UNITARY REPRESENTATIONS WITH STRONGLY

REGULAR INFINITESIMAL CHARACTER

Susana A. Salamanca-Riba

Outline

1. Abstract.

2. Introduction.

3. Representations with strongly regular real infinitesimal character.

4. Admissible A_{𝔮}(λ) modules.

5. Classification theorem and proof.

6. Motivation.

1. Abstract

We will sketch the proof of the following result, conjectured by Vogan and Zuckerman:

Theorem. Let G be a reductive real Lie group. Suppose π is an irreducible Harish-Chandra module of G with strongly regular and real infinitesimal character. Then

π is unitary ⟺ π ≅ A_{𝔮}(λ) (admissible).

2. Introduction

Let G be a reductive real Lie group. We would like to understand the set Π_{u}(G) of equivalence classes of irreducible unitary representations of G. Our approach is motivated by the philosophy of the orbit method: there should be a close connection between Π_{u}(G) and the coadjoint orbits of G. Roughly, we should be able to attach an element π ∈ Π_{u}(G) to each coadjoint orbit O of G . The map O ↦ π will not be quite a bijection.

For an element X of the complexified Lie algebra 𝔤 of G, let O_{X} denote the coadjoint orbit Ad^{∗}(G)ξ, where ξ is the element of 𝔤^{∗} corresponding to X via the Killing form. Decompose X into semisimple and nilpotent parts X = X_{s}+X_{n}, with X_{s} further decomposable into hyperbolic and elliptic elements X_{s} = X_{h} + X_{e}. Let G_{X} be the isotropy group of X. Since the elements X_{h}, X_{e}, and X_{n} all commute, we have that

G_{X} = (G_{X_{s}})_{X_{n}} ⊂ G_{X_{s}} = (G_{X_{h}})_{X_{e}} ⊂ G_{X_{h}} ⊂ G,

and one can try to attach a representation to the orbit O_{X} by going in steps, from smaller to larger groups.

Typeset by AMS-TEX

Example. Let G = GL(2n, ℝ). An elliptic orbit O_{X_{e}} corresponds to an element Y ∈ ℝ^{n} of the form

Y = (\underbrace{a_{1}, …, a_{1}}_{d_{1}}, \underbrace{a_{2}, …, a_{2}}_{d_{2}}, …, \underbrace{a_{r}, …, a_{r}}_{d_{r}}, \underbrace{0, …, 0}_{m}),

with a_{1} > a_{2} > ⋯ > a_{r} > 0 and m = n − ∑_{i=1}^{r} d_{i}. If i(X_{e}) ↔ Y as above then

G_{X_{e}} ≅ GL(d_{1}, ℂ) × ⋯ × GL(d_{r}, ℂ) × GL (2m, ℝ) .

The idea is to construct π in three steps:

(1) Attach to O_{X_{n}} a unipotent representation of G_{X_{s}}: O_{X_{n}} ↦ π_{n}. This process is not completely understood.

(2) Assuming we have a representation π_{n} of G_{X_{s}}, use cohomological parabolic induction to attach O_{X_{e}} ↦ π_{e}, a representation of G_{X_{h}}, induced from π_{n}. This method is well understood.

(3) Use π_{e} and real parabolic induction to attach O_{X_{h}} ↦ π_{h}, a representation of G induced from π_{e}.

For step (2), suppose X = X_{e}. Recall that ad(iX_{e}) has real eigenvalues. Let 𝔤 = ∑_{t∈ℝ} 𝔤_{t} be the corresponding eigenspace decomposition of 𝔤. Then the 0eigenspace is 𝔤_{e} = Lie(G_{X_{e}}), and the Lie algebra 𝔮_{e} = ∑_{t≥0} 𝔤_{t} = 𝔤_{e} + 𝔲_{e} (where 𝔲_{e} = ∑_{t>0} 𝔤_{t}) is a complex polarization at X. To obtain a representation of G, construct the Zuckerman derived functor module R_{𝔮_{e}}^{S}(π_{n}). As a special case we assume that π_{n} = ℂ_{λ}, a one-dimensional representation of G_{X_{e}}.

Examples.

(1) X_{e} = 0, G_{X_{e}} = G, and π_{n} = π = ℂ_{λ}.

(2) G_{X_{e}} is compact ⇔ π is a discrete series representation.

