{"id":"f81f58fc-3956-40fc-a6d0-270196ba47ba","arxiv_id":"2507.18285","paper_version":2,"verdict":"ACCEPT","confidence":"MODERATE","novelty_score":5.0,"correctness_risk":"medium","formal_verification":"none","parameter_count":0,"one_line_summary":"For regular dominant weights λ, the kλ-th equivariant pieces of the Szegő and Poisson kernels on the Grauert tube boundary admit explicit near-diagonal scaling asymptotics, with Gaussian decay away from the coadjoint-orbit locus.","lead":"This paper derives exact asymptotic formulas for high-weight components of two operator kernels associated to a compact Lie group and its complexification. The result tells you precisely where complexified eigenfunctions concentrate, a building block for nodal and concentration studies.","discovery_kind":"extension","skeptic_critique":{"model":"deepseek-v4-flash","headline":"Theorem 1.9 omits a nontrivial Reeb phase that the proof's own stationary-phase calculation produces; as stated, the near-diagonal asymptotics are false for θ1≠θ2.","rationale":"The reader's ACCEPT verdict rests on Theorem 1.9, which is the paper's central asymptotic statement. The proof's own Eq. (125) and Lemma 5.15 lead to a stationary-phase critical value (∥λ∥/τ)(θ1−θ2), so the leading asymptotics must contain e^{i√k(∥λ∥/τ)(θ1−θ2)} unless θ1=θ2 is assumed. The theorem as printed omits this factor, and the omission cannot be absorbed into the polynomial remainder series because √k(θ1−θ2) can grow like k^ε while the series only contains powers k^{−j/2} times polynomials in variables of size k^{ε−1/2}. This is an internal inconsistency of the stated theorem, not a disagreement with an outside consensus. The imported NHLC phase expansion is a legitimate second concern, but even granting that expansion, the final stationary-phase step already produces a factor that Theorem 1.9 does not display. The evidence for the paper's broader program remains strong: the torus example, the detailed derivations, and the applications all suggest the intended formula can be repaired by restoring the Reeb phase. Because the theorem as stated is false but evidently fixable, the appropriate verdict is CONDITIONAL rather than outright rejection.","tokens_in":41630,"tokens_out":15201,"duration_ms":163503,"concrete_test":"Use the closed-form torus example of §3 with G=T^d, where P^τ_{kλ}(x1,x2)=e^{-2τ c_{kλ}}\\tildeφ_{kλ}(x1)\\overline{\\tildeφ_{kλ}(x2)}. Fix x∈X_τ^O and take pure Reeb displacements x_{jk}=x+(θ_j/√k,0,0) with θ1≠θ2 and |θj|≤Ck^{ε−1/2}. Compute the ratio P^τ_{kλ}(x_{1k},x_{2k})/P^τ_{kλ}(x,x) explicitly from the formula for \\tildeφ_{kλ}; it contains the phase e^{i√k(∥λ∥/τ)(θ1−θ2)}. Setting n_j=s_j=0 in Theorem 1.9 gives instead a leading term independent of θ1−θ2, hence a ratio 1+o(1), contradicting the explicit torus computation. This check would settle whether the missing Reeb phase is a genuine omission.","verdict_should_be":"CONDITIONAL","load_bearing_attack":"The proof's phase in Eq. (125), Ψ_r = u(θ1−θ2)+(uτ−∥λ∥)b − γ·Zλρ + R2(...), has by Lemma 5.15 the unique critical point P0 = (∥λ∥/τ, (θ2−θ1)/τ, 0, 0) and critical value (∥λ∥/τ)(θ1−θ2). The stationary phase evaluation in (u,b,ρ,γ) therefore contributes an explicit oscillatory factor e^{i√k(∥λ∥/τ)(θ1−θ2)}. This factor is absent from the displayed formula after Eq. (131) and from the statement of Theorem 1.9. Since θj is allowed to be of size O(k^{ε−1/2}), the exponent √k(θ1−θ2) is O(k^ε)→∞, so this is a genuinely oscillatory leading-order phase, not a negligible remainder. Moreover, no asymptotic series of the form 1+Σ_{j≥1} k^{−j/2}R_j(θ1,θ2,...) with polynomial R_j can reproduce