{"id":"09e3fbf9-30d9-445c-939c-c3eb874ff8b3","arxiv_id":"2506.13530","paper_version":2,"verdict":"CONDITIONAL","confidence":"MODERATE","novelty_score":7.0,"correctness_risk":"medium","formal_verification":"none","parameter_count":0,"one_line_summary":"The paper determines the spectrum of Schwartz functions on Mat_{n,m} and the symplectic Grassmannian using rho-derivatives and the Local Structure Theorem, yielding new proofs and explicit local lifts.","lead":"This paper develops a derivative-based method to compute which representations occur in functions on matrix spaces and symplectic Grassmannians over p-adic fields. It uses the method to reprove Howe duality, describe local Miyawaki lifts, and link L-function poles to geometry.","discovery_kind":"new_method","skeptic_critique":{"model":"deepseek-v4-flash","headline":"Corollary 3.1 is load-bearing for T^m_n(π,μ), but its proof is an asserted induction relying on an unproved symplectic analogue and a blanket transfer to Mp_n.","rationale":"The central results on SGr and Miyawaki lifts (Theorem 5.2, Proposition 5.1, Theorems 5.4 and 5.7) all depend on the well-definedness of T^m_n(π,μ), which Corollary 3.1 provides. The proof of Corollary 3.1 is a sketch: the non-self-dual case invokes a socle formula from Lemma 3.12, whose proof is omitted, and the metaplectic case is asserted without proof. These are not mere technicalities because the symplectic derivative theory in §3.2.2 explicitly excludes self-dual characters, and the metaplectic cover has additional structure that may affect socle decompositions. I agree with the reader's conditional verdict: the theorems are plausible and supported by precedent, but full proofs are needed before acceptance.","tokens_in":36733,"tokens_out":17381,"duration_ms":162978,"concrete_test":"Compute the socle of Ind_{GL_1×Sp_1}^{Sp_2}(ρ⊗π) for ρ=1 and ρ=ν (the unramified character |·|) and each π∈Irr(Sp_1), using the Geometric Lemma and Frobenius reciprocity. Verify the conclusions of Corollary 3.1: for ρ=1 the socle has length ≤2 and determines π; for ρ=ν it is irreducible and determines π. If either fails, the definition of T^m_n(π,μ) is invalid. Then repeat the ρ=1 case for Mp_1 to test the asserted Mp_n transfer; if the metaplectic socle length differs, the blanket transfer fails.","verdict_should_be":"UNCHANGED","load_bearing_attack":"The definition of T^m_n(π,μ) in §3.3 (end) is the cornerstone of Theorem 5.2 and Proposition 5.1. It depends on Corollary 3.1: for a segment Δ=[a,b]_ρ with self-dual ρ, soc(Z(Δ)⋊π) is multiplicity-free of length ≤2 and any irreducible subrepresentation determines π uniquely. The proof of Corollary 3.1 for non-self-dual segments uses the identity soc(Z(Δ)⋊π)=soc((ρν^a_ρ)^{1+d_{ρν^a_ρ,max}(π)}⋊soc(Z(−Δ)⋊D^max_{ρν^a_ρ}(π))), justified by \"we saw in the proof of Lemma 3.12\"; yet Lemma 3.12 is stated only in the GL_n×Sp_r setting and its proof is said to be \"analogous to Lemma 3.8\", not given. The induction step then asserts the conclusion \"follows ... and the induction hypothesis\" without demonstrating that the displayed socle formula preserves multiplicity-freeness and the uniqueness property. Moreover, §3.3 opens by asserting all proofs for Sp_n \"carry out\" for Mp_n by \"just a matter of adjusting the notation\", and §3.2.2 ends with \"all results here have analogous statements when one replaces Spn by Mpn\". This blanket transfer is load-bearing for Theorems 5.4 and 5.7, but the metaplectic cover introduces Weil indices and genuine-representation constraints that could alter socle decompositions.","agreement_with_reader":"agree"},"referee_report":{"model":"deepseek-v4-flash","summary":"The paper proposes a general strategy for computing the spectrum of spherical varieties over non-archimedean local fields, combining rho-derivatives, the Local Structure Theorem, and Frobenius reciprocity. It applies this strategy to the space of matrices