{"id":"8e91e8e3-afea-45e4-a850-79a153f6cc96","arxiv_id":"2501.04621","paper_version":1,"verdict":"CONDITIONAL","confidence":"MODERATE","novelty_score":5.0,"correctness_risk":"medium","formal_verification":"none","parameter_count":0,"one_line_summary":"The Mellin transform of a Radon-transformed (auxiliary) four-point function reproduces the Lorentzian OPE partial wave amplitudes, giving a geometric projection-slice interpretation.","lead":"This paper recasts the Lorentzian OPE inversion formula of conformal field theory as a problem in tomography: an auxiliary four-point function is defined by a Radon transform, and its Mellin transform gives the same partial wave amplitudes as the standard spherical transform. The result connects conformal field theory to classical harmonic analysis and provides explicit calculations for 1D and 2D CFTs.","discovery_kind":"new_application","skeptic_critique":{"model":"deepseek-v4-flash","headline":"The reconstruction of f from a(λ) in Sec. 4.1.3 relies on an unproven analyticity assumption for a(λ) in the right half-plane; without it Eqs. 4.15 and 4.20 fail.","rationale":"The reader's weakest_assumption identifies exactly the same load-bearing point: the analyticity of the partial wave amplitudes a(λ) in the right half-plane, introduced as an assumption after Eq. 4.14 and used to drop the boundary term in Eq. 4.15 and to symmetrize the integrand in Eq. 4.20. This is not a minor technicality but a necessary spectral condition for the inverse Mellin transform to recover the original correlator. The paper does not prove this condition from CFT principles; it is particularly non-trivial because Lorentzian correlators are distributions and the relevant functions are H-bi-invariant rather than K-bi-invariant, so the usual Euclidean spherical-transform framework does not apply. The d=2 worked example only covers Re(Δ)<1, which leaves the physically important regime Δ>1 untreated. Because the reader already marks the paper CONDITIONAL and identifies this concern, the stress-test does not change the verdict. No ad hominem or theatrical language is needed; the issue is a concrete gap in the analytic justification of the inversion procedure.","tokens_in":178,"tokens_out":6877,"duration_ms":109519,"concrete_test":"Evaluate the generalized free-field dDisc in d=1 (e.g., f(cosh η) ∝ (cosh η − 1)^{−2Δ} with a suitable regularization) for Δ=1 and Δ=2. Compute the Radon transform Ff via Eq. 4.10, then the partial wave amplitude a(λ) via Eq. 4.11 as a function of eλ. Check whether a(λ) is analytic and decays for Re(eλ)>1/2, and whether the integral in Eq. 4.15 tends to zero as η_I→∞. If it does not, the reconstruction Eq. 4.20 is not valid and the inversion formula would need a different spectral assumption.","verdict_should_be":"UNCHANGED","load_bearing_attack":"Section 4.1.3 (Eqs. 4.14–4.20) reconstructs the reduced four-point function f from partial wave amplitudes a(λ) by an inverse Laplace transform followed by an inverse Radon transform. The passage from Eq. 4.18 to Eq. 4.20 inserts a zero term that is justified only by the statement after Eq. 4.14: 'if we assume that the partial wave amplitudes a(λ) are analytic to the right...' This same assumption is used to discard Eq. 4.15. For Lorentzian CFT correlators, which are distributions on causal semigroups, the paper offers no argument that a(λ) inherits this analyticity from the CFT; it is not a consequence of the group-theoretic construction in Sec. 3. The condition is genuinely load-bearing: if a(λ) has any singularity or non-decaying behavior for Re(eλ)>1/2, the boundary term in Eq. 4.15 does not vanish and the symmetric integrand in Eq. 4.20 is not equivalent to the original inverse Laplace integral. The d=2 example in Sec. 4.2.2 sidesteps this because it is computed only for Re(Δ)<1, and the paper does not state what happens for physical operators with Δ>1. Thus the claimed 'solves the problem of inversion' is not established for generic unitary CFTs.","agreement_with_reader":"agree"},"referee_report":{"model":"deepseek-v4-flash","summary":"This paper proposes a geometric reinterpretation of the Lorentzian OPE inversion formula of Caron-Huot and of the authors' earlier work [3]. The reduced four-point function f(u,v), or its double discontinuity, is treated as an H-bi-invariant function on the conformal group, and a new \"auxiliary\" four-point function Ff is defined as its horocyclic (Radon/Abel) transform, Eq. (4.5). The main claim, Eq. (4.6), is that the Mellin/Laplace transform of Ff equals the conventional partial wave amplitudes a(λ) of f, presented as a CFT analogue of the projection-slice theorem (Fig. 1.1). The forward equivalence is verified explicitly for d = 1 in Eq. (4.12), and the generalized free-field example is computed for d = 2 in Eq. (4.29), where the SL(2,R) × SL(2,R) structure and the root-system symmetry of the partial waves are discussed (Sec. 4.2.3). The paper then attempts the inverse step, reconstructing f from a(λ), in Sec. 4.1.3 via an inverse Laplace transform followed by an inverse Abel transform, and connects the result to the conformal block expansion (Eqs. (4.20)–(4.22)).","tokens_in":21249,"tokens_out":35764,"duration_ms":301289,"significance":"If fully established, the framework provides a genuinely different and geometrically transparent view of the known Lorentzian inversion formula: partial wave amplitudes are Mellin transforms of the horocyclic Radon transform of the double discontinuity, and the Minkowski conformal blocks are recovered from the Radon kernel (Eq. (4.13)). The paper contains several concrete and verifiable computations, notably the d = 1 equivalence (Eq. (4.12)), the Radon transform of the d = 2 generalized free field (Eq. (4.26)), and the partial wave amplitude (Eq. (4.29)); its identification of the discrete symmetry of the d = 2 partial waves with the D2 root-system symmetry (Sec. 4.2.3) is a nice structural insight, and the connection to the SYK and 2d SYK analyses [40, 43] anchors the construction. These are real strengths. However, the paper is explicitly a proof of concept (Sec. 5): the general-d statement is acknowledged to be open, the d = 2 example is restricted to Re(∆) < 1, and the inverse reconstruction of Sec. 4.1.3 relies on an analyticity assumption not derived from CFT properties.","major_comments":[{"comment":"The reconstruction of the reduced four-point function from the partial wave amplitudes is load-bearing for the paper's claim that the framework \"allows one to address the issue of inversion for causal CFT partial wave amplitudes\" (Sec. 1). The argument drops Eq. (4.15) and inserts the symmetric cosh term in Eq. (4.20) solely on the basis of the sentence after Eq. (4.14): \"if we assume that the partial wave amplitudes a(λ) are analytic to the right...\". No argument is given that the partial wave amplitudes of a Lorentzian CFT satisfy this analyticity or the needed decay at infinity for closing the contour, and the paper itself notes in Sec. 3 that Minkowski four-point functions are distributions on causal semigroups. The assumption does not follow from the group-theoretic construction of Sec. 3 and is not cosmetic: if a(λ) has any singularity for Re(eλ) > 1/2 (Re(λ) > 0), the boundary term in Eq. (4.15) contributes and Eq. (4.20) is not equivalent to Eq. (4.18). Note also that the identity operator (∆ = 0) sits exactly on the contour Re(eλ) = 1/2, so the contour itself needs a defining prescription. The authors should either prove the required analyticity and decay from the Regge growth and OPE convergence of the double discontinuity, or state them explicitly as hypotheses and delimit the class of theories for which the inversion is claimed; as written, the inversion result is conditional on an unverified property.","section":"Sec. 4.1.3, Eqs. (4.14)–(4.20)"},{"comment":"The d = 2 example, one of the two advertised explicit cases, has two gaps. First, Eq. (4.26) is stated to hold \"only if R(∆) < 1\" (i.e., Re(∆) < 1), which is a genuine convergence bound for the Abel integral; however, the physically relevant generalized free-field double-trace operators of an external primary of dimension Δ_φ ≥ 1/2 have ∆ = 2Δ_φ ≥ 1, so the example does not cover typical unitary cases. The paper does not give an analytic continuation in ∆ (e.g., a meromorphic continuation of Eq. (4.26)) and does not state what happens in the regime Re(∆) ≥ 1. Second, the steps from the Laplace-transform integral (4.27) to the final amplitude (4.29) are not shown: the integral evaluation, the convergence conditions, and the resulting pole structure — which underlies the symmetry discussion in Sec. 4.2.3 — are simply asserted. Since Eq. (4.29) is the main quantitative output for d = 2, the derivation should be presented, or at minimum its convergence domain and analytic structure should be stated explicitly.","section":"Sec. 