{2}+ω_{8}) contains δ_{1} 112 times, and combined with (5.27.2) we get

(ω_{2} + ω_{8}) ⟷ 112_{x},  (ω_{3} + ω_{7}) ⟷ 160_{z}.  (5.27.4)

5.28. [sec:5.5] We apply the previous reasoning to

\begin{matrix}35_{x} ⊗ 35_{x} = & 210_{x} + 84_{x} + 567_{x} + 300_{x} + 35_{x} + 28_{x} + 1_{x}, \\ (ω_{3}) ⊗ (ω_{3}) = & (2ω_{3}) + (ω_{1} + ω_{3}) + (ω_{1} + ω_{5}) + (2ω_{1})+ \\ + (2ω_{2}) + (ω_{2} + ω_{4}) + (ω_{2}) + (ω_{4}) + (ω_{6}) + (0). \end{matrix} (5.28.1)

Then

(ω_{1}) ⊗ (ω_{5}) = (ω_{1} + ω_{5}) + (ω_{6}) + (ω_{4}),  (5.28.2)

and the dimension of ω_{5} being 273 · 16, we get that the multiplicity of δ_{1} in (ω_{6}) + (ω_{1} + ω_{5}) is 238. The tensor product decomposes

(ω_{2})⊗(ω_{4}) = (ω_{2} +ω_{4})+(ω_{1} +ω_{3})+(ω_{1} +ω_{5})+(ω_{2})+(ω_{4})+(ω_{6}), (5.28.3)

Using the fact that ω_{4} = 35δ_{1} +7δ_{120} +7δ_{135}, we conclude that δ_{1} must occur 567 times in (ω_{2}+ω_{4}). The formula for (ω_{4}) is a consequence of (5.6.7) which yields the system

\begin{matrix}1820 = 35 + 120a_{1} + 135b_{1}, \\ 7020 = 120a_{2} + 135b_{2}, \\ a_{1} + a_{2} = 34, \\ b_{1} + b_{2} = 35. \end{matrix}  (5.28.4)

The numbers 1820, 7020 are the dimensions of ω_{4} and (ω_{1} + ω_{3}), and the two last equations come from

\begin{matrix}(ω_{1}) ⊗ (ω_{1}) = (2ω_{1}) + (ω_{2}) + (0) \\ δ_{16} ⊗ δ_{16} = δ_{135} + δ_{120} + δ_{1}. \end{matrix}  (5.28.5)

Furthermore,

(ω_{8}) ⊗ (ω_{8}) = (2ω_{8}) + (ω_{6}) + (ω_{4}) + (ω_{2}) + (0)  (5.28.6)

and δ_{1} occurs 184 times. Since the dimension of (ω_{6}) is 8008 = a + 120b + 135c, it follows that a > 0, and a cannot be divisible by 5. We conclude that the only choice from (5.28.3) given that 84_{x}, 567_{x} are not available, is

\begin{matrix}(ω_{1} + ω_{5}) ⟷ 210_{x}, \\ (ω_{6}) ⟷ 28_{x}, \\ (ω_{2} + ω_{4}) ⟷ 567_{x}, \\ (2ω_{3}) ⟷ 300x, \\ (ω_{5}) ⟷ 28_{x} + 35_{x} + 210_{x}. \end{matrix}  (5.28.7)

5.29. [sec:5.6] The previous arguments also imply that

(3ω_{1}) ⟷ 50_{x}.  (5.29.1)

Then

\begin{matrix}50_{x} ⊗ 8_{x} = 400_{x}, \\ (3ω_{1}) ⊗ (ω_{7}) = (3ω_{1} + ω_{7}) + (2ω_{1} + ω_{8}). \end{matrix}  (5.29.2)

But

(ω_{1} + ω_{8}) ⊗ (ω_{1}) = (2ω_{1} + ω_{8}) + (ω_{1} + ω_{7}) + (ω_{2} + ω_{8}) + (ω_{8}). (5.29.3)

Since dim(ω_{1} +ω_{8}) = 120·16, the previous results show that δ_{1} cannot occur in (2ω_{1} + ω_{8}), and we conclude

