Annals of Mathematics, 0 (1996), 1–67

On the classification of unitary representations of reductive Lie groups

By Susana A. Salamanca-Riba and David A. Vogan, Jr.*

Table of Contents

Introduction

1. Projections on convex cones

2. Classification by lowest K-types

3. Relation to the unitary classification

4. Relation to ordinary parabolic induction

5. The reduction step in the classification

6. The set B_{u}^{z}(G) of unitarily small K-types

7. Unitarily small K-types and the Dirac operator

8. Topology of the unitary dual References

Introduction

Suppose G is a real reductive Lie group in Harish-Chandra’s class (see [6], section 3). We propose here a structure for the set Π_{u}(G) of equivalence classes of irreducible unitary representations of G. (The subscript u will be used throughout to indicate structures related to unitary representations.) We decompose Π_{u}(G) into disjoint subsets with a (very explicit) discrete parameter set Λ_{u}:

(0.1)  Π_{u}(G) = ⋃_{λ_{u}∈Λ_{u}} Π_{u}^{λ_{u}}(G).

Each subset is identified conjecturally (Conjecture 0.6) with a collection of unitary representations of a certain subgroup G(λ_{u}) of G. (We will give strong evidence and partial results for this conjecture.) In this way the problem of classifying Π_{u}(G) would be reduced (by induction on the dimension of G) to the case G(λ_{u}) = G.

*Research by Susana Salamanca-Riba is partially supported by NSF Grant #DMS-9510608.

Research by David Vogan is partially supported by NSF Grant #DMS-9402994.

Before considering the general program in more detail, we describe it in the familiar case G = SL(2, ℝ). (This example will be treated more completely in Example 6.2 below.) The parameter set Λ_{u} is ℤ. If n is a non-zero integer, then G(n) = SO(2), and Π_{u}^{n}(G) consists of a single discrete series representation (with Harish-Chandra parameter n + sgn n and lowest SO(2)-type n + 2 sgn n). The corresponding representation of SO(2) is the character usually parametrized by n. If n = 0, then G(0) = SL(2, ℝ), and Π_{u}^{0}(G) consists of all the remaining unitary representations: those containing an SO(2) character parametrized by 0, ±1, or ±2. (These are the principal series, the complementary series, the trivial representation, and the first two discrete series.)

At first glance it might appear natural to separate also the first two discrete series as isolated points in the parametrization. But recall that these representations are not isolated in the unitary dual of SL(2, ℝ); rather they are limits of complementary series. The decomposition we have given is consistent with the topology of Π_{u}(G), and separating the first two discrete series would not be.

As the example of SL(2, ℝ) indicates, our “reduction” to the case G(λ_{u}) = G is in some ways not very powerful; almost all of the interesting phenomena are swept into that case. Nevertheless there is some content in these ideas. In a separate paper, the first author will show how to use them to deduce an old conjecture classifying unitary representations with sufficiently regular infinitesimal character.

We turn now to a description of the decomposition 0.1 in general. Choose a Cartan involution θ of G, with fixed points K a maximal compact subgroup of G. The real Lie algebra 𝔤_{0} then acquires a Cartan decomposition

(0.2a)  𝔤_{0} = 𝔨_{0} + 𝔭_{0},

with 𝔭_{0} the −1 eigenspace of (the differential of) θ. The complexification of 𝔤_{0} is written

(0.2b)  𝔤 = 𝔤_{0} ⊗_{ℝ} ℂ;

similar notation is used for other groups. Fix a non-degenerate G-invariant, θ-invariant symmetric bilinear form ⟨ , ⟩ on 𝔤_{0}, negative definite on 𝔨_{0} and positive definite on 𝔭_{0}. We will use the same notation for the complexification of ⟨ , ⟩ (a complex-valued form on 𝔤) as well as its various restrictions and dualizations.

Our partition of representations of G is in terms of their restrictions to K; so we begin by recalling the highest weight theory for representations of K.

Fix a maximal torus T^{c} ⊂ K. The abelian group T̂^{c} of characters of T^{c} will be identified (by passing to differentials) with a lattice of imaginary-valued linear functionals on the real Lie algebra:

(0.3a)  T̂^{c} ⊂ i(𝔱_{0}^{c})^{∗}

Because the form ⟨ , ⟩ was chosen to be negative definite on 𝔨_{0}, it is positive definite on i(𝔱_{0}^{c})^{∗}. The roots of T^{c} in 𝔨 and the weights of T^{c} in 𝔤 are characters of T^{c}, and so will be regarded as elements of i(𝔱_{0}^{c})^{∗}. Choose a system of positive roots

(0.3b)  Δ^{+}(𝔨, 𝔱^{c}) ⊂ Δ(𝔨, 𝔱^{c}).

The Weyl group of T^{c} in K is by definition the normalizer of T^{c} in K divided by the centralizer:

(0.3c)  W (K, T^{c}) = N_{K}(T^{c})/Z_{K}(T^{c}).

Since K need not be connected, this group may be larger than the Weyl group W (𝔨, 𝔱^{c}) of the root system. Obviously W (K, T^{c}) acts on the root system Δ(𝔨, 𝔱^{c}), so we can define

(0.3d)  R(G) = {w ∈ W (K, T^{c}) | wΔ^{+}(𝔨, 𝔱^{c}) = Δ^{+}(𝔨, 𝔱^{c})}

(see [16], Definition 5.1.2). The notation is chosen to reflect the fact that R(G) is closely related to the R-group of Knapp and Stein. Notice that if G is connected, then R(G) is trivial. There is a semidirect product decomposition

(0.3e)  W (K, T^{c}) = R(G)W (𝔨, 𝔱^{c})

with the second factor normal. Each root α in Δ(𝔨, 𝔱^{c}) corresponds to an eigenspace 𝔨_{α} of Ad(T^{c}) on the complexified Lie algebra. We define

(0.3f)  𝔫_{𝔨} = ∑_{α∈Δ^{+}(𝔨,𝔱^{c})} 𝔨_{α},  𝔟_{𝔨} = 𝔫_{𝔨} + 𝔱^{c};

this is a Borel subalgebra of 𝔨.

Lemma 0.1. (See [16], Proposition 5.1.9 ). Using the notation of 0.3, suppose V is a finite-dimensional complex representation of the Lie algebra 𝔨. The set of highest weights of V is by definition the set of weights of 𝔱^{c} acting on V^{𝔫_{𝔨}}, a subset of (𝔱^{c})^{∗}.

a) If V is an irreducible representation of 𝔨, then V^{𝔫_{𝔨}} has dimension one, so V has exactly one highest weight.

b) The element μ ∈ (𝔱^{c})^{∗} is the highest weight of an irreducible representation of 𝔨 if and only if μ is dominant integral for 𝔨: that is,

2⟨μ, α⟩/⟨α, α⟩ ∈ ℕ

for all α ∈ Δ^{+}(𝔨, 𝔱^{c}).

c) Suppose V is a finite-dimensional irreducible representation of K. Then the collection of highest weights of V is a single orbit of R(G) on T̂^{c} ⊂ i(𝔱_{0}^{c})^{∗} (cf. 0.3(d)).

Write

(0.4a)  𝔥_{0}^{c} = 𝔤_{0}^{T^{c}} = 𝔨_{0}^{T^{c}} + 𝔭_{0}^{T^{c}} = 𝔱_{0}^{c} + 𝔞_{0}^{c},

a fundamental Cartan subalgebra of 𝔤_{0}. We will be interested not so much in the roots of 𝔥^{c} in 𝔤 as in their restrictions to 𝔱^{c}: the non-zero weights of T^{c} on 𝔤. Define

(0.4b)  Δ(𝔭, 𝔱^{c}) = set of non-zero weights of T^{c} on 𝔭.

These weights all occur with multiplicity one. Define

(0.4c)  Δ(𝔤, 𝔱^{c}) = Δ(𝔭, 𝔱^{c}) ∪ Δ(𝔨, 𝔱^{c}),

a subset with multiplicities in i(𝔱_{0}^{c})^{∗}. One can find fairly detailed structural information about this set for example in [15], section 2. We mention here only a few highlights. First, Δ(𝔤, 𝔱^{c}) is a (possibly non-reduced) root system. The elements having multiplicity one are the restrictions of imaginary roots of 𝔥^{c}; each is compact or noncompact. The roots having multiplicity two are the restrictions of complex roots; each corresponds to one root in 𝔨 and one in 𝔭. The Weyl group W (𝔤, 𝔱^{c}) of Δ(𝔤, 𝔱^{c}) may be identified with the stabilizer of 𝔱^{c} in W (𝔤, 𝔥^{c}); it contains W (K, T^{c}).

Whenever 𝔲 ⊂ 𝔤 is a T^{c}-invariant subspace, we write Δ(𝔲, 𝔱^{c}) for the set of weights of T^{c} in 𝔲, counted with multiplicities. Define

(0.4d)  2ρ(𝔲) = ∑_{γ∈Δ(𝔲,𝔱^{c})} γ ∈ i(𝔱_{0}^{c})^{∗}.

As a character of T^{c}, this is the determinant of the adjoint action on 𝔲. As a weight for the Lie algebra 𝔱^{c}, it is the trace of the adjoint action on 𝔲. We will make particular use of

(0.4e)  2ρ(𝔫_{𝔨}) = 2ρ_{c},

the sum of the positive roots of T^{c} in 𝔨 (see 0.3).

We turn now to the definition of the partition 0.1 of the unitary dual. We will say more about the motivations behind the construction in section 2, but here we mention just three primary sources: the notion of lowest Ktype in [15]; the geometric reformulation of that notion by Carmona in [3]; and a characterization used by the first author of the lowest K-types of the representations A_{𝔮}(λ) studied in [18].

Fix a highest weight μ ∈ i(𝔱_{0}^{c})^{∗}. Choose a system of positive roots Δ^{+}(𝔤, 𝔱^{c}) for Δ(𝔤, 𝔱^{c}) making μ + 2ρ_{c} dominant:

(0.5a)  ⟨μ + 2ρ_{c}, γ⟩ ≥ 0,  γ ∈ Δ^{+}(𝔤, 𝔱^{c}).

Because μ is dominant for K and 2ρ_{c} is dominant and regular, Δ^{+}(𝔤, 𝔱^{c}) necessarily contains Δ^{+}(𝔨, 𝔱^{c}). Define

(0.5b)  2ρ = ∑_{γ∈Δ^{+}(𝔤,𝔱^{c})} γ,

the sum of the positive roots. This positive system defines a positive Weyl chamber

(0.5c)  C = {ϕ ∈ i(𝔱_{0}^{c})^{∗} | ⟨ϕ, γ⟩ ≥ 0 (γ ∈ Δ^{+}(𝔤, 𝔱^{c}))}.

This is a closed convex cone in the Euclidean space i(𝔱_{0}^{c})^{∗}. We may therefore define a projection P from i(𝔱_{0}^{c})^{∗} onto C: P ϕ is the unique element of C closest to ϕ. (This classical construction will be considered carefully in section 1.) Using P, we can now define

(0.5d)  λ_{u}(μ) = P (μ + 2ρ_{c} − 2ρ).

This is an element of the positive Weyl chamber C ⊂ i𝔱_{0}^{∗}; we will see in Proposition 1.4 that it is independent of the choice of positive root system subject to 0.5(a). By 0.5(a), μ + 2ρ_{c} lies in C; so by definition of P,

|λ_{u}(μ) − (μ + 2ρ_{c})| ≤ |2ρ|.

Consequently the distance from μ to λ_{u}(μ) may be bounded independently of μ (say by |2ρ|+|2ρ_{c}|). Because the weights μ lie in a (discrete) lattice in i(𝔱_{0}^{c})^{∗}, it follows that the set

(0.5e)  Λ_{u} = {λ_{u}(μ), μ ∈ T̂^{c} dominant}

is discrete in i(𝔱_{0}^{c})^{∗}. It is very easy to check that the group R(G) acts on Λ_{u}. We can now define the decomposition 0.1.

Definition 0.2. Fix λ_{u} ∈ Λ_{u}. The set of K-types attached to λ_{u} is

B_{u}^{λ_{u}}(G) = {δ ∈ K̂ | δ has a highest weight μ ∈ T̂^{c} such that λ_{u}(μ) = λ_{u}}.

The set of unitary representations of G attached to λ_{u} is

Π_{u}^{λ_{u}}(G) = {(π, H_{π}) ∈ Π_{u}(G) | π has a lowest K-type in B_{u}^{λ_{u}}(G)}.

It will sometimes be convenient (especially in our consideration of the Fell topology in section 8) to abuse notation by writing

λ_{u}(π) = λ_{u}

when π ∈ Π_{u}^{λ_{u}}(G). The difficulty is that π may belong to several different Π_{u}^{λ′_{u}}(G). Corollary 0.4 below says that λ_{u}(π) is well-defined as an orbit of R(G), but not (unless G is connected) as an element of i(𝔱_{0}^{c})^{∗}.

The definition of lowest K-type is given in [15], Definition 5.1, or [16], Definition 5.4.18; we will recall it in section 2.

Proposition 0.3. Suppose μ and μ′ are highest weights of lowest Ktypes of a single irreducible representation of G. Then the weights λ_{u}(μ) and λ_{u}(μ′) lie in the same orbit of R(G) (cf. 0.3 ).

We will prove this at the end of section 3.

Corollary 0.4. Suppose λ_{u} and λ′_{u} belong to Λ_{u} (cf. 0.5 ). If λ_{u} and λ′_{u} belong to the same orbit of R(G), then Π_{u}^{λ_{u}}(G) = Π_{u}^{λ′_{u}}(G). If not, then Π_{u}^{λ_{u}}(G) and Π_{u}^{λ′_{u}}(G) are disjoint.

To get a partition as described in 0.1, we could of course replace Λ_{u} by a fundamental domain for the action of R(G); but since there is no natural choice for such a fundamental domain, it is more convenient for us to leave matters in this form.

Next, we want to describe the group G(λ_{u}) discussed after 0.1. The root space decomposition of 𝔨 implies that

𝔨_{0} = 𝔱_{0}^{c} ⊕ [𝔱_{0}^{c}, 𝔨_{0}],

and therefore that

(0.6a)  𝔤_{0} = 𝔱_{0}^{c} ⊕ [𝔱_{0}^{c}, 𝔨_{0}] ⊕ 𝔭_{0}.

By means of this decomposition any linear functional on 𝔱_{0}^{c} extends canonically to 𝔤_{0}, by making it zero on the other summands. In this way we can identify i(𝔱_{0}^{c})^{∗} naturally as a subspace of i𝔤_{0}^{∗}:

(0.6b)  i(𝔱_{0}^{c})^{∗} ↪ i𝔤_{0}^{∗}.

The group G acts on i𝔤_{0}^{∗} by the coadjoint action. In the same way T^{c} (or even its normalizer in G) acts on i𝔱_{0}^{∗}, and the inclusion 0.6(b) is T^{c}-equivariant. Since T^{c} is abelian, it follows that T^{c} acts trivially on the image. For any λ ∈ i(𝔱_{0}^{c})^{∗} ⊂ i𝔤_{0}^{∗}, define

(0.6c)  G(λ) = isotropy group at λ for the G action;

this is a subgroup of G containing T^{c}. It is easy to check that the Cartan involution θ preserves G(λ); and that G(λ) is a reductive group in Harish-Chandra’s class, with Cartan involution θ|_{G(λ)}.

The set of roots of T^{c} in G(λ) is

(0.6d)  Δ(𝔤(λ), 𝔱^{c}) = {γ ∈ Δ(𝔤, 𝔱^{c}) | ⟨γ, λ⟩ = 0}.

We can therefore construct a θ-stable parabolic subalgebra 𝔮(λ) = 𝔤(λ) + 𝔲(λ) by the following requirement (compare [16], Definition 5.2.1, and [9], Proposition 4.76):

(0.6e)  Δ(𝔲(λ), 𝔱^{c}) = {γ ∈ Δ(𝔤, 𝔱^{c}) | ⟨γ, λ⟩ > 0}.

Using the parabolic subalgebra 𝔮(λ), we define functors of cohomological parabolic induction L_{j}(λ) carrying (𝔤(λ), G(λ)∩K)-modules to (𝔤, K)-modules (see [9], section 5.1, or [16], Definition 6.3.1). These are most interesting in the degree

(0.6f)  S = dim(𝔲(λ) ∩ 𝔨).

At the same time, we have L_{j}^{K}(λ) carrying (G(λ)∩K)-modules to K-modules ([9], section 5.6).

Proposition 0.5. With notation as in Definition 0.2 and 0.6, the functor L_{S}^{K}(λ_{u}) implements a bijection from B_{u}^{λ_{u}}(G(λ_{u})) onto B_{u}^{λ_{u}}(G).

We will prove this in section 3 (Proposition 3.1). The aim of this paper is a corresponding result for unitary representations:

Conjecture 0.6. With notation as in Definition 0.2 and 0.6, the functor L_{S}(λ_{u}) implements a bijection from Π_{u}^{λ_{u}}(G(λ_{u})) onto Π_{u}^{λ_{u}}(G).

In section 5 we will explain a program for proving this conjecture, reducing it to the more elementary Conjecture 5.7′. We present strong evidence for the validity of this latter conjecture, including Proposition 7.18 (which is a partial result towards it) and Theorem 5.9 (which provides moral support for the formulation of the conjecture). The program leads to an alternative description of the subsets Π_{u}^{λ_{u}}(G) (Theorem 5.8(c)). This description will imply that each is an open and closed subset of Π_{u}(G), and that the bijection in Conjecture 0.6 is a homeomorphism in the Fell topology. We will examine these issues in section 8.

1. Projections on convex cones

In this section we recall some elementary general results about projections on convex cones. They will be applied to positive Weyl chambers in i(𝔱_{0}^{c})^{∗}, with notation as described in the introduction. The ideas and the exposition are taken from [3]. Throughout this section, V will be a finite-dimensional real vector space, endowed with a positive definite inner product ⟨ , ⟩. Write |v| = ⟨v, v⟩^{1/2} for the corresponding Hilbert space norm on V. Let C be a closed convex cone in V. Write

(1.1)  C^{o} = {v ∈ V | ⟨v, c⟩ ≥ 0, all c ∈ C}

for the dual cone; it is again a closed convex cone in V.

Proposition 1.1. (See [3], Proposition 1.2.) Suppose v ∈ V. Then there is a unique element c_{0} of C closest to v. It may be characterized by any of the following equivalent conditions.

a) For any c ∈ C, we have |v − c_{0}| ≤ |v − c|.

b) For any c ∈ C, we have ⟨v − c_{0}, c − c_{0}⟩ ≤ 0.

c) The vector c_{0} − v belongs to the dual cone C^{o}, and ⟨c_{0} − v, c_{0}⟩ = 0.

We omit the elementary proof.

Definition 1.2. In the setting of Proposition 1.1, the element c_{0} of C satisfying the equivalent conditions (a)-(c) is written P v, the projection of v on C.

We will need some additional properties of the projection P that depend on special properties of Weyl chambers. We therefore assume for the rest of this section that we are given a (possibly non-reduced) root system

(1.2a)  Δ ⊂ V,

and a set of positive roots

(1.2b)  Δ^{+} ⊂ Δ.

We do not assume that Δ spans V. Write V_{s} for the linear span of Δ in V, and V_{z} for its orthogonal complement. (The subscripts s and z stand for “semisimple” and “central.”) Write

(1.2c)  Π = {α_{1}, …, α_{l}} ⊂ Δ^{+}

for the simple roots; these form a basis of V_{s}. Write ξ_{i} for the dual basis of fundamental weights:

(1.2d)  ⟨ξ_{i}, α_{j}⟩ = δ_{ij},  ⟨ξ_{i}, v⟩ = 0 (v ∈ V_{z}).

The closed positive Weyl chamber is by definition

(1.2e)  \begin{aligned}C & = \{v ∈ V | ⟨v, α⟩ ≥ 0 (α ∈ Δ^{+})\} \\ & = V_{z} + ∑_{i=1}^{l} ℝ^{≥0}ξ_{i}. \end{aligned}

Occasionally it will be useful to emphasize the dependence on the choice of positive root system; in that case we will write C(Δ^{+}). The dual cone (cf. 1.1) is

(1.2f)  C^{o} = ∑_{i=1}^{l} ℝ^{≥0}α_{i},

the cone of positive roots. We will refer to this collection of notation and assumptions as “the setting 1.2.”

The main construction we use in this paper is 0.5(d): to begin with a dominant weight, subtract a dominant weight, then project onto the positive Weyl chamber. The following lemma will lead to several important properties of this construction.

Lemma 1.3. In the setting 1.2, suppose γ ∈ C is a dominant weight, α ∈ Π is a simple root, and v ∈ V. If ⟨v, α⟩ ≤ 0, then

⟨P (v − γ), α⟩ = 0.

Proof. By Proposition 1.1, we may write

(1.3a)  v − γ = P (v − γ) − e(v, γ),

with e(v, γ) ∈ C^{o} orthogonal to P (v − γ). By 1.2(f), this means

(1.3b)  e(v, γ) = ∑_{i} c_{i}α_{i}  (c_{i} ≥ 0);

and c_{i} > 0 implies ⟨α_{i}, P (v − γ)⟩ = 0. Now form the inner product of 1.3(a) with α. The hypothesis on v and the dominance of γ make the left side nonpositive. The first term on the right is non-negative. If it is zero we are done; so we may assume it is positive. Then

(1.3c)  ⟨α, e(v, γ)⟩ > 0.

Distinct simple roots have non-positive inner product; so 1.3(b) and 1.3(c) imply that the coefficient c_{i} of α in 1.3(b) must be strictly positive. We have already seen that this implies α is orthogonal to P (v − γ), as we wished to show.

Proposition 1.4. In the setting 1.2, fix a dominant weight γ ∈ C. Define a map T_{γ} from V to V as follows. Given v ∈ V, choose a positive root system Δ^{+}(v) making v dominant; write Δ^{+}(v) = σΔ^{+}, with σ ∈ W (Δ) (the Weyl group). Define

T_{γ}(v) = P (Δ^{+}(v))(v − σγ).

Then T_{γ}(v) is well-defined (independent of the choice of positive root system making v dominant).

Recall that this result was used in 0.5(d).

Proof. Suppose wΔ^{+}(v) is another positive system making v dominant. Then v and w^{−1}v are both dominant for Δ^{+}(v), and are conjugate by the Weyl group; so they coincide. Consequently w fixes v. Using wΔ^{+}(v) to define T_{γ}(v) leads to

P (wΔ^{+}(v))(v − wσγ) = P (wΔ^{+}(v))(wv − wσγ) = w(P (Δ^{+}(v))(v − σγ)).