Mat_{n,m} and to the symplectic Grassmannian SGr_{n,m}, claiming multiplicity-freeness, the lifting property, and explicit descriptions of their spectra: for Mat_{n,m} the spectrum is {pi tensor T^m_n(pi)} and for SGr_{n,m} it is described in terms of pi tensor T^m_n(pi,mu). From these results the paper derives a new proof of Howe duality in type II, an explicit description of local Miyawaki liftings in the Hilbert-Siegel case, and a formula for the order of poles of Godement-Jacquet L-functions at s=-(n-1)/2. It further claims that the arguments extend to metaplectic covers of GL_n and Sp_n and to inner forms of GL_n.","tokens_in":37087,"tokens_out":6201,"duration_ms":59449,"significance":"If the main theorems are correct, this is a substantial contribution: it gives a unified, derivative-theoretic derivation of two nontrivial spectra, a new proof of Howe duality in type II, explicit local Miyawaki lifts, and a new route to the Godement-Jacquet L-factor pole formula that does not use the functional equation. The paper also proposes a general framework that could apply to other spherical varieties. The claimed extensions to metaplectic covers and division algebras are potentially significant for the local theta correspondence. However, several load-bearing steps are compressed or asserted by analogy, in particular the symplectic Delta-derivative theorem (Corollary 3.1), the transfer to metaplectic groups, and the equality statement in the symplectic Grassmannian spectrum (Proposition 5.1). The current manuscript therefore does not yet fully substantiate its main claims; it is a promising but incomplete draft.","major_comments":[{"comment":"The definition of T^m_n(pi,mu) immediately before Section 3.4 relies on Corollary 3.1 for well-definedness, and Theorem 5.2 and Proposition 5.1 rely on it for the spectrum of SGr_{n,m}. The proof of Corollary 3.1 for non-self-dual segments is an induction that invokes the identity soc(Z(Delta)⋊pi) = soc((rho nu^a_rho)^{1+d_{rho nu^a_rho,max}(pi)} ⋊ (soc(Z(-Delta)⋊ D^max_{rho nu^a_rho}(pi)))) with the justification \"we saw in the proof of Lemma 3.12\". However, Lemma 3.12 is not proved in the text (its proof is said to be analogous to Lemma 3.8, a GL_n statement), and the displayed formula does not by itself demonstrate that the resulting socle remains multiplicity-free and that an embedding into both Z(Delta)⋊pi and Z(Delta)⋊pi' forces pi ≅ pi'. Please provide a complete proof of Corollary 3.1, or precise references for every step of the induction.","section":"Section 3.3, Corollary 3.1"},{"comment":"The transfer of all Sp_n results to the metaplectic group Mp_n is asserted in one sentence (\"it is just a matter of adjusting the notation\"), and similar blanket assertions are made for GL'_n and for the metaplectic GL_n. This transfer is load-bearing for Theorems 5.3-5.5 and especially for Theorem 5.7 on local Miyawaki liftings, where the condition on the mu_2-action appears explicitly. On metaplectic covers, genuine representations and Weil indices can affect Jacquet modules and socle decompositions; the paper should either prove the needed analogues or give a precise statement of which results carry over and why. As written, this is an unverified assumption rather than a proved extension.","section":"Sections 3.2.2, 3.3, 5.3.3"},{"comment":"Lemma 5.6, the symplectic analogue of Lemma 5.3, is the key input for Theorem 5.2, but its proof is summarized as \"We proceed analogously to Lemma 5.3\". The symplectic case has the additional feature of self-dual segments and length-two socles, so the analogy is not immediate and needs to be spelled out. Moreover, Proposition 5.1 upgrades the inclusion in Theorem 5.2 to an equality by a limiting argument in s; the proof asserts that the order of the pole of f_s at s=0 equals the order of the associated intertwining operator, and then identifies cosoc(Im(f_0)) with soc(Pi'_0) using