4.2.2, Eqs. (4.25)–(4.29)"},{"comment":"The main equivalence (4.6) is derived by interchanging the N+ integral and the AI integral in Eq. (4.4) and identifying the result with the spherical transform (4.2), with the paper itself noting after Eq. (4.2) that this \"assumes square integrability of the basis functions\". For the double-discontinuity functions of interest, which are distributions with power-law singularities on the lightcones (e.g., Eq. (4.25) behaves as (cosh(y ± η) − 1)^{−∆} near contact) and Regge growth at large rapidity, neither the existence of the integrals nor the Fubini step is automatic; indeed, the same convergence issue forces the restriction Re(∆) < 1 in the d = 2 example (Eq. (4.26)). The explicit d = 1 verification in Eq. (4.12) supports the equivalence at a formal level for that case, and the paper is admirably explicit that the general-d problem is open (Secs. 1 and 5), but the statement \"This is our main result\" (Sec. 1) for Eq. (4.6) should be accompanied by a precise statement of the function space or growth conditions under which the Radon and Mellin transforms and their interchange are defined.","section":"Sec. 4, Eqs. (4.4)–(4.6)"}],"minor_comments":[{"comment":"The exponent on (cosh ηI − cosh η) is printed as +1/2 in Eqs. (4.8) and (4.9); consistency with the Radon transform in Eq. (4.10) and with App. C, Eq. (C.8), requires −1/2. As printed, the measure contradicts the transform that immediately follows.","section":"Sec. 4.1.2, Eqs. (4.8)–(4.9)"},{"comment":"The notation is very hard to follow: λ, λ*, eλ, and e∆ are used with overlapping meanings, with \"λ = −1/2 − e∆\", \"λ* = −1/2 + e∆\", and \"∆ = 1/2 − eλ\" all in play within a few equations. A summary table of transform variables and conjugate integration contours would remove persistent sign ambiguities.","section":"Sec. 4.1, Eqs. (4.11)–(4.15)"},{"comment":"The definition of (w, σ) is typeset without parentheses and is ambiguous as printed; presumably the intended definitions are w = (1 − √v)/√u and σ = (1 + √v)/√u.","section":"Sec. 2, Eq. (2.2)"},{"comment":"On p. 2, \"the well known F HAcycle\" should be \"the well-known FHA cycle\" (Fourier–Hankel–Abel cycle), and reference [17] for the projection-slice theorem is a Wikipedia article; a monograph (e.g., Helgason's work, already cited as [41]) would be more appropriate.","section":"Sec. 1 and References"},{"comment":"The subscript ℓ in F_{∆,ℓ}(yI, ηI) is introduced without definition; the generalized free-field double discontinuity is ℓ-independent, and ℓ enters only through the Laplace labels in Eq. (4.28), so the notation is misleading. Also, \"R(∆)\" in the sentence after Eq. (4.26) should be \"Re(∆)\".","section":"Sec. 4.2.2, Eq. (4.26)"},{"comment":"The sentence \"When the contour is closed in the above representation, one recovers the conformal block expansion of the fourpoint function\" is not demonstrated; in light of the analyticity assumption discussed in the major comments, this step deserves a full contour-closing argument with the pole locations and the discrete-series contributions spelled out.","section":"Sec. 4.1.3, after Eq. (4.22)"}],"recommendation":"major_revision","confidential_remarks":"The paper is a sequel to the authors' own [3] and leans heavily on it for the Master Equation, the zonal spherical functions, and the semigroup kinematics; readers unfamiliar with [3] will find much of Secs. 2–3 opaque, and the manuscript is not fully self-contained. The novelty is a structural and geometric reformulation of a known physical result, not a new prediction, which is a legitimate contribution for a journal like PRD or JHEP provided the analyticity gap in the inverse direction is dealt with. The stress-test concern from the reader's report is well founded: the analyticity assumption after Eq. (4.14) is real, flagged by the authors themselves, and load-bearing. I would support publication after a revision that clearly separates the proven forward direction from the conditional inverse direction and completes the d = 2 Laplace transform computation."