By Chevalley’s theorem (see for example [9], Proposition 4.146) w is a product of reflections in simple roots orthogonal to v. By Lemma 1.3, each such simple root is orthogonal to P (Δ^{+}(v))(v − σγ). So w fixes P (Δ^{+}(v))(v − σγ), and we get finally

P (wΔ^{+}(v))(v − wσγ) = P (Δ^{+}(v))(v − σγ),

as we wished to show.

The next result will be used in section 2 to relate the parameter λ_{u}(μ) defined in 0.5 to the parameter λ(μ) constructed in [15], Proposition 4.1 or [16], Proposition 5.3.3. This will lead directly to Proposition 0.3, and to a variety of stronger properties of the sets Π_{u}^{λ_{u}}(G).

Proposition 1.5. In the setting 1.2, suppose v, γ, and δ are all dominant weights in C. Then

P (v − γ − δ) = P (P (v − γ) − δ)).

Proof. Write

(1.4a)  P (v − γ) = (v − γ) + e(v, γ),

with e(v, γ) ∈ C^{o} orthogonal to P (v − γ) (Proposition 1.1(c)). Next, write

(1.4b)  P (P (v − γ) − δ)) = (P (v − γ) − δ) + e(P (v − γ), δ),

with e(P (v − γ), δ) ∈ C^{o} orthogonal to P (P (v − γ) − δ). Combining these two equations gives

(1.4c)  P (P (v − γ) − δ)) = (v − γ − δ) + e(v, γ) + e(P (v − γ), δ).

The last term on the right is orthogonal to P (P (v − γ) − δ) by 1.4(b). The second is orthogonal to P (v−γ) by 1.4(a), and therefore also to P (P (v−γ)−δ) by Lemma 1.3. We have therefore shown that e(v, γ) + e(P (v − γ), δ) is an element of C^{o} orthogonal to P (P (v−γ)−δ). By Proposition 1.1(c), the formula in 1.4(c) now shows that

P (v − γ − δ) = P (P (v − γ) − δ),

as we wished to show.

Corollary 1.6. In the setting 1.2, suppose γ and δ are dominant weights in C. Define maps T_{γ}, etc., as in Proposition 1.4. Then

T_{γ+δ} = T_{γ} ∘ T_{δ}.

The next result will be used in the characterization of the K-types attached to the parameter 0 (Definition 0.2), for which Conjecture 0.6 gives no information. It is exactly analogous to [3], Corollaire 2.13.

Proposition 1.7. In the setting 1.2, suppose v and γ are dominant weights in C. The following conditions are equivalent.

a) P (v − γ) = 0.

b) The weight γ − v belongs to C^{o}; that is, it is a sum of positive roots with non-negative coefficients.

c) The weight v belongs to the convex hull of the Weyl group orbit of γ.

Proof. The equivalence of (a) and (b) follows from Proposition 1.1(c). That (b) and (c) are equivalent is a well known aspect of the Cartan-Weyl theory of finite-dimensional representations; but since it is not easy to find a convenient reference, we give a proof. The main step is

Lemma 1.8. ([16], Lemma 6.3.28.) Suppose γ ∈ C is dominant and σ ∈ W (Δ). Then there is an element e(σ) ∈ C^{o} so that

σ · γ = γ − e(σ).

Using this lemma, we proceed with the proof of the proposition. Suppose first that (c) holds; say v = ∑_{σ∈W} c_{σ}σ · γ, with c_{σ} ≥ 0 and ∑ c_{σ} = 1. Adding the equations in Lemma 1.8 gives

v = γ − ∑_{σ} c_{σ}e(σ).

Since C^{o} is a convex cone, the last sum belongs to C^{o}, as we wished to show.

Conversely, suppose v is not in the convex hull of the Weyl group orbit of γ. Then it is separated from the convex hull by a hyperplane. That is, there is an element λ′ ∈ V so that

⟨v, λ′⟩ > ⟨σ · γ, λ′⟩ (σ ∈ W).

This inequality may be written as

⟨v, λ′⟩ > ⟨γ, σ · λ′⟩ (σ ∈ W).

(We have replaced the variable σ by σ^{−1}; since the inequality holds for all σ ∈ W, this is permissible.) Now write λ for the dominant element of the Weyl group orbit of λ′, and λ′ = σ_{0} · λ. Then we have

⟨v, σ_{0} · λ⟩ > ⟨γ, σ · λ⟩ (σ ∈ W).

(This time we have replaced σ by σσ_{0}^{−1}.) Applying Lemma 1.8 to the dominant weight λ, we find σ_{0} · λ = λ − e(σ_{0}), with e(σ_{0}) ∈ C^{o}. Since v is dominant, it has a non-negative inner product with e(σ_{0}). So we get

⟨v, λ⟩ > ⟨γ, σ · λ⟩ (σ ∈ W).

Taking σ = 1, we find in particular

⟨v − γ, λ⟩ > 0.

Since λ is dominant, this inequality implies that γ − v ∉ C^{o}, as we wished to show.

We can now give a convenient characterization of the map v ↦ P (v − γ).

Corollary 1.9. In the setting 1.2, suppose v and γ are dominant weights in C. Let W_{0} be the Weyl group of roots orthogonal to P (v − γ) (the stabilizer in W of P (v − γ)). Then v − P (v − γ) is dominant, and belongs to the convex hull of the W_{0} orbit of γ.

Conversely, suppose that v_{0} and γ are dominant weights in C. Let W_{0} be the Weyl group of the roots orthogonal to v_{0}. Suppose γ_{0} is a dominant weight in the convex hull of the W_{0} orbit of γ. Then v = v_{0} + γ_{0} is dominant, and P (v − γ) = v_{0}.

Proof. Write Δ_{0} for the system of roots orthogonal to P (v − γ). By Proposition 1.1(c), v − γ − P (v − γ) is a sum of negative roots orthogonal to P (v − γ); that is, a sum of roots in Δ_{0}^{−}. Now apply Proposition 1.7 to the root system Δ_{0} and the Δ_{0}^{+}-dominant weight v − P (v − γ). The conclusion is that v − P (v − γ) belongs to the convex hull of the W_{0} orbit of γ. Clearly v − P (v − γ) is dominant for Δ_{0}^{+}. The remaining set Δ^{+} − Δ_{0}^{+} is permuted by W_{0}. Since γ is dominant for these roots, the convex hull of W_{0} · γ must be as well.

The converse is similar but easier, and we leave it to the reader.

Because we will most often use these results in the case γ = ρ (half the sum of the positive roots), we will need a precise description of the convex hull in that case.

Proposition 1.10. ([10], Lemma 5.9.) In the setting of 1.2, write ρ for half the sum of the positive roots. Then the convex hull of the Weyl group orbit of ρ coincides with the set R of all weights of the form

r = ∑_{α∈Δ^{+}} c_{α}α, (−1/2 ≤ c_{α} ≤ 1/2).

Equivalently, these are the weights

r = ρ − ∑_{α∈Δ^{+}} b_{α}α, (0 ≤ b_{α} ≤ 1).

2. Classification by lowest K-types

In this section we recall part of the Langlands classification of admissible irreducible representations of G, as reformulated in [15], [16], and [3].

Definition 2.1. Suppose G is a reductive group in Harish-Chandra’s class. The admissible dual of G is the set Π_{a}(G) of infinitesimal equivalence classes of irreducible admissible representations of G. Equivalently, Π_{a}(G) may be identified with the set of equivalence classes of irreducible (𝔤, K)-modules ([16], Definition 0.3.8; here K is a maximal compact subgroup of G as in 0.2).

Most of the equivalence of these two definitions of Π_{a}(G) is proved in [5]. What is missing there is the proof that every irreducible (𝔤, K)-module appears as the space of K-finite vectors in an admissible representation. This was proved for linear groups by Harish-Chandra, and in general by Lepowsky and Rader independently (see for example [19], Theorem 3.5.6).

Definition 2.2. ([15], Definition 5.1 or [16], Definition 5.4.18). Suppose X ∈ Π_{a}(G) is an irreducible (𝔤, K)-module. A lowest K-type of X is an irreducible representation δ ∈ K̂ such that

a) δ appears in the restriction of X to K, and

b) if μ ∈ i(𝔱_{0}^{c})^{∗} is a highest weight of δ (Lemma 1.4), then ⟨μ + 2ρ_{c}, μ + 2ρ_{c}⟩ (notation 0.4(e)) is minimal subject to (a).

Obviously the norm in Definition 2.2(b) is independent of the choice of highest weight of δ.

The classification of admissible representations in [15] is based on the following fact.

Proposition 2.3. ([15], Proposition 4.1 or [16], Proposition 5.3.3.) Suppose μ ∈ i(𝔱_{0}^{c})^{∗} is dominant integral for Δ^{+}(𝔨, 𝔱^{c}). Choose a positive root system Δ^{+} = Δ^{+}(𝔤, 𝔱^{c}) making μ + 2ρ_{c} dominant (cf. 0.5(a)). Write ρ for the corresponding half sum of positive roots. Then there is a weight λ_{a}(μ) ∈ i𝔱_{0}^{∗} with the following properties.

a) The weight λ_{a}(μ) is dominant for Δ^{+}.

b) There is an orthogonal collection β_{i} of positive imaginary roots such that i) λ_{a}(μ) = μ + 2ρ_{c} − ρ + ∑_{i} c_{i}β_{i} (0 ≤ c_{i} ≤ 1/2); and ii) ⟨λ_{a}(μ), β_{i}⟩ = 0.

Corollary 2.4. ([3], Théorème 2.10 ). In the notation of Proposition 2.3, the weight λ_{a}(μ) is the projection of μ+2ρ_{c} −ρ on the cone C of dominant weights. In the notation of Proposition 1.4, λ_{a}(μ) = T_{ρ}(μ+2ρ_{c}). Consequently λ_{a}(μ) is independent of the choice of Δ^{+} making μ + 2ρ_{c} dominant.

Proof. Proposition 2.3 shows that λ_{a}(μ) satisfies the conditions in Proposition 1.1(c) characterizing P (μ+2ρ_{c}−ρ). The remaining assertions now follow from Proposition 1.4.

In analogy with 0.5(e), we may now define

(2.1)  Λ_{a} = {λ_{a}(μ), μ ∈ T̂^{c} dominant}

The analogue of Definition 0.2 for admissible representations is

Definition 2.5. Fix λ_{a} ∈ Λ_{a}. The set of K-types attached to λ_{a} is

B_{a}^{λ_{a}}(G) = {δ ∈ K̂ | δ has a highest weight μ ∈ T̂^{c} such that λ_{a}(μ) = λ_{a}}.

The set of admissible representations of G attached to λ_{a} is

Π_{a}^{λ_{a}}(G) = {π ∈ Π_{a}(G) | π has a lowest K-type in B_{a}^{λ_{a}}(G)}.

Just as in the case of Definition 0.2, we will abuse notation in section 8 by writing

λ_{a}(π) = λ_{a}

when π ∈ Π_{a}^{λ_{a}}(G). Again the difficulty is that π may belong to several different Π_{a}^{λ′_{a}}(G). It follows from Theorem 2.9 below that λ_{a}(π) is well-defined as an orbit of R(G).

Proposition 2.6. With notation as in Definition 2.5, 0.6, and 0.4(d), write S_{a} = dim 𝔲(λ_{a}) ∩ 𝔨. Then the functor L_{S_{a}}^{K}(λ_{a}) implements a bijection from B_{a}^{λ_{a}−ρ(𝔲(λ_{a}))}(G(λ_{a})) onto B_{a}^{λ_{a}}(G).

This is essentially proved in [16], although some careful inspection of definitions is needed to verify that. Typical of the complications is the fact that the functors L_{S}^{K}(λ) are not defined in [16]. Fortunately they are easy to describe in terms of highest weights: roughly speaking, they add 2ρ(𝔲(λ) ∩ 𝔭) to the highest weight. Here is a more precise statement.

Lemma 2.7. ([9], Corollary 5.72 ). Suppose λ ∈ i(𝔱_{0}^{c})^{∗} is dominant for Δ^{+}(𝔨, 𝔱^{c}), so that 𝔲(λ) ∩ 𝔨 ⊂ 𝔫_{𝔨} (notation 0.6(e) and 0.3(f )). Write R = dim 𝔲(λ)∩𝔭, so that ∧^{R}(𝔲(λ)∩𝔭) is a one-dimensional representation of G(λ)∩ K. Suppose Z is an irreducible representation of G(λ) ∩ K, of highest weight denoted μ_{G(λ)} ∈ T̂^{c}, and V is an irreducible representation of K. Then

Hom_{K}(L_{S}^{K}(λ)(Z), V) ≃ Hom_{G(λ)∩K}(Z ⊗ ∧^{R}(𝔲(λ) ∩ 𝔭), V^{𝔲(λ)∩𝔨}).

Define μ_{G} = μ_{G(λ)} + 2ρ(𝔲(λ) ∩ 𝔭). If μ_{G} is dominant for K, then every irreducible constituent of L_{S}^{K}(λ)(Z) has highest weight μ_{G}. If μ_{G} is not dominant for K, then L_{S}^{K}(λ)(Z) = 0.

Given this description of the functor L_{S}^{K}(λ), the translation from Lemma 6.5.4 in [16] to Proposition 2.6 above is more or less straightforward; we omit the details.

In order to formulate the classification of admissible representations, we need one more ingredient.

Proposition 2.8. ([9], Theorem 5.80 ). In the setting 0.6, suppose Z is a (𝔤(λ), G(λ)∩K)-module. Then there is a natural injective map of K-modules

B_{Z}: L_{S}^{K}(λ)(Z) → L_{S}(λ)(Z).

This map is called the bottom layer map for Z, for reasons explained in section 5.6 of [9].

Theorem 2.9. ([16], Theorem 6.5.12.) Suppose λ_{a} ∈ Λ_{a} (cf. 2.1 ). Then there is a natural bijection

(2.2)  Π_{a}^{λ_{a}−ρ(𝔲(λ_{a}))}(G(λ_{a})) → Π_{a}^{λ_{a}}(G)

(Definition 2.5 ) defined as follows. Suppose Z_{a} ∈ Π_{a}^{λ_{a}−ρ(𝔲(λ_{a}))}(G(λ_{a})) is an irreducible (𝔤(λ_{a}), G(λ_{a}) ∩ K)-module.

a) The set of lowest G(λ_{a}) ∩ K-types of Z_{a} consists precisely of those in B_{a}^{λ_{a}−ρ(𝔲(λ_{a}))}(G(λ_{a})).

b) The set of lowest K-types of L_{S_{a}}(λ_{a})(Z_{a}) consists precisely of the image of the bottom layer map composed with L_{S_{a}}^{K}(λ_{a}) on the lowest G(λ_{a})∩K-types of Z_{a}.

Let δ_{G(λ_{a})} ∈ B_{a}^{λ_{a}−ρ(𝔲(λ_{a}))}(G(λ_{a})) be an irreducible representation of the group G(λ_{a}) ∩ K appearing in Z_{a}, and let δ = L_{S_{a}}^{K}(λ_{a})(δ_{G(λ_{a})}) be the corresponding representation of K in B_{a}^{λ_{a}}(G) (Proposition 2.6 ). By Proposition 2.8, δ must appear in L_{S_{a}}(λ_{a})(Z_{a}).

c) There is a unique irreducible subquotient X of L_{S_{a}}(λ_{a})(Z_{a}) containing the K-type δ.

d) The multiplicity of δ in X is equal to the multiplicity of δ in L_{S_{a}}(λ_{a})(Z_{a}), and to the multiplicity of δ_{G(λ_{a})} in Z_{a}.

e) The subquotient X is independent of the choice of the representation δ_{G(λ_{a})} in B_{a}^{λ_{a}−ρ(𝔲(λ_{a}))}(G(λ_{a})), subject to the requirement that δ_{G(λ_{a})} occur in Z_{a}. The bijection 2.2 sends Z_{a} to X. To construct the inverse map, suppose X is an irreducible (𝔤, K)-module in

Π_{a}^{λ_{a}}(G), having a lowest K-type δ ∈ B_{a}^{λ_{a}}(G). Let δ_{G(λ_{a})} ∈ B_{a}^{λ_{a}−ρ(𝔲(λ_{a}))}(G(λ_{a})) be the corresponding irreducible representation of G(λ_{a})∩K (Proposition 2.6 ). Write R_{a} = dim(𝔲(λ_{a}) ∩ 𝔭) as in Lemma 2.7. f ) The representation δ_{G(λ_{a})} appears in the cohomology space H^{∗}(𝔲(λ_{a}), X)

(a (𝔤(λ_{a}), G(λ_{a}) ∩ K))-module) exactly as often as δ appears in X; all occurrences are in degree R_{a}.

g) There is a unique irreducible subquotient Z_{a} of H^{R_{a}}(𝔲(λ_{a}), X) containing the G(λ_{a}) ∩ K-type δ_{G(λ_{a})}.

h) The subquotient Z_{a} is independent of the choice of δ ∈ B_{a}^{λ_{a}}(G), subject to the requirement that δ occur in X. The inverse of 2.2 sends X to Z_{a}.

Just as for Proposition 2.6, some translation is required to pass from the results in [16] to this one; but we omit the details.

Because we are interested in unitary representations, we need to know how the bijection of Theorem 2.9 affects Hermitian forms. Here is the underlying machinery.

Proposition 2.10. ([9], Theorem 6.34.) In the setting of 0.6, suppose Z is a representation of G(λ) ∩ K. Then any invariant Hermitian form on Z induces in a natural way a K-invariant Hermitian form on L_{S}^{K}(λ)(Z). This correspondence carries positive definite forms to positive definite forms, and negative definite forms to negative definite forms.

Suppose Z is actually a (𝔤(λ), G(λ) ∩ K)-module. Then any invariant Hermitian form on Z induces in a natural way a (𝔤, K)-invariant Hermitian form on L_{S}(λ)(Z). The bottom layer map B_{Z} of Proposition 2.8 is unitary; that is, it respects these forms.

Theorem 2.11. The bijection of Theorem 2.9 preserves Hermitian representations. Explicitly, suppose Z_{a} is an irreducible Hermitian (𝔤(λ_{a}), G(λ_{a})∩ K)-module. Then the induced Hermitian form on L_{S_{a}}(λ_{a})(Z_{a}) descends to a non-degenerate form on the subquotient X containing the lowest K-types.

Unfortunately, the analogue of Theorem 2.11 with “Hermitian” replaced by “unitary” is false. We know no examples in which Z_{a} is unitary and X is not; but the other direction fails spectacularly and often. One of the simplest examples has G = SO_{e}(4, 1) and μ = 0 the highest weight of the trivial representation of K. In this case G(λ_{a}) = SO(2) × SO_{e}(2, 1), and Theorem 2.9 relates spherical representations of G to representations of G(λ_{a}) containing the SO(2) × SO(2) representation ℂ_{−1} ⊗ ℂ_{0}. (The subscripts refer to the representations of SO(2), which we index by ℤ as usual.) Then the representation Z_{a} = ℂ_{−1} ⊗ ℂ^{3} of G(λ_{a}) carries a Hermitian form of signature (1, 2); but the corresponding irreducible representation X of G is the trivial representation.

Here is how we propose to circumvent this problem. Suppose 𝔮 = 𝔩 + 𝔲 is a θ-stable parabolic subalgebra containing 𝔮(λ_{a}). Using “induction by stages” theorems, we can get analogues of Theorems 2.9 and 2.11 relating certain representations of G to certain representations of L. These results are in some sense weaker than those above, because they reduce problems about G not to G(λ_{a}), but only to the larger group L. On the other hand, and for the same reason, a representation of L “remembers” more about the corresponding representation of G than the representation of G(λ_{a}) does. For that reason, a unitary version of Theorem 2.11 has a better chance to be true. An extreme possibility is to take 𝔮 = 𝔤; then L = G, so the “reduction” from G to L is perfectly behaved but worthless. Our goal is to choose 𝔮 large enough that the reduction is well-behaved, but small enough that it is interesting.

We begin with formalities. Fix λ_{a} ∈ Λ_{a}. A weight λ ∈ i(𝔱_{0}^{c})^{∗} is called a singularization of λ_{a} if for every root α ∈ Δ(𝔤, 𝔱^{c}), we have

(2.3a)  ⟨λ, α⟩ > 0 ⇒ ⟨λ_{a}, α⟩ > 0.

That is, we require that λ belong to the same closed positive Weyl chamber as λ_{a}, and that it lie on at least as many walls. Another way to say the same thing is that λ should belong to every closed positive Weyl chamber containing λ_{a}. A consequence of 2.3(a) is

(2.3b)  ⟨λ_{a}, α⟩ = 0 ⇒ ⟨λ, α⟩ = 0.

By 0.6,

(2.3c)  𝔲(λ) ⊂ 𝔲(λ_{a}),  𝔤(λ_{a}) ⊂ 𝔤(λ),  𝔮(λ_{a}) ⊂ 𝔮(λ).

Because G is in Harish-Chandra’s class, it follows also that

(2.3d)  G(λ_{a}) ⊂ G(λ).

(To work with groups not in Harish-Chandra’s class, this condition should be included as an additional assumption on λ. We will eventually apply these formalities with λ = λ_{u}, the weight defined by 0.5 for some μ giving rise to λ_{a}. In that case 2.3(d) is automatically satisfied.) We have direct sum decompositions like

(2.3e)  𝔲(λ_{a}) = 𝔲(λ) + (𝔲(λ_{a}) ∩ 𝔤(λ)),

and similarly after intersection with 𝔨. We need two consequences of this decomposition. First,

(2.3f)  ρ(𝔲(λ_{a})) = ρ(𝔲(λ)) + ρ(𝔲(λ_{a}) ∩ 𝔤(λ))

as weights in i(𝔱_{0}^{c})^{∗} (cf. 0.4(d)). Next, if we define

(2.3g)  S_{a} = dim 𝔲(λ_{a}) ∩ 𝔨,  S = dim 𝔲(λ) ∩ 𝔨,

then

(2.3h)  dim(𝔲(λ_{a}) ∩ 𝔤(λ) ∩ 𝔨) = S_{a} − S.

We are going to be interested in cohomological induction functors related to three parabolic subalgebras:

(2.4a)  L_{S_{a}}(λ_{a}): (𝔤(λ_{a}), G(λ_{a}) ∩ K)-modules → (𝔤, K)-modules)

(defined in terms of the parabolic subalgebra 𝔮(λ_{a}));

(2.4b)  L_{S}(λ): (𝔤(λ), G(λ) ∩ K)-modules → (𝔤, K)-modules

(defined in terms of the parabolic subalgebra 𝔮(λ)); and

(2.4c) L_{S_{a}−S}(λ_{a}, λ): (𝔤(λ_{a}), G(λ_{a})∩K)-modules → (𝔤(λ), G(λ)∩K)-modules

(defined in terms of the parabolic subalgebra 𝔮(λ_{a}) ∩ 𝔤(λ)). Similarly, we have the corresponding functors for compact group representations.

