{"id":"5eb53389-9799-46f7-8c47-8ce54f0a362a","arxiv_id":"2412.03234","paper_version":6,"verdict":"CONDITIONAL","confidence":"MODERATE","novelty_score":7.0,"correctness_risk":"medium","formal_verification":"none","parameter_count":0,"one_line_summary":"The mixed Poincaré series of generic PGL_n-character stacks is conjectured, proved at Euler specialization, and shown to interpolate structure coefficients of orbital complexes on PGL_n(F_q) and character sheaves on SL_n(F_q).","lead":"This paper gives a conjectural formula for the intersection cohomology of generic PGL_n(C)-character stacks with coefficients in local systems attached to conjugacy classes with disconnected stabilizers, and proves it after specializing to the Euler characteristic.","discovery_kind":"extension","skeptic_critique":{"model":"deepseek-v4-flash","headline":"The displayed proof of Theorem 8.5.1 omits the character χ(y) in the projection from the H(C)-cover, so the trace identity identifying the convolution product with IE(M_C,E_χ) is not justified as written.","rationale":"The paper's central claim is a conjectural mixed Poincaré formula together with two proven structure-coefficient identifications. I looked first for a gap in the long diagram. The strongest internal stress point I found is not the well-known but only sketched specialization theorem 2.2.5, on which I agree with the reader, but a concrete algebraic omission in the proof of Theorem 8.5.1. Formula (8.5.1) computes the trace of Frobenius on IH(M_C,E_χ) by averaging Tr(yF) over y∈H(C) without the character χ(y). This selects the trivial character, not χ. The paper's own formula (5.5.17), which is the same projection before specialization, contains χ(y); therefore the proof is internally inconsistent at this point. The theorem may still be true, since Remark 8.5.2 sketches a GL_n-reduction, but that reduction is not carried out. A referee should request either inserting χ(y) in the trace computation or writing out the alternate proof in full. I do not treat this as evidence of dishonesty; it is a fixable but real gap. The reader's verdict of CONDITIONAL is correct, and my concrete test is designed to settle the issue by direct computation in the smallest nontrivial case.","tokens_in":48862,"tokens_out":22946,"duration_ms":217275,"concrete_test":"Re-derive the trace identity in §8.5 directly from (5.5.16): apply the projector (1/|H(C)|) Σ_y χ(y) y to both sides before taking the trace, and check whether the displayed formula becomes formula (5.5.17) at t=-1. For a numerical settlement, take a generic degenerate tuple with H(C) of order 2 and nontrivial χ (e.g. n=2, k=2, C_1=C_2=C_{-1}, χ=χ_A with A={1}), compute the convolution Q directly via Proposition 7.1.2 and Theorem 8.3.1, and compare with the RHS of (8.5.2). The proof's displayed average without χ(y) will differ from both; this confirms the gap.","verdict_should_be":"UNCHANGED","load_bearing_attack":"The most load-bearing gap I find is in §8.5. After formula (8.5.1), the proof asserts Tr(F | IH^i_c(M_C,E^C_χ)) = (1/|H(C)|) Σ_{y∈H(C)} Σ_{ζ∈I(C)} Tr(yF | IH^i_c(M_C(ζ))). This is the projection formula for the trivial isotypic component, not the χ-component. From the isomorphism (5.5.16), the projector onto the χ-isotypic summand is (1/|H(C)|) Σ_y χ(y) y, with the paper's own normalization in (2.3.4) and (5.5.17); hence the factor χ(y) is required. As written, the chain of equalities cannot yield IE(M_C,E^C_χ); it yields the corresponding trivial-character quantity unless χ=1. Since Theorem 8.5.1 is one of the two proven legs of the central interpolation diagram, this is a real correctness gap in the proof of the main claim. The alternate proof sketched in Remark 8.5.2 via reduction to GL_n