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Fell continuity of the unitary principal series in the parameter

Statement

Assume the Axiom of Choice (The Axiom of Choice). Let $G = {S L}_{2} (R)$ , $ε \in {0, 1}$ , and $s_{0} \in R$ . Write $H_{ε} = L_{ε}^{2} (K)$ and let $Π_{ε, s}$ be the compact-picture representation of $I_{ε, i s}$ on $H_{ε}$ (The compact picture of the SL2(R) principal series). For every $ξ \in H_{ε}$ and compact $Q \subseteq G$ , $\sup_{g \in Q} ∣ ⟨ Π_{ε, s} (g) ξ, ξ ⟩ - ⟨ Π_{ε, s_{0}} (g) ξ, ξ ⟩ ∣ ⟶ 0 (s \to s_{0}) .$ For $δ > 0$ , set $J_{ε, s_{0}, δ} = {s \in R : 0 < ∣ s - s_{0} ∣ < δ, (ε, s) \neq (1, 0)}$ and $S_{ε, s_{0}, δ} = {[I_{ε, i s}] : s \in J_{ε, s_{0}, δ}} \subseteq \hat{G}$ (Generic irreducibility and the exceptional parameter lattice). Then $Π_{ε, s_{0}}$ is weakly contained both in the parameter-indexed direct sum ${⨁^{^}}_{s \in J_{ε, s_{0}, δ}} Π_{ε, s}$ and in the direct sum of one representative of each class in $S_{ε, s_{0}, δ}$ (Weak containment of unitary representations, Hilbert direct sums of unitary representations). If $(ε, s_{0}) \neq (1, 0)$ , the irreducible class $[I_{ε, i s_{0}}]$ lies in the Fell closure of $S_{ε, s_{0}, δ}$ ; at $(ε, s_{0}) = (1, 0)$ , the reducible $I_{1, 0} = D_{1}^{+} \oplus D_{1}^{-}$ is not a point of $\hat{G}$ , but both irreducible summands $D_{1}^{+}$ and $D_{1}^{-}$ lie in the Fell closure of $S_{1, 0, δ}$ (The Fell topology on the unitary dual, The unitary dual of a locally compact group, The two limits of discrete series, Unitarity and irreducibility of the limits of discrete series).

Facts & Assumptions

Given: AC; $ε \in {0, 1}$ ; $s \in R$ ; and the compact-picture family of The compact picture of the SL2(R) principal series.

[F1]

Every $Π_{ε, s}$ acts unitarily and strongly continuously on the same $H_{ε}$ . In the compact picture, for $a (k, g) = ∣ α (p (k, g)) ∣ > 0$ and $k g = p (k, g) κ (k, g)$ in canonical $A N \times K$ coordinates, $(Π_{ε, s} (g) f) (k) = a (k, g)^{1 + i s} f (κ (k, g)),$ and $a, κ$ depend continuously on $(k, g)$ (The compact picture of the SL2(R) principal series).

[F2]

The finite linear combinations of $f_{n} (k_{θ}) = e^{i n θ}$ with $n \equiv ε (m o d 2)$ are K-finite and dense in $H_{ε}$ (K-type decomposition of the SL2(R) principal series).

[F3]

$c_{ξ, η} (g) = ⟨ Π (g) ξ, η ⟩$ is the matrix coefficient convention; weak containment means compact-uniform approximation of each diagonal coefficient by finite sums of diagonal coefficients (Matrix coefficient of a unitary representation, Weak containment of unitary representations).

[F4]

The Hilbert direct sum of any set-indexed family of strongly continuous unitary representations is a strongly continuous unitary representation, and each summand embeds as a closed invariant subspace (Hilbert direct sums of unitary representations).

[F5]

If $s \in J_{ε, s_{0}, δ}$ , then $I_{ε, i s}$ is irreducible; if $(ε, s_{0}) \neq (1, 0)$ , so is $I_{ε, i s_{0}}$ (Generic irreducibility and the exceptional parameter lattice).

[F6]

At $(ε, s_{0}) = (1, 0)$ , $Π_{1, 0} = D_{1}^{+} \oplus D_{1}^{-}$ orthogonally, and each $D_{1}^{\pm}$ is an irreducible strongly continuous unitary representation (The two limits of discrete series, Unitarity and irreducibility of the limits of discrete series).

