{"id":"66f5f056-e4c0-4dbb-a0e3-f6c67a23617e","arxiv_id":"2412.18382","paper_version":2,"verdict":"CONDITIONAL","confidence":"MODERATE","novelty_score":7.0,"correctness_risk":"medium","formal_verification":"none","parameter_count":0,"one_line_summary":"Sharp L^2-L^{2n} Wehrl inequalities hold for matrix coefficients of vector-valued holomorphic discrete series, with maximizers exactly the reproducing kernels.","lead":"This paper proves sharp L2-to-Lp bounds for matrix coefficients of vector-valued holomorphic discrete series, with p even, and identifies the exact maximizers. It gives explicit constants in terms of Harish-Chandra formal degrees, extending classical Wehrl and Lieb inequalities from the Heisenberg group, SU(2), and SU(1,1) to general Hermitian Lie groups.","discovery_kind":"extension","skeptic_critique":{"model":"deepseek-v4-flash","headline":"Proposition 4.3's Sp(n,R) constant is internally inconsistent and appears wrong: the stated c_G is twice the value forced by the paper's own formulas for Sp(2,R), so the sharp Wehrl constant and equality classification fail for that family.","rationale":"The reader's weakest assumption was Proposition 5.1, the external tensor-product branching theorem. That is a legitimate concern, but it is a citation-dependent one. The more load-bearing problem is internal: the proof of Proposition 4.3 contains two mutually unequal expressions for c_G in the Sp(r,R) case, and substituting the paper's own formulas for d_Λ and d^H_Λ in the concrete case Sp(2,R) gives a different constant from the one stated in the proposition. Since Theorem 5.3 and Corollary 5.4 use c_G directly, a wrong c_G invalidates the sharp constant and the equality statement for an infinite family of groups. The error is localized and likely repairable by correcting Proposition 4.3 and re-checking the equality argument, so a conditional verdict is appropriate rather than a rejection of the method. My disagreement with the reader is therefore about where the central risk sits: not primarily in the external branching theorem, but in the algebraic derivation of the normalization constant in Proposition 4.3. The concrete test above (Sp(2,R), λ=3, n=2) settles the dispute computationally from the paper's own formulas.","tokens_in":24288,"tokens_out":57448,"duration_ms":485487,"concrete_test":"Take G=Sp(2,R), scalar λ=3, n=2. From Proposition 4.2, d_Λ = π^{-3} Γ(3)Γ(5/2)/(Γ(3/2)Γ(1)) = 3π^{-3} and d_{2Λ} = 90π^{-3}. From equation (12) with roots {2ε_i, ε_i±ε_j}, d^H_Λ = (λ-1)(λ-2)(2λ-3)/6 = 1 and d^H_{2Λ} = 30. Thus c_G = d_Λ/d^H_Λ = 3π^{-3}, whereas Proposition 4.3 states 6π^{-3}. Now set f=1 in Theorem 5.3: LHS = ||1||^2_{H_{2Λ}} = d_{2Λ}^{-1} = π^3/90; RHS with c_G=3π^{-3} equals (3π^{-3})(1/30)(d_Λ^{-1})^2 = π^3/90, giving equality, while the stated c_G=6π^{-3} makes the RHS twice the LHS, contradicting the equality classification.","verdict_should_be":"CONDITIONAL","load_bearing_attack":"In Proposition 4.3, Case 1 (Sp(r,R)), the proof asserts c_G = π^{-N} r! ∏_{i<j}(r+1-i+j/2) = π^{-N} 2^{-(r-1)r/2} r! ∏_{i=1}^{r-1}(2+i)^i. For r=2 the two right-hand sides evaluate to 6π^{-3} and 3π^{-3}, and the proposition keeps the former. But the paper's own definitions give the latter: for scalar λ>2, Proposition 4.2 gives d_Λ = π^{-3}(λ-1)(λ-2)(λ-3/2), while equation (12) with the root data displayed in the proof gives d^H_Λ = (λ-1)(λ-2)(2λ-3)/6. Hence c_G = d_Λ/d^H_Λ = 3π^{-3}, not 6π^{-3}. With the stated larger constant, the equality claim in Theorem 5.3 fails at coherent states: for f=1 (a scalar multiple of K_0), ||P(f^{⊗n})||^2/||f||^{2n} equals d_Λ^n/d_{nΛ}, which is half the RHS when the stated c_G is used. The displayed comparison in Proposition 4.3 is therefore algebraically inconsistent, and the sharp Wehrl constant for the entire Sp(n,R) family is wrong as written.","agreement_with_reader":"disagree"},"referee_report":{"model":"deepseek-v4-flash","summary":"The paper proves Wehrl-type L^2(G)--L^{2n}(G) inequalities for matrix coefficients of vector-valued holomorphic discrete series of Hermitian Lie groups. The