is not written out, so the theorem is only conditionally supported. The reader's concern about Theorem 2.2.5 is independent and remains valid, but this missing character is a more immediate algebraic obstruction in a displayed proof.","agreement_with_reader":"partial"},"referee_report":{"model":"deepseek-v4-flash","summary":"The paper studies the mixed Poincaré series of intersection cohomology of generic PGL_n-character stacks M_C, with coefficients in local systems E^C_χ arising from PGL_n-conjugacy classes with possibly non-connected stabilizers. Its main conjecture (Conjecture 5.5.11) expresses IH_c(M_C,E^C_χ;q,t) as a finite sum of HLRV-type rational functions H_{ω_r} with explicit coefficients Δ^{s_χ}_r. The paper proves the conjecture after the Euler specialization t ↦ -1 (Theorem 5.5.10), and proves that the same specialized sum equals the structure coefficient of (i) a convolution product of orbital intersection complexes on PGL_n(F_q) (Theorem 8.5.1) and (ii) a pointwise product of Lusztig character sheaves on SL_n(F_q) (Theorem 8.2.6). These two identifications realize the announced interpolation between the based rings (C(PGL_n(F_q)), Loc, ∗) and (C(SL_n(F_q)), CS, ·). The case n = 2 is worked out explicitly, with the character-sheaf multiplicities verified directly (Section 9).","tokens_in":49067,"tokens_out":19020,"duration_ms":148784,"significance":"If the results hold, this is a substantial contribution to the arithmetic harmonic analysis of character stacks: it extends the Hausel–Letellier–Rodriguez-Villegas/Letellier program from GL_n to PGL_n in the genuinely more delicate setting of degenerate conjugacy classes (non-connected stabilizers), and it gives a geometric 'non-abelian Fourier transform' bridge between convolution on PGL_n(F_q) and pointwise multiplication on SL_n(F_q). The paper is scrupulous in separating theorems from conjectures: the full mixed Poincaré series is only conjectural (5.5.11), the Euler specialization is proved, and the two structure-coefficient theorems are derived from prior GL_n results [27, 28]. Notable strengths are the explicit derivations of Theorems 5.5.10, 8.2.6 and 8.5.1 from the cited prior work, the fully worked n = 2 verification (Theorem 9.6.2 with the explicit table of values), and the authors' candid disclosure of the unproved specialization theorem 2.2.5. The main weaknesses are (a) a missing character factor in the displayed proof of Theorem 8.5.1, and (b) the sketched nature of the foundational comparison Theorem 2.2.5 on which the complex-side interpretation rests.","major_comments":[{"comment":"The projection step drops the character χ(y). After the trace formula the proof asserts Q = Σ_i (-1)^i (1/|H(C)|) Σ_{y∈H(C)} Σ_{ζ∈I(C)} Tr(yF | IH^i_c(M_C(ζ))) and identifies this with IE(M_C,E^C_χ;q). By the isomorphism (5.5.16) and the inversion formula (5.5.17), with the normalization of (2.3.4), the projector onto the χ-isotypic summand of ⊕_{ζ∈I(C)} IH^*_c(M_C(ζ)) is (1/|H(C)|) Σ_{y∈H(C)} χ(y) y; as written, the displayed chain computes the trivial-character quantity IE(M_C,E^C_1;q) when χ ≠ 1. Inserting the factor χ(y) and applying (5.5.17) at t = -1 gives exactly the claimed identification, so the statement is consistent with Theorem 5.5.10, but the displayed proof must be corrected. The alternative route indicated in Remark 8.5.2 (carrying χ through (7.1.2) and Theorem 8.3.1) is only sketched and does not repair the displayed argument as it stands.","section":"§8.5, proof of Theorem 8.5.1"},{"comment":"The specialization comparison IH_c(X;q,t) = IH_c(X_f;q,t) is load-bearing for the complex-side statements: it enters Remark 1.2.2 and, via Theorem 2.7.3 and Theorem 4.3.3, the K = C case of Theorem 5.5.10. The proof is