Proposition 2.12. ([9], Theorem 11.77.) In the setting of 2.3 and 2.4, there are natural equivalences

L_{S_{a}}(λ_{a}) ≃ L_{S}(λ) ∘ L_{S_{a}−S}(λ_{a}, λ), L_{S_{a}}^{K}(λ_{a}) ≃ L_{S}^{K}(λ) ∘ L_{S_{a}−S}^{G(λ)∩K}(λ_{a}, λ).

These equivalences respect the bottom layer maps of Proposition 2.8 and the Hermitian forms of Proposition 2.10.

What is established in [9] is a composition of functors spectral sequence. The equivalence here appears as a corner of that spectral sequence, because the various cohomological induction functors all vanish above the degrees in the proposition. (A careful reader may notice that the assertions about bottom layer maps and Hermitian forms are not actually stated or proved in [9]. In any case they are stated here only for moral support; we will make no use of them.)

Using Proposition 2.12, it is a simple matter to extend Proposition 2.6 and Theorems 2.9 and 2.11 to apply to 𝔮(λ) instead of 𝔮(λ_{a}).

Theorem 2.13. Suppose λ_{a} ∈ Λ_{a}, and that we have fixed a singularization λ of λ_{a} (cf. 2.3 and 2.4 ).

a) The functor L_{S}^{K}(λ) implements a bijection from B_{a}^{λ_{a}−ρ(𝔲(λ))}(G(λ)) onto B_{a}^{λ_{a}}(G).

b) There is a natural bijection

Π_{a}^{λ_{a}−ρ(𝔲(λ))}(G(λ)) → Π_{a}^{λ_{a}}(G)

defined as follows. Suppose that Z ∈ Π_{a}^{λ_{a}−ρ(𝔲(λ))}(G(λ)) is an irreducible (𝔤(λ), G(λ) ∩ K)-module. Then the corresponding (𝔤, K)-module X is the unique irreducible subquotient of L_{S}(λ)(Z) containing K-types in B_{a}^{λ_{a}}(G). Conversely, given X ∈ Π_{a}^{λ_{a}}(G), put R = dim(𝔲(λ)∩𝔭). Then we assign to X the unique irreducible subquotient Z of H^{∗}(𝔲(λ), X) containing G(λ) ∩ K-types in B_{a}^{λ_{a}−ρ(𝔲(λ))}(G(λ)); it appears in degree R. c) The bijection of (b) preserves Hermitian representations.

d) Suppose Z ∈ Π_{a}^{λ−ρ(𝔲(λ))}(G(λ)) is an irreducible Hermitian (𝔤(λ), G(λ) ∩ K)-module. Then the image of the bottom layer map B_{Z} (Proposition 2.8 ) maps injectively to the irreducible subquotient X described in (b). On these K-types, the signature of the Hermitian form on X therefore corresponds precisely to the signature on Z.

e) Suppose Z ∈ Π_{a}^{λ−ρ(𝔲(λ))}(G(λ)) is an irreducible (𝔤(λ), G(λ) ∩ K)-module, and the infinitesimal character of Z corresponds by the Harish-Chandra isomorphism to a weight ϕ ∈ 𝔥^{∗}. Assume that

Re ⟨ϕ + ρ(𝔲(λ)), α⟩ ≥ 0 (α ∈ Δ(𝔲(λ), 𝔥)).

Then L_{S}(λ)(Z) is irreducible. f ) Under the assumptions of (e), suppose also that Z is unitary. Then

L_{S}(λ)(Z) is unitary.

g) Under the assumptions of (e), suppose also that

⟨ϕ + ρ(𝔲(λ)), α⟩ ≠ 0 (α ∈ Δ(𝔲(λ), 𝔥))

and that L_{S}(λ)(Z) is unitary. Then Z is unitary.

Sketch of proof. For (a), Proposition 2.6 (applied to the groups G and G(λ)) provides bijections

L_{S_{a}}^{K}: B_{a}^{λ_{a}−ρ(𝔲(λ_{a}))}(G(λ_{a})) → B_{a}^{λ_{a}}(G),
L_{S_{a}−S}^{G(λ)∩K}: B_{a}^{λ_{a}−ρ(𝔲(λ_{a}))}(G(λ_{a})) → B_{a}^{λ_{a}−ρ(𝔲(λ))}(G(λ)).

Now Proposition 2.12 implies that L_{S}^{K}(λ) provides a bijection between the ranges of these two maps. The proofs of (b) and (c) are similar; we omit the details. Part (d) is the Signature Theorem of [9], Corollary 11.228; (e) is the Irreducibility Theorem of [9], Theorem 8.7; and (f) is the Unitarizability Theorem of [9], Theorem 9.1 or [17], Theorem 1.3(a). Part (g) is [17], Theorem 1.3(b).

3. Relation to the unitary classification

In this section we will relate the various sets defined for unitary representations in Definition 0.2 with the classification by lowest K-types described in section 2. We begin with a K-dominant weight μ ∈ T̂^{c} as in 0.5, and a positive root system Δ^{+}(𝔤, 𝔱^{c}) making μ + 2ρ_{c} dominant. Define ρ as in 0.5(b), half the sum of the positive roots. Recall from Corollary 2.4 that in the notation of Proposition 1.4,

(3.1a)  λ_{a}(μ) = T_{ρ}(μ + 2ρ_{c})

The definition in 0.5(d) may be written as

(3.1b)  λ_{u}(μ) = T_{2ρ}(μ + 2ρ_{c})

By Corollary 1.6,

(3.1c)  λ_{u}(μ) = T_{ρ} ∘ T_{ρ}(μ + 2ρ_{c}) = T_{ρ}(λ_{a}(μ))

Lemma 1.3 (or Proposition 1.4) now imply that

(3.1d)  λ_{u}(μ) is a singularization of λ_{a}(μ) (notation 2.3).

This information leads immediately to a relationship between the sets of Ktypes defined in Definitions 0.2 and 2.5.

Proposition 3.1. Use the notation of 0.5, 2.1, and Proposition 1.4.

a) Λ_{u} = T_{ρ}(Λ_{a}).

b) B_{u}^{λ_{u}}(G) = ⋃_{λ_{a}∈Λ_{a}, T_{ρ}(λ_{a})=λ_{u}} B_{a}^{λ_{a}}(G).

c) In the notation of Theorem 2.13, the functor L_{S_{u}}^{K}(λ_{u}) implements a bijection from B_{u}^{λ_{u}}(G(λ_{u})) onto B_{u}^{λ_{u}}(G).

Proof. Parts (a) and (b) are immediate from 3.1(c) and the definitions. For (c), we combine the description of B_{u}^{λ_{u}}(G) in (b) with Theorem 2.13(a) (which applies because of 3.1(d)). The conclusion is that L_{S_{u}}^{K}(λ_{u}) implements a bijection from

⋃_{{λ_{a}∈Λ_{a} \atop T_{ρ}(λ_{a})=λ_{u}}} B_{a}^{λ_{a}−ρ(𝔲(λ_{u}))}(G(λ_{u}))

onto B_{u}^{λ_{u}}(G). To complete the proof of (c), we need

Lemma 3.2. In the setting of Proposition 3.1, suppose λ′_{a} ∈ Λ_{a}(G(λ_{u})). Then the following two properties are equivalent.

a) T_{ρ(𝔤(λ_{u}))}(λ′_{a}) = λ_{u}.

b) There is a λ_{a} ∈ Λ_{a}(G) such that T_{ρ}(λ_{a}) = λ_{u}, and λ′_{a} = λ_{a} − ρ(𝔲(λ_{u})).

Assuming this lemma for a moment, we see that the last union may be rewritten as

⋃_{{λ′_{a}∈Λ_{a}(G(λ_{u})) \atop T_{ρ(𝔤(λu))}(λ′_{a})=λ_{u}}} B_{a}^{λ′_{a}}(G(λ_{u})).

According to Proposition 3.1(b) (which we have already proved) this is precisely B_{u}^{λ_{u}}(G(λ_{u})), as we wished to show.

Proof of Lemma 3.2. Suppose first that (b) holds. Fix a positive root system Δ^{+}(𝔤, 𝔱^{c}) making λ_{a} dominant, and write ρ for half the sum of the positive roots. Then

(3.2a)  ρ = ρ(𝔲(λ_{u})) + ρ(𝔤(λ_{u}))

According to Proposition 1.1(c), the hypothesis that T_{ρ}(λ_{a}) = λ_{u} means that λ_{u} is dominant for Δ^{+}(𝔤, 𝔱^{c}), and that

(3.2b)  λ_{u} = (λ_{a} − ρ(𝔲(λ_{u}))) − ρ(𝔤(λ_{u})) + w,

with w a non-negative combination of positive roots orthogonal to λ_{u}. But this formulation also shows that if we set λ′_{a} = λ_{a} − ρ(𝔲(λ_{u})), then

T_{ρ(𝔤(λ_{u}))}(λ′_{a}) = λ_{u}.

So (a) is satisfied.

Conversely, assume (a) holds. Choose a positive root system Δ^{+}(𝔤(λ_{u}), 𝔱^{c}) making λ′_{a} dominant. Define

(3.2c)  Δ^{+}(𝔤, 𝔱^{c}) = Δ^{+}(𝔤(λ_{u}), 𝔱) ∪ Δ(𝔲(λ_{u}), 𝔱^{c}).

Then 3.2(a) still holds. Define

(3.2d)  λ_{a} = λ′_{a} + ρ(𝔲(λ_{u})).

By Proposition 1.1(c), the assumption (a) may be written as in 3.2(b). This looks at first like it immediately implies (b) of the proposition. What is missing is this. To compute T_{ρ}(λ_{a}), we must subtract from λ_{a} half the sum of the positive roots for a system making λ_{a} dominant. We do not yet know that λ_{a} is dominant for Δ^{+}(𝔤, 𝔱^{c}). Here is a proof of that. Since λ_{u} is central in G(λ_{u}), we can apply Proposition 1.7 almost directly to the hypothesis (a). The conclusion is

λ′_{a} − λ_{u} belongs to the convex hull of the Weyl group orbit of ρ(𝔤(λ_{u})).

Of course the Weyl group in question is for Δ(𝔤(λ_{u}), 𝔱^{c}). This Weyl group fixes ρ(𝔲(λ_{u})). We conclude that λ_{a} may be written as a convex combination

λ_{a} = ∑ c_{w}(λ_{u} + wρ),

with w running over the Weyl group of Δ(𝔤(λ_{u}), 𝔱^{c}). Each summand here has positive inner product with the roots in Δ(𝔲(λ_{u}), 𝔱^{c}), so λ_{a} does as well. On the other hand it is immediate from 3.2(d) that λ_{a} is dominant for Δ^{+}(𝔤(λ_{u}), 𝔱); so we conclude that λ_{a} is actually dominant for Δ^{+}(𝔤, 𝔱^{c}), as we wished to show. Now 3.2(b) does imply that T_{ρ}(λ_{a}) = λ_{u}, which is (b).

Proof of Proposition 0.3. Write δ and δ′ for lowest K-types of π having highest weights μ and μ′. Define λ_{a} = λ_{a}(μ), and λ′_{a} = λ_{a}(μ′). Use Theorem 2.9 and the K-type δ to write π as a subquotient of some L_{S_{a}}(λ_{a})(Z_{a}). Then δ′ must be a lowest K-type of this cohomologically induced representation. By Theorem 2.9(b), δ′ must also belong to B_{a}^{λ_{a}}(G). It is clear from Definition 2.5 (and Lemma 1.4) that this implies that λ′_{a} = r · λ_{a}, for some r ∈ R(G). Consequently 3.1(c) implies that

λ_{u}(μ′) = T_{ρ}(λ′_{a}) = T_{ρ}(r · λ_{a}) = r · T_{ρ}(λ_{a}) = r · λ_{u}(μ′),

as we wished to show.

4. Relation to ordinary parabolic induction

In this section we consider the relationship between our results about cohomological induction and those formulated with ordinary parabolic induction. We begin with the case to which Theorem 2.9 reduces the classification of admissible representations.

Proposition 4.1. Suppose λ_{a} ∈ Λ_{a} (cf. 2.1 ), and that G(λ_{a}) = G. Then G is quasisplit; that is, there is a Borel subgroup B^{q} = T^{q}A^{q}N^{q} of G. The Levi factor H^{q} = T^{q}A^{q} is a maximally split Cartan subgroup of G. After replacing B^{q} by a conjugate, we may assume that the identity component T_{0}^{q} is contained in T^{c} (cf. 0.3 ). Then the linear functional λ_{a} vanishes on the orthogonal complement of 𝔱^{q} ⊂ 𝔱^{c}, so it makes sense to write λ_{a} ∈ i(𝔱_{0}^{q})^{∗}. An irreducible representation γ ∈ T̂^{q} is said to be fine of type λ_{a} if its differential is a multiple of λ_{a}.

a) Suppose δ ∈ B_{a}^{λ_{a}}(G). Then the restriction of δ to T^{q} consists of fine representations of T^{q} of type λ_{a}, belonging to a single orbit of W (G, H^{q}).

b) The set B_{a}^{λ_{a}}(G) consists of all lowest K-types of the induced representations Ind_{T^{q}}^{K}(γ), as γ runs over fine representations of T^{q} of type λ_{a}.

c) Suppose δ ∈ B_{a}^{λ_{a}}(G), and X ∈ Π_{a}^{λ_{a}}(G) is an irreducible admissible representation of lowest K-type δ. Let γ be any representation of T^{q} appearing in the restriction of δ. Then there is a character ν ∈ Â^{q} so that X is a Langlands subquotient of the principal series representation Ind_{T^{q}A^{q}N^{q}}^{G}(γ ⊗ ν). In particular, the infinitesimal character of X has Harish-Chandra parameter (λ_{a}, ν) ∈ (𝔱^{q})^{∗} + (𝔞^{q})^{∗}.

This is [16], Proposition 5.3.26, Theorem 4.3.16, and Theorem 4.4.8. Inspection of the proofs shows that what is claimed here does not use the assumption of abelian Cartan subgroups. (That is needed for example to show that the restriction of δ to T^{q} decomposes with multiplicity one.)

We turn now to general representations. So let λ_{a} ∈ Λ_{a}(G). Proposition 4.1 guarantees that G(λ_{a}) is quasisplit. A little more precisely, Proposition 2.6 guarantees that

(4.1a)  λ′_{a} = λ_{a} − ρ(𝔲(λ_{a}))

belongs to Λ_{a}(G(λ_{a})); and evidently

(4.1b)  G(λ_{a})(λ′_{a}) = G(λ_{a}),

so we can apply Proposition 4.1. Pick a quasisplit Cartan subgroup

(4.1c)  H(λ_{a}) = T (λ_{a})A(λ_{a}) ⊂ G(λ_{a})

as in Proposition 4.1. Just as in the proposition, we may regard λ′_{a}, λ_{a}, and ρ(𝔲(λ_{a})) as weights in i𝔱(λ_{a})_{0}^{∗}. Now consider the centralizer of A(λ_{a}) in G. Because A(λ_{a}) is the vector part of the Cartan subgroup H(λ_{a}), this centralizer has a Langlands decomposition

(4.1d)  G^{A(λ_{a})} = M(λ_{a})A(λ_{a}).

Here M(λ_{a}) is a θ-stable reductive subgroup of G containing T (λ_{a}) as a compact Cartan subgroup.

Proposition 4.2. In the setting 4.1, there are natural bijections among the following three sets.

a) Fine representations of T (λ_{a}) of type λ′_{a} (Proposition 4.1 ); that is, irreducible representations of T (λ_{a}) of differential equal to a multiple of λ′_{a}.

b) Irreducible representations of T (λ_{a}) of differential equal to a multiple of λ_{a} − ρ(𝔪(λ)). Here the positive root system is the one for 𝔱(λ_{a}) in 𝔪(λ_{a}) making λ_{a} dominant.

c) Discrete series representations of M(λ_{a}) of Harish-Chandra parameter λ_{a}.

Proof. This result is analogous to Proposition 6.6.2 in [16]. To construct the bijection between (a) and (b), consider the set of roots Δ(𝔲(λ_{a}), 𝔥(λ_{a})). The imaginary roots among these are precisely the positive roots of 𝔱(λ_{a}) in 𝔪(λ_{a}). The remaining roots occur in pairs (α, θα). Choose one root from each of these pairs, and list the choices as {α_{1}, …, α_{r}}. Define

τ = ∑_{i} α_{i} ∈ T (λ_{a}) .

Because α and θα have the same restriction to T (λ_{a}), the character τ is independent of choices. Furthermore

ρ(𝔲(λ_{a})) = ρ(𝔪(λ_{a})) + dτ.

The bijection from (a) to (b) is just tensoring with τ. The bijection from (b) to (c) is Harish-Chandra’s parametrization of the discrete series. In the context of section 2, the set in (c) turns out to be Π_{a}^{λ_{a}}(M(λ_{a})), and the bijection with (b) is precisely 2.2; that is, it is given by cohomological induction from T (λ_{a}) to M(λ_{a}). Details may be found in [9], section 11.8.

We can now formulate the relationship between Theorem 2.9 and the parabolic induction version of the Langlands classification.

Theorem 4.3. ([9], Theorem 11.225.) Suppose we are in the setting of 2.2, and that the irreducible (𝔤, K)-module X corresponds to Z_{a}. Use the notation of 4.1, and fix a character γ^{q}⊗ν of H(λ_{a}) as in Proposition 4.1 so that Z_{a} is a Langlands quotient of Ind_{H(λ_{a})N^{q}}^{G(λ_{a})}(γ ⊗ ν). Let γ be the discrete series representation of M(λ_{a}) with Harish-Chandra parameter λ_{a} that corresponds to γ_{q} in the bijection from (a) to (c) of Proposition 4.2. Choose a real parabolic subgroup P = M(λ_{a})A(λ_{a})N of G making Re ν dominant for the 𝔞(λ_{a})-weights in 𝔫; and define N^{q} = N ∩ G(λ_{a}). Then

L_{S_{a}}(λ_{a}) (Ind_{H(λ_{a})N^{q}}^{G(λ_{a})}(γ ⊗ ν)) ≃ Ind_{P}^{G}(γ ⊗ ν).

In particular, X is a Langlands quotient of Ind_{P}^{G}(γ ⊗ ν).

This is stated without complete proof in [16], Theorem 6.6.15; it was essentially first proved in the unpublished second half of [15].

Recall that the tempered representations constitute the closure (in the Fell topology) of the support of the Plancherel measure. A tempered irreducible representation is unitary, but unitary representations need not be tempered.

Corollary 4.4. The bijection of Theorem 2.9 preserves tempered representations. Explicitly, suppose Z_{a} ∈ Π_{a}^{λ_{a}−ρ(𝔲(λ_{a}))}(G(λ_{a})) is an irreducible tempered (𝔤(λ_{a}), G(λ_{a}) ∩ K)-module. Then L_{S_{a}}(λ_{a})(Z_{a}) is an irreducible tempered (𝔤, K)-module in Π_{a}^{λ_{a}}(G); and every such tempered representation arises in this way.

Proof. The nature of the Langlands classification implies that a Langlands quotient is tempered if and only if the continuous parameter ν is purely imaginary (see for example [8], Theorem 8.53). Theorem 4.3 says that the bijection of Theorem 2.9 preserves continuous parameters, so temperedness is also preserved. For the rest, the infinitesimal character of Z_{a} is given by the weight (λ_{a} −ρ(𝔲(λ_{a})), ν) ∈ 𝔥(λ_{a})^{∗}. When ν is purely imaginary, the dominance of λ_{a} implies immediately that the condition in Theorem 2.13(e) is satisfied; so L_{S_{a}}(λ_{a})(Z_{a}) is irreducible.

5. The reduction step in the classification

Theorem 2.13 and Proposition 3.1 together allow us to approach Conjecture 0.6. To see what we have, we consider first the case of Hermitian representations.

Definition 5.1. Suppose G is a reductive group in Harish-Chandra’s class. The Hermitian dual of G is the set Π_{h}(G) of equivalence classes of irreducible (𝔤, K) modules endowed with non-degenerate invariant Hermitian forms. Here K is a maximal compact subgroup of G; and an equivalence is required to preserve the Hermitian form.

Suppose λ_{u} ∈ Λ_{u} (cf. 0.5). The set of Hermitian representations of G attached to λ_{u} is

Π_{h}^{λ_{u}}(G) = {π ∈ Π_{h}(G) | π has a lowest K-type in B_{u}^{λ_{u}}(G)}

(cf. Definition 0.2).

Proposition 0.3 immediately shows that these sets partition the Hermitian dual:

Proposition 5.2. Suppose λ_{u} and λ′_{u} belong to Λ_{u} (cf. 0.5 ). If λ_{u} and λ′_{u} belong to the same orbit of R(G), then Π_{h}^{λ_{u}}(G) = Π_{h}^{λ′_{u}}(G). If not, then Π_{h}^{λ_{u}}(G) and Π_{h}^{λ′_{u}}(G) are disjoint.

At this point it is useful to be a little more precise than we have been about the notion of signature.

Definition 5.3. (See [17], Definition 1.4, or [9], Proposition 6.12.) Suppose X is an admissible representation of K (that is, locally finite with finite multiplicities), and ⟨ , ⟩ is a K-invariant Hermitian form on X. The signature of ⟨ , ⟩ is a set of three functions (p, q, z) from K̂ to the non-negative integers, defined as follows. Suppose (δ, V_{δ}) is an irreducible representation of K. Fix a positive-definite invariant Hermitian form ⟨ , ⟩_{δ} on V_{δ}. Define

X^{δ} = Hom_{K}(V_{δ}, X).

This is a finite-dimensional vector space, of dimension equal to the multiplicity m(δ) of δ in X. There is a unique Hermitian form ⟨ , ⟩^{δ} on X^{δ} characterized by the property

⟨T v, Sw⟩ = ⟨T, S⟩^{δ}⟨v, w⟩_{δ} (T, S ∈ X^{δ}, v, w ∈ V_{δ}).

The form ⟨ , ⟩^{δ} may be diagonalized; and we define p(δ) (respectively q(δ), z(δ)) to be the number of positive (respectively negative, zero) diagonal entries. These numbers are of course independent of the choice of diagonalization, and characterize the form ⟨ , ⟩^{δ} up to equivalence. Notice that

m(δ) = p(δ) + q(δ) + z(δ).

The form ⟨ , ⟩ on X is non-degenerate if and only if z = 0, and positive definite if and only if q = z = 0.

