THE LANGLANDS CLASSIFICATION

FOR UNITARY REPRESENTATIONS

David Vogan, MIT

This is joint work with Susana Salamanca-Riba [SV].

Setting: G is a real reductive group; Ĝ_{t} ⊂ Ĝ_{u} ⊂ Ĝ_{a} are the tempered, unitary, and admissible duals of G; P = MAN ⊂ G is a parabolic subgroup, τ ∈ M̂_{t}, ν ∈ Â_{a}; Re < ν, α >> 0 for all weights α of 𝔞 on 𝔫.

Theorem 1. (Langlands)

(1) Ind_{MAN}^{G}(τ ⊗ ν ⊗ 1) = I_{P} (τ, ν) has a unique irreducible quotient J_{P} (τ, ν) ∈ Ĝ_{a}.

(2) J_{P} (τ, ν) ≃ J_{P′}(τ′, ν′) ↔ (P, τ, ν) ∼ (P′, τ′, ν′) (conjugacy by G on the right hand side)

(3) Every irreducible admissible representation of G is isomorphic to J_{P} (τ, ν) for some (P, τ, ν).

The tempered dual Ĝ_{t} has been classified, so this is a good reduction, and determines Ĝ_{a}.

We now consider Ĝ_{u} ⊂ Ĝ_{h} ⊂ Ĝ_{a} where Ĝ_{h} is the Hermitian dual, i.e. irreducible representations which carry an invariant (not necessarily positive definite) Hermitian form. Setting: P = MAN; ρ ∈ M̂_{h,Re}; ν ∈ Â_{u} with Im < ν, α >> 0 for all weights α of 𝔞 on 𝔫. Here M_{h,Re} is the Hermitian representations with real infinitesimal character (defined below). The following Theorem is essentially due to Harish-Chandra — see Knapp [K].

Theorem 2.

(1) I_{P} (ρ, ν) is an irreducible Hermitian representation, and is unitary (respectively tempered) if and only if ρ is unitary (respectively tempered).

(2) I_{P} (ρ, ν) ≃ I_{P′}(ρ′, ν′) ↔ (P, ρ, ν) ∼ (P′, ρ′, ν′)

(3) Every irreducible Hermitian representation of G is isomorphic to J_{P} (τ, ν) for some (P, τ, ν).

Note:. This reduces the classification of Ĝ_{u} to the case of real infinitesimal character.

Fix a Cartan involution θ of G with fixed points K. Let H = T A be a θ-stable Cartan subgroup with Lie algebra 𝔥_{0}, and complexified Lie algebra 𝔥 = 𝔱 ⊕ 𝔞. Let

𝔥^{∗} = 𝔱^{∗} ⊕ 𝔞^{∗} be the duals. Then 𝔥^{∗} ⊃ 𝔥_{ℝ}^{∗} =^{def.} i𝔱_{0}^{∗} ⊕ 𝔞_{0}^{∗}, and 𝔥^{∗} ⊃ i𝔥_{ℝ}^{∗} = 𝔱_{0}^{∗} ⊕ i𝔞_{0}^{∗}. We

call 𝔥_{ℝ} (respectively i𝔥_{ℝ}^{∗}) the canonical real (respectively imaginary) part of 𝔥^{∗}. For λ ∈ 𝔥^{∗} write λ = Re(λ) + iIm(λ) with respect to this decomposition.

Typeset by AMS-TEX

Note that 𝔥_{ℝ}^{∗} is invariant by W = W (𝔤, 𝔥). Recall by the Harish-Chandra homomorphism 𝔷(𝔤) ≃ S(𝔥)^{W} and Hom(𝔷(𝔤), ℂ) ≃ 𝔥^{∗}/W.

Definition. An infinitesimal character λ, i.e. an element of Hom(𝔷(𝔤), ℂ), is real if it corresponds to an element of 𝔥_{ℝ}^{∗} under the Harish-Chandra homomorphism.

For example the infinitesimal character of the discrete series representation are real (contained in i𝔱_{o}^{∗}), and the infinitesimal character of the (tempered) minimal principal series are imaginary (in i𝔞_{0}^{∗}).

—————————————————————— Cohomological Parabolic Induction

This produces unitary representations from the centralizer of a compact torus. Idea: This should produce all representation of real infinitesimal character from representations of infinitesimal character 0 (although this is not strictly true) or from representations with imaginary infinitesimal character.

The work of Salamanca-Riba and Vogan is concerned with trying to find a version of this idea which is true.

Start with 𝔮 = 𝔩 ⊕ 𝔲 ⊂ 𝔤, \overline{𝔮} ∩ 𝔮 = 𝔩, and L = Norm_{G}(𝔮). (For notation and more details on this construction, see [K].) For example if G = U(p, q) then L = ∏ U(p_{i}, q_{i}). Let S = dim(𝔲 ∩ 𝔨), and consider the functor L_{𝔮}^{S}, which takes (𝔩, L ∩ K)-modules to (𝔤, K)-modules. It takes infinitesimal character γ_{L} for L to infinitesimal character γ_{L} +ρ(𝔲) for 𝔤. If π_{L} has lowest L∩K-type μ_{L} then L_{𝔮}^{S}(π_{L}) may contain the K-type μ_{L} + 2ρ(𝔲 ∩ 𝔭) (note that 2ρ(𝔲 ∩ 𝔭) is the weight of the one-dimensional representation Λ^{top}(𝔲 ∩ 𝔭).)