only a sketch, and the authors state that they 'were not able to locate a proof in the literature.' The delicate steps are (i) the claim that the isomorphism (2.2.7), obtained from the constancy of σ_!Q_ℓ over an open U ⊆ Spec(R), preserves the weight filtration through [11, Théorème 14], and (ii) the identification IH^*_c(X) ≅ ⋂_{T≠X} Ker(r^*_T) inside H^*_c(X̃) and its transfer to X_f through [14] and [3, Lemma 6.2.6], including the weight-preserving isomorphism (2.2.14). I recommend either completing this proof in the text or explicitly stating the complex-geometric theorems as conditional on this comparison.","section":"§2.2, Theorem 2.2.5"}],"minor_comments":[{"comment":"The displayed formula IE(M_C;q) = n(qt^2)^{d_ω} H_ω(√q, 1/√q) cannot be right as printed: the Euler specialization removes the t^2 factor, so this should read n q^{d_ω} H_ω(√q, 1/√q), matching Theorem 5.5.10 and the t = -1 specialization of Conjecture 5.5.2.","section":"§5.5.1, Theorem 5.5.3"},{"comment":"The pure part is displayed with a factor (qt^2)^{k-3}; consistent with (1.2.3) and with the adjacent formula (9.5.1), the pure part contains no t, so the factor should be q^{k-3}.","section":"§9.5, Formula (9.5.3)"},{"comment":"Given the gap in the displayed proof of Theorem 8.5.1, the alternative proof sketched in Remark 8.5.2 should either be written out in full or replaced by a corrected version of the main argument; as it stands the remark does not by itself establish the theorem.","section":"§8.5, Remark 8.5.2"},{"comment":"The character group of A(C) is rendered inconsistently: expressions like 'χ∈[A(C)' appear where \\(\\widehat{A(C)}\\) is clearly intended; if this reflects the typeset version, the notation must be fixed, since \\(\\widehat{A(C)}\\) denotes the character group while A(C) denotes the finite component group.","section":"throughout (e.g., §5.5.2, §8.2)"}],"recommendation":"major_revision","confidential_remarks":"To the editor: the manuscript's reduction of the PGL_n statements to the authors' earlier GL_n results via the H(C)-Galois covering is structural rather than circular, but the citation pattern is heavily self-referential ([19, 27, 28] and companions), which is worth bearing in mind for the provenance review. The paper is explicit that its headline formula for the full mixed Poincaré series is conjectural; the proved content is the Euler specialization and the two structure-coefficient identifications, plus a convincing explicit n = 2 check. The two issues identified above are locally fixable within scope: the missing χ(y) in the proof of Theorem 8.5.1 is an immediate algebraic correction, and Theorem 2.2.5 needs a complete proof or a conditional formulation. I do not see grounds for rejection."},"author_rebuttal":null,"desk_editor":{"model":"deepseek-v4-flash","letter":"The genuinely new content is the PGL_n/SL_n story with non-connected stabilizers: the conjectural mixed series for IH_c(M_C,E_χ), the proved Euler specialization, and the two structure-coefficient theorems. The n=2 section is a real check, not a formality. The paper builds heavily on the authors' earlier GL_n work, but the PGL_n quantities are not assumed inside the H_ω functions, so I see dependence, not circularity.\n\nThe soft spots are real. First and most serious: in §8.5, right after (8.5.1), the displayed chain writes\n\nΣ_i (-1)^i Tr(F | IH^i_c(M_C,E^C_χ)) = (1/|H(C)|) Σ_{y∈H(C)} Σ_{ζ∈I(C)} Σ_i (-1)^i Tr(yF | IH^i_c(M_C(ζ))).\n\nThat is the projector onto the trivial isotypic component of H(C), not the χ-isotypic component. Their own inversion formula (5.5.17) has χ(y) inside the y-sum. So as written, the equality cannot produce IE(M_C,E^C_χ) unless χ=1; it gives the trivial-character quantity. Theorem 8.5.1 is one of the two proven legs of the interpolation diagram, so this is load-bearing. The alternative proof via reduction to GL_n in Remark 8.5.2 is only sketched; it needs to be written out.