Theorem 5.4. Suppose λ_{u} ∈ Λ_{u} (cf. 0.5 ).

a) The functor L_{S_{u}}^{K}(λ_{u}) provides a bijection from B_{u}^{λ_{u}}(G(λ_{u})) onto B_{u}^{λ_{u}}(G).

b) The set Π_{h}^{λ_{u}}(G(λ_{u})) consists of all irreducible (𝔤(λ_{u}), G(λ_{u}) ∩ K)-modules endowed with an invariant Hermitian form, and containing at least one (G(λ_{u}) ∩ K)-type in the set B_{u}^{λ_{u}}(G(λ_{u})). c) There is a natural bijection

Π_{h}^{λ_{u}}(G(λ)) → Π_{h}^{λ_{u}}(G)

defined as follows. Suppose that Z ∈ Π_{h}^{λ_{u}}(G(λ_{u})) is irreducible as a (𝔤(λ_{u}), G(λ_{u}) ∩ K)-module. Then the corresponding (𝔤, K)-module X is the unique irreducible subquotient of L_{S_{u}}(λ_{u})(Z) containing K-types in B_{a}^{λ_{a}}(G). The Hermitian form on X is the one inherited from the one on L_{S_{u}}(λ_{u})(Z) induced from Z by Proposition 2.10.

d) The bijections of (a) and (c) preserve signatures of Hermitian forms; that is, the signature on a K type in B_{u}^{λ_{u}}(G) of the Hermitian form on X is equal to the signature on the corresponding G(λ_{u}) ∩ K-type of the Hermitian form on Z.

e) Suppose Z ∈ Π_{h}^{λ_{u}}(G(λ)) is an irreducible (𝔤(λ_{u}), G(λ) ∩ K)-module, and the infinitesimal character of Z corresponds by the Harish-Chandra isomorphism to a weight ϕ ∈ 𝔥^{∗}. Assume that

Re ⟨ϕ + ρ(𝔲(λ_{u})), α⟩ ≥ 0 (α ∈ Δ(𝔲(λ_{u}), 𝔥).

Then L_{S_{u}}(λ_{u})(Z) is irreducible.

Proof. Part (a) is Proposition 3.1(c). For (b), the group G plays no rôle, so we may as well assume G = G(λ_{u}). One inclusion is obvious; so suppose X is an irreducible (𝔤, K)-module containing a K-type δ ∈ B_{u}^{λ_{u}}(G). We must show that the lowest K-type δ′ also belongs to B_{u}^{λ_{u}}(G). Write Z for the identity component of the compact part of the center of G. Then 𝔱^{c} is the orthogonal direct sum of 𝔷 and the span of the roots. Since G(λ_{u}) = G, the weight λ_{u} is orthogonal to all the roots; so λ_{u} ∈ i𝔷_{0}^{∗} is just the differential of the character by which Z acts on δ. Since Z acts by the same character on δ′, we see that λ′_{u}|_{𝔷} = λ_{u}. This implies that

(5.1)  |λ_{u}| ≤ |λ′_{u}|.

Now we apply

Lemma 5.5. Suppose X is an irreducible (𝔤, K)-module of lowest K-type δ′ ∈ B_{u}^{λ′_{u}}(G), and δ is any other K-type of X. Then δ has a highest weight μ with

λ_{u}(μ) = λ′_{u} + Q′,

with Q′ a sum with non-negative coefficients of roots of 𝔱^{c} in 𝔮(λ′_{u}). In particular,

⟨λ_{u}(μ), λ_{u}(μ)⟩ ≥ ⟨λ′_{u}, λ′_{u}⟩,

with equality only if λ_{u}(μ) = λ′_{u}.

We will give the proof in a moment. Assuming the result, we find that |λ_{u}| ≥ |λ′_{u}|. By 5.1, equality must hold. The condition for equality in the Lemma is λ′_{u} = λ_{u}; so the lowest K-type δ′ of X belongs to B_{u}^{λ_{u}}(G), as we wished to show.

Part (c) is (in light of 3.1) just Theorem 2.13(b); part (d) is Theorem 2.13(d); and part (e) is Theorem 2.13(e).

Proof of Lemma 5.5. This is a variation on Lemma 6.5.6 in [16], which we will also use here. Fix a highest weight μ′ of δ′ so that λ_{u}(μ′) = λ′_{u}, and define λ′_{a} = λ_{a}(μ′). According to the proof of Lemma 6.5.6 and Theorem 6.5.9 in [16] we can find a highest weight μ of δ so that

(5.2a)  λ_{a}(μ) = λ′_{a} + Q′_{1},

with Q′ a sum with non-negative coefficients of roots of 𝔱^{c} in 𝔮(λ′_{u}). We want a relationship between λ_{u}(μ) and λ′_{u}, so we rewrite both sides in those terms. Fix a positive root system (Δ^{+})′ making λ′_{a} dominant, and write ρ′ for half the sum of its positive roots. Then 5.2(a) becomes

(5.2b)  λ_{u}(μ) + ρ_{0} = λ′_{u} + Q′_{1} + ρ′ − w′.

Here w′ is a non-negative combination of roots in (Δ^{+})′ orthogonal to λ′_{u}. On the left we have used instead Corollary 1.9: ρ_{0} is a dominant weight in the convex hull of the W_{0}-orbit of ρ. By Proposition 1.10, we can write

(5.2c)  ρ_{0} = ρ′ − ∑_{α′∈(Δ^{+})′} b_{α′}α′,  (0 ≤ b_{α′} ≤ 1).

Inserting this in 5.2(b) gives

(5.2d)  λ_{u}(μ) = λ′_{u} + Q′_{1} − w′ + ∑_{α′∈(Δ^{+})′} b_{α′}α′.

This has the form required in the lemma. The last inequality is an immediate consequence.

We need one more definition.

Definition 5.6. ([16], Definition 5.4.11). Suppose 𝔥_{0} is a θ-stable Cartan subalgebra of 𝔤_{0}. Write 𝔥_{0} = 𝔱_{0} + 𝔞_{0} for its decomposition into the +1 and −1 eigenspaces of θ: 𝔱_{0} = 𝔥_{0} ∩ 𝔨_{0}, and 𝔞_{0} = 𝔥_{0} ∩ 𝔭_{0}. Any complex-valued linear functional γ ∈ 𝔥^{∗} may then be written as

γ = (λ, ν) ∈ 𝔱^{∗} + 𝔞^{∗}.

Now make a further decomposition of γ as

γ = γ_{1} + iγ_{2} = (λ_{1}, ν_{1}) + i(λ_{2}, ν_{2}),

in such a way that γ_{j} takes purely real values on 𝔞_{0}, and purely imaginary values on 𝔱_{0}. That is,

λ_{j} ∈ i𝔱_{0}^{∗},  ν_{j} ∈ 𝔞_{0}^{∗},  γ_{j} ∈ i𝔱_{0}^{∗} + 𝔞_{0}^{∗}.

Define the canonical real and imaginary parts of γ as

RE γ = γ_{1}, IM γ = γ_{2}.

If α ∈ 𝔥^{∗} is a root, then

Re ⟨γ, α⟩ = ⟨RE γ, α⟩,

and similarly for imaginary parts.

One checks easily that the decomposition into canonical real and imaginary parts commutes with the action of the Weyl group, and in fact with any automorphism that is inner for the complexified Lie algebra ([16], Lemma 5.4.12).

Here is the main conjecture of this paper.

Conjecture 5.7. Suppose λ_{u} ∈ Λ_{u} (cf. 0.5), and that G(λ_{u}) = G. Suppose X ∈ Π_{h}^{λ_{u}}(G) is an irreducible Hermitian (𝔤, K)-module (Definition 5.1). Let 𝔥 be a θ-stable Cartan subalgebra of 𝔤_{0}, and ϕ ∈ 𝔥^{∗} a weight parametrizing the infinitesimal character of X by the Harish-Chandra isomorphism. Assume that the canonical real part RE ϕ does not belong to λ_{u} plus the convex hull of the Weyl group orbit of ρ. Then the signature of the Hermitian form on X must be indefinite on K-types in B_{u}^{λ_{u}}(G).

The statement of this conjecture is complicated somewhat by the central weight λ_{u}. Of course the center of 𝔤 is more or less irrelevant to interesting statements about unitary representations. In particular, Conjecture 5.7 is equivalent to the following statement about semisimple groups. (This kind of reduction is discussed again in 6.1 below.)

Conjecture 5.7′. Suppose G is a connected semisimple group in Harish-Chandra’s class; and suppose X ∈ Π_{h}^{0}(G) is an irreducible Hermitian (𝔤, K)- module (Definition 5.1). Let 𝔥 be a θ-stable Cartan subalgebra of 𝔤_{0}, and ϕ ∈ 𝔥^{∗} a weight parametrizing the infinitesimal character of X by the Harish-Chandra isomorphism. Assume that the canonical real part RE ϕ does not belong to the convex hull of the Weyl group orbit of ρ. Then the signature of the Hermitian form on X must be indefinite on K-types in B_{u}^{0}(G).

We will give some supporting evidence for this conjecture in Proposition 7.18 below, which is based on Parthasarathy’s Dirac operator inequality. We also formulate a conjectural sharpening of the Dirac operator inequality which would imply Conjecture 5.7′ (Conjecture 7.19).

Theorem 5.8. In the setting of Theorem 5.4, assume that Conjecture 5.7 is true for G(λ_{u}). Then the bijection of (c) preserves unitarity. More precisely, suppose 𝔥_{0} is a θ-stable Cartan subalgebra of 𝔤(λ_{u})_{0}. Write W_{0} for the Weyl group of 𝔥 in 𝔤(λ_{u}).

a) The cohomological parabolic induction functor L_{S_{u}}(λ_{u}) implements a bijection from Π_{u}^{λ_{u}}(G(λ_{u})) onto Π_{u}^{λ_{u}}(G).

b) Assume that Z is an irreducible unitary (𝔤(λ_{u}), G(λ)∩K)-module containing a (G(λ_{u}) ∩ K)-type in B_{u}^{λ_{u}}(G(λ_{u})), and that the infinitesimal character of Z corresponds by the Harish-Chandra isomorphism to a weight ϕ(Z) ∈ 𝔥^{∗}. Then RE ϕ(Z) − λ_{u} (cf. Definition 5.6 ) belongs to the convex hull of the W_{0} orbit of ρ(𝔤(λ_{u})).

c) Assume that X ∈ Π_{u}^{λ_{u}}(G), and that the infinitesimal character of X corresponds by the Harish-Chandra isomorphism to a weight ϕ(X) ∈ 𝔥^{∗}. Assume that RE ϕ(X) is dominant for the roots of 𝔥 in 𝔲(λ_{u}). Then RE ϕ(X) − (λ_{u} + ρ(𝔲(λ_{u}))) belongs to the convex hull of the W_{0} orbit of ρ(𝔤(λ_{u})). In particular, T_{ρ}(RE ϕ(X)) = λ_{u}.

The last statement of (c) appears to recover λ_{u} from the infinitesimal character of any representation in Π_{u}^{λ_{u}}(G). It does not quite do that, however, since we needed the weight ϕ(X) to satisfy a positivity condition determined by λ_{u}. At any rate we can recover the Weyl group orbit of λ_{u}.

Before embarking on the proof of this result, we need a complement to Conjecture 5.7.

Theorem 5.9. Suppose λ_{u} ∈ Λ_{u} (cf. 0.5 ), and that G(λ_{u}) = G. Suppose X ∈ Π_{h}^{λ_{u}}(G) is an irreducible Hermitian (𝔤, K)-module. Let 𝔥 be a θ-stable Cartan subalgebra of 𝔤_{0}, and ϕ ∈ 𝔥^{∗} a weight parametrizing the infinitesimal character of X by the Harish-Chandra isomorphism. Assume that the canonical real part RE ϕ belongs to λ_{u} plus the convex hull of the Weyl group orbit of ρ. Then X is unitary if and only if its Hermitian form is positive definite on K-types in B_{u}^{λ_{u}}(G).

Proof. There is no loss of generality in assuming that 𝔥 is the fundamental Cartan subalgebra. We use Theorem 1.5 of [17]. That provides irreducible tempered (𝔤, K)-modules Z_{1}, …, Z_{p} (of real infinitesimal character) and integers r_{1}^{±}, …, r_{p}^{±}, so that the signature of the Hermitian form on X is

(∑ r_{i}^{+}Θ_{K}(Z_{i}), ∑ r_{i}^{−}Θ_{K}(Z_{i})) .

Here Θ_{K}(Z) denotes the formal K-character of Z. Fix a lowest K-type δ_{i} of Z_{i}, and a highest weight μ_{i} of δ_{i}. Write λ_{i} = λ_{a}(μ_{i}).

Because Z_{i} is tempered and has real infinitesimal character, Theorem 4.3 shows that the infinitesimal character is given by the weight λ_{i}. The proof of Theorem 1.5 in [17] shows first of all that each Z_{i} has the same central character as X; and second, that the infinitesimal character λ_{i} must belong to the convex hull of the Weyl group orbit of RE ϕ. From these two facts we deduce that λ_{i} must belong to λ_{u} plus the convex hull of the Weyl group orbit of ρ. Corollary 1.9 therefore implies that (in the notation of Proposition 1.4) T_{ρ}(λ_{i}) = λ_{u}. By Corollary 1.6, it follows that λ_{u}(μ_{i}) = λ_{u}, and therefore that δ_{i} ∈ B_{u}^{λ_{u}}(G).

For the proof of the theorem, “only if” is clear; so suppose X is not positive definite. This means that ∑ r_{i}^{−}Θ_{K}(Z_{i}) ≠ 0. It follows that the lowest K-types of one of the Z_{i} must appear with non-zero multiplicity in this sum; that is, that the form on X is not positive on some δ_{i}. (For this it is enough to take |λ_{i}| minimal with r_{i}^{−} ≠ 0. The absence of cancellation then follows from Theorem 6.5.9(b) in [16].) But we have just seen that δ_{i} belongs to B_{u}^{λ_{u}}(G), as we wished to show.

Proof of Theorem 5.8. Part (b) is an immediate consequence of Conjecture 5.7. For (a), suppose X ∈ Π_{u}^{λ_{u}}(G); write Z for the corresponding Hermitian representation in Π_{h}^{λ_{u}}(G(λ_{u})) (Theorem 5.4(c)). By Theorem 5.4(d), the Hermitian form on Z must be positive on the (G(λ_{u}) ∩ K)-types in B_{u}^{λ_{u}}(G(λ_{u})). By Conjecture 5.7, the infinitesimal character of Z corresponds to a weight ϕ(Z) ∈ 𝔥^{∗} with

(5.3a)  RE ϕ(Z) = λ_{u} + w_{0}.

Here w_{0} is a weight in the convex hull of the W_{0} orbit of ρ(𝔤(λ_{u})). By Theorem 5.9, Z is unitary. Adding ρ(𝔲(λ_{u})) to 5.3(a) gives

(5.3b)  RE ϕ(Z) + ρ(𝔲(λ_{u})) = λ_{u} + ρ(𝔲(λ_{u})) + w_{0}.

The right side here is strictly dominant on roots of 𝔥 in 𝔲(λ_{u}) (compare the argument at the end of the proof of Lemma 3.2). By Theorem 5.4(e), L_{S_{u}}(λ_{u})(Z) is irreducible. For the converse, suppose Z ∈ Π_{u}^{λ_{u}}(G(λ_{u})). By Conjecture 5.7, the infinitesimal character of Z satisfies 5.3(a). We have seen that this implies the positivity condition in Theorem 2.13(e); so by Theorem 2.13(e) and (f), L_{S_{u}}(λ_{u})(Z) is irreducible and unitary, as we wished to show.

For (c), we may assume by (a) that X = L_{S_{u}}(λ_{u})(Z), and that the infinitesimal character of Z corresponds to a weight ϕ(Z) satisfying 5.3(a). Using Corollary 5.25 of [9] (or Proposition 6.3.11 of [16]) we find that the infinitesimal character of X is represented by the weight

(5.3c)  ϕ(X) = ϕ(Z) + ρ(𝔲(λ_{u})).

We have seen above that RE ϕ(X) is strictly dominant for the roots of 𝔥 in 𝔲(λ_{u}); so this ϕ(X) must coincide (up to the action of W_{0}) with the representative chosen in (c). Now 5.3(c) and 5.3(b) together imply that ϕ(X) has the property asserted in (c). This in turn implies that T_{ρ}(RE ϕ(X)) = λ_{u} by (for example) Corollary 1.9.

Corollary 5.10. Conjecture 5.7 (for the subgroup G(λ_{u}) of G) implies Conjecture 0.6.

6. The set B_{u}^{z}(G) of unitarily small K-types

Definition 6.1. Suppose G is a real reductive group in Harish-Chandra’s class, and K is a maximal compact subgroup. A parameter λ_{u} ∈ Λ_{u} (cf. 0.5) is called unitarily small if G(λ_{u}) = G (cf. 0.6). The set of unitarily small parameters is written Λ_{u}^{z}(G); the z stands for central. A highest weight μ ∈ T̂^{c} is called unitarily small if λ_{u}(μ) is unitarily small. A representation δ ∈ K̂ is called unitarily small (with respect to G) if δ has a unitarily small highest weight. The set of all unitarily small K-types is written B_{u}^{z}(G).

This is the analogue of “small” ([16], Definition 5.3.24, or [15], Definition 5.1); the change is that the weight λ_{a}(μ) constructed in Proposition 2.3 has been replaced by λ_{u}(μ). Because of 3.1(d), unitarily small is a weaker condition than small. It is fairly easy to see from the definition and Proposition 1.1 that the trivial representation of K is always unitarily small. By contrast, the trivial representation is small in the sense of [16] if and only if G is quasisplit.

Theorem 5.8 reduces the classification of unitary representations (at least modulo Conjecture 5.7) to the classification of sets Π_{u}^{λ_{u}}(G), with λ_{u} unitarily small. By Theorem 5.4(b), membership in such a set is equivalent to containing a K-type in B_{u}^{λ_{u}}(G). So Conjecture 5.7 reduces the classification of unitary representations to the classification of those containing a unitarily small K-type. That is one reason to understand such K-types as completely as possible. A second reason is that Conjecture 5.7 is again a statement about representations containing unitarily small K-types.

For these reasons, we devote the next two sections to a series of characterizations of unitarily small K-types (Theorem 6.7, Corollary 6.12, Proposition 7.1, and Proposition 7.17). We begin with some examples of the calculation in 0.5.

Example 6.2. Suppose G = SL(2, ℝ), and K = SO(2) = T^{c}, so that
(a)  K̂ = T̂^{c} ≃ ℤ ⊂ ℝ ≃ i(𝔱_{0}^{c})^{∗}.

The set of roots of T^{c} in 𝔤 is

(b)  Δ(𝔤, 𝔱^{c}) = {±2};

both of these are noncompact imaginary. Write μ_{n} for the irreducible representation of K corresponding to the integer n. Because there are no compact roots, 2ρ_{c} = 0. Therefore μ_{n} + 2ρ_{c} = μ_{n} is dominant for the positive root system

(c)  Δ^{+}(𝔤, 𝔱^{c}) = {(sgn n) · 2};

here we define sgn 0 = +1. The corresponding positive Weyl chamber C consists of the half line containing sgn n. The projection P on C is the identity on C, and sends the other half line to zero. Now 2ρ = 2 sgn n, so

(d)  μ_{n} + 2ρ_{c} − 2ρ = n − 2 sgn n.

This belongs to C (and so is equal to λ_{u}(μ_{n})) as long as |n| ≥ 2. For smaller values of n it does not belong to C, so the projection on C is zero. To summarize:

(e)  λ_{u}(μ_{n}) = \{ \begin{matrix}n − 2 sgn n, & if |n| ≥ 2; \\ 0, & if |n| ≤ 2. \end{matrix}

This verifies the assertions made in the introduction after 0.1. In particular, we see that the set of unitarily small K-types (or highest weights) is

(f)  B_{u}^{z}(SL(2, ℝ)) = {μ_{n} | |n| ≤ 2}.

By contrast, the set of small K-types in the sense of [16] consists of the μ_{n} with |n| ≤ 1.

Example 6.3. Suppose G = Sp(4, ℝ), the real symplectic group of rank 2. Then K = U(2) (the rank two unitary group), so T = U(1) × U(1), and

(a)  T̂^{c} ≃ ℤ^{2} ⊂ ℝ^{2} ≃ i(𝔱_{0}^{c})^{∗}.

The set of roots of T^{c} in 𝔤 is

(b)  Δ(𝔤, 𝔱^{c}) = {(±2, 0), (0, ±2), ±(1, 1), ±(1, −1)};

all but the last pair are noncompact. As a positive root in K we choose (1, −1) = 2ρ_{c}; the dominant weights μ_{(p,q)} are then parametrized by decreasing pairs of integers (p, q). We write δ_{(p,q)} for the irreducible representation of U(2) of highest weight μ_{(p,q)}. Calculation of λ_{u}(μ_{(p,q)}) falls into four cases according to the positive root system defined by μ_{(p,q)} +2ρ_{c}. We look at just one of these cases carefully. Suppose that (c) p + 1 ≥ 1 − q ≥ 0, Δ^{+}(𝔤, 𝔱^{c}) = {(2, 0), (0, −2), (1, 1), (1, −1)}, 2ρ = (4, −2).

Then one can calculate

(d) λ_{u}(μ_{(p,q)}) = \{ \begin{matrix}(p − 3, q + 1), & if p − 3 ≥ −1 − q ≥ 0; \\ (p − 3, 0), & if p − 3 ≥ 0 and −1 − q ≤ 0; \\ (\frac{p−q−4}{2}, \frac{−(p−q−4)}{2}), & if p − q ≥ 4 and p − 3 ≤ −1 − q; \\ (0, 0) & if p − 3 ≤ 0 and p − q ≤ 4. \end{matrix}

Combining this information with similar calculations in the other three cases, we find

(e)  B_{u}^{z}(Sp(4, ℝ)) = {δ_{(p,q)} | 3 ≥ p ≥ q ≥ −3, p − q ≤ 4}.

There are 25 representations of K in this set. The set of small representations of K consists of just five of them:

(f)  {δ_{(p,q)} | 1 ≥ p ≥ q ≥ −1, p − q ≤ 1}.

Example 6.4. In order to understand the rôle of the compact part of the center of G, we include one example where that is non-trivial. So let G = U(1, 1), the unitary group of the standard indefinite Hermitian form on ℂ^{2}. We can take K = T^{c} = U(1) × U(1), and

(a)  T̂^{c} ≃ ℤ^{2} ⊂ ℝ^{2} ≃ i(𝔱_{0}^{c})^{∗}.