constant is expressed through Harish-Chandra formal degrees, and the authors claim that the maximizers are exactly the reproducing kernels with highest-weight target vector. The proof combines tensor-product branching for holomorphic discrete series, a normalization of the intertwining projection onto the Cartan component, exact Selberg-integral evaluations of formal degrees, and an appendix giving a full proof of the compact-group Wehrl inequality. The main results are Theorem 5.3 and Corollary 5.4.","tokens_in":24522,"tokens_out":30563,"duration_ms":263432,"significance":"If correct, the paper would settle the sharp L^2--L^{2n} Wehrl inequality for all holomorphic discrete series and give the complete set of maximizers, extending earlier results for SU(1,1) and the Fock space. The paper has real strengths: the formal degree is computed by an independent Selberg integral evaluation rather than fitted, the compact-group inequality of [2] is repaired in Appendix A, and the equality analysis via Toeplitz operators and bounded point evaluations is substantial. These contributions make the manuscript potentially significant. However, the Sp(n,R) constant in Proposition 4.3 is internally inconsistent and appears wrong, and since this constant feeds directly into the main inequality and equality statement, the central claim is not correct as written for that family.","major_comments":[{"comment":"The two displayed formulas for c_G in the Sp(r,R) case are not equal for r=2: the first equals 6π^{-3} while the second equals 3π^{-3}. The value forced by the paper's own definitions is the smaller one. For Sp(2,R), Proposition 4.2 gives d_Λ = π^{-3}(λ-1)(λ-2)(λ-3/2), while equation (12) with the root data displayed in the proof gives d^H_Λ = (λ-1)(λ-2)(2λ-3)/6. Hence d_Λ/d^H_Λ = 3π^{-3}, not 6π^{-3}. This constant enters Theorem 5.3 and Corollary 5.4. With the stated larger constant, equality at a coherent state such as f=1 fails: for n=2 the right-hand side computed from the stated formulas is strictly larger than the left-hand side. The displayed equality between the two product expressions in Proposition 4.3 must be corrected, and the consequences for the sharp constant and equality set in Sections 5 must be rechecked for the whole Sp(n,R) family.","section":"Proposition 4.3, Case 1 (Sp(r,R))"},{"comment":"The proof of the main inequality and the equality classification both rest on the tensor-product branching theorem stated in Proposition 5.1 and cited from Repka [22]: H_Λ⊗H_Λ' decomposes discretely, H_{Λ+Λ'} occurs with multiplicity one, and J_0(F)(z)=P_{Λ+Λ'}F(z,z) is an intertwiner onto it. This result is used to define the partial isometry Q_0 in Proposition 5.2, to compute its normalization, and to identify the leading component in Theorem 5.3. Since the manuscript does not prove this theorem, please state the exact version being used, including the hypotheses on Λ and Λ', and verify that it applies to every Λ satisfying Theorem 3.2. A failure of any of these hypotheses would change both the constant and the maximizer set.","section":"Proposition 5.1"}],"minor_comments":[{"comment":"There are typographical errors such as \"Dete the corresponding co-roots\" in the paragraph defining the strongly orthogonal roots; please correct throughout.","section":"Section 2.1"},{"comment":"The codomain of Q_0 is written as H_Λ^{⊗n}; it should be H_{nΛ} (or the inclusion H_{nΛ}⊆H_Λ^{⊗n} should be made explicit).","section":"Corollary 5.4, proof"},{"comment":"The sentence \"the inequality is an equality if and only if f_{Λ''} ≠ 0 ⇔ Λ''=2Λ\" is confusing; it should state that all components f_{Λ''} with Λ''≠2Λ vanish. The reduction from n=2 to general n is only asserted; please spell out the argument for general n, since the equality classification is a central claim.","section":"Theorem 5.3, equality proof"},{"comment":"In the formal-degree product, the compact-root factors contribute 1 because Λ(h_α)=0 for compact roots such as ε_i-ε_j; stating this explicitly would have made the Sp(r,R) computation easier to check and would have exposed the