},"author_rebuttal":null,"desk_editor":{"model":"deepseek-v4-flash","letter":"The paper is an honest attempt to recast the Lorentzian OPE inversion formula as a two-step transform: a Radon transform on the conformal group followed by a Mellin transform. This is a nice unifying picture, and the d=1 derivation is explicit and convincing. The d=1 check that the Mellin transform of the Radon transform equals the standard spherical transform (Eq. 4.12) is a real result, and the identification with the projection-slice theorem is apt. Credit where due: the authors know the classical harmonic analysis (Lang, Sugiura) and they are clear that the factorization itself is not new; what is new is the application to H-bi-invariant functions and to the double-discontinuity of the four-point function. The d=2 example, despite being limited to Re(Δ)<1, demonstrates the mechanism and shows the Δ vs ℓ symmetry of the amplitudes.\n\nBut the paper overstates what it proves. The inverse reconstruction in Sec. 4.1.3 relies on the partial wave amplitudes a(λ) being analytic in the right half-plane, used both to drop the boundary term (Eq. 4.15) and to add a zero term (Eq. 4.20). This is not a technicality: for Minkowski CFT correlators, which are distributions on causal semigroups, there is no argument that a(λ) has the needed analyticity and decay. The paper simply asserts it after Eq. 4.14. Without it, the claimed inversion formula is not established for generic unitary CFTs. The d=2 example sidesteps this by working only at Re(Δ)<1, and says nothing about the physical region Δ>1. So the headline claim that this 'solves the problem of inversion' is too strong; what is solid is the forward equivalence, which is a useful reformulation.\n\nThe d=2 derivation also skips the Laplace transform details leading to Eq. 4.29, which makes it harder to check, but this is a minor issue. The reliance on the previous paper [3] for the Master Equation and zonal spherical functions means the paper is not self-contained, though that is a matter of presentation rather than a flaw.\n\nWho is this for: bootstrap practitioners who want a geometric handle on the inversion formula and a concrete d=1,d=2 playground. It deserves a serious referee; the gap in the inverse step is repairable, either by proving the analyticity for well-defined classes of correlators (e.g., meromorphic partial waves as in SYK) or by recasting the inversion distributionally. I would send it out, and ask the authors to either prove or clearly flag the analyticity assumption.","headline":"Genuinely useful geometric repackaging of the Lorentzian OPE inversion, but the inverse step has an unproven analyticity assumption that keeps the advertised 'solution' from being a theorem.","tokens_in":21785,"tokens_out":4913,"would_cite":true,"duration_ms":46036,"reading_group":"yes","serious_thinker":"yes","would_accept_peer_review":true},"rs_alignment":null,"lean_confirmation":null,"pith_extraction":{"msc":["81T40","22E70","43A85","44A12"],"pacs":["11.25.Hf"],"model":"deepseek-v4-flash","headline":"Mellin transforms of horocyclic Radon slices equal the partial wave amplitudes of the original CFT four-point function.","keywords":["Lorentzian OPE inversion formula","conformal partial waves","Radon transform","Mellin transform","double discontinuity","causal spherical functions","conformal field theory","harmonic analysis"],"falsifier":"Take a known causal four-point function whose partial wave amplitudes have a singularity or branch structure reaching into the right half-plane $\\mathrm{Re}\\,\\lambda>0$ (for instance a theory with an accumulation of Regge poles), and check whether the contour-closing reconstruction of Eqs. 4.14--4.20 reproduces the original double discontinuity; any pole or cut to the right makes the dropped boundary term nonzero and the reconstruction fails. A simpler version: compute the $d=1$ Abel and Laplace transforms for a compactly supported distributional $f$ whose transform has a right-half-plane singularity and verify that the two-sided inversion no longer returns $f$.","tokens_in":20766,"feed_emoji":"📐","tokens_out":10699,"duration_ms":88500,"temperature":0.7,"pith_summary":"The paper claims that the standard Lorentzian OPE inversion formula—the procedure that extracts the operator spectrum (dimensions and spins) of a conformal field theory from the double discontinuity of a four-point function—can be reorganized as a two-step tomographic procedure. First one integrates the reduced four-point function along horocycles, a Radon/Abel transform; then one takes a Mellin transform of the resulting auxiliary function. The paper's central equation (4.6) says this composition returns exactly the partial wave amplitudes of the original correlator. This matters because Minkowski-space conformal partial waves