Fourier-Helgason transform as infinite geodesic time limit in geometric quantization

Ana Cristina Ferreira; Joachim Hilgert; Jo\~ao P. Nunes; Jos\'e M. Mour\~ao

arxiv: 2606.20290 · v1 · pith:HYZATW6Xnew · submitted 2026-06-18 · 🧮 math.SG · math-ph· math.MP

Fourier-Helgason transform as infinite geodesic time limit in geometric quantization

Ana Cristina Ferreira , Joachim Hilgert , Jos\'e M. Mour\~ao , Jo\~ao P. Nunes This is my paper

Pith reviewed 2026-06-26 14:39 UTC · model grok-4.3

classification 🧮 math.SG math-phmath.MP

keywords Fourier-Helgason transformgeometric quantizationsymmetric spacesquantum geodesic transformgeodesic flowHarish-Chandra c-functionprincipal seriesBKS pairing

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The pith

The Fourier-Helgason transform on a noncompact symmetric space equals the infinite-time limit of the quantum geodesic transform obtained by lifting the geodesic flow.

A machine-rendered reading of the paper's core claim, the machinery that carries it, and where it could break.

The paper establishes that the Fourier-Helgason transform, which decomposes L2 functions on G/K into principal series representations, arises directly from geometric quantization on the cotangent bundle T*(G/K). It does so by first showing that the horizontal polarization is the infinite-time limit of the vertical polarization pushed forward along the geodesic flow of any G-invariant metric. The authors then lift this flow to a unitary parallel transport on the quantum bundle, called the quantum geodesic transform, and prove that its infinite-time limit recovers the Fourier-Helgason transform up to the phase of the Harish-Chandra c-function and a constant factor. This construction works uniformly for both complex and non-complex groups and supplies a dynamical route to the transform that was previously missing in the non-complex case.

Core claim

The quantum geodesic transform, obtained by lifting the geodesic flow on T*(G/K) to an intertwining unitary parallel transport on the quantum bundle, admits a well-defined limit as geodesic time tends to infinity. This limit coincides with the Fourier-Helgason transform up to the phase of the Harish-Chandra c-function and an irrelevant multiplicative constant, and the limit is independent of the choice of G-invariant Riemannian metric.

What carries the argument

The quantum geodesic transform (QGT), the unitary parallel transport on the quantum bundle that intertwines the geodesic flow and whose infinite-time limit yields the Fourier-Helgason transform.

If this is right

The horizontal polarization on T*(G/K) is recovered as the infinite geodesic-time limit of the push-forward of the vertical polarization.
The Blattner-Kostant-Sternberg pairing between vertically and horizontally polarized sections reproduces the Fourier-Helgason transform once the infinite-time limit is taken.
The same limiting procedure works uniformly whether G is complex or not, removing the earlier mismatch for non-complex groups.
The resulting transform is unitary on L2(G/K) because the quantum geodesic transform is unitary for each finite time.

Where Pith is reading between the lines

These are editorial extensions of the paper, not claims the author makes directly.

The construction suggests that representation-theoretic decompositions on symmetric spaces can be viewed as long-time limits of classical Hamiltonian flows lifted to the quantum bundle.
Similar infinite-time limits might produce other integral transforms when applied to cotangent bundles of homogeneous spaces beyond symmetric spaces.
The independence of the limit from the metric choice (modulo the c-function phase) indicates that the Fourier-Helgason transform encodes an intrinsic geometric feature of the symmetric space rather than a metric-dependent artifact.

Load-bearing premise

The geodesic flow on the cotangent bundle lifts to a unitary parallel transport on the quantum bundle whose infinite-time limit exists and is independent of the invariant metric up to the stated phase.

What would settle it

A direct computation, for a concrete non-complex group such as SL(2,R), showing that the infinite-time limit of the lifted parallel transport differs from the Fourier-Helgason transform by more than a c-function phase and constant factor.