inconsistency noted above.","section":"Proposition 4.3, proof"},{"comment":"The proof of boundedness of point evaluations for u outside D relies on Oka-Weil approximation; please clarify the approximation argument and the role of the spectral norm in the final step.","section":"Appendix B, Lemma B.1"}],"recommendation":"major_revision","confidential_remarks":"The Sp(n,R) constant error is localized but load-bearing: it directly affects the sharp constant in Theorem 5.3 and Corollary 5.4 for one of the main families. The rest of the architecture is sound and the error appears fixable by recomputing the formal-degree comparison in Proposition 4.3. Given the manuscript's scope and strengths, I would view a corrected version favorably, but the current version should not be published without this fix."},"author_rebuttal":null,"desk_editor":{"model":"deepseek-v4-flash","letter":"Read the van Haastrecht–Zhang paper. The short version: the method is serious and the vector-valued extension is real, but the paper as written has a load-bearing error in the Sp(n,R) constant, and the main theorem's sharp constant and equality classification are wrong for that whole family.\n\nWhat's genuinely new: the sharp L2–L2n estimates for matrix coefficients of vector-valued holomorphic discrete series, with explicit constants from Selberg integral evaluations and a complete maximizer classification. The scalar case was known from Zhang's earlier work, and the compact and SU(1,1) cases were known, but the vector-valued case for general Hermitian groups is new. The strategy — tensor-product branching, normalization via formal degrees, Toeplitz operators, bounded point evaluations — is coherent, and Appendix A's repair of the compact-group argument is a useful contribution. I did not find a breaking error in the main inequality's proof structure once the correct constant is used.\n\nThe soft spot is not soft. Proposition 4.3, Case 1 (Sp(r,R)), states c_G = π^{−N} r! ∏_{i<j}(r+1−i+j/2) = π^{−N} 2^{−(r−1)r/2} r! ∏_{i=1}^{r−1}(2+i)^i. For r=2 the two right-hand sides are 6π^{−3} and 3π^{−3}; the proposition keeps the former. But their own Proposition 4.2 and equation (12) force c_G = 3π^{−3}. So the displayed equality is false and the kept constant is wrong. For r≥3 both displayed expressions are wrong; the correct value is π^{−N} r! ∏_{i<j}(2r+2−i−j)/2^{(r−1)r/2} (e.g., 45π^{−6} for r=3, not 36π^{−6}). With the stated inflated constant, coherent states no longer attain equality in Theorem 5.3, so the equality classification fails for Sp(n,R). This is not a typo in a lemma; it's the constant in the headline theorem for a classical family.\n\nOther concerns are minor: equation (18) abuses operator domain notation, Appendix A's Weyl-group step is compressed, and the branching theorem (Prop 5.1) is cited rather than proved. The self-citations to [31] and [8] are legitimate methodological precedents.\n\nWho is this for? Harmonic analysts and representation theorists working on coherent-state inequalities. The paper earns a serious referee, but it needs a major correction before the main result can stand. I'd recommend sending to review with a note to check the formal-degree constant for Sp(n,R) and re-evaluate the other two cases; the method is strong enough that the paper is worth saving.","headline":"Genuinely new vector-valued Wehrl-type inequalities with a clean method, but the Sp(n,R) formal-degree constant is internally inconsistent and wrong, which breaks the headline theorem and its equality classification for that whole family.","tokens_in":25086,"tokens_out":18639,"would_cite":false,"duration_ms":137408,"reading_group":"maybe","serious_thinker":"yes","would_accept_peer_review":true},"rs_alignment":null,"lean_confirmation":null,"pith_extraction":{"msc":["22E30","22E45","32A36","43A15"],"pacs":[],"model":"deepseek-v4-flash","headline":"This paper proves sharp $L^2(G)$--$L^{2n}(G)$ Wehrl-type inequalities for matrix coefficients of every vector-valued holomorphic discrete series of a Hermitian Lie group, with equality exactly at the reproducing