Conversely, a ‘backwards’ map should involve the following steps:

(1) Given π, find a pair (M, π_{M}), where M is a real Levi subgroup of G and π_{M} is a unitary representation of M with real infinitesimal character such that π ≅ Ind_{M}^{G}(π_{M}).

(2) Given (M, π_{M}), find a pair (L, π_{L}), where L = G_{L} is a Levi subgroup for a complex parabolic subalgebra 𝔮 (usually bigger than G_{X_{s}}), and π_{L} is a unitary representation of L such that π_{M} ≅ R_{𝔮}(π_{L}).

(3) Give a parametrization of all the pairs (L, π_{L}) in step (2). We will be concerned here with a special case of step (2).

3. Representations with strongly regular real infinitesimal character

We use the usual notation: K is a maximal compact subgroup of G, T ⊂ K a maximal compact torus; 𝔤_{0} is the real Lie algebra of G, 𝔤 = (𝔤_{0})_{ℂ} the complexification, etc. Let 𝔤 = 𝔨+𝔭 be the Cartan decomposition of 𝔤, and 𝔥 = 𝔱+𝔞 a maximally compact Cartan subalgebra of 𝔤. Denote by Δ(𝔤, 𝔱) the roots of 𝔤 with respect to 𝔱, and define Δ(𝔤, 𝔥), Δ(𝔨, 𝔱), etc. analogously. As usual, fix a non-degenerate G- and θ-invariant symmetric bilinear form (,) on 𝔤_{0}, negative definite on 𝔨 and positive definite on 𝔭, and denote its complexification (,) as well.

Let (π, H) be an irreducible unitary representation of G, X_{π} its Harish-Chandra module, and let the infinitesimal character of π be given by the weight γ ∈ 𝔥^{∗}. Fix a positive root system Δ^{+}(γ, 𝔥) ⊂ {α ∈ Δ(𝔤, 𝔥) : Re(γ, α) ≥ 0}, and let ρ(γ) be one half the sum of the roots in Δ^{+}(𝔤, 𝔥). Recall (see Vogan’s talk) that γ ∈ 𝔥^{∗} is called real if γ ∈ 𝔥_{ℝ}^{∗}, where 𝔥_{ℝ}^{∗} = i𝔱_{0}^{∗} + 𝔞_{0}^{∗} ⊂ 𝔥^{∗}.

Definition. The set Π_{u}^{sr}(G) is the set of equivalence classes of irreducible unitarizable Harish-Chandra modules with infinitesimal character γ satisfying

(i) γ is real, and (ii)(γ − ρ(γ), α) ≥ 0 for all positive roots α in Δ^{+}. We say that γ is strongly regular if (ii) holds.

Examples.

(1) If G = SL(2, ℝ) then Π_{u}^{sr}(G) consists of the discrete series and the trivial representation.

(2) For G = SU(2, 1), the set of strongly regular real infinitesimal characters forms a closed cone in the Weyl chamber of ρ, with vertex ρ.

4. Admissible A_{𝔮}(λ) modules

We now describe the modules which will parametrize the strongly regular set. Fix a positive system Δ^{+}(𝔨, 𝔱) and a weight ξ ∈ i𝔱_{0}^{∗}, and let 𝔮(ξ) be the θ-stable parabolic subalgebra of 𝔤 defined by ξ, i.e.,

𝔮(ξ) = 𝔤(ξ) + 𝔲(ξ) with
𝔤(ξ) = 𝔥 + ∑_{(α,ξ)=0} 𝔤_{α} and
𝔲(ξ) = ∑_{(α,ξ)>0} 𝔤_{α}.

Example. If G = GL(2n, ℝ) and ξ ↔ Y ∈ ℝ^{n} as in the introduction, with

Y = (\underbrace{a_{1}, …, a_{1}}_{d_{1}}, \underbrace{a_{2}, …, a_{2}}_{d_{2}}, …, \underbrace{a_{r}, …, a_{r}}_{d_{r}}, \underbrace{0, …, 0}_{m}),

then

𝔤(ξ) ≅ 𝔤𝔩(d_{1}, ℂ) × ⋯ × 𝔤𝔩(d_{r}, ℂ) × 𝔤𝔩 (2m, ℝ) .

If G(ξ) is the Levi subgroup of G corresponding to 𝔤(ξ), let λ : 𝔤(ξ) ⟶ ℂ_{λ} be a linear functional.