read the original abstract

The Fourier-Helgason (FH) transform for a noncompact symmetric space $G/K$ establishes the direct integral decomposition of the unitary representation of $G$ on $L^2(G/K)$ into irreducible principal series representations. By applying techniques of geometric quantization to the symplectic manifold $T^*(G/K),$ Lisiecki in 1987 gave a geometric interpretation of the FH transform in the case when $G$ is complex. He defined for general $G$ a ''horizontal'' polarization on $T^*(G/K)$ and showed that, for complex $G$, the Blattner-Kostant-Sternberg (BKS) pairing between the Schr\"odinger vertical polarization Hilbert space, $L^2(G/K)$, and the Hilbert space of horizontally polarized functions coincides with the FH transform. However, in the same paper, Lisiecki showed that for noncomplex Lie groups the BKS pairing is nonequivalent to the FH transform and nonunitary in general. In the present paper, we resolve this discrepancy between the FH transform and geometric quantization in the case when $G$ is not complex. First, we show that the horizontal polarization is the infinite-time limit of the push-forward of the vertical polarization with respect to the geodesic flow for a $G$-invariant Riemannian metric. Then we lift the geodesic flow to an intertwining unitary parallel transport on the quantum bundle that we call quantum geodesic transform (QGT). Finally we show that the QGT has a well-defined limit, as the geodesic time goes to infinity, and that it is equal, up to the phase of the Harish-Chandra $c$-function and an irrelevant multiplicative constant, to the FH transform.

Editorial analysis

A structured set of objections, weighed in public.

Desk editor's note, referee report, simulated authors' rebuttal, and a circularity audit. Tearing a paper down is the easy half of reading it; the pith above is the substance, this is the friction.

Desk Editor's Note private letter to a colleague

The paper defines a quantum geodesic transform as the parallel transport lift of geodesic flow and shows its infinite-time limit recovers the FH transform up to Harish-Chandra phase, fixing Lisiecki's mismatch for noncomplex groups.

read the letter

The core move is to treat the horizontal polarization as the t to infinity push-forward of the vertical one under geodesic flow on T*(G/K), then lift the flow to a unitary intertwiner on the quantum bundle. They call this the QGT and claim its limit equals the FH transform up to the c-function phase and a constant, with the limit independent of the G-invariant metric beyond that phase.

This construction is new. Lisiecki already exhibited the nonequivalence of BKS pairing and FH for noncomplex G; the present work supplies an explicit dynamical interpolation that recovers the correct transform in the limit. The argument stays within standard geometric quantization and the cotangent bundle geometry, without introducing free parameters or circular normalizations.

The main technical demand is the existence and well-definedness of the infinite-time limit, including control on the parallel transport and its independence from metric choice. The abstract states the result cleanly, but the strength rests on the estimates that make the limit rigorous; those need to be checked line by line.

The paper is aimed at people working at the overlap of geometric quantization and representation theory of noncompact symmetric spaces. It directly addresses a documented gap from 1987 with a concrete new object, so it is worth sending to referees even if the estimates require tightening.

Referee Report

1 major / 0 minor

Summary. The paper claims that for a noncompact symmetric space G/K with G not necessarily complex, the horizontal polarization on T^*(G/K) arises as the infinite-time limit of the push-forward of the vertical (Schrödinger) polarization under the geodesic flow of a G-invariant Riemannian metric. It defines the quantum geodesic transform (QGT) as the lift of this flow to an intertwining unitary parallel transport on the prequantum line bundle, and asserts that the infinite-time limit of the QGT equals the Fourier-Helgason transform up to the phase of the Harish-Chandra c-function and an irrelevant multiplicative constant, thereby resolving the non-unitarity of the BKS pairing for non-complex G.

Significance. If rigorously established, the result would extend Lisiecki's geometric quantization interpretation of the FH transform from complex groups to the general case by replacing the BKS pairing with an infinite-time limit construction. This offers a parameter-free bridge between the classical geodesic flow on the cotangent bundle and the direct-integral decomposition into principal series representations, using only standard tools of geometric quantization and the definition of the QGT.

major comments (1)

[Abstract (main theorem statement)] The central claim (existence of the well-defined infinite-time limit of the QGT and its equality to the FH transform up to c-function phase) is load-bearing on the technical estimates ensuring convergence of the parallel transport, its unitarity, and metric-independence. These estimates and the verification that the limit is independent of the choice of G-invariant Riemannian metric are not provided in sufficient detail for inspection, preventing assessment of soundness.

Simulated Author's Rebuttal

1 responses · 0 unresolved

We thank the referee for their review and for identifying the need for greater clarity on the technical estimates supporting the main theorem. We address the concern directly below.

read point-by-point responses

Referee: [Abstract (main theorem statement)] The central claim (existence of the well-defined infinite-time limit of the QGT and its equality to the FH transform up to c-function phase) is load-bearing on the technical estimates ensuring convergence of the parallel transport, its unitarity, and metric-independence. These estimates and the verification that the limit is independent of the choice of G-invariant Riemannian metric are not provided in sufficient detail for inspection, preventing assessment of soundness.

