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The limits of discrete series are not square-integrable

Statement

Assume the Axiom of Choice (The Axiom of Choice). Define a square-integrable irreducible unitary representation to mean one for which every matrix coefficient lies in $L^{2} (G)$ ; this is the discrete-series convention of Frahm's SL $(2, R)$ Plancherel notes. Let $D_{1}^{-}$ and $D_{1}^{+}$ be the two limits of discrete series of The two limits of discrete series. Neither is square-integrable. In the compact picture of $I_{1, 0}$ , the normalized weight-one vector $f_{1} (k_{θ}) = e^{i θ}$ in $D_{1}^{+}$ has coefficient $⟨ Π_{0} (a_{τ}) f_{1}, f_{1} ⟩ = sech (τ / 2), a_{τ} = diag (e^{τ / 2}, e^{- τ / 2}),$ and $\int_{G} ∣ ⟨ Π_{0} (g) f_{1}, f_{1} ⟩ ∣^{2} d g = 2 π \int_{0}^{\infty} {sech}^{2} (τ / 2) \sinh τ d τ = + \infty .$ Complex conjugation gives the same nonintegrable coefficient modulus on $D_{1}^{-}$ . This statement establishes failure of the all-coefficients criterion; it makes no claim about Plancherel support or occurrence in the regular representation.

Facts & Assumptions

Given: AC; the compact-picture representation $I_{1, 0}$ , its two limit summands, the unitary structure of those limits, the weight-one coefficient formula, and the fixed left Haar measure.

[F1]

The odd compact-picture basis has unit vectors $f_{m} (k_{θ}) = e^{i m θ}$ ; $D_{1}^{+}$ is the closed positive-weight tail beginning at $f_{1}$ , and $D_{1}^{-}$ is the closed negative-weight tail beginning at $f_{- 1}$ . Both are irreducible strongly continuous unitary representations (The two limits of discrete series, Unitarity and irreducibility of the limits of discrete series).

[F2]

At $ν = 0$ , the compact-picture action has the form $(Π_{0} (g) f) (k) = r (k, g) f (κ (k, g))$ with $r (k, g)$ real and positive. Therefore pointwise complex conjugation commutes with $Π_{0} (g)$ and maps $f_{1}$ to $f_{- 1}$ (The compact picture of the SL2(R) principal series, The two limits of discrete series).

[F3]

In this model, $⟨ Π_{0} (a_{τ}) f_{1}, f_{1} ⟩ = sech (τ / 2)$ for every $τ \in R$ (Matrix-coefficient formulas and decay for the discrete and principal series(b)).

[F4]

For a continuous nonnegative K-bi-invariant function $ψ$ , the fixed Haar measure satisfies $\int_{G} ψ (g) d g = 2 π \int_{0}^{\infty} ψ (a_{τ}) \sinh τ d τ$ (KAK integration formula for K-bi-invariant functions on SL2(R)).

[F5]

A matrix coefficient is $c_{v, w} (g) = ⟨ π (g) v, w ⟩$ with pairing linear in the first variable; such coefficients are continuous for strongly continuous unitary representations (Matrix coefficient of a unitary representation).

[A1]

AC is assumed and inherited through the normalized principal-series and fixed-Haar constructions (The Axiom of Choice).

Proof

technique · direct

Given: The assumptions and notation of the Statement.

1.1F1F3

Let $c_{+} (g) : = ⟨ Π_{0} (g) f_{1}, f_{1} ⟩$ . By [F1], $f_{1}$ is a unit K-eigenvector in $D_{1}^{+}$ , and [F3] gives $c_{+} (a_{τ}) = sech (τ / 2)$ .

2.1F1F3F4F5step 1.1algebraA1

Unitarity and the K-character property of $f_{1}$ imply that $ψ_{+} (g) : = ∣ c_{+} (g) ∣^{2}$ is continuous, nonnegative, and K-bi-invariant. Applying [F4] and using $\sinh τ = 2 \sinh (τ / 2) \cosh (τ / 2)$ gives $\int_{G} ψ_{+} (g) d g = 2 π \int_{0}^{\infty} 2 \tanh (τ / 2) d τ = + \infty$ : the integrand $2 \tanh (τ / 2)$ tends to $2$ , hence is at least $1$ for all sufficiently large $τ$ . Thus $c_{+} \notin L^{2} (G)$ .

3.1F1F2F5step 1.1step 2.1algebra∎

Let $C f = \overline{f}$ on $L_{1}^{2} (K)$ . By [F2], $C$ commutes with $Π_{0} (g)$ and maps the positive tail $D_{1}^{+}$ onto $D_{1}^{-}$ . Since $C$ is antiunitary, $⟨ Π_{0} (g) f_{- 1}, f_{- 1} ⟩ = \overline{⟨ Π_{0} (g) f_{1}, f_{1} ⟩}$ , so its modulus also fails to lie in $L^{2} (G)$ by step 2.1. The irreducible unitary representations $D_{1}^{\pm}$ therefore each have a matrix coefficient outside $L^{2} (G)$ , which violates the defining requirement that every matrix coefficient be square-integrable.

Depends on

Used by

A limit of discrete series is not square-integrable Counterexample

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Sources

Jan Frahm, The Plancherel formula for real reductive groups I: Examples (AIM RTG lecture notes) (standard reference, not scraped)
Emmanuel Kowalski, An Introduction to the Representation Theory of Groups (AMS GSM 155; author's PDF) (standard reference, not scraped)