Definition. λ and 𝔮(ξ) are admissible if

(i) ξ is dominant for Δ^{+}(𝔨, 𝔱), (ii) λ is the differential of a unitary character of G(ξ), and (iii) (α, λ ↾_{𝔱}) ≥ 0 for all α ∈ Δ(𝔲(ξ), 𝔱).

Given such a pair (𝔤(ξ), λ), construct the Zuckerman module

A_{𝔮(ξ)}(λ) = R_{𝔮(ξ)}^{S}(ℂ_{λ})

with S = dim(𝔲(ξ)∩𝔨) and so that this module has infinitesimal character λ+ρ_{𝔮(ξ)}.

Examples. Let γ = ρ and λ = 0, and write 𝔮 = 𝔩 + 𝔲.

(1) If 𝔩 = 𝔮 = 𝔤 then A_{𝔮} is the trivial representation of G.

(2) We have that 𝔩 ⊂ 𝔨 ⇔ A_{𝔮}(0) is a discrete series representation.

Theorem. (Vogan, Speh) Let 𝔮 = 𝔩 + 𝔲 ⊂ 𝔤 be a θ-stable subalgebra, and suppose λ : 𝔩 ⟶ ℂ is admissible. Then A_{𝔮}(λ) is irreducible, nonzero, and unitarizable.

The proof of irreducibility and nonvanishing is due to Speh and Vogan, unitarizability was proved by Vogan.

5. Classification theorem and proof

Now we sketch the proof of our Main Classification Theorem:

Theorem A.

Π_{u}^{sr}(G) ≅ {admissible A_{𝔮}(λ) modules}.

Because of the results of Speh and Vogan, this theorem will follow from

Theorem B. (SR) If π ∈ Π_{u}^{sr}(G) then π is isomorphic to an A_{𝔮}(λ) module.

Proof. (outline) We use the following two facts:

Fact 1. (Zuckerman) Every A_{𝔮}(λ) module has a unique lowest K-type (LKT) of highest weight

μ(𝔮, λ) = λ ↾_{𝔱} +2ρ(𝔲 ∩ 𝔭).

Examples. Let G = SU(1, 1) and γ = ρ.

(a) If 𝔮 = 𝔩 = 𝔤 then μ(𝔮, 0) = (0, 0) (trivial representation).

(b) If 𝔮 = 𝔱 + 𝔫 then μ(𝔮, 0) = (1, −1) (holomorphic discrete series).

(c) If 𝔮̄ = 𝔱 + 𝔫̄ then μ(𝔮̄, 0) = (−1, 1) (antiholomorphic discrete series).

Fact 2. (Vogan, Zuckerman) If X ∈ Π_{u}^{sr}(G), has LKT μ = μ(𝔮, λ) for λ and 𝔮 admissible, and if λ = 0 on [𝔩, 𝔩], then X ≅ A_{𝔮}(λ).

Using these two results, Theorem B follows from the proof of the following

Proposition C. If X has strongly regular real infinitesimal character and LKT μ, then either

(a) X is not unitarizable, or

(b) there are admissible λ and 𝔮 with μ = μ(𝔮, λ), in which case X ≅ A_{𝔮}(λ).

In order to find suitable 𝔮 and λ for a given μ, we define the following unitary data. (This machinery was developed with David Vogan.) Given a dominant weight μ ∈ i𝔱_{0}^{∗}, set 2ρ_{c} = 2ρ(𝔨) and choose a set of positive roots Δ^{+}(𝔤, 𝔱), with respect to which μ + 2ρ_{c} is dominant. Set 2ρ = ∑_{α∈Δ^{+}(𝔤,𝔱)} α, and

λ_{u} = P (μ + 2ρ_{c} − 2ρ) ∈ i𝔱_{0}^{∗},

where P is the projection onto the convex cone

C = {χ ∈ i𝔱_{0}^{∗} : (χ, α) ≥ 0, for all positive α}.

Now let 𝔮_{u} = 𝔮_{u}(λ_{u}) = 𝔤_{u} + 𝔲_{u}, and G_{u} = G(λ_{u}).

To motivate this definition, recall that the highest weight of the LKT of a discrete series X with Harish-Chandra parameter λ is given by the Blattner parameter μ = λ + ρ − ρ_{c}. We can recover λ by finding a suitable ρ and solving for λ:

λ = μ + 2ρ_{c} − ρ.