Now suppose 𝔮 = 𝔩 ⊕ 𝔲 is θ-stable, and τ_{L} is a unitary representation of L. Let T_{L} be the compact part of the center of L (analogous to the A in P = MAN). Let λ_{L} ∈ 𝔱_{ℓ}^{∗} be the character of τ_{L} on T_{L}. Assume:

(1) < λ_{L} + ρ(𝔲), α >> 0 for all α ∈ Δ(𝔲) (here ρ(𝔲) is one half the sum of the weights of T_{L} on 𝔲).

(2) (infinitesimal character of τ_{L}) restricted to [ℓ, ℓ] is purely imaginary.

The following theorem is a version of Theorem 2.

Theorem 3.

(1) L_{𝔮}^{S}(τ_{L}) is irreducible and tempered

(2) L_{𝔮}^{S}(τ_{L}) ≃ L_{𝔮′}^{S}(τ′_{L′}) if and only if (L, τ) ∼ (L′, τ′).

(3) Every irreducible tempered representation appears this way.

Note. The conditions imply τ_{L} is tempered.

Note. Theorem 2 (reducing to real infinitesimal character) together with Theorem 3 (reducing to imaginary infinitesimal character) reduces the classification of tempered representations to those with infinitesimal character 0. This forces G to be quasisplit, containing a Borel subgroup B = MAN, and a representation Ind_{B}^{G}(δ ⊗ 1 ⊗ 1). Here δ is a representation of M/M_{0}, and the reducibility of these induced representations is known by work of Knapp and Stein. This gives a complete classification of tempered representations (equivalent to [KZ], but not by the same method).

We try to do something similar for unitary representations.

Let 𝔮 = 𝔩 ⊕ 𝔲, π_{L}, λ_{L} on T_{L} be as before. We now strenthen the positivity conditions: assume

< λ_{L}, α >> 0 for all weights α of 𝔱 in 𝔲

This is stronger than the condition < λ_{L} + ρ(𝔲), α >> 0 from before. We replace the condition of purely imaginary infinitesimal character (i.e. Re(λ) = 0) with:

Re(infinitesimal character of π_{L} on[ℓ, ℓ]) ∈ convex hull of W (ℓ, 𝔥) · ρ_{L}

This allows the real part of λ to be small but non-zero. For example it allows for the trivial representation of L.

Theorem.

(1) L_{𝔮}^{S}(π_{L}) is irreducible and unitary

(2) L_{𝔮}^{S}(π_{L}) ≃ L_{𝔮′}^{S}(π′_{L′}) if and only if (L, π_{L}) ∼ (L′, π′_{L′})

Main Conjecture:. Every unitary representation occurs this way.

This theorem and conjecture, together with the Harish-Chandra/Knapp result, reduce the problem of computing Ĝ_{u} to the case of

real infinitesimal character λ ∈ the convex hull of ρ

This is always the hardest part. One advantage of this conjecture is that it comes with an idea of a proof: Parthasarathy’s Dirac inequality, together with a conjectural strengthening of it would imply the conjecture above.

Recall 𝔤 = 𝔨 ⊕ 𝔭 comes with a non-degenerate symmetric form which is positive definite on 𝔭_{0}, negative definite on 𝔨_{0} and 𝔨_{0} ⊥ 𝔭_{0}. The adjoint action defines a map Ad : K → O(𝔭_{0}). Suppose for simplicity that G is connected. Then the spin cover of SO(𝔭_{0}) pulls back to a two-fold cover K̃ of K. Composing the map K̃ → Spin(𝔭_{0}) with the spin representation, gives an action of K̃ on the spin module Cliff(𝔭_{0}), the Clifford algebra of 𝔭_{0}. Recall Cliff(𝔭_{0}) has a positive definite Spin(𝔭_{0})-invariant Hermition form.

Now if V is irreducible and unitary, the positive definite invariant Hermitian form gives a positive definite invariant Hermition form on V ⊗ S, and we construct the Dirac operator D : V ⊗ S → V ⊗ S. Explicitly

D(v ⊗ s) = ∑_{i=1}^{n} X_{i}v ⊗ X_{i}s

Here {X_{i}} is an orthogonal basis of 𝔭_{0}, and the action on v is the 𝔤-action, the action on s is via the Clifford algebra action.

Now suppose ũ is the highest weight of an irreducible K̃-representation appearing in V ⊗ S. Then

Theorem. (Parthasarathy)

If V has infinitesimal character character γ ∈ 𝔥^{∗}, then

< ũ + ρ_{c}, ũ + ρ_{c} >≥< γ, γ >

Conjecture. The inequality in the previous Theorem may be replaced by:

Re(γ) ∈ the convex hull of W (𝔤, 𝔥) · (ũ + ρ_{c})

Note. This conjecture implies the main conjecture.

References

[K] A. Knapp, Representation Theory of Real Semisimple Groups: an Overview Based on Examples, Princeton University Press, Princeton, New Jersey, 1986.

[KZ] A. Knapp and G. Zuckerman, Classification of irreducible tempered representations of semisimple Lie groups Ann. of Math. 116 (1982), 389–501.

[SV] S. Salamanca-Riba and D. Vogan, On the classification of unitary representations of reductive Lie groups, preprint.