\n\nSecond, Theorem 2.2.5 and its twisted version 2.7.3 assume a weight-preserving comparison between complex and finite-field intersection cohomology, with only a sketch relying on [14] and [3, Lemma 6.2.6]. The authors themselves say they could not locate a proof. If that specialization fails in the stated generality, the bridge from IE computed over F_q to E-polynomials over C is incomplete.\n\nAll of that said, the paper is technically rich, the reductions to prior GL_n results are clearly organized, and Theorem 8.2.6 is proved in detail. The n=2 numerics give independent support. I would send this to a serious referee. The referee should focus on §8.5 and on Theorem 2.2.5; the missing χ(y) is the more immediate obstruction and should be fixed before acceptance.","headline":"A substantial extension of the GL_n character-stack program to PGL_n, with real new content, but the displayed proof of Theorem 8.5.1 drops the character χ(y) from the H(C)-projector, so one of the two proven legs is not justified as written.","tokens_in":49690,"tokens_out":3358,"would_cite":true,"duration_ms":30059,"reading_group":"yes","serious_thinker":"yes","would_accept_peer_review":true},"rs_alignment":null,"lean_confirmation":null,"pith_extraction":{"msc":["14D23","14F20","20G40","20C33"],"pacs":[],"model":"deepseek-v4-flash","headline":"One combinatorial sum is conjectured to compute PGL_n character-stack cohomology and to interpolate two finite-field rings, with the identity proved after Euler specialization.","keywords":["PGL_n character stacks","intersection cohomology of stacks","mixed Poincaré series","Langlands duality over finite fields","character sheaves","orbital complexes","structure coefficients","non-abelian Fourier transform"],"falsifier":"Compute the mixed Poincaré series on both sides of Conjecture 5.5.11 for $n=2$, $k=3$, with one degenerate semisimple $\\mathrm{PGL}_2$-conjugacy class, using the explicit formula (9.5.2) and a direct calculation of the weight filtration on the corresponding character stack; agreement in every $(q,t)$-coefficient would confirm the full conjecture in the first case beyond the Euler specialization, while a single mismatch would refute it.","tokens_in":48575,"feed_emoji":"🔗","tokens_out":12098,"duration_ms":106971,"temperature":0.7,"pith_summary":"The paper proposes a closed formula for the mixed Poincaré series of the intersection cohomology of a generic $\\mathrm{PGL}_n(\\mathbb{C})$-character stack, with coefficients in the local systems attached to $\\mathrm{PGL}_n(\\mathbb{C})$-conjugacy classes whose stabilisers are disconnected. The formula writes the series as a finite combination of the rational functions $H_{\\omega_r}(z,w)$ used in the $\\mathrm{GL}_n$ theory, with combinatorial weights $\\Delta_r^{s_\\chi}$ encoding the local-system characters. The identity is proved after the Euler specialization $t\\mapsto -1$, giving the $IE$-polynomial, and the same specialized combination is shown to be the structure coefficient of a convolution product of intersection-cohomology orbital complexes on $\\mathrm{PGL}_n(\\mathbb{F}_q)$ and of a pointwise product of character sheaves on $\\mathrm{SL}_n(\\mathbb{F}_q)$. If the full conjecture holds, one geometric object would interpolate between two a priori unrelated finite-group rings, which the authors describe as a non-abelian Fourier transform.","feed_headline":"One identity links PGL_n stack cohomology to two finite-field rings","feed_subtitle":"Proved at the Euler specialization; the full conjecture would unify