Theorem 3.3. As written, these steps are too compressed for the main spectrum theorem; please expand the argument.","section":"Section 5.3.2, Lemma 5.6 and Proposition 5.1"},{"comment":"The proof of Lemma 5.11 claims that the pole order of L(pi,s) at s=-(nd-1)/2 equals that of L(rho_{n,alpha},s), based on a factorization of the Godement-Jacquet zeta integral. The displayed identity involves an integral over Mat_{n-|alpha|,|alpha|} × Mat_{|alpha|,n-|alpha|} × GL_{|alpha|}; the argument requires that this integral can be chosen non-zero and that no other contributions affect the pole. This is not fully justified, and Theorem 5.8 depends on it. Please provide the missing quantitative statement.","section":"Section 5.4.3, Lemma 5.11 and Theorem 5.8"}],"minor_comments":[{"comment":"The displayed definitions of pi^{×k} and of the iterated derivatives D^k_{rho,r}(pi) and D^k_rho(pi) contain corrupted symbols (e.g., \"⌟⟪⟨⟪rl⟫l⟩⟩\"); these should be replaced by standard notation.","section":"Section 2.1 and Sections 3.2.1/3.2.2"},{"comment":"There are typos such as \"enabeling\" in the introduction and missing spaces in the sentence \"Itisclearthatthecompositionisgivenbyintegrating...\" in Lemma 5.5; the latter should also be rewritten for clarity.","section":"Introduction and Section 5.3.1"},{"comment":"The chain of equalities in the proof of Lemma 5.5 appears to contain a typo: it should read T(chi_{k-1}) = T(chi_k) + T(chi'), not T(chi_k) = T(chi_{k-1}) = T(chi') + T(chi_k).","section":"Section 5.3.1, Lemma 5.5"},{"comment":"The notation \"Irr̃SGrm,n,Lµ\" in Theorem 5.4 is undefined and should be written consistently with the notation used elsewhere, e.g., Irr_{SGr_{n,m},L_mu}(Mp_n × Mp_m).","section":"Section 5.3.3"}],"recommendation":"major_revision","confidential_remarks":"This is a promising paper with a plausible overall strategy, but the referee cannot currently certify the main spectrum theorems because of the compressed proofs of Corollary 3.1, Lemma 5.6, Proposition 5.1, and the blanket metaplectic transfer. I would encourage the editor to request a revision that gives complete proofs or precise references for these points, rather than accepting the current 'analogous' and 'straightforward' statements. The paper's reliance on the author's previous works [Dro25b] and [Dro25a] is not itself circular, but the metaplectic claims should be checked independently since they appear to be new."},"author_rebuttal":null,"desk_editor":{"model":"deepseek-v4-flash","letter":"You should know about this one: it is not a routine application of known machinery. Droschl combines rho-derivatives with the Local Structure Theorem to compute spectra of Mat_{n,m} and the symplectic Grassmannian, and the method pays off with genuinely new statements: the explicit SGr spectrum with a unitary twist, explicit local Miyawaki lifts, the pole-order formula for Godement-Jacquet L-functions, and a new proof of Howe duality in type II. The novelty is real. The paper also extends the author's earlier conservation-relation work to metaplectic covers.\n\nWhat it does well: the general framework is clear and attractive. The reduction of S(X) to orbit strata and then to smaller matrix spaces is elegant. For Mat_{n,m} the proof is fairly complete and recovers Minguez's Howe duality. The SGr theorem and the applications are the interesting parts.\n\nNow the soft spots, and they are real. The definition of T^m_n(pi,mu) is the cornerstone of the SGr spectrum, and it rests on Corollary 3.1. That corollary is load-bearing and its proof is a sketch. The key display, soc(Z(Delta) rt\\pi) = soc((rho nu^a)^{1+d} rt soc(Z(-Delta) rt D^max(pi))), is something \"we saw in the proof of Lemma 3.12\", but Lemma 3.12 is itself only proved by saying it is analogous to Lemma 3.8, and Lemma 3.8 lives in the GL_n setting. The induction step of Corollary 3.1 is not actually demonstrated; it asserts that multiplicity-freeness and uniqueness survive. That is a genuine gap, not manufactured. Also, the blanket transfer to metaplectic groups — \"just a matter of adjusting the notation\" — covers Theorems 5.3-5.5 and 5.7. For Mpn, Weil indices and genuine-representation constraints can genuinely change socle decompositions, so this needs more than an analogy statement. The division algebra case has the same issue, though it is lower stakes.