TheoremStatement: AI-adaptedProof: AI-adaptedPipeline-generatedprecheck passjudge pass (gpt-6.1-sol)audited 2026-10-08

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Tempered status of the SL2(R) unitary series

Statement

Assume the Axiom of Choice (The Axiom of Choice) and use the normalized parameter convention of The normalized principal series I(epsilon, nu). Every irreducible unitary principal-series class $[I_{ε, i s}]$ , with $ε \in {0, 1}$ and $s \in R$ , is tempered. The odd family at $s = 0$ is reducible, $I_{1, 0} = D_{1}^{+} \oplus D_{1}^{-}$ , and its two irreducible summands are tempered; each $D_{n}^{\pm}$ for $n \geq 2$ is also tempered. No nontrivial spherical complementary-series class $[I_{0, ν}]$ with $0 < ∣ ν ∣ < 1$ , and not the trivial class, is tempered. Here tempered means weak containment in the left regular representation (Tempered unitary representations); the reducible $I_{1, 0}$ is not itself a point of the unitary dual (The unitary dual of a locally compact group).

Facts & Assumptions

Given: AC, the normalized principal-series parameter, the irreducible unitary dual of $G = {S L}_{2} (R)$ , its fixed left Haar measure, and the Plancherel support theorem for this group.

[F1]

A strongly continuous unitary representation is tempered exactly when it is weakly contained in the left regular representation (Tempered unitary representations, Left and right regular unitary representations of an LCH group, Weak containment of unitary representations). The Plancherel supplier proves that the closed support of its actual onto regular disintegration is exactly the irreducible classes weakly contained in that representation (Plancherel support for SL2(R), The Fell topology on the unitary dual).

[F2]

The Plancherel transform identifies the regular representation with an irreducible direct integral over its Plancherel support. The carrier consists of nonzero-parameter unitary principal classes and $D_{n}^{\pm}$ for $n \geq 2$ ; its closed support also contains $[I_{0, 0}]$ and $D_{1}^{\pm}$ , while the spherical complementary classes with $0 < ν < 1$ and the trivial class lie outside the support (Plancherel support for SL2(R)).

[F3]

The compact-picture representations $I_{ε, i s}$ are strongly continuous and unitary. The spherical $I_{0, 0}$ is irreducible, while $I_{1, 0} = D_{1}^{+} \oplus D_{1}^{-}$ is reducible (Unitarity of the unitary principal series, The two limits of discrete series).

[F4]

Each $D_{1}^{\pm}$ is an irreducible strongly continuous unitary limit, and $D_{n}^{\pm}$ for $n \geq 2$ are irreducible unitary discrete-series models (The two limits of discrete series, Unitarity and irreducibility of the limits of discrete series, Irreducibility and K-types of the discrete series).

[F5]

The spherical complementary models $I_{0, ν}$ are irreducible unitary representations for $0 < ∣ ν ∣ < 1$ (Unitarity of the complementary series).

[F6]

The unitary complementary representations with parameters $ν$ and $- ν$ are equivalent for $0 < ν < 1$ ; this follows by extending the normalized intertwiner to a unitary map between their completed Hilbert spaces (The spherical complementary series converge to the trivial representation, proof step 1.2). Its convergence-to-trivial clause is not used here.

[A1]

AC is inherited through the unitary dual, regular representation, Plancherel field, and complementary-series Hilbert models (The Axiom of Choice).

Proof

technique · identify tempered classes by the regular Plancherel support and exclude the complementary family by its explicit sign equivalence

Given: The statement, Facts [F1]–[F6], and AC.

1.1F1F2A1

Let $S$ be the closed support of the Plancherel measure. By [F2], the regular representation has its irreducible direct-integral decomposition over $S$ ; by [F1], the irreducible classes in this support are exactly the tempered classes.

2.1F2F3F4step 1.1

Every nonzero-parameter unitary principal class and every $D_{n}^{\pm}$ for $n \geq 2$ is in the Plancherel carrier, hence in $S$ . The even endpoint $[I_{0, 0}]$ and the two limits $D_{1}^{\pm}$ lie in $S$ by [F2]; [F3] and [F4] make these irreducible unitary dual points. Step 1.1 therefore proves all principal, discrete, and limit claims. At the odd endpoint, [F3] identifies $I_{1, 0}$ with the two summands, so the reducible direct sum is not asserted to be a dual point.

3.1F2F5F6step 1.1A1∎

By [F2], the positive-parameter spherical complementary classes $[I_{0, ν}]$ , $0 < ν < 1$ , and the trivial class are outside $S$ , hence are not tempered by step 1.1. If $- 1 < ν < 0$ , [F6] gives $[I_{0, ν}] = [I_{0, - ν}]$ , and $0 < - ν < 1$ , so the negative-parameter class is outside $S$ as well. This uses the complementary Hilbert-space sign equivalence; no non-temperedness conclusion is drawn from its convergence to the trivial class or from zero Plancherel mass alone.

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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)
Jan Frahm, The Plancherel formula for real reductive groups I: Examples (AIM RTG lecture notes) (standard reference, not scraped)
Peter Hochs, Harish-Chandra's Plancherel formula for SL(2,R) (lecture notes) (standard reference, not scraped)