are distributions and do not form a complete orthonormal basis, which previously blocked a clean inversion; the Radon-then-Mellin route bypasses that obstruction. The claim is established explicitly for spacetime dimensions $d=1$ and $d=2$, with a framework laid out for general $d$.","feed_headline":"Radon plus Mellin: CFT inversion as tomography","feed_subtitle":"For four-point functions, causal inversion is a projection-slice theorem: Radon then Mellin, and the OPE data falls out.","key_machinery":"The load-bearing object is the Master Equation (Eq. 3.2), $N A_I P A_I^T N^T = H A P A^T H^T$, which relates the Iwasawa decomposition of the coset $G/H$ to the Cartan decomposition. It lets integrals over the subgroup $H$ be re-expressed as integrals over the nilpotent subgroup $N_+$ and the maximal abelian subgroup $A_I$, giving the horocyclic Radon transform $Ff(a_I)=\\int_{N_+} f(n_+ a_I)\\,dn_+$. The accompanying Mellin transform $\\widetilde{Ff}(\\lambda)=\\int_{A_I} Ff(a_I)e^{\\lambda^* a_I}\\alpha\\,da_I$ converts the Radon-transformed auxiliary four-point function into the partial wave amplitudes of the original correlator.","core_discovery":"The central claim is that $\\hat{f}(\\lambda)=\\int Ff(a_I)\\, e^{\\lambda^* a_I}\\,\\alpha\\, da_I$ (Eq. 4.6): the spherical transform of an $H$-bi-invariant reduced four-point function equals the Mellin transform of its horocyclic Radon transform, where $Ff(a_I)=\\int_{N_+} f(n_+ a_I)\\,dn_+$ and $\\lambda$ labels the boost and dilatation eigenvalues. Since the $\\hat{f}(\\lambda)$ so obtained are the partial wave amplitudes of the original correlator, the Lorentzian inversion problem becomes a composition of two invertible transforms rather than a search for a complete orthonormal set of blocks. In $d=1$ the authors carry this out explicitly for $SO(1,2)/SO(1,1)$, where the Radon kernel is $(\\cosh\\eta_I-\\cosh\\eta)^{1/2}$, the partial wave amplitude is a Laplace transform of the Radon transform, and the inverse reproduces the four-point function through Legendre functions, recovering both principal and discrete series contributions. In $d=2$, the factorization $SO(2,2)\\cong SL(2,\\mathbb{R})\\times SL(2,\\mathbb{R})$ makes the Radon transform factorize, and the authors compute the full generalized-free-field amplitude (Eq. 4.29), which is symmetric under the scale-shadow flip $\\Delta\\to -\\Delta$. Geometrically, the statement is a generalization of the Projection-Slice Theorem: in the Iwasawa decomposition the transverse integrals are horocycles, and the remaining integration is a Laplace/Mellin transform.","pith_inferences":["The Radon-then-Mellin factorization suggests a directly implementable numerical pipeline: compute or model the double discontinuity on the causal wedge, apply a numerical Radon transform along horocycles, and Mellin transform to read off OPE data—this would bypass constructing conformal blocks entirely, though the paper offers no numerical demonstration.","If the analyticity assumption on $a(\\lambda)$ fails for realistic Minkowski correlators, the contour-closing steps (Eqs. 4.15 and 4.20) would acquire boundary contributions; reformulating the inversion distributionally, using wave-front sets rather than pointwise analyticity, is a natural extension the paper leaves open.","The projection-slice analogy may carry over to crossing: if the Radon transform intertwines the $s$, $t$, and $u$ channels in a simple way, the auxiliary four-point function could make crossing symmetry manifest in Mellin space, which would simplify bootstrap constraints.","The eight-chamber structure predicted for $d>2$ is testable already at the level of generalized free fields: computing the $d=3$ amplitude and checking whether it decomposes into the predicted Weyl-chamber sum would confirm or falsify the group-theoretic picture before any full inversion is attempted."],"forward_implications":["For $d=1$ CFTs, the inversion automatically produces both the principal and discrete series, with the discrete series cancelling spurious principal-series poles exactly as in the SYK conformal limit.","For $d=2$, the generalized free field partial wave amplitude (Eq. 4.29) is recovered, and its scale-shadow symmetry $\\Delta\\to -\\Delta$ reflects the semigroup invariance under $D\\leftrightarrow -D$; it matches the bosonic two-dimensional SYK amplitude under the stated identifications.","Eq. 4.13 gives a new integral representation of