e^{i√k(θ1−θ2)} for θj=O(k^{ε−1/2}). Thus Theorem 1.9 is internally inconsistent unless one imposes θ1=θ2 or restores the missing phase. The imported NHLC phase expansion flagged by the reader is a real concern, but this missing phase is a more immediate and concrete defect in the central claim: it appears from the authors' own stationary-phase computation, independently of the imported Proposition 48.","agreement_with_reader":"partial"},"referee_report":{"model":"deepseek-v4-flash","summary":"The paper studies the k→∞ scaling asymptotics of the λ-equivariant components Πτkλ and Pτkλ of the Szegő and Poisson kernels on the sphere bundle Xτ inside the complexification of a compact Lie group G, for a fixed dominant weight λ. The main results are rapid-decay theorems (Theorems 1.4–1.6) away from a locus ZτO attached to the coadjoint orbit of λ, and a near-diagonal asymptotic expansion (Theorem 1.9) in normal Heisenberg local coordinates, with additional applications to sup-norm bounds for complexified matrix elements, Husimi distributions, and Lp→Lq estimates. The proofs use the Weyl character formula, the Kirillov character formula, the Fourier-integral description of Szegő/Poisson kernels, normal Heisenberg local coordinates, and stationary phase.","tokens_in":41908,"tokens_out":6506,"duration_ms":69601,"significance":"If the central expansion is correct, the paper gives a fairly complete equivariant analogue, in the Grauert tube setting, of previously known line-bundle scaling asymptotics, with all constants expressed in terms of geometric invariants of the coadjoint orbit, the metric, and the stabilizer. The absence of fitted parameters, the use of the Kirillov character formula to avoid ad-hoc normalizations, and the concrete applications to sup-norms and Lp→Lq estimates are genuine strengths. However, the statement of Theorem 1.9 currently omits an oscillatory phase that the paper's own stationary-phase computation produces; as written, the theorem is false in the stated range θj=O(kε−1/2), θ1≠θ2. This makes the result not acceptable in its present form, although the defect appears to be local and repairable.","major_comments":[{"comment":"The stationary-phase evaluation of Ix,k(r) contains an explicit factor e^{i√k(∥λ∥/τ)(θ1−θ2)} coming from the critical value Ψr(P0)=∥λ∥τ−1(θ1−θ2). This factor is dropped in the final displayed expansion for Πτkλ(x1k,x2k) and in the statement of Theorem 1.9. Since θ1−θ2 is allowed to be of size O(kε−1/2), the exponent √k(θ1−θ2) is O(kε)→∞, so the missing factor is a genuine leading-order oscillation, not a remainder. Moreover no asymptotic series of the form 1+Σj≥1 k−j/2Rj(θ1,θ2,s1,s2,n1,n2) with polynomial coefficients can reproduce e^{i√k(θ1−θ2)} on that range. Thus Theorem 1.9 is internally inconsistent as stated unless θ1=θ2 is imposed or the oscillatory factor is restored.","section":"§5.4.2, Eq. (125), Lemma 5.15, and the displayed asymptotic expansion after (131); Theorem 1.9"},{"comment":"The third-order expansion of the Szegő phase in normal Heisenberg local coordinates is imported from Proposition 48 of [P-2024] and Lemma 64 of [GvP-2024] and is not proved or stated in this paper. This expansion is load-bearing: the Gaussian exponent and the polynomial remainder structure in Theorem 1.9 rest on it, and the uniformity in k and in the base point x is essential for the claimed uniform asymptotics. The authors should either give a precise statement of the imported result with the exact hypotheses (including the treatment of the remainder R3(•/√k)), or include a self-contained proof or appendix. I do not regard this as circularity, because the