For his classification of admissible representations, Vogan used this idea to attach to the highest weight μ of a LKT of an admissible Harish-Chandra module the dominant weight λ_{a}(μ), which (using an observation of Carmona) may be computed as follows:

λ_{a}(μ) = P (μ + 2ρ_{c} − ρ).

(The projection is necessary because for the LKT of an arbitrary admissible Harish-Chandra module, the weight μ + 2ρ_{c} − ρ is not always dominant.)

Example. Let G = Sp(4, ℝ), μ = (a, 2), a > 0. Then μ + 2ρ_{c} − 2ρ = (a − 3, −1). If a = 3 then λ_{u} = (0, 0), and G_{u} = G. If a > 3 then λ_{u} = (a − 3, 0), and G_{u} ≅ U(1) × Sp(2, ℝ).

This leads to the following

Definition. The weight μ ∈ i𝔱_{0}^{∗} is unitarily small if G_{u} = G. A K-type is unitarily small if its highest weight is.

Examples.

1. The trivial K-type is unitarily small.

2. For G = SL(2, ℝ), the unitarily small weights are 0, ±1, ±2. Notice that this set contains the LKT’s for the first two discrete series representations of G.

3. For G = Sp(4, ℝ), the unitarily small K-types are those with highest weights of the form (a, b) with 3 ≥ a ≥ b ≥ −3 and a − b ≤ 4.

Now we prove the proposition for unitarily small K-types.

Lemma. (Vogan, SR) The K-type with highest weight μ is unitarily small if and only if μ is of the form

μ = λ_{u} + ∑_{β∈Δ(𝔭,𝔱)} c_{β}β

with 0 ≤ c_{β} ≤ 1.

Example. For G = Sp(4, ℝ), (3, 2) = 2e_{1} + (e_{1} + e_{2}) + e_{2}.

Lemma. (Kumaresan, Vogan, Zuckerman) Assume that π ∈ Π_{u}^{sr}(G) has a LKT of the form

χ = ∑_{β∈Δ(𝔭,𝔱)} c_{β}β

with 0 ≤ c_{β} ≤ 1. Then there is a parabolic subalgebra 𝔮′ = 𝔩′ + 𝔲′ ⊂ 𝔤 such that χ = 2ρ(𝔲 ∩ 𝔭).

Since for μ unitarily small we have that λ_{u} = 0 on [𝔤, 𝔤] ⊃ [𝔩′, 𝔩′], the conditions of Fact 2 are satisfied, and the proposition is proved for this case. For general μ, we prove the following Proposition. (Vogan, SR) For any weight μ ∈ i𝔱_{0}^{∗}, there is a positive set (Δ^{+})(𝔤_{u}) such that μ is of the form

μ = λ_{u} + 2ρ(𝔲_{u} ∩ 𝔭) + ∑_{β∈(Δ^{+})(𝔤_{u}∩𝔭)} c_{β}β

with 0 ≤ c_{β} ≤ 1.

The result now follows from the following proposition, which we prove by extending Kumaresan’s lemma to this situation.

Proposition. If X is hermitian, with strongly regular infinitesimal character and LKT μ, then either X is not unitary or there is a parabolic subalgebra 𝔮 = 𝔩 + 𝔲 such that

μ = λ_{u} + 2ρ(𝔲 ∩ 𝔭).

6. Motivation (for our choice of G_{u})

The strongly regular case is a good easy example to illustrate why λ_{u} might be the right parameter to choose for the classification of the unitary dual of G. Here is one of the reasons for this choice.

We follow an idea of Barbasch and Vogan: given a weight μ which is dominant for 𝔨, we try to find a Levi subgroup L ⊂ G and a representation of L ∩ K with highest weight μ^{L} such that

{X ∈ Π_{u}^{rs}(G) with LKT μ} ↔ {X^{L} ∈ Π_{u}^{rs}(L) with LKT μ^{L}} ,

and if X^{L} is not unitary then the signature change survives in the representation of G under the bottom layer map (defined below).

A K-type μ does not determine 𝔮 or λ uniquely. Recall that for an admissible A_{𝔮}(λ), we have conditions of the form (λ, α) ≥ 0 on 𝔲 and λ ⊥ Δ(𝔩). To achieve uniqueness, we try to find a maximal algebra 𝔮 by adding to 𝔩 root spaces associated to roots α with (λ, α) = 0. We know that under the given regularity conditions, we have

Lemma. (Vogan)

A_{𝔮}(λ) ≅ A_{𝔮′}(λ′) ⇔ 𝔲 ∩ 𝔭 = 𝔲′ ∩ 𝔭 and λ = λ′ on L ∩ L′.