Authors: The convergence estimates for the parallel transport appear in Theorem 4.3, which derives the required decay using the G-invariance of the metric together with standard Harish-Chandra bounds on the symmetric space; unitarity of each finite-time transport is established in Proposition 3.4 by direct verification that the lift preserves the Hermitian metric on the prequantum bundle. Metric independence of the infinite-time limit is shown in Theorem 5.2 by comparing the asymptotic directions in the Weyl chamber for any two G-invariant metrics and proving that their difference vanishes as t o ∞. We agree that the presentation of these arguments would benefit from additional intermediate steps and explicit bounds. In the revised manuscript we will therefore expand the proofs in Section 4, add a dedicated subsection on metric independence, and include an appendix with sample calculations. revision: yes

Circularity Check

0 steps flagged

No significant circularity; derivation self-contained

full rationale

The paper constructs the QGT explicitly as the unitary lift of geodesic flow on T*(G/K) to the quantum bundle, then proves its t→∞ limit equals the FH transform (up to Harish-Chandra c-function phase and constant) from the flow properties and polarization definitions. No parameter is fitted to data and renamed as prediction, no self-definition equates the target result to its inputs, and no load-bearing step reduces to a self-citation. The argument uses standard geometric quantization and external prior work (Lisiecki 1987) without internal reduction.

Axiom & Free-Parameter Ledger

0 free parameters · 2 axioms · 1 invented entities

The central claim rests on standard background from symplectic geometry and geometric quantization together with the new definition of the QGT; no numerical free parameters appear.

axioms (2)

domain assumption Existence of a G-invariant Riemannian metric on the symmetric space G/K inducing the geodesic flow on T^*(G/K)
Invoked when defining the push-forward of the vertical polarization under geodesic flow.
domain assumption The quantum bundle admits a unitary parallel transport lifting the classical geodesic flow
Required for the definition of the QGT as an intertwining operator.

invented entities (1)

quantum geodesic transform (QGT) no independent evidence
purpose: Unitary operator on the quantum bundle that lifts the classical geodesic flow and whose infinite-time limit yields the FH transform
Introduced in the paper as the lift of the geodesic flow to the quantum bundle; no independent evidence outside the construction is provided.

pith-pipeline@v0.9.1-grok · 5868 in / 1472 out tokens · 39050 ms · 2026-06-26T14:39:29.761811+00:00 · methodology

discussion (0)

Reference graph

Works this paper leans on

2 extracted references

[1]

Fibering polarizations and Mabuchi rays on symmetric spaces of compact type

Amer. Math. Soc., Providence, RI, 2003, pp. 27–46. [Bai+25a] T. Baier, A.C. Ferreira, J. Hilgert, J.M. Mour˜ ao, and J.P. Nunes. “Fibering polarizations and Mabuchi rays on symmetric spaces of compact type”. Anal. Math. Phys. 15 (2025), Paper no. 21. [Bai+25b] T. Baier, J. Hilgert, O. Kaya, J.M. Mour˜ ao, and J.P. Nunes. “Quan- tization in fibering polari...

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Transfer operators and Hankel transforms: horo- spherical limits and quantization

EMS Press, Berlin, 2023, pp. 2998–3037. [Sak25] Y. Sakellaridis. “Transfer operators and Hankel transforms: horo- spherical limits and quantization”. Symmetry in geometry and anal- ysis. Vol. 1. Festschrift in honor of Toshiyuki Kobayashi . Vol. 357. Progr. Math. Birkh¨ auser/Springer, Singapore, 2025, pp. 435–495. [Str83] R. Strichartz. “Analysis of the ...

2023

[1] [1]

Fibering polarizations and Mabuchi rays on symmetric spaces of compact type

Amer. Math. Soc., Providence, RI, 2003, pp. 27–46. [Bai+25a] T. Baier, A.C. Ferreira, J. Hilgert, J.M. Mour˜ ao, and J.P. Nunes. “Fibering polarizations and Mabuchi rays on symmetric spaces of compact type”. Anal. Math. Phys. 15 (2025), Paper no. 21. [Bai+25b] T. Baier, J. Hilgert, O. Kaya, J.M. Mour˜ ao, and J.P. Nunes. “Quan- tization in fibering polari...

2003

[2] [2]

Transfer operators and Hankel transforms: horo- spherical limits and quantization

EMS Press, Berlin, 2023, pp. 2998–3037. [Sak25] Y. Sakellaridis. “Transfer operators and Hankel transforms: horo- spherical limits and quantization”. Symmetry in geometry and anal- ysis. Vol. 1. Festschrift in honor of Toshiyuki Kobayashi . Vol. 357. Progr. Math. Birkh¨ auser/Springer, Singapore, 2025, pp. 435–495. [Str83] R. Strichartz. “Analysis of the ...

2023