The set of roots of T^{c} in 𝔤 is

(b)  Δ(𝔤, 𝔱^{c}) = {±(1, −1)};

these roots are noncompact. Of course this example is very close to Example 6.2, and we use parallel notation. A calculation like the one given there shows that

(c)  λ_{u}(μ_{(p,q)}) = \{ \begin{matrix}(p, q) − (sgn (p − q))(1, −1), & if |p − q| ≥ 2; \\ ((p + q)/2, (p + q)/2), & if |p − q| ≤ 2. \end{matrix}

The set of unitarily small parameters is therefore

(d)  Λ_{u}^{z}(U(1, 1)) = {(m/2, m/2) | m ∈ ℤ}.

Notice that these weights are precisely the restrictions to the center of arbitrary weights in T̂^{c}. The unitarily small K-types are

(e)  B_{u}^{z}(U(1, 1)) = {δ_{(p,q)} | |p − q| ≤ 2}.

With these examples in hand, we turn to a general description of unitarily small K-types. In addition to the notation from the introduction, we need to introduce

(6.1a)  Z = identity component of the center of G.

This group is preserved by θ, and so has a direct product decomposition

(6.1b)  Z = Z_{c}Z_{h}  (Z_{c} = Z ∩ K, Z_{h} = exp(𝔷_{0} ∩ 𝔭_{0}).

(The subscript h stands for “hyperbolic.”) Write

(6.1c)  G_{s} = derived group of G_{0}, T_{s} = G_{s} ∩ T^{c}.

On the Lie algebra level we have direct sum decompositions

(6.1d)  𝔤_{0} = 𝔤_{s,0} + 𝔷_{0}, 𝔱_{0}^{c} = 𝔱_{s,0} + 𝔷_{c,0}.

(The subspaces 𝔱_{s,0} and 𝔷_{c,0} correspond to what was called V_{s} and V_{z} in 1.2.) On the group level the corresponding decompositions are not direct: Z∩G_{s} can be non-trivial. (Because we have taken G_{s} to be connected, the multiplication map from G_{s} × Z to G is never surjective unless G is connected. In fact this problem persists even if we replace G_{s} by the full derived group of G.)

We propose to use the decompositions in 6.1 to reduce many problems to the semisimple case. There is a technical complication, however: the subgroup G_{s} need not be closed in G. It is preserved by the Cartan involution, and has a well-behaved Cartan decomposition

G_{s} = (G_{s} ∩ K) · exp(𝔤_{s,0} ∩ 𝔭_{0});

but the (connected reductive) group G_{s} ∩ K need not be compact. When this happens, G_{s} is not in Harish-Chandra’s class. Nevertheless all the results we have proved apply to it, after minor adjustments in the formulations. (For example, one considers only unitary representations of G_{s}∩K, and only unitary characters of T_{s}.) Alternatively, one can strengthen the hypotheses on G to exclude the problem. For example, one can assume that G is a finite cover of a real reductive algebraic group. This assumption is inherited by G_{s}. We will therefore ignore the problem, and speak as if G_{s} were again in Harish-Chandra’s class.

Lemma 6.5. With notation as above, suppose μ ∈ T̂^{c} is a K-dominant weight. Write μ_{z} and μ_{s} for its restrictions to Z_{c} and T_{s}. Then

λ_{u}(G)(μ) = λ_{u}(G_{s})(μ_{s}) + μ_{z} ∈ i𝔱_{s,0}^{∗} + i𝔷_{c,0}^{∗}.

In particular, μ is unitarily small for G if and only if μ_{s} is unitarily small for G_{s}.

This is obvious from the definitions. In the same way we can reduce matters to the various simple factors of G_{s}.

Lemma 6.6. With notation as above, list the simple factors of G_{s} as G_{s}^{1}, … G_{s}^{r}. Put T_{s}^{i} = T_{s} ∩ G_{s}^{i}, so that we have a direct sum decomposition

𝔱_{s}^{∗} = (𝔱_{s}^{1})^{∗} + ⋯ + (𝔱_{s}^{r})^{∗}.

Suppose μ_{s} ∈ T̂_{s} is a (G_{s} ∩ K)-dominant weight. Write μ_{s}^{i} for its restriction to T_{s}^{i}. Then

λ_{u}(G_{s})(μ_{s}) = λ_{u}(G_{s}^{1})(μ_{s}^{1}) + ⋯ + λ_{u}(G_{s}^{r})(μ_{s}^{r}).

In particular, μ_{s} is unitarily small if and only if all the μ_{s}^{i} are.

Again we omit the simple proof. Here is the main result of this section.

Theorem 6.7. Use the notation of Definition 6.1 and 6.1. Suppose δ ∈ K̂ has highest weight μ ∈ T̂^{c}. Write μ_{z} ∈ i𝔷_{c,0}^{∗} for the differential of the restriction of μ to Z_{c}. Define λ_{u}(μ) and λ_{a}(μ) as in 0.5 and Proposition 2.3. Then the following conditions are equivalent.

a) The K-type δ is unitarily small (Definition 6.1 ).

b) The parameter λ_{u}(μ) is equal to μ_{z}.

c) The parameter λ_{a}(μ) belongs to μ_{z} plus the convex hull of the Weyl group orbit of ρ.

d) Let Δ^{+}(𝔤, 𝔱^{c}) be a positive root system making μ + 2ρ_{c} dominant. List the fundamamental weights for Δ^{+} as {ξ_{1}, …, ξ_{l}} (cf. 1.2 ). Write 2ρ_{n} for the sum of the noncompact positive roots. Then

⟨ξ_{i}, μ⟩ ≤ ⟨ξ_{i}, 2ρ_{n}⟩ (1 ≤ i ≤ l).

e) Let (Δ^{+})′(𝔤, 𝔱^{c}) be any positive root system containing Δ^{+}(𝔨, 𝔱^{c}). List the fundamamental weights for (Δ^{+})′ as {ξ′_{1}, …, ξ′_{l}} (cf. 1.2 ). Write 2ρ′_{n} for the sum of the noncompact positive roots. Then

⟨ξ′_{i}, μ⟩ ≤ ⟨ξ′_{i}, 2ρ′_{n}⟩ (1 ≤ i ≤ l).

f ) There is a system of positive roots (Δ^{+})′(𝔤, 𝔱^{c}) containing Δ^{+}(𝔨, 𝔱^{c}), with the property that

μ = μ_{z} + ∑_{β∈(Δ^{+})′(𝔭,𝔱^{c})} c_{β}β (0 ≤ c_{β} ≤ 1).

g) There is an expression

μ = μ_{z} + ∑_{β∈Δ(𝔭,𝔱^{c})} b_{β}β (0 ≤ b_{β} ≤ 1).

Proof. The equivalence of (a) and (b) is clear from Lemma 6.5 and the definitions. For the equivalence of (b) and (c), recall from 3.1(c) that λ_{u}(μ) = T_{ρ}((λ_{a})(μ)). Now the equivalence is clear from Proposition 1.7.

For the rest, the conditions in (d) through (g) are clearly satisfied if and only if the analogous conditions on μ_{s} are satisfied. Lemma 6.5 therefore allows us to assume that G = G_{s} is connected and semisimple. Under this assumption, we will prove that (a) implies (d); that (d) implies (e); that (e) implies (f); that (f) implies (g); and finally that (g) implies (a). The most difficult of these is (e) implies (f).

So suppose that G is semisimple and that μ is unitarily small; that is, that λ_{u}(μ) = 0. According to 0.5 and Proposition 1.7, this means that μ + 2ρ_{c} − 2ρ belongs to −C^{o}. Since the fundamental weights ξ_{i} belong to the positive Weyl chamber C, it follows that

⟨ξ_{i}, μ + 2ρ_{c} − 2ρ⟩ ≤ 0.

Because 2ρ = 2ρ_{c} + 2ρ_{n}, this is equivalent to

⟨ξ_{i}, μ⟩ ≤ ⟨ξ_{i}, 2ρ_{n}⟩.

This is (d).

Next, suppose that (d) holds. Fix (Δ^{+})′ as in (e); say

(6.2a)  (Δ^{+})′ = wΔ^{+},

for some w in the Weyl group. We may assume that the fundamental roots are labelled in such a way that ξ′_{i} = wξ_{i}. Then

(6.2b)  ⟨ξ′_{i}, μ⟩ = ⟨ξ′_{i}, μ + 2ρ_{c}⟩ − ⟨ξ′_{i}, 2ρ_{c}⟩ = ⟨wξ′_{i}, μ + 2ρ_{c}⟩ − ⟨ξ′_{i}, 2ρ_{c}⟩

By Lemma 1.8,

(6.2c)  ⟨wξ_{i}, μ + 2ρ_{c}⟩ ≤ ⟨ξ_{i}, μ + 2ρ_{c}⟩.

By our hypothesis (d), it follows that

(6.2d)  ⟨wξ_{i}, μ + 2ρ_{c}⟩ ≤ ⟨ξ_{i}, 2ρ⟩ = ⟨wξ_{i}, 2wρ⟩ = ⟨ξ′_{i}, 2ρ′⟩.

Inserting the inequality 6.2(d) in 6.2(b) yields

(6.2e)  ⟨ξ′_{i}, μ⟩ ≤ ⟨ξ′_{i}, 2ρ′⟩ − ⟨ξ′_{i}, 2ρ_{c}⟩ = ⟨ξ′_{i}, 2ρ′_{n}⟩.

This is (e).

Because the proof that (e) implies (f) is the most difficult part of the argument, we postpone it to the end. That (f) implies (g) is trivial. Assume therefore that (g) holds (still assuming G is semisimple); we will deduce (a). Choose a positive system Δ^{+} making μ + 2ρ_{c} dominant. The formula for μ may then be written as

μ = ∑_{β∈Δ^{+}(𝔭,𝔱^{c})}(b_{β} − b_{−β})β.

Here the coefficients c_{β} = b_{β} − b_{−β} lie between −1 and 1. We may therefore write

μ = 2ρ_{n} + ∑_{β∈Δ^{+}(𝔭,𝔱^{c})}(c_{β} − 1)β;

the coefficients a_{β} = c_{β} − 1 in this formula lie between 0 and -2. It follows that

μ + 2ρ_{c} − 2ρ = ∑_{β∈Δ^{+}(𝔭,𝔱)} a_{β}β (−2 ≤ a_{β} ≤ 0).

Obviously the right side belongs to −C^{o}, the cone spanned by negative roots. By Proposition 1.7, P (μ + 2ρ_{c} − 2ρ) = 0; that is, λ_{u}(μ) = 0. This is (a).

Finally, we turn to the proof that (e) implies (f). We proceed by induction on the dimension of G. Recall that we are assuming that G = G_{s} is semisimple. Lemma 6.6 actually allows us to assume that G is simple. (More precisely, if 𝔤_{0} has a non-trivial decomposition into simple factors, then the summands have lower dimension, so we already know the implication for them. Lemma 6.6 allows us to deduce the implication for G.)

Lemma 6.8. Suppose G is compact and simple. Then the only weight μ satisfying either condition (e) or condition (f ) of Theorem 6.7 is μ = 0.

Proof. The condition in (e) is

⟨ξ, μ⟩ ≤ 0

for any fundamental weight ξ (with respect to the fixed positive system making μ dominant). Because fundamental weights have non-negative inner product, this implies μ = 0. The argument for (f) is even simpler.

We may therefore assume that G is noncompact and simple.

Lemma 6.9. Suppose G is noncompact and simple. Then the noncompact roots span (𝔱^{c})^{∗}. In particular, in the setting of Theorem 6.7(e), we must have

⟨ξ′_{i}, 2ρ′_{n}⟩ > 0.

Proof. Define

(6.3a)  Δ^{0} = {α ∈ Δ(𝔤,, 𝔱^{c}) | ⟨α, β⟩ = 0 (β ∈ Δ(𝔭, 𝔱))},
(6.3b)  Δ^{1} = roots in the integer span of Δ(𝔭, 𝔱^{c}).

Obviously Δ^{0} and Δ^{1} are mutually orthogonal root subsystems in Δ. We claim that

(6.3c)  Δ = Δ^{0} ∪ Δ^{1}.

Assume this for a moment. The orthogonality of the decomposition allows us to conclude that root vectors for Δ^{0} and Δ^{1} must commute with each other, and therefore that we get a corresponding decomposition

(6.3d)  𝔤_{0} = 𝔤_{0}^{0} + 𝔤_{0}^{1}.

Because 𝔤_{0} is simple, this decomposition must be trivial. Since G is assumed to be noncompact, the second factor is non-trivial; so the first must be trivial. Therefore Δ = Δ^{1}, which is the first claim in the lemma.

To prove 6.3(c), suppose γ is any root belonging neither to Δ^{0} nor to Δ^{1}. Since Δ^{1} includes the noncompact roots, γ must be compact. Since it is not in Δ^{0}, there is a noncompact root β with which it has non-zero inner product. Possibly replacing β by −β, we get

⟨γ, β⟩ < 0.

It follows (by the representation theory of the SU(2) generated by γ) that β + γ is a noncompact root. Therefore γ = (β + γ) − β belongs to Δ^{1}, a contradiction.

To prove the last inequality, notice that ξ′_{i} has non-negative inner product with each positive root. Therefore (6.3e)

Δ^{+}(𝔭, 𝔱^{c}) = {β ∈ Δ(𝔭, 𝔱) | ⟨ξ′_{i}, β⟩ > 0} ∪ { certain roots orthogonal to ξ′_{i} }.

So

(6.3f)  ⟨ξ′_{i}, 2ρ_{n}⟩ = ∑_{{β∈Δ(𝔭,𝔱^{c}) \atop ⟨ξ′_{i},β⟩>0}}⟨ξ′_{i}, β⟩.

This is a sum of strictly positive terms; and it is non-empty by the first part of the lemma.

We continue now with the proof that (e) implies (f) in Theorem 6.7. Recall that we are proceeding by induction on the dimension of G, and that we have reduced to the case when G is simple and noncompact. We propose to apply the inductive hypothesis to a maximal Levi subgroup of G. In order to construct it, we choose a positive root system (Δ^{+})′(𝔤, 𝔱^{c}) ⊃ Δ^{+}(𝔨, 𝔱^{c}) and a fundamental weight ξ′_{i} so as to maximize the quotient

(6.4a)  c_{i} = ⟨ξ′_{i}, μ⟩/⟨ξ′_{i}, 2ρ′_{n}⟩.

Notice that the denominator is strictly positive by Lemma 6.9. We can make the numerator non-negative by an appropriate choice of (Δ^{+})′ (for example by making μ dominant); so our maximum c_{i} must be non-negative. By hypothesis (e) of Theorem 6.7, c_{i} is bounded above by 1. Consequently

(6.4b)  0 ≤ c_{i} ≤ 1.

The weight ξ′_{i} belongs to i(𝔱_{0}^{c})^{∗}, so by 0.6 it defines a proper Levi subgroup

(6.4c)  L_{i} = G(ξ′_{i}) ⊃ T^{c}.

The root system of L_{i} is

(6.4d)  Δ_{i} = {γ ∈ Δ | ⟨ξ′_{i}, γ⟩ = 0} = span of {α′_{j} | j ≠ i}.

Our choice of positive roots for K gives a positive system

(6.4e)  Δ_{i}^{+}(𝔩_{i} ∩ 𝔨, 𝔱^{c}) = Δ_{i} ∩ Δ^{+}(𝔨, 𝔱^{c}).

The construction of 0.6 also provides a unipotent subalgebra

(6.4f)  𝔲_{i} = 𝔲(ξ′_{i}),  Δ(𝔲_{i}, 𝔱) = {γ ∈ Δ | ⟨ξ′_{i}, γ⟩ > 0}.

Recall from 0.4(d) that 2ρ(𝔲_{i} ∩ 𝔭) denotes the sum of the roots of 𝔱^{c} in 𝔲_{i} ∩ 𝔭; that is, the sum of the noncompact roots having positive inner product with ξ′_{i}. Define

(6.4g)  μ_{i} = μ − 2c_{i}ρ(𝔲_{i} ∩ 𝔭).

Lemma 6.10. Use the notation 6.4.

a) The weight μ_{i} belongs to the ℝ-span of the roots of L_{i}.

b) The weight μ_{i} is dominant integral for L_{i} ∩ K. More precisely, if α is any

root in L_{i} ∩ K, then

⟨α, μ_{i}⟩ = ⟨α, μ⟩.

c) The weight μ_{i} satisfies condition (e) of Theorem 6.7 with respect to the group L_{i}.

Let us assume this lemma for a moment, and complete our proof that (e) implies (f). Because L_{i} has strictly lower dimension than G, we may assume by inductive hypothesis that (e) implies (f) for L_{i}. Parts (a) and (b) of Lemma 6.10 guarantee that Theorem 6.7 makes sense for L_{i} and μ_{i}; and part (c) says that Theorem 6.7(e) is satisfied. By induction, Theorem 6.7(f) must be satisfied as well. That is, we can find a positive system (Δ_{i}^{+})′′ for L_{i}, containing Δ_{i}^{+}(𝔩_{i} ∩ 𝔨, 𝔱^{c}), so that

(6.5a)  μ_{i} = ∑_{β∈(Δ_{i}^{+})′′(𝔩_{i}∩𝔭,𝔱^{c})} c_{β}β  (0 ≤ c_{β} ≤ 1).

Now define a positive root system for G by

(6.5b)  (Δ^{+})′′ = (Δ_{i}^{+})′′ ∪ Δ(𝔲_{i}, 𝔱^{c}).

Then 6.4(g) and 6.5(a) show that

(6.5c)  \begin{aligned}μ & = 2c_{i}ρ(𝔲_{i} ∩ 𝔭) + μ_{i} \\ & = ∑_{β∈Δ(𝔲_{i}∩𝔭)} c_{i}β + ∑_{β∈(Δ_{i}^{+})′′(𝔩_{i}∩𝔭,𝔱^{c})} c_{β}β  (0 ≤ c_{β} ≤ 1). \end{aligned}

Because of 6.4(b), this expression has the form required in Theorem 6.7(f).

It remains to prove Lemma 6.10. For (a), 1.2 shows that the ℝ-span of the roots of L_{i} is precisely the orthogonal complement of ξ′_{i}. We calculate

⟨ξ′_{i}, μ_{i}⟩ = ⟨ξ′_{i}, μ⟩ − c_{i}⟨ξ′_{i}, 2ρ(𝔲_{i} ∩ 𝔭)⟩

Using 6.3(f), we see that the second inner product on the right is just ⟨ξ′_{i}, 2ρ′_{n}⟩. Therefore

⟨ξ′_{i}, μ_{i}⟩ = ⟨ξ′_{i}, μ⟩ − c_{i}⟨ξ′_{i}, 2ρ′_{n}⟩

Inserting the definition 6.4(a) of c_{i}, we see immediately that the right side is zero, as we wished to show.

For (b), it is enough to prove the formula. In light of 6.4(g), the formula amounts to

⟨α, 2ρ(𝔲_{i} ∩ 𝔭)⟩ = 0 (α ∈ Δ(𝔩_{i} ∩ 𝔨, 𝔱^{c})).

Because the adjoint action of the reductive group L_{i} ∩ K preserves 𝔲_{i} ∩ 𝔭, the sum of the weights of the latter must be orthogonal to the roots of L_{i} ∩ K.

We turn now to (c). Suppose therefore that (Δ_{i}^{+})′′ is a system of positive roots for L_{i} containing Δ_{i}^{+}(𝔩_{i} ∩ 𝔨, 𝔱^{c}). Just as in 6.5 we define a positive root system for G by

(6.6a)  (Δ^{+})′′ = (Δ_{i}^{+})′′ ∪ Δ(𝔲_{i}, 𝔱^{c}).

Write

(6.6b)  Π′′ = {α′′_{1}, …, α′′_{l}}

for the set of simple roots, and {ξ′′_{j}} for the dual basis of fundamental weights. By construction the two positive root systems (Δ^{+})′ and (Δ^{+})′′ differ by an element of the Weyl group of Δ_{i}. It follows that

(6.6c)  ξ′_{i} = ξ′′_{i}.

Furthermore the constants c_{i} defined using (Δ^{+})′ and (Δ^{+})′′ coincide, as do the weights 2ρ(𝔲_{i} ∩ 𝔭). Consequently there is no loss of generality in assuming henceforth that (Δ^{+})′ = (Δ^{+})′′.

We are trying to establish an inequality relating μ_{i} and the fundamental weights for the positive system (Δ_{i}^{+})′. Let us write (ξ_{j}^{i})′ for these fundamental weights. According to 1.2, they are characterized by belonging to the span of Δ_{i}, and by

(6.7a)  ⟨(ξ_{j}^{i})′, α′_{k}⟩ = δ_{jk}  (k, j ≠ i)

From these requirements we see that (ξ_{j}^{i})′ must be the projection of ξ′_{j} orthogonal to ξ′_{i}. That is,

(6.7b)  (ξ_{j}^{i})′ = ξ′_{j} − (⟨ξ′_{i}, ξ′_{j}⟩/⟨ξ′_{i}, ξ′_{i}⟩)ξ′_{i}.

The sum of the noncompact positive roots for L_{i} is

(6.7c)  2(ρ_{n}^{i})′ = 2ρ′_{n} − 2ρ(𝔲_{i} ∩ 𝔭).

The inequality of Theorem 6.7(e) that we are trying to prove is

(6.7d)  ⟨(ξ_{j}^{i})′, μ_{i}⟩ ≤ ⟨(ξ_{j}^{i})′, 2(ρ_{n}^{i})′⟩.

We begin by examining the right side. Because 2(ρ_{n}^{i})′ is a sum of roots in L_{i}, it is orthogonal to ξ′_{i}. So 6.7(b) allows us to replace (ξ_{j}^{i})′ by ξ′_{j} on the right side. Inserting 6.7(c) gives

(6.7e)  ⟨(ξ_{j}^{i})′, 2(ρ_{n}^{i})′⟩ = ⟨ξ′_{j}, 2ρ′_{n} − 2ρ(𝔲_{i} ∩ 𝔭)⟩.

Next we examine the left side of 6.7(d). By construction μ_{i} is orthogonal to ξ′_{i}, so 6.7(b) allows us to replace (ξ_{j}^{i})′ by ξ′_{j} here as well. Just as in 6.4(a), let us define

(6.7f)  c_{j} = ⟨ξ′_{j}, μ⟩/⟨ξ′_{j}, 2ρ′_{n}⟩.

Using this definition and 6.4(g), we get

(6.7g)  \begin{aligned}⟨(ξ_{j}^{i})′, μ_{i}⟩ & = ⟨ξ′_{j}, μ⟩ − c_{i}⟨ξ′_{j}, 2ρ(𝔲_{i} ∩ 𝔭)⟩ \\ & = c_{j}⟨ξ′_{j}, 2ρ′_{n}⟩ − c_{i}⟨ξ′_{j}, 2ρ(𝔲_{i} ∩ 𝔭)⟩. \end{aligned}

Recall now that we chose i to maximize c_{i}. In particular we must have c_{j} ≤ c_{i}. Since the coefficient of c_{j} on the right in 6.7(g) is positive (Lemma 6.9), we deduce

(6.7h) ⟨(ξ_{j}^{i})′, μ_{i}⟩ ≤ c_{i} (⟨ξ′_{j}, 2ρ′_{n}⟩ − ⟨ξ′_{j}, 2ρ(𝔲_{i} ∩ 𝔭)⟩) = c_{i}⟨ξ′_{j}, 2ρ′_{n} − 2ρ(𝔲_{i} ∩ 𝔭)⟩.