cited results are stated as proved elsewhere, but the dependence should be made explicit and verifiable in the present setting.","section":"§5.4.2, Eqs. (94) and (111)"}],"minor_comments":[{"comment":"The reference “[GP24]” should be “[GvP-2024]”; the same paper is also referred to as “[GvP-2024]” elsewhere. The notation for [P-2024] is also inconsistent: the text uses “[P2024]”, “[Pao2024]”, and “[P-2024]” for the same reference.","section":"§5.2.3, Eq. (70) and reference list"},{"comment":"The word “irreducuble” in the abstract is a typo and should be “irreducible”.","section":"Abstract"},{"comment":"The sentence “we shall equivalently write ψh2 = ψφ2 = ψγ2” is redundant and slightly confusing; the equivalence of the three notations should be stated once, and the notation ψh2 should then be used consistently.","section":"Definition 1.7"},{"comment":"The phrase “the previous expression being a polynomial in k” is imprecise: the displayed identity expresses dkλ as a factor times vol(Oλ)(1+O(k−1)), and the sentence should say that the bracket is an asymptotic expansion in powers of k−1 rather than calling the whole expression a polynomial in k.","section":"§5.4.2, Eq. (119)"}],"recommendation":"major_revision","confidential_remarks":"The missing oscillatory phase in Theorem 1.9 is a concrete error in the central statement, but the paper's own proof contains the phase factor in the stationary-phase evaluation, so the issue appears to be a repairable omission rather than a fatal flaw in the overall strategy. The referee recommends that the authors add the factor e^{i√k(∥λ∥/τ)(θ1−θ2)} to both expansions in Theorem 1.9, or explicitly restrict to θ1=θ2, and clarify the imported NHLC phase expansion before the paper is reconsidered."},"author_rebuttal":null,"desk_editor":{"model":"deepseek-v4-flash","letter":"Thanks for the report. I checked the stationary-phase computation in the proof of Theorem 1.9, and the stress-test note is right: the theorem is missing an oscillatory factor that the proof's own calculation produces. In the phase Ψ_r of Eq. (125), the critical point is P0=(∥λ∥/τ,(θ2−θ1)/τ,0,0) with critical value (∥λ∥/τ)(θ1−θ2). The expression for I_{x,k}(r) after Eq. (131) carries e^{i√k(∥λ∥/τ)(θ1−θ2)}, and the r-integration does not touch it. The final displayed expansion and the statement of Theorem 1.9 drop it. For θ1≠θ2 of size O(k^{ε−1/2}) this is a genuine leading-order oscillation, not a remainder; no expansion 1+Σk^{-j/2}R_j can produce it. So the main theorem is false as stated unless one imposes θ1=θ2 or restores the phase.\n\nThis is a pity, because the paper has real substance. The compact Lie group setting, the kλ-ladder, the rapid decay results (Theorems 1.4–1.6), and the applications to L∞ and Lp bounds are worked out carefully, with detailed character and Kirillov computations and no fitted parameters. The self-citations to [P-2024] and [GvP-2024] are legitimate, not circular; the imported NHLC phase expansion flagged in the reader’s report is a technical reliance, but secondary to the concrete phase defect.\n\nOther soft spots are minor: the Pτ norm bound is left to the reader, and the regular-weight assumption is stated honestly. The main issue is the missing phase.