\n\nThe citation pattern is fine: the author's prior papers are used as starting tools and are explicitly extended. No circularity in the main argument.\n\nBottom line: the method is promising and the main structural argument is probably right, but the manuscript is not finished. A serious editor should send it to peer review, and the referee should ask for full proofs of Corollary 3.1 and the metaplectic/division algebra transfers. I would read a revised version with interest. Worth bringing to the reading group once the proofs are filled in.","headline":"A genuinely new derivative-plus-Local-Structure method with real payoffs, but the load-bearing Corollary 3.1 and the metaplectic transfer need full proofs before the results can be trusted.","tokens_in":37576,"tokens_out":3118,"would_cite":false,"duration_ms":28046,"reading_group":"maybe","serious_thinker":"yes","would_accept_peer_review":true},"rs_alignment":null,"lean_confirmation":null,"pith_extraction":{"msc":["22E50","14M27","20G25"],"pacs":[],"model":"deepseek-v4-flash","headline":"This paper computes the full multiplicity-free spectrum of the matrix space and the symplectic Grassmannian, and derives Howe duality in type II, explicit local Miyawaki lifts, and a pole formula for Godement–Jacquet L-functions.","keywords":["spherical varieties","rho-derivatives","multiplicity-free spectrum","symplectic Grassmannian","space of matrices","Howe duality","Miyawaki liftings","Godement–Jacquet L-functions"],"falsifier":"Find non-isomorphic $\\pi,\\pi'\\in\\mathrm{Irr}(\\mathrm{Sp}_n)$ and a self-dual segment $\\Delta$ such that $\\mathrm{soc}(Z(\\Delta)\\rtimes\\pi)$ and $\\mathrm{soc}(Z(\\Delta)\\rtimes\\pi')$ share an irreducible constituent. Corollary 3.1 forbids this, and since the definition of $T^m_n(\\pi,\\mu)$ uses exactly that uniqueness, such an example would collapse the symplectic spectrum formula.","tokens_in":2632,"feed_emoji":"🧮","tokens_out":6011,"duration_ms":113302,"temperature":0.7,"pith_summary":"This paper gives a unified way to compute the full spectrum—the list of irreducible representations that can be realized on Schwartz functions—of two spherical varieties over a local non-archimedean field: the space $\\mathrm{Mat}_{n,m}$ of $n\\times m$ matrices under $\\mathrm{GL}_n\\times\\mathrm{GL}_m$, and the symplectic Grassmannian $\\mathrm{SGr}_{n,m}$ under $\\mathrm{Sp}_n\\times\\mathrm{Sp}_m$. The main theorems assert that both spaces are multiplicity-free, have the lifting property, and that their spectra have explicit descriptions in terms of the socle of a single induced representation built from a rank datum attached to $\\pi$. From these descriptions the paper derives a new proof of Howe duality in type II, an explicit formula for local Miyawaki lifts in the Hilbert–Siegel case, and a pole formula for Godement–Jacquet $L$-functions at the point $s=-(n-1)/2$. The method combines the theory of $\\rho$-derivatives with the Local Structure Theorem for spherical varieties, and the same arguments are stated to extend to metaplectic covers and inner forms.","feed_headline":"Matrix space and symplectic Grassmannian spectra are computed exactly","feed_subtitle":"A single rho-derivative argument yields explicit spectra, Howe duality in type II, and local Miyawaki lifts.","key_machinery":"The load-bearing mechanism is an induction on rank strata rooted in the theory of $\\rho$-derivatives and the Local Structure Theorem for spherical varieties. For each orbit stratum $\\mathrm{Mat}^r_{n,m}$ or $\\mathrm{SGr}^r_{n,m}$, the theorem gives an equivariant isomorphism $N_Y\\times S_Y\\to X^\\circ_Y$ with $S_Y\\cong\\mathrm{Mat}_{n-r,m-r}\\times\\mathrm{GL}_r$ or $S_Y\\cong\\mathrm{Mat}_{n-r,m-r}\\times\\mathrm{Sp}_r$, which transfers the spectrum problem from the whole variety to its boundary strata; the lifting property is then proven by Frobenius reciprocity and a genericity argument. The single most important auxiliary result is Corollary 3.1: for a self-dual segment $\\Delta$, $\\mathrm{soc}(Z(\\Delta)\\rtimes\\pi)$ is multiplicity-free of length at most two, and any embedding of $\\tau$ into both $Z(\\Delta)\\rtimes\\pi$ and $Z(\\Delta)\\rtimes\\pi'$ forces $\\pi\\cong\\pi'$. This makes the map $T^m_n(\\pi,\\mu)$ well-defined, and the paper states that the same arguments carry over to $\\mathrm{Mp}_n$ by adjusting notation.","core_discovery":"The central discovery is that the spectrum of $\\mathrm{Mat}_{n,m}$ is exactly $\\{\\pi\\otimes T^m_n(\\pi):\\pi\\in\\mathrm{Irr}(\\mathrm{GL}_n)\\}$, where for the minimal rank $r$ such that $\\pi$ embeds into $\\nu^{r/2}_{n-r}\\times\\tau$, one sets $T^m_n(\\pi)=\\mathrm{soc}(\\tau^\\vee\\nu^{(m-n)/2}\\times\\nu^{-r/2}_{m-r})$; similarly, the spectrum of $(\\mathrm{SGr}_{n,m},L_\\mu)$ is exactly $\\{\\pi\\otimes\\pi':\\pi\\otimes\\pi'\\subseteq\\pi\\otimes T^m_n(\\pi,\\mu)\\}$, where $T^m_n(\\pi,\\mu)=\\mathrm{soc}(\\mu(\\det_{m-n+r})^{-1}\\nu^{-r/2}_{m-n+r}\\rtimes\\tau^\\vee)$ for the maximal $r$ with $\\pi\\hookrightarrow\\mu(\\det_r)^{-1}\\nu^{-(m-n+r)/2}_r\\rtimes\\tau$. The paper proves both spaces are multiplicity-free and have the lifting property, and shows that this implies Howe duality in type II, that the local Miyawaki lift $M_\\mu^m(\\pi)$ has cosocle $\\mathrm{soc}(\\mu_\\psi(\\widetilde{\\det}_{m-n+r})\\nu^{r/2}_{m-n+r}\\rtimes\\tau)$ of length at most two, and that $\\mathrm{ord}_{s=-(n-1)/2}L(\\pi,s)=\\Lambda(\\nu^{r/2}_{n-r},\\tau)+1$.","pith_inferences":["If the same two-step scheme—ordered orbit stratification plus derivative-based genericity—applies to other spherical varieties whose Local Structure Theorem slices are products of matrix spaces and smaller copies of the same group, the spectra of those varieties would be explicitly computable by the same induction.","The explicit pole formula suggests a testable pattern beyond $\\mathrm{Mat}_{n,n}$: for more general spherical varieties, the order of a pole of a relative $L$-function may be read off from the first nonvanishing orbit stratum of the canonical period map.","The extension to metaplectic groups is asserted by analogy rather than written out in full; since Corollary 3.1 is the hinge, a careful written verification for $\\mathrm{Mp}_n$ would be the natural next check.","The Miyawaki description may allow one to compute the full big lift $M_\\mu^m(\\pi)$ rather than only its cosocle, because the proof obtains the cosocle from the image of an intertwining operator whose poles are already controlled."],"forward_implications":["Every irreducible smooth representation of $\\mathrm{GL}_n$ appears exactly once in the spectrum of $\\mathrm{Mat}_{n,m}$, paired with the explicit partner $T^m_n(\\pi)$, so the matrix space is multiplicity-free with a rank-by-rank construction.","Every $\\pi\\in\\mathrm{Irr}(\\mathrm{Sp}_n)$ has at most two possible partners in the spectrum of $(\\mathrm{SGr}_{n,m},L_\\mu)$, both contained in $\\pi\\otimes T^m_n(\\pi,\\mu)$, and each orbit stratum contributes a definite tensor product of socles.","Howe duality in type II follows as a corollary: the big theta lift $\\Theta_{n,m}(\\pi)$ has irreducible cosocle $\\theta_{n,m}(\\pi)$, and $\\theta_{n,m}(\\pi)\\cong\\theta_{n,m}(\\pi')\\neq0$ implies $\\pi\\cong\\pi'$.","Local Miyawaki lifts in the Hilbert–Siegel case become explicit: $M_\\mu^m(\\pi)$ is of finite length, nonzero for $m\\geq n$, and its cosocle is the socle of a single explicitly written