Now 6.7(e) and the fact that c_{i} ≤ 1 (cf. 6.4(b)) show that 6.7(h) implies the inequality 6.7(d) that we want. This completes the proof of Lemma 6.10, and therefore of Theorem 6.7.

We want next to explore some formal consequences of the characterizations of unitarily small K-types in Theorem 6.7. We will apply the following lemma to the set of roots of T^{c} in 𝔭.

Lemma 6.11. Use the notation 0.3. Suppose S ⊂ i(𝔱_{0}^{c})^{∗} is an arbitrary finite subset invariant under the Weyl group of K:

w · S ⊂ S (w ∈ W (K, T)).

Define

R(S) = {∑_{s∈S} b_{s}s (0 ≤ b_{s} ≤ 1)} ⊂ i(𝔱_{0}^{c})^{∗}.

Then R(S) is convex and invariant under W (K). The following conditions on an irreducible representation δ ∈ K̂ are equivalent. a) Every weight of δ belongs to R(S).

b) Every extremal weight of δ belongs to R(S).

c) Every highest weight of δ belongs to R(S).

d) Some highest weight of δ belongs to R(S).

e) Some extremal weight of δ belongs to R(S).

We will say that a representation of K (even reducible) is of type S if these conditions are satisfied.

Proof. That R(S) is convex and W (K) invariant is obvious, as are the implications (a) ⇒ (b) ⇒ (c) ⇒ (d) ⇒ (e). We will complete the proof by showing that (e) ⇒ (b) ⇒ (a). So assume (e) holds. The set of extremal weights of δ is a single orbit of W (K), so (b) follows by the W (K) invariance of R(S). Now assume (b). Every weight must lie in the convex hull of the extremal weights. (For connected K this is a consequence of Proposition 1.7, and the general case follows at once.) So the convexity of R(S) shows that (b) implies (a).

Corollary 6.12. Suppose G is a real reductive group in Harish-Chandra’s class, and K is a maximal compact subgroup; use the notation of 6.1. The following conditions on an irreducible representation δ ∈ K̂ are equivalent.

a) The K-type δ is unitarily small.

b) The restriction δ_{s} of δ to G_{s} ∩ K is of type Δ(𝔭, 𝔱^{c}) (Lemma 6.11 ).

c) Write μ_{z} ∈ i𝔷_{c,0}^{∗} for the differential of the character of Z_{c} on δ. Then δ is of type {μ_{z}} ∪ Δ(𝔭, 𝔱^{c}).

Proof. In light of Lemmas 6.11 and 6.5, this is a reformulation of the equivalence of (a) and (g) in Theorem 6.7.

Before we embark on a more serious investigation of unitarily small Ktypes, we want to describe the unitarily small parameters of Definition 6.1. The result (Proposition 6.13) is fairly simple, but the proof is a little involved. We need to extend somewhat the notation introduced in 6.1. Recall that 𝔤_{s,0} = [𝔤_{0}, 𝔤_{0}] is the derived algebra of 𝔤_{0}. It is a direct sum of simple ideals (Lemma 6.6). Write

(6.8a)  𝔤_{c,0} = sum of the compact simple factors of 𝔤_{s,0}.

The corresponding subgroup G_{c} of G_{s} is compact, connected, and semisimple. Therefore it has finite center; so in particular the intersection

(6.8b)  F_{c} = Z(G_{c}) ∩ Z ⊂ Z_{c}

of its center with the identity component of the center of G must be finite.

Proposition 6.13. Suppose we are in the setting of Definition 6.1; use also the notation of 6.1 and 6.8. The set Λ_{u}^{z}(G) of unitarily small parameters for G consists precisely of the differentials of characters of Z_{c}/F_{c}.

Proof. Suppose δ ∈ B_{u}^{z}(G) is a unitarily small representation of K. It is clear from conditions (f) or (g) of Theorem 6.7 that δ must be trivial on G_{c}. (This was essentially proved in Lemma 6.8.) The restriction of δ to Z_{c} therefore defines a character μ_{z} of Z_{c}/F_{c}, and Lemma 6.5 shows that the corresponding unitarily small parameter is the differential of μ_{z}.

Conversely, suppose μ_{z} is a character of Z_{c}/F_{c}. We must show how to extend μ_{z} to a unitarily small representation of K. The problem is easily reduced to the identity component of G, so we assume henceforth that G is connected. Define

(6.9a)  G_{n} = identity component of the centralizer in G of G_{c}.

Clearly this is a closed connected reductive subgroup of G. On the level of Lie algebras it is easy to prove a direct sum decomposition

(6.9b)  𝔤_{0} = 𝔤_{c,0} + 𝔤_{n,0}.

From this it follows that the multiplication map defines a surjective covering homomorphism

(6.9c)  G_{c} × G_{n} → G.

Define

(6.9d)  F = G_{c} ∩ G_{n} = Z(G_{c}) ∩ Z(G_{n}).

This is a finite central subgroup of both factors. Its intersection with the identity component of the center of G_{n} is just the group F_{c} defined in 6.8(b). The kernel of the multiplication map is

(6.9e)  F_{Δ} = {(f, f^{−1}) | f ∈ F } ⊂ G_{c} × G_{n}.

Now 6.9 allows us to describe the representation theory of G completely in terms of the representation theory of G_{c} and G_{n} separately. Here is an example.

Lemma 6.14. Suppose G is connected; use the notation of 6.9. The irreducible representations δ of K are in one-to-one correspondence with pairs

(δ_{c}, δ_{n}) ∈ Ĝ_{c} × (G_{n} ∩ K) ,

subject to the requirement that

δ_{c}|_{F} = δ_{n}|_{F} .

Proof. Because of 6.9, the multiplication map defines an isomorphism

K ≃ (G_{c} × (G_{n} ∩ K))/F_{Δ}.

The representations of the direct product G_{c} × (G_{n} ∩ K) are tensor products, and so correspond to pairs (δ_{c}, δ_{n}). The requirement in the lemma is exactly what is needed to make δ_{c} ⊗ δ_{n} trivial on F_{Δ}.

We now set out to construct a unitarily small representation of K from our character μ_{z} of Z_{c}/F_{c}. As a first step, consider the subgroup F Z_{c} of G_{n}. Because F ∩ Z_{c} = F_{c}, we have

(6.10a)  F Z_{c}/F ≃ Z_{c}/F_{c}.

So we begin by regarding μ_{z} as a character of F Z_{c}/F. Because this is a compact central subgroup of G_{n}, we can find an irreducible representation δ_{n}^{1} of G_{n} ∩ K such that

(6.10b)  δ_{n}^{1}|_{F Z_{c}} = μ_{z}.

If δ_{n}^{1} were unitarily small, then ℂ⊗δ_{n}^{1} would be a unitarily small representation of K restricting to μ_{z}, and we would be done. But δ_{n}^{1} may not be unitarily small, so we need to work a little harder.

Write T_{n} for our fixed maximal torus in G_{n} ∩ K (the intersection of T^{c} with G_{n}), and μ_{n}^{1} ∈ T̂_{n} for any weight of δ_{n}^{1}. By hypothesis

(6.10c)  μ_{n}^{1}|_{F Z_{c}} = μ_{z}.

On the other hand, G_{n} has no compact simple factors. By Lemma 6.9, this implies that the noncompact roots span the semisimple part of i𝔱_{n,0}^{∗}. We may therefore write

(6.10d)  μ_{n}^{1} = μ_{z} + ∑_{β∈Δ(𝔭,𝔱_{n})} b_{β}^{1}β  (b_{β}^{1} ∈ ℝ)

For each noncompact root β, choose an integer n_{β} so that

(6.10e)  b_{β} = b_{β}^{1} + n_{β} ∈ [0, 1].

Now define μ_{n} ∈ T̂^{c} to be μ_{n}^{1} twisted by the sum of the various n_{β}β:

(6.10f)  μ_{n} = μ_{n}^{1} + ∑_{β∈Δ(𝔭,𝔱_{n})} n_{β}β.

Because the roots are characters of T_{n}, this is well-defined. Because the center of G_{n} acts trivially on the roots, 6.10(c) implies

(6.10g)  μ_{n}|_{F Z_{c}} = μ_{z}.

Finally, 6.10(d)–(f) give

(6.10h)  μ_{n} = μ_{z} + ∑_{β∈Δ(𝔭,𝔱_{n})} b_{β}β  (0 ≤ b_{β} ≤ 1)

Now define δ_{n} to be the unique irreducible representation of G_{n}∩K of extremal weight μ_{n}. By Lemma 6.11, the highest weight of δ_{n} has an expression of the same form as in 6.10(h). By Theorem 6.7, δ_{n} is unitarily small. By Lemma 6.14, δ = ℂ ⊗ δ_{n} is a unitarily small representation of K, and the restriction of δ to Z_{c} is μ_{z} by 6.10(g).

7. Unitarily small K-types and the Dirac operator

Proposition 7.1. In the setting of Definition 6.1, any irreducible representation of K appearing in the exterior algebra ⋀ 𝔭 (notation 0.2 ) is unitarily small. More precisely, the representation ⋀ 𝔭 is of type Δ(𝔭, 𝔱^{c}) (cf. Lemma 6.11 ).

Proof. The weights of the exterior algebra of 𝔭 are those of the form

∑_{β∈Δ(𝔭,𝔱^{c})} b_{β}β, (b_{β} = 0 or 1).

Therefore ⋀ 𝔭 is of type Δ(𝔭, 𝔱). Now apply Corollary 6.12.

In the theory of unitary representations, the interaction between the Ktypes of ⋀ 𝔭 and certain spinor representations plays an important rôle. In this section we will outline this theory, and show how to extend parts of it to all unitarily small representations.

Our development is taken from [4], Chapter II and [2], II.6. We will indicate particularly large departures from their conventions. We begin with a finite-dimensional real inner product space (V_{0}, ⟨ , ⟩). Set

(7.1a)  SO(V_{0}) = special orthogonal group of V_{0},  V = V_{0} ⊗_{ℝ} ℂ

The Clifford algebra C(V_{0}) is the real associative algebra with unit generated by V_{0} and subject to the relations

(7.1b)  v^{2} = −⟨v, v⟩ · 1  (v ∈ V_{0}).

(For Chevalley this is the Clifford algebra of the form −⟨ , ⟩.) Sometimes it is convenient to write the defining relations in the equivalent form

(7.1c)  vw + wv = −2⟨v, w⟩  (v, w ∈ V_{0}).

Obviously SO(V_{0}) (and even the full orthogonal group O(V_{0})) acts on C(V_{0}) by algebra automorphisms. The action of −1 ∈ O(V_{0}) defines a ℤ/2ℤ-grading

(7.1d)  C(V_{0}) = C(V_{0})_{even} + C(V_{0})_{odd}.

There is an algebra anti-automorphism transpose of C(V_{0}), characterized by

(7.1e)  ^{t}v = −v,  ^{t}(ab) = ^{t}b^{t}a  (v ∈ V_{0}, a, b ∈ C(V_{0})).

(Chevalley emphasizes the antiautomorphism α that acts by +1 on V_{0}. It differs from transpose by the action of −1 ∈ O(V_{0}).) The Clifford algebra is filtered by assigning the generators degree 1. Write C_{p}(V_{0}) for the pth level of the filtration. Then there is a natural isomorphism

(7.1f)  C_{p}(V_{0})/C_{p−1}(V_{0}) ≃ ⋀^{p} V_{0}

respecting the algebra structure and the ℤ/2ℤ-grading. It follows easily that there is a natural isomorphism

(7.1g)  C_{1}(V_{0})_{odd} ≃ V_{0};

the map from right to left just sends the space V_{0} of generators into C(V_{0}).

Write C(V_{0})^{×} for the group of invertible elements of C(V_{0}). Because C(V_{0}) is a finite-dimensional associative algebra with a unit, C(V_{0})^{×} is an open subset. It is a Lie group; the Lie algebra is C(V_{0}) with the commutator bracket. This group acts on C(V_{0}) by conjugation: (7.2a) Ad: C(V_{0})^{×} → Aut(C(V_{0})), Ad(u)(a) = uau^{−1} (u ∈ C(V_{0})^{×}, a ∈ C(V_{0})).

The differential of this representation is the action of C(V_{0}) on itself by bracket. The subgroup C(V_{0})_{even}^{×} acts in Ad to preserve the ℤ/2ℤ grading. We now define the spin group of V_{0} as

(7.2b)  Spin(V_{0}) = {g ∈ C(V_{0})_{even}^{×} | Ad(g)(V_{0}) ⊂ V_{0}, ^{t}g = g^{−1}}.

This is the group that Chevalley calls the reduced Clifford group Γ_{0}^{+}. The first condition here may also be expressed as requiring Ad(g) to preserve the filtration of 7.1(f). By construction Spin(V_{0}) is an algebraic group, and the adjoint action provides an algebraic homomorphism

τ: Spin(V_{0}) → GL(V_{0}).

Because Ad preserves the algebra structure, the relation 7.1(b) shows that the image of τ is contained in O(V_{0}):

(7.2c)  τ: Spin(V_{0}) → O(V_{0}).

Lemma 7.2. The image of the map τ of 7.2(c) is the special orthogonal group SO(V_{0}). The kernel consists of the two scalar elements ±1 in the center of C(V_{0}). If dim V_{0} ≥ 2, this kernel is contained in the identity component of Spin(V_{0}); so in that case Spin(V_{0}) is a connected double cover of SO(V_{0}).

Proposition 7.3. The Clifford algebra C(V_{0}) is a semisimple algebra of dimension 2^{dim V_{0}}. Write Z = Z(C(V_{0})) for the center.

Suppose that dim V_{0} = 2r is even. Then Z consists only of the scalars C_{0}. Consequently C(V_{0}) and its complexification are both central simple algebras. In particular, C(V_{0}) has a unique complex simple module (γ, S) (up to equivalence), of dimension 2^{r}. We may regard γ as a representation of the group C(V_{0})^{×}. Write σ for the restriction of γ to the subgroup Spin(V_{0}). If r ≥ 1, then σ is the direct sum of two inequivalent irreducible representations (σ_{±}, S_{±}), each of dimension 2^{r−1}.

Suppose that dim V_{0} = 2r + 1 is odd. Then Z is two-dimensional; it is spanned by the scalars and an odd element z_{−}, satisfying

(a)  (z_{−})^{2} = ±1 ∈ C_{0}.

The subalgebra C(V_{0})_{even} has dimension 2^{2r} and one-dimensional center; it is central simple, and generated as an algebra by Spin(V_{0}). The entire Clifford algebra is a tensor product

(b)  C(V_{0}) ≃ Z ⊗_{ℝ} C(V_{0})_{even}.

There is a unique complex simple module (γ_{even}, S) for C(V_{0})_{even} (up to equivalence), of dimension 2^{r}. The restriction of γ to Spin(V_{0}) is an irreducible representation (σ, S). There are exactly two extensions γ and γ′ of γ_{even} to a module for C(V_{0}) (on the same space S); they are related by

(c)  γ(v) = −γ′(v)  (v ∈ V_{0}).

Any complex simple module for C(V_{0}) is isomorphic to exactly one of γ and γ′.

In either case, suppose (γ, S) is a simple module for C(V_{0}). Then there is a positive definite invariant Hermitian form ⟨ , ⟩ on S. This means that γ(a)^{∗} = γ(^{t}a), or explicitly that

(d)  ⟨γ(a)s, s′⟩ = ⟨s, γ(^{t}a)s′⟩.

This form is unique up to a positive scalar multiple. In particular, the representation (σ, S) of Spin(V_{0}) is unitary.

We should make two remarks about the notation introduced in this proposition. There is no preferred choice between the two irreducible constituents of the spin representation in the even-dimensional case; to call one of them σ_{+} is to make a choice. Similarly, there is no preferred extension of γ_{even} in the odd-dimensional case; calling one γ is making a choice.

This proposition summarizes several results in Chapter 2 of [4]. (The formulation in II.6.2 of [2] is not quite correct in the odd-dimensional case, when it overlooks the non-uniqueness of the extension of γ_{even} to C(V_{0}).)

Definition 7.4. Suppose we are in the setting 7.1–7.2. A space of spinors for V_{0} is a complex simple module (γ, S) for the Clifford algebra C(V_{0}), endowed with a positive definite invariant Hermitian form ⟨ , ⟩, as described in Proposition 7.3. We write (σ, S) for the corresponding unitary representation of Spin(V_{0}); we will sometimes call σ a spin representation. By 7.2(a) and the definition of σ, we have

γ(Ad(g)(a)) = σ(g)γ(a)σ(g^{−1}) (g ∈ Spin(V_{0}), a ∈ C(V_{0})).

Example 7.5. Suppose V_{0} is a one-dimensional space ℝv_{0}, and that v_{0} has length one. Then SO(V_{0}) is trivial. The Clifford algebra C(V_{0}) is generated by v_{0} with the relation v_{0}^{2} = −1; that is, C(V_{0}) is isomorphic to the algebra ℂ of complex numbers. There are two possible isomorphisms, sending v_{0} to i or to −i. The even subalgebra is

C(V_{0})_{even} = ℝ ⊂ ℂ,

and the odd subspace is iℝ. The transpose automorphism is complex conjugation, so we see that

Spin(V_{0}) = ±1 ⊂ ℝ^{×}.

The unique complex simple module for C(V_{0})_{even} is just ℂ; the two possible extensions to C(V_{0}) have v_{0} acting by ±i. (The distinction between these is not just a matter of choosing i ∈ ℂ; we must also choose between v_{0} and −v_{0} as a basis element of V_{0}.)

Example 7.6. Suppose V_{0} is two-dimensional, with an orthonormal basis {j, k}. Then SO(V_{0}) consists of the 2 × 2 matrices

(a)  SO(V_{0}) = \{r(θ) = ( \begin{matrix}cos θ & − sin θ \\ sin θ & cos θ \end{matrix} )\} .

The Clifford algebra has basis {1, j, k, jk}, with relations

(b)  j^{2} = k^{2} = −1, jk = −kj.

This is evidently isomorphic to the algebra of quaternions

(c)  ℍ = {a + bi + cj + dk}

by sending jk to i. The transpose antiautomorphism is the usual conjugation of quaternions (which is −1 on i, j, and k). The even subalgebra is

(d)  C(V_{0})_{even} = ℂ ⊂ ℍ.

It is not difficult to check from (d) and the definition 7.2(b) that

(e)  Spin(V_{0}) = {s(ϕ) = cos ϕ + i sin ϕ | ϕ ∈ ℝ} .

The conjugation action on V_{0} is

(f)  τ(s(ϕ)) = r(2ϕ).

To see that, one calculates for example

τ(s(ϕ))j = (cos ϕ + i sin ϕ)j(cos ϕ − i sin ϕ)
= j cos^{2} ϕ − iji sin^{2} ϕ + (ij − ji) sin ϕ cos ϕ
= j(cos^{2}ϕ − sin^{2}ϕ) + k(2 sin ϕ cos ϕ)
= j cos 2ϕ + k sin 2ϕ
= r(2ϕ)j.

Therefore Spin(V_{0}) is the double cover of the circle SO(V_{0}). The unique complex simple module for C(V_{0}) may be realized on S = ℂ^{2} by

(g)  γ(j) = ( \begin{matrix}0 & 1 \\ −1 & 0 \end{matrix} ) , γ(k) = ( \begin{matrix}0 & i \\ i & 0 \end{matrix} ) , γ(i) = ( \begin{matrix}i & 0 \\ 0 & −i \end{matrix} ) .

The natural decomposition ℂ^{2} = ℂ + ℂ gives a Spin(V_{0})-invariant decomposition S = S_{+} + S_{−}. Combining (e) with the third formula in (g), we see that

(h)  σ_{±}(r(ϕ)) = exp(±iϕ).

That is, the two half-spin representations σ_{±} are the first two characters of the circle Spin(V_{0}).

One reason for considering these two examples in such detail is that many aspects of general Clifford algebras and spin representations can be reduced to dimensions one and two. In order to understand how, we need a little more notation.

Suppose first that W_{0} ⊂ V_{0} is any subspace. We immediately get a natural map of Clifford algebras

(7.3a)  C(W_{0}) → C(V_{0})

respecting ℤ/2ℤ gradings, filtrations, and the transpose. By 7.1(f) this map is one-to-one; we use it to identify C(W_{0}) as a subalgebra of C(V_{0}). It is clear from the definition in 7.2 that 7.3(a) induces an inclusion of groups

(7.3b)  Spin(W_{0}) ↪ Spin(V_{0}).

Suppose now that U_{0} is a second subspace of V_{0}, and that

(7.3c)  U_{0} is orthogonal to W_{0}.

Then 7.1(c) says that

(7.3d)  wu + uw = 0  (u ∈ U_{0}, w ∈ W_{0}).

From this we can get complete commutation or anti-commutation rules for even and odd elements of C(U_{0}) and C(V_{0}); but we will be content with

(7.3e)  ab − ba = 0  (a ∈ C(W_{0})_{even}, b ∈ C(U_{0})).

In particular,

(7.3f)  Spin(W_{0}) and Spin(U_{0}) commute.

The isomorphism 7.1(f) also shows that C(W_{0}) and C(U_{0}) meet only in their common scalars. Lemma 7.2 therefore shows that

(7.3g)  \begin{matrix}Spin(W_{0}) ∩ Spin(U_{0}) & = \{±1\}, \\ Spin(W_{0}) ×_{\{±1\}} Spin(U_{0}) & ↪ Spin(V_{0}). \end{matrix}

Here the notation means that we divide the product group by the diagonal copy of {±1}.

The formula 7.3(f) suggests that an orthogonal decomposition of V_{0} ought to give rise to a tensor product decomposition of the Clifford algebra. This is complicated by the strange shape of the commutation relation 7.3(d). The following bit of extra structure will allow us to circumvent this.

Definition 7.7. Suppose we are in the setting 7.1–7.2, and that dim V_{0} is even. A special element ϵ ∈ Spin(V_{0}) is one satisfying

τ(ϵ) = −1 ∈ SO(V_{0}).

Such an element exists and is unique up to sign by Lemma 7.2. Because τ(ϵ^{2}) = 1, we must have ϵ^{2} = ±1. In Example 7.6, ϵ = s(π/2) = i is a special element of square −1. If V_{0} = 0, then ϵ = ±1 are special elements of square 1. It turns out that ϵ^{2} = (−1)^{r} when dim V_{0} = 2r, but we will not need this.