convolution on PGL_n(F_q) with character sheaves on SL_n(F_q).","key_machinery":"The device that carries the argument is the finite Galois covering\n$$p:\\bigsqcup_{\\zeta\\in I(\\mathbf C)}\\mathcal M_{\\mathbf C(\\zeta)}\\to\\mathcal M_{\\mathbf C}$$\nwith Galois group $H(\\mathbf C)=\\{(y_1,\\dots,y_k)\\in\\prod_i A(C_i): y_1\\cdots y_k=1\\}$; it expresses each local-system coefficient $E_\\chi$ as the $\\chi$-isotypic component of the pushforward of the constant sheaf, so the cohomology of $(\\mathcal M_{\\mathbf C},E_\\chi)$ becomes an $H(\\mathbf C)$-twisted average of cohomologies of ordinary $\\mathrm{GL}_n$-character stacks. Those $\\mathrm{GL}_n$ stacks are governed by the rational functions $H_\\omega(z,w)$ of the $\\mathrm{GL}_n$ theory, while the weights $\\Delta_r^{s_\\chi}$ are cyclotomic sums that record the restriction of $\\chi$ to the cyclic groups $A(C_i)$. On the finite-field side, the Langlands correspondence for $(\\mathrm{PGL}_n,\\mathrm{SL}_n)$ matches orbital intersection-cohomology complexes on $\\mathrm{PGL}_n(\\mathbb{F}_q)$ with character sheaves on $\\mathrm{SL}_n(\\mathbb{F}_q)$, so the same $H_\\omega$-sum appears once as a convolution structure constant and once as a pointwise-product structure constant.","core_discovery":"For a generic $k$-tuple $\\mathbf C=(C_1,\\dots,C_k)$ of $\\mathrm{PGL}_n$ conjugacy classes and each character $\\chi=\\chi_1\\boxtimes\\dots\\boxtimes\\chi_k$ of $A(\\mathbf C)=\\prod_i A(C_i)$, the theorem proved after $t\\mapsto -1$ is\n$$\\operatorname{IE}(\\mathcal M_{\\mathbf C},E_\\chi;q)=\\frac{\\iota(\\mathbf C)}{|A(\\mathbf C)|}$q^{{d_\\omega}}$\\sum_{r\\in R_{d_1,\\dots,d_k}}\\$Delta_r^{{s_\\chi}}$H_{\\omega_r}(\\sqrt q,1/\\sqrt q).$$\nEquivalently, this number is the convolution coefficient of the intersection-cohomology orbital complexes on $\\mathrm{PGL}_n(\\mathbb{F}_q)$ (Theorem 8.5.1) and, up to a normalizing factor, the pointwise-product coefficient of the corresponding character sheaves on $\\mathrm{SL}_n(\\mathbb{F}_q)$ (Theorem 8.2.6). The unspecialized formula (Conjecture 5.5.11) is the only statement that would give the full mixed Hodge series, and it is reduced to the analogous $\\mathrm{GL}_n$ conjecture, for which the paper cites substantial evidence. The authors read the resulting triangle as a finite-field manifestation of Langlands duality for the dual pair $(\\mathrm{PGL}_n,\\mathrm{SL}_n)$.","pith_inferences":["The interpolation picture suggests that the mixed Hodge series of $\\mathrm{PGL}_n$-character stacks could serve as a geometric origin for the structure constants of both finite-group rings, effectively a categorification of the non-abelian Fourier transform the authors invoke.","Extending the same reduction to non-generic conjugacy classes, as the paper says the $\\mathrm{GL}_n$ case now allows, would likely replace 'generic tuple' by a level-of-genericity condition and produce conjectural formulae for all $\\mathrm{PGL}_n$ character stacks, with the pure-part statement no longer tracking character multiplicities.","A concrete check of the $n=2$ formulas against the weight filtration computed by standard resolution methods would test the full conjecture in the first case where the Euler specialization has already been proved.","The $H(\\mathbf C)$-action on $\\mathrm{GL}_n$-stack cohomology should agree with the Weyl-group action introduced by analytic methods; proving that equality would connect the combinatorial weights to Hecke correspondences on parabolic Higgs bundles."],"forward_implications":["For every generic $\\mathbf C$ and $\\chi$, the Euler-specialized polynomial $\\operatorname{IE}(\\mathcal M_{\\mathbf C},E_\\chi;q)$ is now a computable rational expression; in the $n=2$ case