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The complementary form loses positivity beyond the unitary interval

Statement refuted

Assume the Axiom of Choice (The Axiom of Choice). For every real $ν \geq 1$ , the normalized spherical invariant form $B_{ν}$ on the full even K-finite principal-series module $I_{0, ν}^{K}$ is positive definite.

Facts & Assumptions

Given: AC, the spherical compact-picture principal series and its normalized invariant form, and the allowed even K-types $f_{n} (k_{θ}) = e^{i n θ}$ .

[F1]

In spherical parity $a_{0} (ν) = 1$ and $(1 + ν) a_{2} (ν) = (1 - ν) a_{0} (ν)$ whenever the quotient is regular; $f_{0}, f_{2}$ are distinct nonzero K-type vectors (K-type eigenvalues of A(nu): recurrence, closed form and nonvanishing, K-type decomposition of the SL2(R) principal series).

[F2]

For every real $ν$ except negative odd integers, $B_{ν}$ is a finite continuous G-invariant form with Fourier weights $a_{0} = 1$ and $a_{\pm 2 j} = \prod_{l = 1}^{j} (2 l - 1 - ν) / (2 l - 1 + ν)$ . At $ν = 1$ , every nonzero even weight vanishes (Unitarity of the complementary series).

[A1]

AC is declared by the principal-series and invariant-form constructions and supplies the normalized Haar setup; the two coefficient evaluations here make no further choice (The Axiom of Choice).

Counterexample

Use the normalized spherical form from [F2]; the parameter $ν = 3$ is regular and the endpoint $ν = 1$ gives a separate degeneracy witness.

1.1F1F2A1algebra

At $ν = 3$ , the normalized weights are finite. By [F1], $a_{0} (3) = 1$ and $4 a_{2} (3) = - 2 a_{0} (3)$ , so $a_{2} (3) = - 1 / 2$ . Thus $B_{3} (f_{0}, f_{0}) = 1 > 0$ and $B_{3} (f_{2}, f_{2}) = - 1 / 2 < 0$ : this regular invariant form is indefinite and refutes positive definiteness. The parameter $3$ is a reducibility point, but regularity of the normalized form does not require irreducibility.

2.1F2A1step 1.1algebra∎

At $ν = 1$ , [F2] gives $a_{0} = 1$ and $a_{2} = 0$ . Hence $B_{1} (f_{0}, f_{0}) = 1$ , while $B_{1} (f_{2}, h) = 0$ for every smooth $h$ by the Fourier-diagonal formula; $f_{2} \neq 0$ , so the endpoint form is nonzero and degenerate. This also contradicts the refuted claim at its boundary and confirms that the positive-definite range $∣ ν ∣ < 1$ is strict.

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Sources

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