Proposition 7.8. ([4], II.2.5.) In the setting 7.1–7.2, suppose that we have an orthogonal decomposition

(a)  V_{0} = W_{0} + U_{0},

and that dim W_{0} is even. Choose a special element ϵ ∈ Spin(W_{0}) ⊂ C(W_{0})_{even}. Suppose first that ϵ^{2} = 1. Then there is an isomorphism of algebras

(b)  j_{ℝ}: C(W_{0}) ⊗_{ℝ} C(U_{0}) → C(V_{0}),

defined by

(c)  j_{ℝ}(a ⊗ b) = \{ \begin{matrix}ab, & if a ∈ C(W_{0}) and b ∈ C(U_{0})_{even}; \\ aϵb, & if a ∈ C(W_{0}) and b ∈ C(U_{0})_{odd}; \end{matrix}

This isomorphism complexifies to

(d)  j_{ℂ}: C(W_{0})_{ℂ} ⊗_{ℂ} C(U_{0})_{ℂ} → C(V_{0})_{ℂ}.

Suppose next that ϵ^{2} = −1. Then there is an isomorphism of complexified algebras as in (d), given by

(e)  j_{ℂ}(a ⊗ b) = \{ \begin{matrix}ab, & if a ∈ C(W_{0})_{ℂ} and b ∈ C(U_{0})_{ℂ,even}; \\ iaϵb, & if a ∈ C(W_{0})_{ℂ} and b ∈ C(U_{0})_{ℂ,odd}; \end{matrix}

Suppose (γ(W_{0}), S(W_{0})) and (γ(U_{0}), S(U_{0})) are complex simple modules for C(W_{0}) and C(U_{0}) (Proposition 7.3 ). Then

(f)  S = S(W_{0}) ⊗ S(U_{0})

is a simple module for C(V_{0}) under the isomorphism j_{ℂ} of (d). In this way the restriction of the spin representation σ(V_{0}) of Spin(V_{0}) to the subgroup Spin(W_{0})Spin(U_{0}) (Definition 7.4 and 7.3(g)) is identified with the tensor product of the spin representations of the factors:

(g)  σ(V_{0})|_{Spin(W_{0})Spin(U_{0})} ≃ σ(W_{0}) ⊗ σ(U_{0}).

Corollary 7.9. In the setting 7.1–7.2, suppose dim V_{0} = 2r +δ, with δ equal to 0 or 1. Choose a collection of r orthogonal two-dimensional subspaces W_{1}, …, W_{r} of V_{0}. Recall from Example 7.6 that each group Spin(W_{i}) is a double cover of the circle group SO(W_{i}), with two-element kernel ±1_{i}:

1 → {±1_{i}} → Spin(W_{i}) → SO(W_{i}) → 1.

Write

T (W_{1}, …, W_{r}) = SO(W_{1}) × ⋯ × SO(W_{r}),
T̃(W_{1}, …, W_{r}) = Spin(W_{1}) ×_{{±1}} ⋯ ×_{{±1}} Spin(W_{r}).

Here the second definition means the quotient of the direct product by the subgroup generated by all elements (−1_{i})(−1_{j}) (1 ≤ i, j ≤ r). This is a double cover of the corresponding product of orthogonal groups:

1 → {±1} → T̃(W_{1}, …, W_{r}) → T (W_{1}, …, W_{r}) → 1.

a) The group T (W_{1}, …, W_{r}) is a maximal torus in SO(V_{0}).

b) The group T̃(W_{1}, …, W_{r}) may be naturally identified as a subgroup of Spin(V_{0}); it is a maximal torus.

c) Write ±β_{i} for the two weights of SO(W_{i}) on (W_{i})_{ℂ}; extend them to all of T (W_{1}, …, W_{r}) by making them trivial on the other factors SO(W_{j}). Then the weights of T (W_{1}, …, W_{r}) on (V_{0})_{ℂ} are {±β_{1}, …, ±β_{r}}, together with 0 if δ = 1.

d) The weights of the spin representation σ(V_{0}) of Spin(V_{0}) are the 2^{r} weights {1/2(±β_{1} ± ⋯ ± β_{r})}.

Corollary 7.10. ([2], Lemma II.6.5.) In the setting 7.1–7.2, choose a spin representation (σ, S) (Definition 7.4 ). Then the tensor product representation σ ⊗ σ of Spin(V_{0}) descends to SO(V_{0}) by the map of 7.2(c). As representations of SO(V_{0}), we have

⋀(V_{0})_{ℂ} ≃ \{ \begin{matrix}σ ⊗ σ & if dim V_{0} is even; \\ σ ⊗ σ + σ ⊗ σ & if dim V_{0} is odd. \end{matrix}

Finally we introduce the Dirac operator. In the setting 0.2, recall that our fixed bilinear form ⟨ , ⟩ is positive definite on 𝔭_{0}. We will make the constructions based on 7.1–7.2 with 𝔭_{0} playing the rôle of V_{0}. The adjoint action of K on 𝔭_{0} preserves the form, and so defines a homomorphism

(7.4a)  Ad: K → O(𝔭_{0}).

Of course the identity component K_{0} is mapped to SO(𝔭_{0}). Recall from 7.2(c) and Lemma 7.2 that we also have a spin double cover of SO(𝔭_{0}). This covering pulls back by Ad to a double cover of K_{0}. More precisely, we define

(7.4b)  K̃_{0} = {(k, g) ∈ K_{0} × Spin(𝔭_{0}) | Ad(k) = τ(g)}.

Then projection on the first factor defines a short exact sequence

(7.4c)  1 → {±1} → K̃_{0} → K_{0} → 1.

Similarly, projection on the second factor defines a homomorphism

(7.4d)  Ãd: K̃_{0} → Spin(𝔭_{0}).

Fix now a space of spinors (γ, S) for 𝔭_{0} (Definition 7.4). By composition with Ãd, the spin representation defines a representation

(7.4e)  (σ ∘ Ãd, S)

of K̃_{0} on S. We call this the spin representation of K̃_{0}, and often denote it simply σ.

Suppose now that X is a (𝔤, K)-module. Often we can understand very well the action of K on X, in terms of the Cartan-Weyl theory of finitedimensional representations. What remains, therefore, is to understand the action of 𝔭_{0} on X. The Dirac operator provides a way to compare this action with the action of 𝔭_{0} on S by Clifford multiplication. Because the latter action is also well understood, the problem of understanding X is to some extent reduced to understanding the Dirac operator.

Definition 7.11. Suppose G is a reductive group as in 0.2. Use the notation of 7.4, so that in particular we have chosen a space (γ, S) of spinors for 𝔭_{0}. Suppose that (π, X) is a representation of 𝔤. The Dirac operator for X is an endomorphism D of X ⊗ S, defined as follows. Let {Y_{i}} be a basis of 𝔭_{0}, and {Y^{j}} the dual basis:

(a)  ⟨Y_{i}, Y^{j}⟩ = δ_{ij}.

Then the Dirac operator on X ⊗ S is

(b)  D = ∑_{i} π(Y_{i}) ⊗ γ(Y^{i}).

The construction of a Dirac operator therefore requires a choice of a space of spinors. We saw in Proposition 7.3 that if 𝔭_{0} is odd-dimensional, there are two inequivalent choices for such a space. It is natural to ask how changing the choice affects the Dirac operator. Proposition 7.3 shows that the two choices γ and γ′ may be realized on a common space S, and that they are related by γ(Y) = −γ′(Y) (for Y ∈ 𝔭_{0}). The two Dirac operators D and D′ on X ⊗ S are then related by D = −D′. We will be interested mostly in the eigenvalues of D^{2}; they are of course unaffected by such a change.

Proposition 7.12. (Parthasarathy [14], Lemma 2.5; see also [13], Proposition 3.1, and [2], Lemma II.6.11.)

a) In the setting of Definition 7.11, the operator D is independent of the choice of basis {Y_{i}}.

b) Write Ω_{G} for the Casimir operator of G, and Ω_{K} for the Casimir operator of K ([8], page 209 ). Then

D^{2} = −π(Ω_{G}) ⊗ 1 + (π ⊗ σ)(Ω_{K}) − (⟨ρ, ρ⟩ − ⟨ρ_{c}, ρ_{c}⟩).

Here ρ_{c} is defined in 0.4(e), and ρ is half the sum of any set of positive roots for 𝔤.

c) Suppose that X is a Hermitian representation of 𝔤_{0}. Then D is formally self-adjoint with respect to the tensor product Hermitian structure on X ⊗ S:

⟨Dx, y⟩ = ⟨x, Dy⟩ (x, y ∈ X ⊗ S).

d) Suppose X is a (𝔤, K)-module. Then X ⊗ S is a locally finite representation of K̃_{0} for the tensor product action π ⊗ σ. The Dirac operator D intertwines this action.

e) Suppose X is a unitary (𝔤, K)-module, and that π(Ω_{G}) is a scalar operator. Then D^{2} has non-negative eigenvalues on X ⊗ S. In particular, suppose that X has an infinitesimal character corresponding by the Harish-Chandra parametrization to a weight ϕ(X) ∈ 𝔥^{∗}. Let μ̃ ∈ (𝔱^{c})^{∗} be the highest weight of any representation of K̃_{0} on X ⊗ S. Then

⟨μ̃ + ρ_{c}, μ̃ + ρ_{c}⟩ ≥ ⟨ϕ(X), ϕ(X)⟩.

Proof. Part (a) is entirely elementary; the argument is omitted in any of the references mentioned. A version of (b) (essentially for X equal to C^{∞}(G)) is proved in [13]; the observation that Parthasarathy’s argument applies in the present setting appears in all the other references. For part (c), the Hermitian property of π means that the operators π(Y_{i}) are skew-adjoint. By 7.1(e) and 7.3(d), the Clifford multiplication operators γ(Y^{i}) are skew-adjoint. By (b) of Definition 7.11, the Dirac operator D is self-adjoint.

For (d), the first assertion is trivial. For the second, suppose k̃ ∈ K̃_{0}; write k ∈ K_{0} for its image by 7.4(c). Then

(7.5a) (π(k) ⊗ σ(k̃)) D (π(k^{−1}) ⊗ σ(k̃^{−1})) = ∑_{i} π(Ad(k)Y_{i}) ⊗ γ(Ad(k)Y^{i}).

Here we have used for π the compatibility of the group and Lie algebra representations, and for γ the condition in Definition 7.4. Because {Ad(k)Y_{i}} is a basis of 𝔭_{0} and {Ad(k)Y^{i}} is the dual basis (by the K-invariance of our bilinear form), it follows from (a) that the right side of 7.5(a) is D, proving (d).

For (e), part (b) shows that D^{2} is diagonalizable. If x is a non-zero eigenvector of eigenvalue c, then it follows from (c) that

(7.5b)  c = ⟨Dx, Dx⟩/⟨x, x⟩.

Because the form on X is positive, the form on X ⊗S is positive as well; so the denominator in 7.5(b) is positive and the numerator non-negative. Therefore c ≥ 0, as we wished to show. For the last inequality, the eigenvalues of D^{2} are computed by (b). Under the assumptions in (e), we find first of all that

(7.5c)  π(Ω_{G}) = ⟨ϕ(X), ϕ(X)⟩ − ⟨ρ, ρ⟩.

(This standard formula is proved for example in [7], Exercise 4 on page 134.) Let δ̃ be a representation of K̃_{0} of highest weight μ̃; then for the same reason we have

(7.5d)  δ̃(Ω_{K}) = ⟨μ̃ + ρ_{c}, μ̃ + ρ_{c}⟩ − ⟨ρ_{c}, ρ_{c}⟩.

Suppose now that x ∈ X ⊗ S transforms according to the representation δ̃. Then the formulas in (b) and 7.5(c) and 7.5(d) show that x is an eigenvector for D^{2}. The eigenvalue c is equal to (7.5e)

\begin{matrix}− (⟨ϕ(X), ϕ(X)⟩ − ⟨ρ, ρ⟩) + (⟨μ̃ + ρ_{c}, μ̃ + ρ_{c}⟩ − ⟨ρ_{c}, ρ_{c}⟩) − (⟨ρ, ρ⟩ − ⟨ρ_{c}, ρ_{c}⟩) \\ = ⟨μ̃ + ρ_{c}, μ̃ + ρ_{c}⟩ − ⟨ϕ(X), ϕ(X)⟩. \end{matrix}

The non-negativity of c is therefore the last inequality in (e).

The inequality in Proposition 7.12(e) is sufficiently important that we state it separately, in a slightly strengthened form.

Proposition 7.13. (Parthasarathy’s Dirac operator inequality; see [14], Lemma 2.5 ). Suppose G is a reductive group as in 0.2, and X is an irreducible Hermitian (𝔤, K)-module. Suppose that X has infinitesimal character corresponding by the Harish-Chandra parametrization to a weight ϕ(X) ∈ 𝔥^{∗}. Recall from Definition 5.6 the canonical real part RE ϕ(X) ∈ i𝔱_{0}^{∗} + 𝔞_{0}^{∗}. Fix a space S of spinors for 𝔭_{0}, and let μ̃ be the highest weight of a representation δ̃ of K̃_{0} occurring in X ⊗ S.

List the representations of K_{0} occurring in δ̃⊗ σ as {δ_{1}, …, δ_{r}}. Assume that

the Hermitian form on X is positive on the K_{0}-types δ_{i}.

(This is automatic if X is unitary.) Then

⟨μ̃ + ρ_{c}, μ̃ + ρ_{c}⟩ ≥ ⟨RE ϕ(X), RE ϕ(X)⟩.

The improvement over the inequality in Proposition 7.12(e) takes two forms. First, the positivity of the Hermitian form on X is assumed only on a specific finite set of K_{0}-types, and not on the entire representation. Second, we have replaced ϕ(X) by its canonical real part. To see that this is an improvement, write ϕ(X) = RE ϕ(X) + iIM ϕ(X). Then the squared length of ϕ(X) is

(7.6a) ⟨RE ϕ(X), RE ϕ(X)⟩ − ⟨IM ϕ(X), IM ϕ(X)⟩ + 2i⟨RE ϕ(X), IM ϕ(X)⟩.

The inner product is real and positive definite on the image of RE and IM; so the first term here is positive, the second negative, and the third purely imaginary. On a Hermitian representation the Casimir operator must have a real eigenvalue, so the third term is zero; and we get

(7.6b)  ⟨ϕ(X), ϕ(X)⟩ ≤ ⟨RE ϕ(X), RE ϕ(X)⟩.

This says that the inequality in Proposition 7.13 implies the one in Proposition 7.12(e).

Proof. Write X_{0} for the K_{0}-invariant subspace of X generated by the isotypic subspaces of type δ_{i}. Suppose x ∈ X⊗S is a non-zero vector transforming according to the representation δ̃. Then it follows from the definition of {δ_{i}} that x ∈ X_{0} ⊗ S. By Proposition 7.12(d), Dx also transforms according to δ̃, so Dx ∈ X_{0} ⊗ S. By hypothesis, our Hermitian form is positive definite on X_{0} ⊗ S; so

(7.7a)  ⟨Dx, Dx⟩/⟨x, x⟩ ≥ 0.

Now the same calculation as in 7.5(b) and 7.5(e) shows that

(7.7b)  ⟨μ̃ + ρ_{c}, μ̃ + ρ_{c}⟩ ≥ ⟨ϕ(X), ϕ(X)⟩.

To get the canonical real part into the picture, we use Theorem 16.10 in [8]. The result is of sufficient interest to state separately.

Proposition 7.14. ([8], Theorem 16.10.) Suppose G is a real reductive group in Harish-Chandra’s class, and (X, ⟨ , ⟩) is an irreducible Hermitian (𝔤, K)-module. Suppose H is a θ-stable Cartan subgroup of G, and that the infinitesimal character of X is given in the Harish-Chandra parametrization by a weight ϕ(X) ∈ 𝔥^{∗}. Then there exist a parabolic subgroup P_{1} = M_{1}A_{1}N_{1} of G, an irreducible Hermitian Harish-Chandra module X_{1} for M_{1}, and a unitary character ν_{1} ∈ i𝔞_{1,0}^{∗}, with the following three properties.

a) The infinitesimal character of X_{1} has a Harish-Chandra parameter that is conjugate by Ad(𝔤) to RE ϕ(X).

b) The weight ν_{1} is conjugate by Ad(𝔤) to iIM ϕ(X).

c) There is an isomorphism of Hermitian representations

X ≃ Ind_{M_{1}A_{1}N_{1}}^{G}(X_{1} ⊗ e^{ν_{1}} ⊗ 1).

Consider the Hermitian representation

X′ = Ind_{M_{1}A_{1}N_{1}}^{G}(X_{1} ⊗ 1 ⊗ 1);

this is a (𝔤, K)-module of finite length.

d) The representations X and X′ are isomorphic as Hermitian representations of K. In particular, their signature characters coincide (Definition 5.3 ).

e) The infinitesimal character of X′ is just that of X_{1} ⊗ 1 composed with a Harish-Chandra homomorphism, and is therefore given in the Harish-Chandra parametrization by the weight

ϕ(X′) = RE ϕ(X).

Proof. Parts (a)–(c) are essentially proved in [8], Theorem 16.10. Part (d) follows immediately from the construction of the Hermitian form on a parabolically induced representation in the “compact picture.” Part (e) is a standard calculation of infinitesimal characters of induced representations ([8], Proposition 8.22).

Returning to the proof of Proposition 7.13, we now apply Proposition 7.14 to the representation X. By (d), the representation X′ inherits the positivity assumption made on the K-types of X. We may also identify the element x with an element of X′ ⊗ S, still transforming by δ̃. Applying the inequality (8.23)(b) to the representation X′ therefore gives the conclusion we want.

Now we will relate these ideas to the notion of unitarily small K-types. Here is a classical fact that hints at what we want.

Proposition 7.15. Suppose G is a connected reductive group as in 0.2, and (σ, S) is a spin representation for 𝔭_{0} (a representation of K̃_{0}; see 7.4 ). Suppose δ is a finite-dimensional irreducible representation of K. Then the following conditions on δ are equivalent.

a) The representation δ occurs in ⋀ 𝔭. b) There is an irreducible constituent σ_{1} of σ so that δ⊗σ_{1} contains a K̃_{0}-type

in S.

c) There are irreducible constituents σ_{1} and σ_{2} of σ so that δ occurs in σ_{1}⊗σ_{2}.

This is an easy consequence of Corollary 7.10; we omit the argument. By Proposition 7.1, a representation δ satisfying these conditions is unitarily small. What we want to do is weaken the conditions until they are precisely equivalent to unitarily small. We begin with some properties of the spin representation.

Lemma 7.16. ([2], Lemma II.6.9 ). Suppose G is a is a connected reductive group as in 0.2, and (σ, S) is a spin representation for 𝔭_{0} (a representation of K̃_{0}; see 7.4 ).

a) The representation σ is of type \frac{1}{2}Δ(𝔭, 𝔱^{c}) (Lemma 6.11 ). That is, every weight of σ is a sum of noncompact roots with coefficients between −1/2 and 1/2.

b) Suppose Δ^{+}(𝔤, 𝔱^{c}) is a system of positive roots containing Δ^{+}(𝔨, 𝔱^{c}). Then

ρ_{n} = 1/2 ∑_{β∈Δ^{+}(𝔭,𝔱^{c})} β

is a highest weight of σ. Similarly −ρ_{n} is a lowest weight of σ.

c) Every highest weight of σ is of the form in (b).

Proposition 7.17. Suppose G is a semisimple group as in 0.2, and (σ, S) is a spin representation for 𝔭_{0}. Suppose δ is an irreducible representation of K. In the notation of Definition 6.1, the following conditions on δ are equivalent.

a) The representation δ is unitarily small.

b) There is an irreducible constituent σ_{1} of σ and an irreducible constituent σ_{2} of δ ⊗ σ_{1} so that σ_{2} is of type \frac{1}{2}Δ(𝔭, 𝔱_{c}) (Lemma 6.11 ). That is, every weight of σ_{2} is a sum of noncompact roots with coefficients between −1/2 and 1/2.

c) There is an irreducible constituent σ_{1} of σ and an irreducible representation σ_{2} of type \frac{1}{2}Δ(𝔭, 𝔱_{c}), with the property that δ|_{K_{0}} contains an irreducible constituent of σ_{1} ⊗ σ_{2}.

d) There are irreducible representations σ_{1} and σ_{2} of K̃_{0}, of type \frac{1}{2}Δ(𝔭, 𝔱_{c}), with the property that δ|_{K_{0}} contains an irreducible constituent of σ_{1} ⊗ σ_{2}.

Proof. We will prove that (a)⇒(b)⇒(c)⇒(d)⇒(a). So suppose (a) holds. Let μ ∈ i(𝔱_{0}^{c})^{∗} be a highest weight of δ. By Theorem 6.7(f), we can find a positive root system Δ^{+}(𝔤, 𝔱^{c}) so that

μ = ∑_{β∈Δ^{+}(𝔭,𝔱^{c})} c_{β}β (0 ≤ c_{β} ≤ 1).

Write δ_{0} for an irreducible constituent of δ (as a representation of K_{0}) of highest weight μ, and σ_{1} for an irreducible constituent of σ of lowest weight −ρ_{n} (Lemma 7.16(b)). By a theorem of Parthasarathy, Rao, and Varadarajan ([12], Corollary 1 to Theorem 2.1), there is an irreducible constituent σ_{2} of δ_{0} ⊗ σ_{1} of extremal weight

μ − ρ_{n} = ∑_{β∈Δ^{+}(𝔭,𝔱^{c})}(c_{β} − 1/2)β (−1/2 ≤ c_{β} − 1/2 ≤ 1/2).

By Lemma 6.11, σ_{2} is of type \frac{1}{2}Δ(𝔭, 𝔱_{c}). This is (b). The implication (b)⇒(c) is formal from the self-duality of S. That (c)⇒(d) is a formal consequence of Lemma 7.16(a). Finally, assume (d). The weights of σ_{1} ⊗σ_{2} are all of the form γ_{1} + γ_{2}, with γ_{i} a weight of σ_{i}. It follows immediately that σ_{1} ⊗ σ_{2} is of type Δ(𝔭, 𝔱_{c}). Now (a) follows from Theorem 6.7(g).

We can now prove a result in the direction of Conjecture 5.7; more precisely, of Conjecture 5.7′.

Proposition 7.18. Suppose G is a semisimple group, and suppose X ∈ Π_{h}^{0}(G) is an irreducible Hermitian (𝔤, K)-module (Definition 5.1 ). Let 𝔥 be a θ-stable Cartan subalgebra of 𝔤, and ϕ ∈ 𝔥^{∗} a weight parametrizing the infinitesimal character of X. Assume that

⟨RE ϕ, RE ϕ⟩ > ⟨ρ, ρ⟩.