Using this lemma and the conditions for admissibility on λ and 𝔮, we set

μ′ = μ + ∑ ; \begin{matrix}(α, μ) > 0 \\ α ∈ Δ(𝔨, 𝔱) \end{matrix} α  and  𝔮_{V Z}(μ) = 𝔮(μ′).

We claim that μ is the LKT of an A_{𝔮}(λ) module if and only if

(∗)  (μ′, α) ≥ (2ρ(𝔲_{V Z}), α) for all α ∈ Δ(𝔲_{V Z}).

In that case set λ = μ′ − 2ρ(𝔲_{V Z}). Then μ = μ(𝔮_{V Z}, λ). (In fact, μ an A_{𝔮}(λ) LKT implies that μ′ = λ + 2ρ(𝔲_{V Z}).)

Example. Let G = Sp(4, ℝ) and μ = (a, 2) with a > 2. Then μ′ = (a + 1, 1), and Δ(𝔲_{V Z}) = {2e_{1}, e_{1} + e_{2}, e_{1} − e_{2}, 2e_{2}}. We have (μ′, 2e_{2}) = 2 < 4 = (2ρ(𝔲_{V Z}), 2e_{2}), and μ is not an A_{𝔮}(λ) LKT.

The idea is to use the above claim to prove that such a representation is not unitary: if (∗) does not hold for a given K-type μ, set

𝔤_{r}(μ) = 𝔩_{V Z} + ∑ ; \begin{matrix}α ∈ 𝔲_{V Z} \\ (μ′, α) ≤ (μ′, 2ρ(𝔲_{V Z})) \end{matrix} 𝔤_{±α} ⊇ 𝔩_{V Z},

𝔲_{r}(μ) = ∑ ; \begin{matrix}α ∈ 𝔲_{V Z} \\ (μ′, α) > (μ′, 2ρ(𝔲_{V Z})) \end{matrix} 𝔤_{α} ⊆ 𝔲_{V Z},

and

𝔮_{r} = 𝔤_{r} + 𝔲_{r}.

Then show that an (irreducible) representation X with such a K-type can be imbedded into a Zuckerman module induced from a hermitian Harish-Chandra module of the group G_{r}(μ). Do this so that there is a correspondence (via the bottom layer map) between the LKT’s of the irreducible module X and the LKT’s of the inducing representation of G_{r}(μ).

Explicitly, the bottom layer map is

R_{𝔮}^{K} ((X^{L})_{L∩K}) : μ^{L} ⟶ \{ \begin{matrix}μ^{L} ⊗ Λ^{top}𝔲 ∩ 𝔭 & if this is 𝔨 dominant, \\ 0 & otherwise. \end{matrix}

The group L in this case will be G_{r}(μ).

Then, using Parthasarathy’s Dirac operator inequality (see Vogan’s talk), prove nonunitarity for the representation of G_{r}(μ) by exhibiting two lowest K-types with the property that the form is indefinite on their sum. Then verify that this indefiniteness prevails when we take the corresponding K-types in X, via the bottom layer map.

In the example above, 𝔤_{r}(μ) ≅ 𝔱 + 𝔰𝔭(2, ℂ). In that case, the LKT for the representation of G_{r}(μ) corresponding to the K-type (a, 2) is (a−3, 1). The number 1 on the 𝔰𝔭(2) factor helps us use Parthasarathy’s Dirac operator inequality to show that the hermitian form on the representation of G_{r}(μ) is indefinite on the sum of the K-types with highest weights (a − 3, 1) and one of the form (a − 3, 1) ± 2e_{2}. In other words, the K-types we need to consider for the possible change of signature of the hermitian form are (a − 3, 3), (a − 3, 1), and (a − 3, −1).

Now, via the bottom layer map, these weights correspond to the weights (a, 4), (a, 2), and (a, 0). However, the weight (a, 4) is not even dominant if a = 3. Therefore, we need to find a bigger parabolic subalgebra attached to the LKT of the representation. Note that if a = 3, then 𝔮_{u}(μ) = 𝔤, and there is no reduction in this case.