ExampleConstruction: Literature-sourcedVerification: AI-adaptedPipeline-generatedprecheck passjudge pass (gpt-6.1-sol)audited 2026-10-08

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Intertwiner eigenvalues in the spherical complementary range

Example

Assume the Axiom of Choice (The Axiom of Choice) and work in spherical parity. At regular real parameters the first normalized eigenvalues of the standard intertwiner are ${\hat{c}}_{0} (ν) = 1$ , ${\hat{c}}_{\pm 2} (ν) = \frac{1 - ν}{1 + ν}$ , and ${\hat{c}}_{\pm 4} (ν) = \frac{(1 - ν) (3 - ν)}{(1 + ν) (3 + ν)}$ ; the normalized values at $ν = 0$ are given by regular continuation. They are positive for $∣ ν ∣ < 1$ . At $ν = (1 + 3) / 2 = 2$ , ${\hat{c}}_{2} (2) = - 1 / 3 \neq 0$ ; the coefficient is positive on $(- 1, 1)$ , zero at $1$ , and negative throughout $(1, 3)$ .

Facts & Assumptions

Given: AC, the spherical parity $ε = 0$ , and the normalized meromorphic eigenvalues of the standard intertwiner.

[F1]

The even K-type eigenvalues satisfy the cross-multiplied recurrence $(2 j + 1 + ν) c_{2 j + 2} = (2 j + 1 - ν) c_{2 j}$ and the symmetry $c_{- 2 j} = c_{2 j}$ . These identities continue meromorphically, and at regular parameters dividing by the base scalar gives the recurrence for ${\hat{c}}_{2 j} = c_{2 j} / c_{0}$ with ${\hat{c}}_{0} = 1$ (K-type eigenvalues of A(nu): recurrence, closed form and nonvanishing, The standard intertwining operator A(nu)).

[F2]

The normalized spherical weights are ${\hat{c}}_{\pm 2 j} (ν) = \prod_{l = 1}^{j} (2 l - 1 - ν) / (2 l - 1 + ν)$ whenever regular. On $(- 1, 1)$ every weight is regular, including the normalized meromorphic continuation at $ν = 0$ (Unitarity of the complementary series).

[A1]

AC supplies the normalized Haar measure on $K$ used by the principal-series and complementary-form setup; through $A C \Rightarrow {A C}_{ω}$ it also supplies the countable-choice hypotheses used in deriving the recurrence and Fourier-form suppliers. The finite recurrence iteration and sign checks make no further choice (The Axiom of Choice).

Verification

technique · iterate the even-parity eigenvalue recurrence and inspect the signs of its first factors

1.1F1F2algebra

Normalize the meromorphic recurrence in [F1] by its base scalar wherever the quotient is initially regular. The $j = 0$ and $j = 1$ instances give ${\hat{c}}_{2} = (1 - ν) / (1 + ν)$ and ${\hat{c}}_{4} = {\hat{c}}_{2} (3 - ν) / (3 + ν) = (1 - ν) (3 - ν) / ((1 + ν) (3 + ν))$ . The negative-index symmetry gives ${\hat{c}}_{- 2} = {\hat{c}}_{2}$ and ${\hat{c}}_{- 4} = {\hat{c}}_{4}$ . These ratios extend meromorphically; [F2] identifies their regular values at $ν = 0$ and agrees with the displayed products.

1.2F1F2algebra

If $∣ ν ∣ < 1$ , then for every $l \geq 1$ both $2 l - 1 - ν$ and $2 l - 1 + ν$ are positive. Thus ${\hat{c}}_{0} = 1$ , ${\hat{c}}_{\pm 2} > 0$ , and ${\hat{c}}_{\pm 4} > 0$ throughout the open interval, including $ν = 0$ . More generally every factor in the finite product for any fixed even K-type is positive there.

1.3F1F2algebra

On $1 < ν < 3$ , the numerator $1 - ν$ is negative and the denominator $1 + ν$ positive, so ${\hat{c}}_{2} (ν) < 0$ ; at the midpoint $ν = (1 + 3) / 2 = 2$ its value is $- 1 / 3 \neq 0$ . On $- 1 < ν < 1$ the coefficient is positive, and it vanishes at $ν = 1$ , so its sign changes at that endpoint.

2.1A1given∎

AC enters through the normalized Haar construction and the countable-choice hypotheses of the cited suppliers described in [A1]; this finite computation makes no additional selection.

Depends on

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Sources

Matt Kerr, Notes on the Representation Theory of SL2(R) (CBMS workshop writeup) (standard reference, not scraped)
Pavel Etingof, Representations of Lie Groups (MIT 18.757 lecture notes, Fall 2023) (standard reference, not scraped)