the paper writes it out explicitly for all local systems on degenerate semisimple classes.","The convolution algebra of class functions on $\\mathrm{PGL}_n(\\mathbb{F}_q)$ generated by IC-orbital complexes and the pointwise algebra on $\\mathrm{SL}_n(\\mathbb{F}_q)$ generated by character sheaves share structure coefficients, at least for generic inputs paired with the trivial character.","If Conjecture 5.5.11 is correct, the full mixed Poincaré series—not just its Euler specialization—would be the common origin of both structure coefficients, with the pure-part specialization corresponding to the pointwise product on $\\mathrm{SL}_n$.","The identity element of the convolution ring (the function $1_1$) corresponds under the stated bijection to the identity element of the pointwise ring (the trivial character of $\\mathrm{SL}_n(\\mathbb{F}_q)$), making the dictionary genuinely one of rings with unit.","Because the $\\mathrm{PGL}_n$ conjecture reduces to the known $\\mathrm{GL}_n$ mixed-Poincaré conjecture, a proof of the latter would automatically upgrade all the finite-field structure-coefficient identities in this paper to identities of full mixed Hodge series."],"supporting_citations":[{"why":"Introduces the rational functions $H_\\omega(z,w)$ and the arithmetic-harmonic method for $\\mathrm{GL}_n$ character stacks; supplies the central functions used in the $\\mathrm{PGL}_n$ formula.","marker":"[19]"},{"why":"Extends the $\\mathrm{GL}_n$ theory to conjugacy classes with Zariski closures; its Theorem 4.2.7 and Theorem 4.14 are the $\\mathrm{GL}_n$ results behind Theorem 4.3.3 and Theorem 5.5.10.","marker":"[27]"},{"why":"Proves that trivial-character multiplicities in tensor products of $\\mathrm{GL}_n(\\mathbb{F}_q)$ characters equal $H_\\omega(0,\\sqrt q)$, the input for the $\\mathrm{SL}_n$ structure-coefficient theorem.","marker":"[28]"},{"why":"Computes intersection cohomology of $\\mathrm{GL}_n$ character varieties and introduces the Weyl-group action to which the $H(\\mathbf C)$-action is compared.","marker":"[2]"},{"why":"Proves the Poincaré-polynomial version of the underlying $\\mathrm{GL}_n$ character-variety conjecture in the semisimple case, cited as the main evidence for Conjecture 5.5.11.","marker":"[38]"},{"why":"Establishes the derived Lusztig correspondence and geometric induction that defines the bijection between $\\mathrm{PGL}_n$ orbital complexes and $\\mathrm{SL}_n$ character sheaves.","marker":"[25]"},{"why":"Sets up Langlands duality for finite groups of Lie type, including torus duality and Kummer local systems, on which the correspondence $c_G$ is built.","marker":"[13]"},{"why":"Provides the motivic projectors used in the sketched proof of the weight-preserving specialization theorem comparing complex and finite-field intersection cohomology.","marker":"[14]"}],"fun_headline_variants":["Euler specialisation proves identity bridging PGL_n and SL_n rings","One formula interpolates PGL_n and SL_n finite-field rings","Non-abelian Fourier transform links PGL_n and SL_n rings","Single identity ties PGL_n stack cohomology to SL_n character sheaves","Langlands duality in finite fields via character stacks"],"cache_read_input_tokens":3200,"weakest_assumption_plain":"The load-bearing premise is that intersection cohomology with weights computed over $\\mathbb C$ agrees, after specializing a finite-ring model, with the corresponding computation over $\\mathbb F_q$ (Theorem 2.2.5); the paper gives only a sketch of this comparison and states that it could not locate a proof in the literature, so the entire bridge from the finite-field identities to the complex $IE$-polynomials would collapse if that specialization failed.","fun_headline_variants_meta":{"raw":{"variants":["Euler