Then the Hermitian form on X must be indefinite on K-types in B_{u}^{0}(G).

Proof. Suppose not. Then we may assume that the form is positive definite on the K-types in B_{u}^{0}(G). By hypothesis X contains a unitarily small K-type δ ∈ B_{u}^{0}(G). By Proposition 7.17, we can find a representation σ_{2} of type \frac{1}{2}Δ(𝔭, 𝔱^{c}) occurring in δ ⊗ σ, and therefore in X ⊗ S. We are going to apply Proposition 7.13. We therefore list the representations of K containing constituents of σ_{2} ⊗ σ as {δ_{1}, …, δ_{r}}. By Proposition 7.17(d), these are all unitarily small; so by hypothesis the Hermitian form on X is positive on them. Write μ_{2} for the highest weight of σ_{2}. According to Proposition 7.13,

(7.8a)  ⟨μ_{2} + ρ_{c}, μ_{2} + ρ_{c}⟩ ≥ ⟨RE ϕ, RE ϕ⟩.

Fix a positive root system Δ^{+}(𝔤, 𝔱^{c}) containing Δ^{+}(𝔨, 𝔱^{c}), and write ρ_{n} for half the sum of the corresponding noncompact positive roots. Because σ_{2} is of type \frac{1}{2}Δ(𝔭, 𝔱^{c}), we can write

(7.8b)  \begin{aligned}μ_{2} & = ∑_{β∈Δ^{+}(𝔭,𝔱^{c})}(c_{β} − c_{−β})β  (0 ≤ c_{±β} ≤ 1/2) \\ & = ∑_{β∈Δ^{+}(𝔭,𝔱^{c})} a_{β}β  (−1/2 ≤ a_{β} ≤ 1/2) \\ & = ρ_{n} − ∑_{β∈Δ^{+}(𝔭,𝔱^{c})} b_{β}β  (0 ≤ b_{β} ≤ 1). \end{aligned}

Adding ρ_{c} gives

(7.8c)  μ_{2} + ρ_{c} = ρ − ∑_{β∈Δ^{+}(𝔭,𝔱^{c})} b_{β}β  (0 ≤ b_{β} ≤ 1).

By Proposition 1.10, it follows that μ_{2} + ρ_{c} belongs to the convex hull of the Weyl group orbit of ρ. Consequently

(7.8d)  ⟨μ_{2} + ρ_{c}, μ_{2} + ρ_{c}⟩ ≤ ⟨ρ, ρ⟩.

Combining this with 7.8(a) gives ⟨ρ, ρ⟩ ≥ ⟨RE ϕ, RE ϕ⟩, contradicting the hypothesis in the proposition. This contradiction completes the proof.

This argument seems to us to be rather natural and powerful; so it is disappointing that we cannot deduce Conjecture 5.7′ from it. Perhaps the problem lies in Proposition 7.13. The following sharpening of that proposition would (by the argument above) immediately imply Conjecture 5.7′.

Conjecture 7.19. Suppose we are in the setting of Proposition 7.13; make the same hypotheses as for that proposition. Then RE ϕ(X) is conjugate by Ad(𝔤) to a weight in the convex hull of the Weyl group orbit of μ̃ + ρ_{c}.

We are being a little bit vague about the Weyl group in this conjecture; probably it is enough to use the Weyl group W (𝔤, 𝔱^{c}) of the restricted root system, but the conjecture would still imply Conjecture 5.7′ even if we take the full Weyl group W (𝔤, 𝔥^{c}).

8. Topology of the unitary dual

In this section we consider the relationship between the parametrization of the unitary dual given (conjecturally) by Theorem 5.8 and the Fell topology. A careful treatment of this topic would require a careful foundational discussion of the Fell topology, which we prefer to avoid. Our arguments will therefore be sketchy at several crucial points.

We begin with a discussion of infinitesimal characters. Fix a θ-stable positive root system Δ^{+}(𝔤, 𝔥^{c}) for the Cartan subalgebra 𝔥^{c} of 0.4(a); this choice is equivalent to the choice of a system Δ^{+}(𝔤, 𝔱^{c}) of positive restricted roots. We may assume that Δ^{+}(𝔤, 𝔱^{c}) contains our fixed system Δ^{+}(𝔨, 𝔱^{c}) of positive roots for 𝔨.

Definition 8.1. With notation as above, suppose ϕ ∈ (𝔥^{c})^{∗} is a weight. We say that ϕ is complex dominant if it satisfies either of the following equivalent conditions.

a) The canonical real part RE ϕ (Definition 5.6) is dominant for Δ^{+}(𝔤, 𝔥^{c}); and the canonical imaginary part IM ϕ is dominant for those positive roots orthogonal to RE ϕ.

b) For every positive root α ∈ Δ^{+}(𝔤, 𝔥^{c}), we have either Re ⟨ϕ, α⟩ > 0; or Re ⟨ϕ, α⟩ = 0, and Im ⟨ϕ, α⟩ > 0; or ⟨ϕ, α⟩ = 0. The set of complex dominant weights is denoted

C_{ℂ}(Δ^{+}(𝔤, 𝔥^{c})) ⊂ (𝔥^{c})^{∗}.

It is a fundamental domain for the action of W (𝔤, 𝔥^{c}) on (𝔥^{c})^{∗}:

C_{ℂ}(Δ^{+}(𝔤, 𝔥^{c})) ≃ (𝔥^{c})^{∗}/W (𝔤, 𝔥^{c}).

We endow C_{ℂ}(Δ^{+}(𝔤, 𝔥^{c})) with the quotient topology from this bijection (and not with the subspace topology from 𝔥^{c}, which has more neighborhoods of purely imaginary elements).

The ordinary positive Weyl chamber C_{ℝ}(Δ^{+}(𝔤, 𝔥^{c})) consisting of dominant elements of (i𝔱_{0}^{c} + 𝔞_{0}^{c})^{∗} is a subset of C_{ℂ}(Δ^{+}(𝔤, 𝔥^{c})). We will use the fact that the topology on C_{ℝ} coming from this inclusion (that, is the quotient topology from

C_{ℝ}(Δ^{+}(𝔤, 𝔥^{c})) ≃ (i𝔱_{0}^{c} + 𝔞_{0}^{c})^{∗}/W (𝔤, 𝔥^{c})

coincides with the subspace topology from the inclusion

C_{ℝ}(Δ^{+}(𝔤, 𝔥^{c})) ⊂ (i𝔱_{0}^{c} + 𝔞_{0}^{c})^{∗}.

We now write

(8.1a)  ℨ(𝔤) = center of U(𝔤).

The Harish-Chandra isomorphism is

(8.1b)  ℨ(𝔤) ≃ S(𝔥^{c})^{W (𝔤,𝔥^{c})}.

An infinitesimal character is by definition a homomorphism from ℨ(𝔤) to the complex numbers, or equivalently a maximal ideal in ℨ(𝔤). By analogy with the notation we are using for representations, the set of infinitesimal characters is written

(8.1c)  Π(ℨ(𝔤)) = Max ℨ(𝔤).

The isomorphism 8.1(b) identifies these infinitesimal characters with W orbits on 𝔥^{∗}:

(8.1d)  Π(ℨ(𝔤)) ≃ (𝔥^{c})^{∗}/W (𝔤, 𝔥^{c}) ≃ C_{ℂ}(Δ^{+}(𝔤, 𝔥^{c}));

the last equivalence comes from Definition 8.1. We will describe this map a little more carefully in 8.2 below.

The set of infinitesimal characters carries a “Fell topology” of its own: the weakest topology making each of the evaluation maps

(8.2a)  e_{z}: Π(ℨ(𝔤)) → ℂ,  e_{z}(χ) = χ(z)  (z ∈ ℨ(𝔤))

continuous. (We use the usual analytic topology on ℂ; if instead we used the Zariski topology, we would get the restriction to Max ℨ(𝔤) of the Zariski topology on Spec ℨ(𝔤).) We can restate the definition of the topology as follows. A set U ⊂ Π(ℨ(𝔤)) is a neighborhood of the element χ_{0} ∈ U if and only if there are elements z_{1}, …, z_{n} ∈ ℨ(𝔤) and positive numbers ϵ_{1}, …, ϵ_{n} with the property that U contains

(8.2b) N(z, ϵ; χ_{0}) = {χ ∈ Π(ℨ(𝔤)) | |χ(z_{i}) − χ_{0}(z_{i})| < ϵ_{i} (i = 1, …, n)}.

Now Chevalley’s theorem says that ℨ(𝔤) is isomorphic to a polynomial ring ℂ[x_{1}, …, x_{m}]. (The elements x_{1}, …, x_{m} may be thought of as generators of ℨ(𝔤); by the Harish-Chandra isomorphism 8.1(b), they correspond to generators for the algebra of W -invariants in S(𝔥^{c}).) Each element ξ = (ξ_{1}, …, ξ_{m}) ∈ ℂ^{m} defines an element χ_{ξ} ∈ Π(ℨ(𝔤)); the characteristic property is χ_{ξ}(x_{i}) = ξ_{i}. By the Nullstellensatz, this correspondence gives an isomorphism

(8.2c)  Π(ℨ(𝔤)) ≃ ℂ^{m}.

It is very easy to check (using the description of neighborhoods in 8.2(b)) that this isomorphism sends the topology on Π(ℨ(𝔤)) to the standard topology on ℂ^{m}. We want to know something slightly more subtle, however. We may identify S(𝔥^{c}) with the algebra of polynomial functions on (𝔥^{c})^{∗}. Each λ ∈ (𝔥^{c})^{∗} therefore defines an algebra homomorphism

(8.2d)  ξ_{λ}: S(𝔥^{c}) → ℂ,  p ↦ p(λ).

We use the same symbol to denote the composition with the Harish-Chandra homomorphism:

(8.2e)  ξ_{λ}: ℨ(𝔤) → ℂ.

In this way we get a map

(8.2f)  (𝔥^{c})^{∗} → Π(ℨ(𝔤)),  λ ↦ ξ_{λ}.

This is the map of 8.1(d): it is surjective, and the fibers are precisely the orbits of W (𝔤, 𝔥^{c}).

It is more or less obvious from the definitions that the map of 8.2(f) is continuous (from the standard topology on (𝔥^{c})^{∗} to the “Fell topology” on Π(ℨ(𝔤))). What we need is the stronger assertion that the topology on Π(ℨ(𝔤)) is actually the quotient topology; equivalently, that the map 8.2(f) is open. Here is an explicit formulation.

Lemma 8.2. In the setting of 8.2, suppose (p_{1}, …, p_{m}) is a collection of generators for S(𝔥^{c})^{W (𝔤,𝔥^{c})}. Fix λ_{0} ∈ (𝔥^{c})^{∗} and ϵ > 0. Then we can find a positive number δ so that if λ ∈ (𝔥^{c})^{∗} and

|p_{i}(λ) − p_{i}(λ_{0})| < δ (i = 1, …, m),

then there is a w ∈ W (𝔤, 𝔥^{c}) such that

|λ − w · λ_{0}| < ϵ.

In the case of GL(n), this lemma in turn can be reformulated as follows. Suppose f_{0} is a monic polynomial of degree n with roots (λ_{0}^{1}, …, λ_{0}^{n}), and ϵ > 0. Then there is a δ > 0 so that if f is a monic polynomial whose coefficients are within δ of those of f_{0}, then the roots (λ^{1}, …, λ^{n}) of f are each (after permutation) within ϵ of the corresponding roots of f_{0}. That is, it says that the collection of roots of a polynomial depends continuously on the coefficients. This is of course well known. In keeping with the incomplete nature of this section, we omit the argument for the general case.

Corollary 8.3. The bijection of 8.1(d) is a homeomorphism from the “Fell topology” to the quotient topology from (𝔥^{c})^{∗}.

Definition 8.4. The infinitesimal character map

Π_{u}(G) → Π(ℨ(𝔤)), π ↦ ξ(π)

sends an irreducible unitary representation π to the homomorphism ξ(π) by which ℨ(𝔤) acts on the space of smooth vectors of π.

Theorem 8.5. (Bernat-Dixmier [1].) The infinitesimal character map is continuous from the Fell topology on the unitary dual to the “Fell topology” on Π(ℨ(𝔤)) (cf. 8.2 ).

Corollary 8.6. The map

Π_{u}(G) → C_{ℂ}(Δ^{+}), π ↦ ϕ(π; Δ^{+})

(obtained by composing the infinitesimal character map of Theorem 8.5 with the isomorphisms of 8.1(d)) is continuous from the Fell topology on the unitary dual to the quotient topology (Definition 8.1 ) on the cone of complex dominant weights.

This is immediate from Theorem 8.5 and Corollary 8.3.

We are now in a position to draw some conclusions about the topology of the unitary dual. A little more notation is helpful. In the setting of Definition 8.1, suppose λ ∈ i(𝔱_{0}^{c})^{∗}. Define D_{Δ^{+}}(λ) to be the unique W (𝔤, 𝔱^{c}) conjugate of λ belonging to the positive Weyl chamber C of 0.5(c). (Notice that C is contained in C_{ℝ}(Δ^{+}).) The map

(8.3a)  i(𝔱_{0}^{c})^{∗} → C,  λ ↦ D_{Δ^{+}}(λ)

is continuous and finite-to-one. On the other hand, taking the canonical real part (Definition 5.6) defines a continuous map

(8.3b)  (𝔥^{c})^{∗} → i(𝔱_{0}^{c})^{∗} + (𝔞_{0}^{c})^{∗},  ϕ ↦ RE ϕ.

This map is W -equivariant, and therefore descends to a continuous map for the quotient topologies

(8.3c)  C_{ℂ}(Δ^{+}) → C_{ℝ}(Δ^{+}),  ϕ ↦ RE ϕ.

Similarly, the map T_{ρ} of Proposition 1.4 is continuous:

(8.3d)  i(𝔱_{0}^{c})^{∗} + (𝔞_{0}^{c})^{∗} → i(𝔱_{0}^{c})^{∗} + (𝔞_{0}^{c})^{∗},  ϕ ↦ T_{ρ}(ϕ).

To see this, recall that T_{ρ} is defined separately on each closed Weyl chamber C′: if ρ′ is half the sum of the positive roots for C′, and P′ is the projection on C′ (Definition 1.2), then

(8.3e)  T_{ρ}(ϕ) = P′(ϕ − ρ′).

This mapping is obviously continuous on C′, because P′ is a contraction. Because C′ is closed, and the definitions agree on the intersections of different chambers (Proposition 1.4), T_{ρ} is continous.

Theorem 8.7. In the setting of Definition 8.1, suppose π is an irreducible unitary representation, and

ϕ(π; Δ^{+}) ∈ C_{ℂ}(Δ^{+}) ⊂ (𝔥^{c})^{∗}

comes from the infinitesimal character as in Corollary 8.6. Define

λ(π; Δ^{+}) = T_{ρ}(RE ϕ(π; Δ^{+}))

(notation as in Proposition 1.4 ).

a) The weight λ(π; Δ^{+}) belongs to the positive Weyl chamber C_{ℝ}(Δ^{+}) ⊂ i(𝔱_{0}^{c})^{∗} + (𝔞_{0}^{c})^{∗}.

b) The map

Π_{u}(G) → C_{ℝ}(Δ^{+}), π ↦ λ(π; Δ^{+})

is continuous from the Fell topology to the standard topology on the cone of dominant weights. c) Assume that Conjecture 5.7 holds, and that π ∈ Π_{u}^{λ_{u}}(G) (Definition 0.2 ).

Then

D_{Δ^{+}}(λ_{u}) = λ(π; Δ^{+})

(notation as in 8.3(a)).

Proof. Because RE ϕ(π; Δ^{+}) is dominant for Δ^{+}(𝔤, 𝔥^{c}), T_{ρ} may be computed by subtracting ρ and projecting on the positive Weyl chamber. This proves (a). Part (b) follows from Corollary 8.6, the continuity of the maps in 8.3(c) and (d), and the identification noted in Definition 8.1 of the quotient and subspace topologies on C_{ℝ}(Δ^{+}). Part (c) is a reformulation of Theorem 5.8(c).

Corollary 8.8. Fix λ_{u} ∈ Λ_{u} (cf. 0.5(e)). Choose representatives λ_{u}^{1}, …, λ_{u}^{r} for the orbits of R(G) on

W (𝔤, 𝔱^{c}) · λ_{u} ∩ Λ_{u}.

Assume that Conjecture 5.7 holds.

a) The set ⋃_{i=1}^{r} Π_{u}^{λ_{u}^{i}}(G) of unitary representations is open and closed in the Fell topology.

b) Suppose i ≠ j, and π ∈ Π_{u}^{λ_{u}^{i}}(G). Then π contains no K-types in B_{u}^{λ_{u}^{j}}(G).

c) Each set Π_{u}^{λ_{u}^{i}}(G) is open and closed in the Fell topology.

d) The mapping

Π_{u}(G) → Λ_{u}/R(G), π ↦ λ_{u}(π)

(notation as in Definition 0.2 ) is continuous from the Fell topology to the natural (discrete) topology on the image.

The statement of this corollary is a little cumbersome; part (c), for example, is obviously stronger than (a). The formulation is chosen to make the proof clearer.

One might expect that (d) should be more or less a formal consequence of the definitions. To see why it is not, consider the analogous map

(8.4)  Π_{u}(G) → Λ_{a}/R(G),  π ↦ λ_{a}(π)

(notation as in 2.1 and Definition 2.5). The definition of λ_{a} is formally very similar to that of λ_{u}; but the mapping in 8.4 is not continuous in the Fell topology. To see why, suppose G = SL(2, ℝ), and let π be the first holomorphic discrete series representation. In the notation of Example 6.2, the lowest Ktype of π is μ_{2}, and λ_{a}(π) = 1. But there is a sequence {π_{j}} of spherical complementary series representations having π as a limit point. (The reason is that π appears as a composition factor in the reducible spherical principal series representation at the end of the complementary series.) Each π_{j} has lowest K-type μ_{0}, so λ_{a}(π_{j}) = 0. So λ_{a}(π_{j}) fails to converge to λ_{a}(π). (On the other hand, Example 6.2 shows that λ_{u}(π) = 0 = λ_{u}(π_{j}), so there is no contradiction to (d) in the corollary.)

Proof. By Theorem 8.7(c), we have a continuous map

Π_{u}(G) → D_{Δ^{+}}(Λ_{u}), π ↦ λ(π, Δ^{+}) = D_{Δ^{+}}(λ_{u}(π)).

The range is discrete because Λ_{u} is (cf. 0.5(e)), and so carries the discrete topology. The set of unitary representations in (a) is just the preimage of D_{Δ^{+}}(λ_{u}), and so must be open and closed. Part (b) is immediate from Lemma 5.5.

For (c), it suffices to prove that each Π_{u}^{λ_{u}^{i}}(G) is closed. So suppose {π_{n}} is a sequence of representations in Π_{u}^{λ_{u}^{i}}(G), and that π is a limit point. By (a), π must belong to some Π_{u}^{λ_{u}^{j}}(G); we want to show that i = j. Suppose not. By Definition 0.2, π must contain a K-type τ in B_{u}^{λ_{u}^{j}}(G). We can therefore form a corresponding non-zero matrix coefficient f_{π}; it transforms on the right and left by τ under K. By the definition of the Fell topology, f_{π} may be uniformly approximated on compact sets by matrix coefficients of the representations π_{n}. By (b), these representations do not contain τ; and it follows easily that their matrix coefficients cannot approximate f_{π}. This contradiction proves (c). Part (d) is immediate from (c).

Corollary 8.9. In the setting of Theorem 5.8, assume that Conjecture 5.7 holds. Then the bijection

L_{S_{u}}(λ_{u}): Π_{u}^{λ_{u}}(G(λ_{u})) → Π_{u}^{λ_{u}}(G)

is a homeomorphism in the Fell topology; the image is open and closed in Π_{u}(G).

Sketch of proof. According to fundamental (and partly unpublished) work of Miličić, the Fell topology may be described in terms of the reducibility of certain parabolically induced representations. To be a little more precise, suppose P = MAN is a parabolic subgroup of G, δ ∈ Π_{a}(M) is an admissible representation, and

(8.5a)  ν: [0, 1] → 𝔞^{∗}

is a continuous map. Define a family of admissible representations by

(8.5b)  I(t) = Ind_{P}^{G}(δ ⊗ ν(t) ⊗ 1),  (t ∈ [0, 1]).

Now we make the following assumption:

(8.5c) For all t ∈ [0, 1), the representation I(t) is irreducible and unitary.

Here by “unitary” we mean “infinitesimally equivalent to a unitary representation.” List the irreducible composition factors of the representation I(1) as I_{1}, I_{2}, …, I_{r}. As a consequence of assumption 8.5(c) and a theorem of Miličić, (see [11]), all of the representations I_{j} are unitary; and they are precisely the limit points of I(t) as t approaches 1:

(8.5d)  lim_{t→1} I(t) = {I_{1}, …, I_{r}}.

Finally, every convergent sequence in the Fell topology has a subsequence constructed in this way.

Using this description of the Fell topology, and the good behavior of L_{S_{u}}(λ_{u}) (in particular Theorem 5.4(e)), it is not difficult to prove the corollary. We omit the details.

We make one final remark. All of the interesting results in this section depend on Conjecture 5.7; it is natural to ask what can be said absolutely. In the setting of Theorems 5.4 and 5.8, we can consider a certain subset

(8.6a)  Π_{u,good}^{λ_{u}}(G(λ_{u})) ⊂ Π_{u}^{λ_{u}}(G(λ_{u}))

consisting of unitary representations for which the infinitesimal character ϕ(Z) satisfies

(8.6b)  RE ϕ(Z) ∈ λ_{u} + {convex hull of W · ρ};

here both the Weyl group and ρ come from G(λ_{u}). The content of Conjecture 5.7 is that this set is all of Π_{u}^{λ_{u}}(G(λ_{u})); but in any case it is a well-defined closed subset. The proof of Theorem 5.8 shows that L_{S_{u}}(λ_{u}) defines a bijection from this set to a closed subset

(8.6c)  Π_{u,good}^{λ_{u}}(G) ⊂ Π_{u}^{λ_{u}}(G)

We define

(8.6d)  Π_{u,good} = ∪_{λ_{u}∈Λ_{u}}Π_{u,good}^{λ_{u}}(G),

a closed subset of Π_{u}(G). Everything we have said about Π_{u}(G) using Conjecture 5.7 applies unconditionally to Π_{u,good}(G).

Mathematical Sciences, New Mexico State University, Las Cruces, NM 88003 Department of Mathematics, Massachusetts Institute of Technology, Cambridge,

MA 02139

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