induced representation of length at most two; for $n\\geq m$ the correspondence is injective.","The pole of the Godement–Jacquet $L$-function at $s=-(n-1)/2$ is computed geometrically: its order equals $\\Lambda(\\nu^{r/2}_{n-r},\\tau)+1$, where $r$ is the first rank stratum on which the canonical map $T_\\pi$ does not vanish."],"supporting_citations":[{"why":"Supplies the Luna–Vust classification and the Local Structure Theorem used to describe the orbit neighborhoods of both varieties.","marker":"[LV83]"},{"why":"Provides the Geometric Lemma for the Jacquet filtration of parabolically induced representations, used throughout the derivative computations.","marker":"[BZ77]"},{"why":"Establishes the uniqueness of right and left $\\rho$-derivatives for $\\mathrm{GL}_n$, the base of the derivative induction.","marker":"[M´09]"},{"why":"Supplies the socle-irreducibility result for parabolic induction on classical $p$-adic groups used in the same step.","marker":"[Jan00]"},{"why":"Extends the $\\rho$-derivative theory to symplectic groups, giving the uniqueness of $D_\\rho(\\pi)$ for $\\mathrm{Sp}_n$.","marker":"[AM23]"},{"why":"Gives the $\\Delta$-derivative formalism and the irreducibility criterion that underpin the proof of Corollary 3.1.","marker":"[LM25]"},{"why":"Provides the uniqueness and irreducibility facts for $T^m_n(\\pi)$ and for $\\square$-irreducible representations that the definitions rely on.","marker":"[LM18]"},{"why":"Is the earlier proof of Howe duality in type II that Theorem 5.6 reproves and makes explicit via the new method.","marker":"[M´08]"},{"why":"Defines the local Miyawaki liftings and supplies the flat-section and doubling constructions used to extend the relevant maps to $s=0$.","marker":"[Ato19]"},{"why":"Describes the equivariant structure of the symplectic Grassmannian strata and supplies the doubling-method map used in Proposition 5.1.","marker":"[KR05]"}],"fun_headline_variants":["Explicit spectra for matrix space and symplectic Grassmannian","Rho-derivatives give exact spectra and Howe duality","New proof of Howe duality and explicit spectra","From rho-derivatives to Howe duality and Miyawaki lifts"],"cache_read_input_tokens":39680,"weakest_assumption_plain":"The argument depends on Corollary 3.1, that for a self-dual segment $\\Delta$ and any irreducible $\\pi$ of $\\mathrm{Sp}_n$ the socle of $Z(\\Delta)\\rtimes\\pi$ is multiplicity-free of length at most two and determines $\\pi$ uniquely; the proof is a compressed induction, and its stated analog for the metaplectic cover $\\mathrm{Mp}_n$ is asserted without a written proof.","fun_headline_variants_meta":{"raw":{"variants":["Explicit spectra for matrix space and symplectic Grassmannian","Rho-derivatives give exact spectra and Howe duality","New proof of Howe duality and explicit spectra","From rho-derivatives to Howe duality and Miyawaki lifts"]},"model":"deepseek-v4-flash","effort":"low","cost_usd":0.000555,"raw_usage":{"total_tokens":2747,"prompt_tokens":1150,"completion_tokens":1597,"prompt_tokens_details":{"cached_tokens":384},"prompt_cache_hit_tokens":384,"prompt_cache_miss_tokens":766,"completion_tokens_details":{"reasoning_tokens":1528}},"tokens_in":766,"tokens_out":1597,"duration_ms":10363,"temperature":1.0,"reasoning_tokens":1528,"cache_read_input_tokens":384,"cache_creation_input_tokens":0},"cache_creation_input_tokens":0},"created_at":"2026-08-15T19:59:32.621102+00:00","model_set":{"reader":"deepseek-v4-flash"},"falsifier":"Find non-isomorphic $\\pi,\\pi'\\in\\mathrm{Irr}(\\mathrm{Sp}_n)$ and a self-dual segment $\\Delta$ such that $\\mathrm{soc}(Z(\\Delta)\\rtimes\\pi)$ and $\\mathrm{soc}(Z(\\Delta)\\rtimes\\pi')$ share an irreducible constituent. Corollary 3.1 forbids this, and since the definition of $T^m_n(\\pi,\\mu)$ uses exactly that uniqueness, such an example would collapse the symplectic spectrum formula.","supporting_citations":[],"review_version":2}