specialisation proves identity bridging PGL_n and SL_n rings","One formula interpolates PGL_n and SL_n finite-field rings","Non-abelian Fourier transform links PGL_n and SL_n rings","Single identity ties PGL_n stack cohomology to SL_n character sheaves","Langlands duality in finite fields via character stacks"]},"model":"deepseek-v4-flash","effort":"low","cost_usd":0.000623,"raw_usage":{"total_tokens":2968,"prompt_tokens":1109,"completion_tokens":1859,"prompt_tokens_details":{"cached_tokens":384},"prompt_cache_hit_tokens":384,"prompt_cache_miss_tokens":725,"completion_tokens_details":{"reasoning_tokens":1777}},"tokens_in":725,"tokens_out":1859,"duration_ms":13219,"temperature":1.0,"reasoning_tokens":1777,"cache_read_input_tokens":384,"cache_creation_input_tokens":0},"cache_creation_input_tokens":0},"created_at":"2026-08-11T22:38:10.845203+00:00","model_set":{"reader":"deepseek-v4-flash"},"falsifier":"Compute the mixed Poincaré series on both sides of Conjecture 5.5.11 for $n=2$, $k=3$, with one degenerate semisimple $\\mathrm{PGL}_2$-conjugacy class, using the explicit formula (9.5.2) and a direct calculation of the weight filtration on the corresponding character stack; agreement in every $(q,t)$-coefficient would confirm the full conjecture in the first case beyond the Euler specialization, while a single mismatch would refute it.","supporting_citations":[{"cited_title":"and Rodriguez-Villegas, F","cited_arxiv_id":null,"evidence_quote":"Introduces the rational functions $H_\\omega(z,w)$ and the arithmetic-harmonic method for $\\mathrm{GL}_n$ character stacks; supplies the central functions used in the $\\mathrm{PGL}_n$ formula."},{"cited_title":": Character varieties with Zariski closures of _n -conjugacy classes at punctures, Selecta Math","cited_arxiv_id":null,"evidence_quote":"Extends the $\\mathrm{GL}_n$ theory to conjugacy classes with Zariski closures; its Theorem 4.2.7 and Theorem 4.14 are the $\\mathrm{GL}_n$ results behind Theorem 4.3.3 and Theorem 5.5.10."},{"cited_title":": Quiver varieties and the character ring of general linear groups over finite fields, J","cited_arxiv_id":null,"evidence_quote":"Proves that trivial-character multiplicities in tensor products of $\\mathrm{GL}_n(\\mathbb{F}_q)$ characters equal $H_\\omega(0,\\sqrt q)$, the input for the $\\mathrm{SL}_n$ structure-coefficient theorem."},{"cited_title":": Intersection cohomology of character varieties for punctured Riemann surfaces, J","cited_arxiv_id":null,"evidence_quote":"Computes intersection cohomology of $\\mathrm{GL}_n$ character varieties and introduces the Weyl-group action to which the $H(\\mathbf C)$-action is compared."},{"cited_title":": Poincar\\'e polynomials of character varieties, Macdonald polynomials and affine Springer fibers, Ann","cited_arxiv_id":null,"evidence_quote":"Proves the Poincaré-polynomial version of the underlying $\\mathrm{GL}_n$ character-variety conjecture in the semisimple case, cited as the main evidence for Conjecture 5.5.11."},{"cited_title":"and Letellier, E","cited_arxiv_id":null,"evidence_quote":"Establishes the derived Lusztig correspondence and geometric induction that defines the bijection between $\\mathrm{PGL}_n$ orbital complexes and $\\mathrm{SL}_n$ character sheaves."},{"cited_title":"and Lusztig, G","cited_arxiv_id":null,"evidence_quote":"Sets up Langlands duality for finite groups of Lie type, including torus duality and Kummer local systems, on which the correspondence $c_G$ is built."},{"cited_title":null,"cited_arxiv_id":null,"evidence_quote":"Provides the motivic projectors used in the sketched proof of the weight-preserving specialization theorem comparing complex and finite-field intersection cohomology."}],"review_version":1}