\n\nI would send this to a serious referee rather than desk-reject, because the framework is valuable and the flaw looks fixable. But the current version should not be accepted: the central asymptotic expansion is inconsistent with its own proof. I would not cite it in its present form.","headline":"Theorem 1.9 loses a √k oscillatory phase that the proof's own stationary-phase calculation produces, so the main asymptotic expansion is false as stated.","tokens_in":42485,"tokens_out":5283,"would_cite":false,"duration_ms":51446,"reading_group":"maybe","serious_thinker":"no","would_accept_peer_review":true},"rs_alignment":null,"lean_confirmation":null,"pith_extraction":{"msc":["53D50","58J40","22E30","32T15"],"pacs":[],"model":"deepseek-v4-flash","headline":"This paper derives complete near-diagonal asymptotic expansions for the equivariant Szegő and Poisson kernel components on the Grauert tube of a compact Lie group as the weight drifts to infinity along a ray in weight space.","keywords":["Grauert tube","Szegő kernel","Poisson kernel","equivariant asymptotics","compact Lie group","coadjoint orbit","normal Heisenberg local coordinates","scaling asymptotics"],"falsifier":"Take $G=\\mathrm{SU}(2)$ with a regular dominant weight $\\lambda$, compute $\\Pi^\\tau_{k\\lambda}(x_{1k},x_{2k})$ for large $k$ directly from the Weyl character formula as an oscillatory integral, and compare the $k$-exponent and the prefactor $\\left(\\operatorname{vol}(O_\\lambda)/\\operatorname{vol}_\\kappa(G)\\right)^2 \\operatorname{vol}_\\kappa(T)/(D_\\kappa(x)\\det(S_\\lambda))$ with Theorem 1.9 at $s_1=s_2=0$, $n_1=n_2=0$; a mismatch in either would disprove the expansion.","tokens_in":41340,"feed_emoji":"🎯","tokens_out":10842,"duration_ms":101162,"temperature":0.7,"pith_summary":"The paper studies the pieces of the Szegő and Poisson kernels attached to the ladder of irreducible representations $k\\lambda$ of a compact Lie group $G$, for $k\\to\\infty$. It establishes that these kernels are rapidly decaying away from a locus determined by the coadjoint orbit of $\\lambda$, and that near that locus, on the $k^{-1/2}$ scale, they admit a full asymptotic expansion. The leading term is an explicit Gaussian in the directions normal to the locus multiplied by an oscillatory factor in the remaining directions, with constants built from the volumes of the coadjoint orbit, the group, and a maximal torus, and from a metric determinant. A sympathetic reader would care because this gives the sharp concentration profile of complexified matrix elements, with consequences for sup-norm bounds of complexified eigenfunctions and for $L^p$--$L^q$ estimates of the equivariant projectors.","feed_headline":"Complexified eigenfunctions concentrate on coadjoint orbits","feed_subtitle":"New scaling asymptotics determine the Szegő and Poisson kernel profile for high weights of a compact Lie group.","key_machinery":"The central machinery is the Fourier integral operator representation of the Szegő projector with a complex phase of positive type, together with the matching expansion for the Poisson kernel. The equivariant components are extracted by integrating the character $\\chi_{k\\lambda}$ against the full kernel over $G$; the Weyl and Kirillov character formulas turn that character into oscillatory integrals over the coadjoint orbit $O_\\lambda$. Everything is evaluated in normal Heisenberg local coordinates (coordinates adapted to the contact and CR structure of the sphere bundle), where the phase $\\psi^\\tau$ has a third-order expansion with explicitly controlled linear, quadratic, and remainder terms. The resulting phase has a unique non-degenerate critical point whose Hessian determinant is $-\\tau^2\\det(S_\\lambda)^2$, and stationary phase produces the Gaussian profile and the volume constants.","core_discovery":"For a fixed regular dominant weight $\\lambda$ and a point $x\\in X^\\tau_O$ lying over the coadjoint orbit $O_\\lambda$, write nearby points in normal Heisenberg local coordinates as $x_{j,k}=x+(\\theta_j/\\sqrt{k},\\, n_j/\\sqrt{k}+s_j)$. Then, uniformly for $\\|(\\theta_j,n_j,s_j)\\le Ck^{\\epsilon-1/2}$, the $k\\lambda$-component of the Szegő kernel satisfies $$\\Pi^\\tau_{k\\$\\lambda$}(x_{1k},x_{2k}) \\sim \\left(\\frac{k\\|\\$\\lambda$\\|}{2\\pi\\tau}\\right)^{d-1+(1-r_G)/2} \\left(\\frac{\\operatorname{vol}(O_\\$\\lambda$)}{\\operatorname{vol}_\\kappa(G)}\\right)^2 \\frac{\\operatorname{vol}_\\kappa(T)}{D_\\kappa(x)\\,\\det(S_\\$\\lambda$)} \\exp\\left(\\frac{\\|\\$\\lambda$\\|}{\\tau}\\left(\\$psi^{{\\omega_x}}$_2(s_1,s_2)-\\|n_1\\|^2_{\\tilde\\kappa_x}-\\|n_2\\|^2_{\\tilde\\kappa_x}\\right)\\right) \\left(1+\\sum_{j\\ge1} $k^{{-j/2}}$R_j(\\cdot)\\right),$$ and the Poisson kernel $P^\\tau_{k\\lambda}$ obeys the same expansion with an extra prefactor $(1/2)^{(d-1)/2}$ and with $k$-power $(d-r_G)/2$ instead of $d-1+(1-r_G)/2$. The exponential factor is a Gaussian in the normal variables $n_j$ and an oscillatory factor $\\psi^{\\omega_x}_2$ in the tangential variables $s_j$; the constants involve the symplectic volume of the coadjoint orbit, the Riemannian volumes of $G$ and $T$, the metric determinant $D_\\kappa(x)$, and the determinant of the skew-adjoint map $S_\\lambda=\\operatorname{ad}_{\\lambda_\\kappa}$ on $T^\\perp$.","pith_inferences":["The same mechanism suggests a microlocal description of the complexified isotypical projectors as quantizations of the coadjoint orbit $O_\\lambda$ inside the Grauert tube, so the leading constant $\\operatorname{vol}(O_\\lambda)/\\operatorname{vol}_\\kappa(G)$ may be read as a semiclassical density of states per unit symplectic volume.","One testable extension is to non-regular weights, where the stabilizer is larger than $T$: the proof structure suggests a similar expansion with $r_G$ replaced by the stabilizer dimension and with a modified normal-space dimension, something the present theorems do not cover.","Because the torus case is checked explicitly in the paper, the same asymptotics could be verified numerically for small-rank groups such as $\\mathrm{SU}(2)$ by evaluating the character integral directly, providing a low-cost check of the $k$-power and prefactor.","The near-diagonal Gaussian shape implies that complexified eigenfunctions at high frequency are concentrated in a tube of radius $O(k^{-1/2})$ around the coadjoint-orbit locus, which may feed into nodal-set or restriction estimates for eigenfunctions of compact Lie groups."],"forward_implications":["Complexified eigenfunctions concentrate: the kernels are $O(k^{-\\infty})$ away from $Z^\\tau_O$, so the mass of the $k\\lambda$-equivariant kernels localizes on a locus determined by the coadjoint orbit of $\\lambda$.","At the $k^{-1/2}$ scale the concentration profile is sharp: Gaussian decay in the normal directions with width set by $\\|\\lambda\\|/\\tau$, and an oscillatory phase in the tangential directions governed by the symplectic form $\\omega_x$.","Sup-norm bounds follow for complexified eigenfunctions: $|\\varphi^\\tau(x)|\\le C e^{\\tau c_{k\\lambda}}(c_{k\\lambda})^{(d-r_G)/4}$, and the associated Husimi distributions satisfy a matching bound.","The equivariant Szegő projectors obey explicit $L^p\\to L^q$ operator bounds of the form $\\|\\Pi^\\tau_{k\\lambda}\\|_{L^p\\to L^q}\\le C k^{\\frac{1}{2R}[(d-1)(R-1)+R(d-r_G)]+\\epsilon}$ with $1/R=1-1/p+1/q$."],"supporting_citations":[{"why":"Supplies the third-order expansion of the Szegő phase in normal Heisenberg local coordinates (Proposition 48) and the value of the leading amplitude coefficient $s^\\tau_0(x,x)=\\tau/(2\\pi)^d$ (Theorem 51), both used in the stationary phase evaluation.","marker":"[P-2024]"},{"why":"Supplies