kernels.","keywords":["Wehrl inequality","holomorphic discrete series","matrix coefficients","Harish-Chandra formal degree","reproducing kernels","Hermitian symmetric spaces","tensor product decomposition","Bergman spaces"],"falsifier":"Take a concrete rank-two Hermitian group such as $G=SU(2,2)$, choose a scalar holomorphic discrete series with integer parameter $\\lambda$, and compute the ratio $\\|F_f\\|_{2n}^{2n}/\\|F_f\\|_2^{2n}$ for $F_f(g)=\\langle\\pi_\\Lambda(g)f,v_\\Lambda\\rangle$, with $f$ the constant function $v_\\Lambda$ and with a non-kernel function such as $f(z)=v_\\Lambda+z_1v_\\Lambda$: exceeding $d_\\Lambda^n/d_{n\\Lambda}$ for any $n\\ge2$ would refute the theorem, while inequality failure at the reproducing kernels would invalidate the constant $c_G$. Alternatively, compute the decomposition of $H_\\Lambda\\otimes H_\\Lambda$ for the same group and verify that $H_{2\\Lambda}$ has multiplicity one; any multiplicity different from one would refute Proposition 5.1 and with it the proof.","tokens_in":24028,"feed_emoji":"🎯","tokens_out":17273,"duration_ms":143060,"temperature":0.7,"pith_summary":"The paper establishes a sharp $L^2(G)$-to-$L^{2n}(G)$ Wehrl-type inequality for the matrix coefficients $\\langle\\pi_\\Lambda(g)f,v_\\Lambda\\rangle$ of every vector-valued holomorphic discrete series of a Hermitian Lie group, for every integer $n\\ge2$. The optimal constant is expressed as a ratio of Harish-Chandra formal degrees, with a group-dependent normalization factor $c_G$ evaluated explicitly by Selberg integrals. The maximizers are exactly the reproducing kernels $f(z)=cK(z,w)\\tau_\\Lambda(k)v_\\Lambda$; equivalently, coherent states saturate the inequality and no other vectors do. This sharpens the orthogonality relations for discrete series into a complete extremal statement and identifies the coherent states as the extremal vectors. The proof works by comparing the norm of $f^{\\otimes n}$ with its projection onto the Cartan component $H_{n\\Lambda}$, the part that controls the $L^{2n}$ norm of the matrix coefficient.","feed_headline":"Coherent states are the sole maximizers of the sharp Wehrl bound","feed_subtitle":"The optimal constant is a ratio of Harish-Chandra formal degrees, and equality holds exactly at coherent states.","key_machinery":"The load-bearing device is the Cartan-component projection $Q_0=C_{\\Lambda,n}J_0$, a partial isometry (a norm-preserving map on the orthogonal complement of its kernel) from $H_\\Lambda^{\\otimes n}$ onto the leading tensor component $H_{n\\Lambda}$, built from the diagonal evaluation $J_0(F)(z)=P_{n\\Lambda}(F(z,\\dots,z))$. Proposition 5.2 fixes its normalization by formal degrees, so applying it to $f^{\\otimes n}$ converts the desired inequality into a comparison of $\\|f^{\\otimes n}\\|$ with $\\|P_{n\\Lambda}(f^{\\otimes n})\\|$. The equality analysis then shows that any maximizer is a joint eigenvector of the Toeplitz operators $T_{z_i}$, which forces $f$ to be a reproducing kernel with a highest-weight vector value at $0$; the bounded point evaluations for the vector-valued Bergman space ensure that the evaluation point lies in $D$.","core_discovery":"The central claim is that for every integer $n\\ge2$, every Hermitian Lie group $G$, and every vector-valued holomorphic discrete series $(\\pi_\\Lambda,H_\\Lambda)$ of $G$ with highest weight $\\Lambda$, the projection $P_{n\\Lambda}$ onto the leading (Cartan) component of $V_\\Lambda^{\\otimes n}$ satisfies\n$$\\|P_{n\\Lambda}($f^{{\\otimes n}}$)\\|^2_{H_{n\\Lambda}}\\le $c_G^{{n-1}}$\\frac{(d^H_\\Lambda)^n}{d^H_{n\\Lambda}}\\|f\\|^{2n}_{H_\\Lambda},$$\nand the equivalent matrix-coefficient estimate\n$$\\int_G |\\langle\\pi_\\Lambda(g)f,v_\\Lambda\\rangle|^{2n}\\,dg \\le \\frac{d_\\Lambda^n}{d_{n\\Lambda}}\\left(\\int_G|\\langle\\pi_\\Lambda(g)f,v_\\Lambda\\rangle|^2\\,dg\\right)^n,$$\nwhere $d_\\Lambda=c_G d^H_\\Lambda$ and $v_\\Lambda$ is a unit highest-weight vector. Equality holds exactly for $f(z)=cK(z,w)\\tau_\\Lambda(k)v_\\Lambda$ with $w\\in D$, $k\\in K$, $c\\in\\mathbb{C}$; that is, the reproducing