(3ω_{1} + ω_{7}) ⟷ 400_{x}.  (5.29.4)

5.30. [sec:5.7] Recall again that

(3ω_{1}) ⟷ 50_{x},  (ω_{3}) ⟷ 35_{x}.  (5.30.1)

The tensor product decomposes

50_{x} ⊗ 35_{x} = 1050_{x} + 700_{x}.  (5.30.2)

On the other hand,

(3ω_{1}) ⊗ (ω_{3}) = (3ω_{1} + ω_{3}) + (2ω_{1} + ω_{2}) + (2ω_{1} + ω_{4}) + (ω_{1} + ω_{3}),
(ω_{1}) ⊗ (ω_{1} + ω_{4}) =  (5.30.3)
(2ω_{1} + ω_{4}) + (ω_{2} + ω_{4}) + (ω_{1} + ω_{5}) + (ω_{1} + ω_{3}) + (ω_{4}).

In the first equation 2(ω_{1} + ω_{4}) + (3ω_{1} + ω_{3}) contains all the M-spherical vectors, and their dimensions add up to 1750. Counting dimensions of Mfixed vectors in the second equation, we find that the dimension of the M-fixed vectors in (2ω_{1} + ω_{4}) is 700. Thus (2ω_{1} + ω_{4}) must contain 700_{x} and (3ω_{1} + ω_{3}) must contain 1050_{x}.

This completes the proof of corollary 4.11.

5.31. [sec:5.8] Recall that 50_{x} ⟷ (3ω_{1}). The tensor product decomposes

50_{x} ⊗ 50_{x} = 1_{x} + 50_{x} + 84_{x} + 168_{y} + 525_{x} + 700_{xx} + 972_{x}.  (5.31.1)

It corresponds to

\begin{matrix}(3ω_{1}) ⊗ (3ω_{1}) = \\ (6ω_{1}) + (4ω_{1} + ω_{2}) + (4ω_{1}) + (2ω_{1} + 2ω_{2}) + (2ω_{1} + ω_{2})+ \\ (2ω_{1}) + (3ω_{2}) + (2ω_{2}) + (ω_{2}) + (0). \end{matrix}  (5.31.2)

Since

\begin{matrix}(ω_{1} + ω_{2}) ⊗ ω_{1} = (2ω_{1} + ω_{2}) + (2ω_{2}) + (ω_{1} + ω_{3}) + (2ω_{1}) + (ω_{2}), \\ (ω_{1}) ⊗ (ω_{3}) = (ω_{4}) + (ω_{2}) + (ω_{1} + ω_{3}), \end{matrix} (5.31.3)

and taking the earlier results into account, we find that (ω_{1} +ω_{3}) and (2ω_{1} + ω_{2}) cannot have M-fixed vectors. This shows that the sum of Weyl group representations in (5.31.1) corresponds to the M-fixed vectors in (5.31.2). We omit the details of how they match the irreducible modules. Finally, recall that (ω_{1} + ω_{2}) corresponds to 84_{x}, while ω_{3} corresponds to 35_{x}. Then

84_{x} ⊗ 35_{x} = 1344_{x} + 700_{x} + 210_{x} + 567_{x} + 84_{x} + 35_{x},  (5.31.4)

while

\begin{matrix}(ω_{1} + ω_{2}) ⊗ (ω_{3}) = \\ (2ω_{1} + ω_{2}) + (2ω_{1} + ω_{4}) + (2ω_{1}) + (ω_{1} + ω_{2} + ω_{3}) + 2(ω_{1} + ω_{3})+ \\ + (ω_{1} + ω_{5}) + (2ω_{2}) + (ω_{2} + ω_{4}) + (2ω_{2}) + (ω_{2}) + (ω_{4}). \end{matrix} (5.31.5)

The fact that the representation of W (E_{8}) on the M-fixed vectors of (ω_{1} + ω_{2} + ω_{3}) is 1344_{x} follows from these two equations.

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(D. Barbasch) Dept. of Mathematics, Cornell University, Ithaca, NY 14850

   E-mail address: barbasch@math.cornell.edu