the refinement (Lemma 64) for how left translations act in normal Heisenberg local coordinates, used to expand $\\mu^\\tau_{e^{-\\xi/\\sqrt{k}}}(x_{1k})$ in the proof of Theorem 1.9.","marker":"[GvP-2024]"},{"why":"Establishes the Szegő projector as a Fourier integral operator with complex phase, giving the representation (44) on which the whole asymptotic analysis begins.","marker":"[BdM-S]"},{"why":"Provides the description of $P^\\tau$ as a complexified Poisson-wave/Toeplitz operator and the $L^2$-restriction norm estimate used in the sup-norm applications.","marker":"[Z-2020]"},{"why":"Supplies the Kirillov character formula, which converts the character $\\chi_{k\\lambda}$ into an oscillatory integral over the coadjoint orbit $O_\\lambda$.","marker":"[Kir]"},{"why":"Provides Grauert-tube scaling asymptotics for Szegő kernels and an a priori bound used to pass from diagonal rapid decay to off-diagonal rapid decay.","marker":"[CR1]"},{"why":"Gives the analogous equivariant asymptotics in the line-bundle setting whose proof architecture and stationary-phase strategy are adapted here.","marker":"[P-2022]"},{"why":"Establishes the global adapted complex structure on $TG$ and the biholomorphism to the complexified group, the geometric setup that identifies the sphere bundle with the tube boundary.","marker":"[Sz-1998]"}],"fun_headline_variants":["High-weight eigenfunctions localize on coadjoint orbits","Szegő and Poisson kernels exhibit Gaussian asymptotics on coadjoint orbits","Coadjoint orbit concentration for high-weight eigenfunction kernels","Gaussian localization of high-weight kernels on coadjoint orbits","High-weight kernel asymptotics: Gaussian concentration on coadjoint orbits"],"cache_read_input_tokens":3200,"weakest_assumption_plain":"Everything rests on a third-order expansion of the Szegő phase in normal Heisenberg local coordinates that the paper imports from the authors' earlier work and does not prove here; if that expansion or its remainder estimate is wrong, the critical-point computation and the Gaussian exponents in Theorem 1.9 do not follow.","fun_headline_variants_meta":{"raw":{"variants":["High-weight eigenfunctions localize on coadjoint orbits","Szegő and Poisson kernels exhibit Gaussian asymptotics on coadjoint orbits","Coadjoint orbit concentration for high-weight eigenfunction kernels","Gaussian localization of high-weight kernels on coadjoint orbits","High-weight kernel asymptotics: Gaussian concentration on coadjoint orbits"]},"model":"deepseek-v4-flash","effort":"low","cost_usd":0.001272,"raw_usage":{"total_tokens":5266,"prompt_tokens":1073,"completion_tokens":4193,"prompt_tokens_details":{"cached_tokens":384},"prompt_cache_hit_tokens":384,"prompt_cache_miss_tokens":689,"completion_tokens_details":{"reasoning_tokens":4105}},"tokens_in":689,"tokens_out":4193,"duration_ms":32577,"temperature":1.0,"reasoning_tokens":4105,"cache_read_input_tokens":384,"cache_creation_input_tokens":0},"cache_creation_input_tokens":0},"created_at":"2026-08-15T18:16:24.378458+00:00","model_set":{"reader":"deepseek-v4-flash"},"falsifier":"Take $G=\\mathrm{SU}(2)$ with a regular dominant weight $\\lambda$, compute $\\Pi^\\tau_{k\\lambda}(x_{1k},x_{2k})$ for large $k$ directly from the Weyl character formula as an oscillatory integral, and compare the $k$-exponent and the prefactor $\\left(\\operatorname{vol}(O_\\lambda)/\\operatorname{vol}_\\kappa(G)\\right)^2 \\operatorname{vol}_\\kappa(T)/(D_\\kappa(x)\\det(S_\\lambda))$ with Theorem 1.9 at $s_1=s_2=0$, $n_1=n_2=0$; a mismatch in either would disprove the expansion.","supporting_citations":[],"review_version":1}