kernels, up to translation and scale, are the unique maximizers.","pith_inferences":["The paper proves the inequality only at even integers $p=2n$ and only for holomorphic discrete series; interpolating between even values would give a conjectural sharp $L^2$--$L^p$ inequality for all $p\\ge2$, and differentiating at $p=2$ would then yield a Wehrl entropy bound for these representations. Neither step is carried out here.","The proof's reliance on a multiplicity-one branching theorem suggests a concrete test for wider applicability: representations whose tensor products lack a unique leading component would need different constants and may have additional extremizers, which would delimit how far the coherent-state phenomenon extends.","The explicit three-case formula for $c_G$ is independently testable: on a rank-two group such as $SU(2,2)$, reproducing kernels should attain equality to machine precision, verifying the Selberg-integral normalization without repeating the symbolic derivation."],"forward_implications":["The constant in the main inequality is sharp: every reproducing kernel $K(\\cdot,w)\\tau(k)v_\\Lambda$ attains equality, so no smaller constant can work.","The equality set is exactly the $G$-orbit of the highest-weight vectors; no other vector in $H_\\Lambda$ saturates the $L^2$--$L^{2n}$ estimate.","The matrix-coefficient estimate sharpens Schur orthogonality ($n=1$) into a full family of even-order interpolation inequalities for discrete-series matrix coefficients.","For the unit disk, the improved inequality adds an explicit positive remainder term to the left-hand side, and the remainder vanishes only at the reproducing kernels, so the unimproved bound is strict away from the extremizers."],"supporting_citations":[{"why":"It supplies the tensor-product branching theorem (Proposition 5.1) asserting that the leading component appears with multiplicity one via the diagonal projection, the load-bearing step for constructing the partial isometry.","marker":"[22]"},{"why":"It provides the Bergman-space realization of holomorphic discrete series and the reproducing-kernel formula $K(z,z)=C(\\Lambda)\\tau(B(z,z))$ that fixes the normalization.","marker":"[13]"},{"why":"It gives Harish-Chandra's formal-degree formula that enters the sharp constant and the Haar-measure normalization.","marker":"[9]"},{"why":"It evaluates the invariant integrals over the bounded symmetric domain that determine the constant $c_G$ in Proposition 4.3.","marker":"[4]"},{"why":"It supplies the Selberg integral evaluation used in Proposition 4.2 to compute $d_\\Lambda^{-1}$.","marker":"[1]"},{"why":"It proves the scalar-case Wehrl inequality and introduces the Toeplitz-operator argument used to force maximizers to be reproducing kernels.","marker":"[31]"},{"why":"It states the compact-group Wehrl inequality for matrix coefficients that the proof generalizes; Appendix A repairs the earlier proof, which the paper identifies as incomplete.","marker":"[2]"},{"why":"It states the tensor-power version of the compact-group inequality that the argument applies for general $n\\ge2$.","marker":"[26]"},{"why":"It supplies the Jordan-triple description of the bounded symmetric domain and the Bergman-operator transformation rules used throughout.","marker":"[19]"}],"fun_headline_variants":["Coherent states uniquely solve the sharp Wehrl inequality","Sharp Wehrl bound: only coherent states attain equality","Wehrl inequality sharp for discrete series, maximizers coherent states","Harish-Chandra degrees fix optimal Wehrl constant, kernels maximize","Exact Wehrl maximizers: reproducing kernels of discrete series"],"cache_read_input_tokens":3200,"weakest_assumption_plain":"The argument rests on a cited branching theorem---that $H_\\Lambda\\otimes H_{\\Lambda'}$ contains the leading component $H_{\\Lambda+\\Lambda'}$ exactly once via the diagonal projection---and if that theorem failed for some highest weight, both the sharp constant and the equality classification would collapse.","fun_headline_variants_meta":{"raw":{"variants":["Coherent states uniquely solve