[F7]

For $S \subseteq \hat{G}$ , an irreducible $π \in \hat{G}$ lies in $\overline{S}$ exactly when $π ≺ {⨁^{^}}_{σ \in S} σ$ (The Fell topology on the unitary dual, The unitary dual of a locally compact group, Fell closure is characterized by weak containment).

[A1]

AC is inherited from the compact-picture, dual, and Hilbert-direct-sum suppliers (The Axiom of Choice).

Proof

technique · direct

Given: The assumptions and notation in the Statement.

1.1F1F3algebra

If $Q = \emptyset$ the coefficient assertion is immediate. Otherwise, by [F1] the positive function $a (k, g)$ is continuous on the compact set $K \times Q$ , so $A_{Q} = \sup_{K \times Q} a < \infty$ and $M_{Q} = \sup_{K \times Q} ∣ \log a ∣ < \infty$ . For a K-finite $f$ , the compact-picture formula gives $(Π_{ε, s} (g) f) (k) = a (k, g) e^{i s \log a (k, g)} f (κ (k, g))$ . Using $∣ e^{i t} - 1 ∣ \leq ∣ t ∣$ for real $t$ , we obtain $\sup_{g \in Q} ∥ Π_{ε, s} (g) f - Π_{ε, s_{0}} (g) f ∥_{2} \leq A_{Q} M_{Q} ∣ s - s_{0} ∣ ∥ f ∥_{\infty} .$ Cauchy–Schwarz then gives uniform convergence of the diagonal coefficients for this $f$ .

2.1F1F2step 1.1algebra

Let $ξ \in H_{ε}$ and choose K-finite $f$ with $∥ ξ - f ∥_{2}$ arbitrarily small using [F2]. For every $s$ and $g$ , unitarity [F1] gives $∣ ⟨ Π_{ε, s} (g) ξ, ξ ⟩ - ⟨ Π_{ε, s} (g) f, f ⟩ ∣ \leq (∥ ξ ∥_{2} + ∥ f ∥_{2}) ∥ ξ - f ∥_{2} .$ The same bound holds at $s_{0}$ , uniformly in $g$ . Combining these two bounds with step 1.1 and first choosing $f$ close to $ξ$ , then $s$ close to $s_{0}$ , proves the claimed compact-uniform convergence for every $ξ$ .

3.1F1F3F4step 2.1algebraA1

Fix $δ > 0$ . For any diagonal coefficient of $Π_{ε, s_{0}}$ , compact $Q$ , and tolerance $η > 0$ , step 2.1 gives a parameter $s \in J_{ε, s_{0}, δ}$ whose coefficient differs by less than $η$ on $Q$ ; such parameters exist arbitrarily close to $s_{0}$ while avoiding the finitely many excluded points. Embedding the vector into the $s$ -summand realizes that coefficient in the parameter-indexed direct sum of [F4]. The class $[I_{ε, i s}]$ is also in $S_{ε, s_{0}, δ}$ ; transporting the vector through a unitary equivalence to the chosen representative realizes the same coefficient in the class-indexed direct sum. Thus both weak-containment assertions follow.

4.1F5F7step 3.1

If $(ε, s_{0}) \neq (1, 0)$ , [F5] puts $[I_{ε, i s_{0}}]$ in $\hat{G}$ , and every class indexed by $J_{ε, s_{0}, δ}$ is in $\hat{G}$ as well. Apply [F7] to step 3.1 to see that $[I_{ε, i s_{0}}]$ lies in the Fell closure of those classes.

5.1F6F7step 3.1∎

At $(ε, s_{0}) = (1, 0)$ , every vector in either $D_{1}^{+}$ or $D_{1}^{-}$ is a vector of $H_{1}$ , so each of its diagonal coefficients for the restricted representation is also a coefficient of $Π_{1, 0}$ . Step 3.1 therefore gives $D_{1}^{\pm} ≺ {⨁^{^}}_{s \in J_{1, 0, δ}} Π_{1, s}$ ; [F6] makes these irreducible dual points, and [F7] puts each class in the Fell closure. Since $D_{1}^{+}$ and $D_{1}^{-}$ are nonzero orthogonal summands, $Π_{1, 0}$ is reducible and is not itself a point of $\hat{G}$ .

Depends on

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Sources

Emmanuel Kowalski, An Introduction to the Representation Theory of Groups (AMS GSM 155; author's PDF) (standard reference, not scraped)
Matt Kerr, Notes on the Representation Theory of SL2(R) (CBMS workshop writeup) (standard reference, not scraped)