the zonal spherical functions (the principal-series continuation of Minkowski conformal blocks) as a Radon transform followed by an exponential integral.","The inversion can be run in reverse: inverse Mellin transform of the amplitude, then inverse Radon transform, reconstructs the double discontinuity and, by contour closure, the conformal block expansion with all physical residues.","For $d>2$ the restricted root system is of type $B_2$ and the positive roots have no reflection symmetry, so the paper expects a full partial wave basis to require eight symmetry-related terms, one per Weyl chamber."],"supporting_citations":[{"why":"It supplies the Lorentzian OPE inversion formula whose partial wave amplitudes this paper recovers via Radon plus Mellin transforms.","marker":"[1]"},{"why":"It provides the spacetime derivation of the same inversion formula and the double-discontinuity setup that the paper reinterprets.","marker":"[2]"},{"why":"It establishes the $H$-bi-invariant causal spherical functions and the Master Equation that the auxiliary-function construction builds on.","marker":"[3]"},{"why":"It defines Minkowski conformal blocks and the $(q,\\bar{q})$ variables used for the $d=2$ root-structure and symmetry analysis.","marker":"[20]"},{"why":"It gives the $SL(2,\\mathbb{R})$ harmonic-analysis framework, spherical transform as Radon plus Laplace/Mellin, that the paper generalizes.","marker":"[21]"},{"why":"It supplies the SYK conformal-limit partial wave computation whose discrete and principal series structure the $d=1$ result reproduces.","marker":"[40]"},{"why":"It offers the bosonic two-dimensional SYK amplitude to which the generalized free field result of Eq. 4.29 is compared.","marker":"[43]"}],"fun_headline_variants":["CFT inversion as a projection-slice theorem","Radon transform for OPE inversion","Tomography view of CFT four-point functions","Projection-slice theorem meets CFT","Lorentzian OPE as Radon-Mellin composition"],"cache_read_input_tokens":3200,"weakest_assumption_plain":"The argument requires the partial wave amplitudes $a(\\lambda)$ to be analytic in the right half-plane, with all physical singularities to the left, so that a boundary term at infinity (Eq. 4.15) vanishes and a zero can be inserted in Eq. 4.20; this analyticity is assumed, not proven, for the distributional correlators that occur in Minkowski CFT.","fun_headline_variants_meta":{"raw":{"variants":["CFT inversion as a projection-slice theorem","Radon transform for OPE inversion","Tomography view of CFT four-point functions","Projection-slice theorem meets CFT","Lorentzian OPE as Radon-Mellin composition"]},"model":"deepseek-v4-flash","effort":"low","cost_usd":0.00107,"raw_usage":{"total_tokens":4487,"prompt_tokens":958,"completion_tokens":3529,"prompt_tokens_details":{"cached_tokens":384},"prompt_cache_hit_tokens":384,"prompt_cache_miss_tokens":574,"completion_tokens_details":{"reasoning_tokens":3457}},"tokens_in":574,"tokens_out":3529,"duration_ms":22047,"temperature":1.0,"reasoning_tokens":3457,"cache_read_input_tokens":384,"cache_creation_input_tokens":0},"cache_creation_input_tokens":0},"created_at":"2026-08-10T21:29:27.939725+00:00","model_set":{"reader":"deepseek-v4-flash"},"falsifier":"Take a known causal four-point function whose partial wave amplitudes have a singularity or branch structure reaching into the right half-plane $\\mathrm{Re}\\,\\lambda>0$ (for instance a theory with an accumulation of Regge poles), and check whether the contour-closing reconstruction of Eqs. 4.14--4.20 reproduces the original double discontinuity; any pole or cut to the right makes the dropped boundary term nonzero and the reconstruction fails. A simpler version: compute the $d=1$ Abel and Laplace transforms for a compactly supported distributional $f$ whose transform has a right-half-plane singularity and verify that the two-sided inversion no longer returns $f$.","supporting_citations":[{"cited_title":"Minkowski Conformal Blocks and the Regge Limit for SYK-like Models","cited_arxiv_id":"1801.04208","evidence_quote":"It defines Minkowski conformal blocks and the $(q,\\bar{q})$ variables used for the $d=2$ root-structure and symmetry analysis."},{"cited_title":"Lang, SL2(R), Springer-Verlag (1974)","cited_arxiv_id":null,"evidence_quote":"It gives the $SL(2,\\mathbb{R})$ harmonic-analysis framework, spherical transform as Radon plus Laplace/Mellin, that the paper generalizes."}],"review_version":1}