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

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 $L<sup>2(G/K)$ into irreducible principal series representations. By applying techniques of geometric quantization to the symplectic manifold $T<sup>*(G/K),$ 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 $T<sup>*(G/K)$ and showed that, for complex $G$, the Blattner-Kostant-Sternberg (BKS) pairing between the Schrödinger vertical polarization Hilbert space, $L<sup>2(G/K)$, 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.

• The paper establishes that infinite-time geodesic flow in geometric quantization yields a unitary realization of the Fourier–Helgason transform, resolving discrepancies for noncomplex groups.
• It employs a novel quantum geodesic transform by evolving the Schrödinger polarization via real-time Hamiltonian flow, creating a bridge to the Fourier (horizontal) polarization.
• Key results include explicit asymptotic convergence with phase correction using the Harish-Chandra c-function and implications for spectral decomposition in representation theory.

Summary of "Fourier-Helgason transform as infinite geodesic time limit in geometric quantization" (2606.20290)

Context and Motivation

The paper examines the interplay between representation theory, symmetric spaces, and geometric quantization. The Fourier–Helgason (FH) transform is a fundamental tool for decomposing unitary representations of noncompact semisimple Lie groups $G$ on $L^2(G/K)$, where $K$ is a maximal compact subgroup, into irreducible principal series representations. Prior work (notably Lisiecki, 1987) provided a geometric quantization interpretation when $G$ is complex but left unresolved issues for non-complex $G$: the Blattner–Kostant–Sternberg (BKS) pairing does not align with the FH transform nor is it unitary in general.

Main Contributions

The paper resolves the aforementioned discrepancy for non-complex $G$ by employing real-time Hamiltonian evolution within geometric quantization. The authors construct a continuous deformation of polarizations on $T^*(G/K)$ via geodesic flow, culminating in the FH transform as the infinite geodesic time limit. Key innovations include:

• Characterizing Polarization Evolution: The vertical (Schrödinger) polarization evolves under geodesic flow, interpolating continuously to the Fourier (horizontal) polarization.
• Quantum Geodesic Transform (QGT): Lifting the classical geodesic flow to the quantum bundle yields a unitary correspondence between polarized Hilbert spaces, analogous to generalized coherent state transforms for Hamiltonian flows in imaginary time but realized here in real time.
• Infinite-Time Limit: The QGT admits a well-defined infinite-time limit, which matches the FH transform up to the phase of the Harish-Chandra $c$-function and normalization.

Technical Details

Geometric Quantization Framework

Let $X = G/K$ be a symmetric space of noncompact type. $T^*X$ is realized as the symplectic reduction of $L^2(G/K)$0 with respect to the right $L^2(G/K)$1 action, and polarizations are considered as foliations on $L^2(G/K)$2. The Schrödinger polarization corresponds to $L^2(G/K)$3. The Fourier polarization, found by Lisiecki, yields functions on $L^2(G/K)$4, mirroring the image of FH.

Geodesic Flow and Polarizations

The geodesic flow generated by a $L^2(G/K)$5-invariant metric induces a family of polarizations $L^2(G/K)$6. The push-forward of the vertical polarization under this flow converges, as $L^2(G/K)$7, to the Fourier polarization. This convergence is rigorously established using representation-theoretic and symplectic geometric arguments.

Quantum Geodesic Transform

The QGT, defined via a combination of the prequantum operator associated to the quadratic Casimir and the quantized Hamiltonian evolution, lifts the geodesic flow to the quantum bundle: $L^2(G/K)$8 where $L^2(G/K)$9 and $K$0 are the Kostant-Souriau prequantum operator and the quantum operator, respectively.

This operator intertwines the Hilbert spaces associated with different polarizations, respecting $K$1-equivariance and unitarity for all $K$2.

Infinite-Time Limit and FH Transform

Analyzing the asymptotics, the authors show: $K$3 where $K$4 is the FH transform (up to phase and normalization, specifically the Harish-Chandra $K$5-function). The image is the Hilbert space of sections polarized with respect to the Fourier polarization, $K$6.

The analysis includes explicit calculations of asymptotic behavior of canonical forms, normalization constants, and volume measures relevant for the induced transformations.

Numerical and Structural Results

• Unitarity: The QGT and its limit are unitary (up to constant), confirming equivalence of quantizations for the vertical and Fourier polarizations.
• FH Transform Recovery: The infinite-time limit recovers the FH transform, including its spectral decomposition and Plancherel measure.
• Phase Correction: The phase of the Harish-Chandra $K$7-function is explicitly identified as a necessary correction absent in prior geometric quantization interpretations.

Implications and Future Directions

The results clarify the geometric origin of the FH transform for general semisimple Lie groups, providing a template for geometric quantization interpretations of harmonic analysis transforms beyond the complex group case. This has implications in the following domains:

• Langlands Program: The identification of the FH transform as a polarization limit offers an avenue for interpreting transfer operators and horospherical limits in the geometric Langlands context, particularly for symplectic quotients and quantum Hamiltonian reduction of homogeneous spaces.
• Generalized Quantization Schemes: The real-time flow approach suggests new families of unitary quantizations for other homogeneous symplectic spaces and may be extensible to infinite-dimensional settings.
• Representation Theory and Transfer Operators: The equivalence between Schrödinger models for different polarizations has direct utility for intertwining operators and spectral decomposition methods (e.g., transfer operators arising in automorphic representation theory).

Conclusion

This paper rigorously establishes equivalence between geometric quantizations associated to vertical and Fourier polarizations on $K$8, resolving previous gaps for non-complex Lie groups. The quantum geodesic transform delivers a unitary route to the Fourier–Helgason transform as an infinite-time geodesic limit, enabling new perspectives in harmonic analysis, geometric quantization, and their applications in representation theory.

The methodology and results offer pathways for further exploration of geometric quantization in connection with harmonic analysis and automorphic representation theory, potentially influencing developments in symplectic geometry, quantum transfer operators, and the geometric Langlands program.