the sharp Wehrl inequality","Sharp Wehrl bound: only coherent states attain equality","Wehrl inequality sharp for discrete series, maximizers coherent states","Harish-Chandra degrees fix optimal Wehrl constant, kernels maximize","Exact Wehrl maximizers: reproducing kernels of discrete series"]},"model":"deepseek-v4-flash","effort":"low","cost_usd":0.000761,"raw_usage":{"total_tokens":3344,"prompt_tokens":877,"completion_tokens":2467,"prompt_tokens_details":{"cached_tokens":384},"prompt_cache_hit_tokens":384,"prompt_cache_miss_tokens":493,"completion_tokens_details":{"reasoning_tokens":2379}},"tokens_in":493,"tokens_out":2467,"duration_ms":19268,"temperature":1.0,"reasoning_tokens":2379,"cache_read_input_tokens":384,"cache_creation_input_tokens":0},"cache_creation_input_tokens":0},"created_at":"2026-08-11T04:43:51.254590+00:00","model_set":{"reader":"deepseek-v4-flash"},"falsifier":"Take a concrete rank-two Hermitian group such as $G=SU(2,2)$, choose a scalar holomorphic discrete series with integer parameter $\\lambda$, and compute the ratio $\\|F_f\\|_{2n}^{2n}/\\|F_f\\|_2^{2n}$ for $F_f(g)=\\langle\\pi_\\Lambda(g)f,v_\\Lambda\\rangle$, with $f$ the constant function $v_\\Lambda$ and with a non-kernel function such as $f(z)=v_\\Lambda+z_1v_\\Lambda$: exceeding $d_\\Lambda^n/d_{n\\Lambda}$ for any $n\\ge2$ would refute the theorem, while inequality failure at the reproducing kernels would invalidate the constant $c_G$. Alternatively, compute the decomposition of $H_\\Lambda\\otimes H_\\Lambda$ for the same group and verify that $H_{2\\Lambda}$ has multiplicity one; any multiplicity different from one would refute Proposition 5.1 and with it the proof.","supporting_citations":[{"cited_title":"Repka, Tensor products of holomorphic discrete series representa tions, Canadian J","cited_arxiv_id":null,"evidence_quote":"It supplies the tensor-product branching theorem (Proposition 5.1) asserting that the leading component appears with multiplicity one via the diagonal projection, the load-bearing step for constructing the partial isometry."},{"cited_title":"A simplified approach to the holomorphic discrete series","cited_arxiv_id":"2312.16350","evidence_quote":"It provides the Bergman-space realization of holomorphic discrete series and the reproducing-kernel formula $K(z,z)=C(\\Lambda)\\tau(B(z,z))$ that fixes the normalization."},{"cited_title":null,"cited_arxiv_id":null,"evidence_quote":"It gives Harish-Chandra's formal-degree formula that enters the sharp constant and the Haar-measure normalization."},{"cited_title":"Faraut and A","cited_arxiv_id":null,"evidence_quote":"It evaluates the invariant integrals over the bounded symmetric domain that determine the constant $c_G$ in Proposition 4.3."},{"cited_title":"Andrews, R","cited_arxiv_id":null,"evidence_quote":"It supplies the Selberg integral evaluation used in Proposition 4.2 to compute $d_\\Lambda^{-1}$."},{"cited_title":"Zhang, Wehrl-type inequalities for Bergman spaces on domains in Cd and completely positive maps, in The Bergman kernel and related topics , 343-355, Springer Proc","cited_arxiv_id":null,"evidence_quote":"It proves the scalar-case Wehrl inequality and introduces the Toeplitz-operator argument used to force maximizers to be reproducing kernels."},{"cited_title":"Delbourgo and J","cited_arxiv_id":null,"evidence_quote":"It states the compact-group Wehrl inequality for matrix coefficients that the proof generalizes; Appendix A repairs the earlier proof, which the paper identifies as incomplete."},{"cited_title":"Sugita, Proof of the generalized Lieb-Wehrl conjecture for integer indices larger than one , J","cited_arxiv_id":null,"evidence_quote":"It states the tensor-power version of the compact-group inequality that the argument applies for general $n\\ge2$."},{"cited_title":"Loos, Bounded symmetric domains and Jordan pairs , University of California, Irvine (1977)","cited_arxiv_id":null,"evidence_quote":"It supplies the Jordan-triple description of the bounded symmetric domain and the Bergman-operator transformation rules used throughout."}],"review_version":1}